PARTICULAR ASPECTS IN THE DESIGN OF COLD FORMED STEEL MEMBERS AND SHEETING
11 Introduction 12 Industrial production of cold formed thin gauge sections13 The steel used for cold formed thin gauge members Characteristics for
design14 Influence of cold forming (hammering)15 Maximum Width-to-Thickness Ratios21Specific Features of the Cross Sections of Cold Formed Thin Gauge Shapes 22 Calculation of Sectional Properties
11 INTRODUCTION
bull Sheet strip plates or flat bars fabricated in roll-forming machines or by press brake operations
bull The thickness of the steel sheets or strips excluding the coating 05 mm to 4 mm for sheeting and from 1 mm to 8 mm for profiles also steel plates and bars as 25 mm may be cold formed into structural shapes
bull Some important advantagesbull a) cold formed light members are manufactured for relatively light loads andor short
spansbull b) various and intricate sectional configurations are produced economically by cold
forming operations favourable strength-to-weight ratios may be obtainedbull c) nestable sections are produced compact packaging and shippingbull d) load carrying panels and decks are able to provide useful surfaces for floors roofs
and wall constructions and in other cases they can also provide enclosed cells for electrical and other conduits
bull e) panels and decks not only withstand loads normal to their surfaces but they can also act as shear diaphragms to resist force in their own plans if they are adequately interconnected to each other and to the supporting members
12 INDUSTRIAL PRODUCTION OF COLD FORMED THIN GAUGE SECTIONS
A Continuous process important series of sections by continuous forming in rolling mills The coil is unrolled and the steel sheet passes through successive pairs of roles and after that the sections are cut at the desired length Stripped steel may be processed with thickness between 03 mm and 18 mm and width between 20 mm and 2000 mm
B Discontinuous process small series of sections either a leaf press brake (folding) of the steel sheets or a coin press brake (press braking) are commonly used for pressing the steel strip in a mould The thickness of the of the shapes obtained by press folding is relatively small under 3 mm and the length of the elements is between 15 m and 40 m The shapes obtained by pressing in moulds have the thickness under 16 mm and 6 m length
Types of structural elements Cold-formed structural members can be classified into two major types
a)- individual structural framing members used in buildings as beams columns trusses and in the workshop design as purlins skylights bracing structural elements for walls transmission towers etcb)- panels and decks (corrugated shells) used in facades as external layer for curtain walls diaphragms roofs floors and permanent shuttering
COLD FORMING OF THIN GAUGE SECTIONS
Forme diverse realizate din prfile cu pereti subtiri (Trebilcock 1994)
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
11 INTRODUCTION
bull Sheet strip plates or flat bars fabricated in roll-forming machines or by press brake operations
bull The thickness of the steel sheets or strips excluding the coating 05 mm to 4 mm for sheeting and from 1 mm to 8 mm for profiles also steel plates and bars as 25 mm may be cold formed into structural shapes
bull Some important advantagesbull a) cold formed light members are manufactured for relatively light loads andor short
spansbull b) various and intricate sectional configurations are produced economically by cold
forming operations favourable strength-to-weight ratios may be obtainedbull c) nestable sections are produced compact packaging and shippingbull d) load carrying panels and decks are able to provide useful surfaces for floors roofs
and wall constructions and in other cases they can also provide enclosed cells for electrical and other conduits
bull e) panels and decks not only withstand loads normal to their surfaces but they can also act as shear diaphragms to resist force in their own plans if they are adequately interconnected to each other and to the supporting members
12 INDUSTRIAL PRODUCTION OF COLD FORMED THIN GAUGE SECTIONS
A Continuous process important series of sections by continuous forming in rolling mills The coil is unrolled and the steel sheet passes through successive pairs of roles and after that the sections are cut at the desired length Stripped steel may be processed with thickness between 03 mm and 18 mm and width between 20 mm and 2000 mm
B Discontinuous process small series of sections either a leaf press brake (folding) of the steel sheets or a coin press brake (press braking) are commonly used for pressing the steel strip in a mould The thickness of the of the shapes obtained by press folding is relatively small under 3 mm and the length of the elements is between 15 m and 40 m The shapes obtained by pressing in moulds have the thickness under 16 mm and 6 m length
Types of structural elements Cold-formed structural members can be classified into two major types
a)- individual structural framing members used in buildings as beams columns trusses and in the workshop design as purlins skylights bracing structural elements for walls transmission towers etcb)- panels and decks (corrugated shells) used in facades as external layer for curtain walls diaphragms roofs floors and permanent shuttering
COLD FORMING OF THIN GAUGE SECTIONS
Forme diverse realizate din prfile cu pereti subtiri (Trebilcock 1994)
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
12 INDUSTRIAL PRODUCTION OF COLD FORMED THIN GAUGE SECTIONS
A Continuous process important series of sections by continuous forming in rolling mills The coil is unrolled and the steel sheet passes through successive pairs of roles and after that the sections are cut at the desired length Stripped steel may be processed with thickness between 03 mm and 18 mm and width between 20 mm and 2000 mm
B Discontinuous process small series of sections either a leaf press brake (folding) of the steel sheets or a coin press brake (press braking) are commonly used for pressing the steel strip in a mould The thickness of the of the shapes obtained by press folding is relatively small under 3 mm and the length of the elements is between 15 m and 40 m The shapes obtained by pressing in moulds have the thickness under 16 mm and 6 m length
Types of structural elements Cold-formed structural members can be classified into two major types
a)- individual structural framing members used in buildings as beams columns trusses and in the workshop design as purlins skylights bracing structural elements for walls transmission towers etcb)- panels and decks (corrugated shells) used in facades as external layer for curtain walls diaphragms roofs floors and permanent shuttering
COLD FORMING OF THIN GAUGE SECTIONS
Forme diverse realizate din prfile cu pereti subtiri (Trebilcock 1994)
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
COLD FORMING OF THIN GAUGE SECTIONS
Forme diverse realizate din prfile cu pereti subtiri (Trebilcock 1994)
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Forme diverse realizate din prfile cu pereti subtiri (Trebilcock 1994)
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
13 The steel used for cold formed thin gauge members Characteristics for design
Continuously hot-dip metal coated sheeting with nominal thickness supplied with half of the normal standard tolerances the design thickness t may be taken as the nominal core thickness tcnom
In case of continuously hot-dip metal coated steel sheet and strip the core thickness is
tz the thickness of the zinc protection usually 004 mm both sides of the sheet and 275 gm2
ldquoStandardrdquo grades of steel shall have the properties that conform to the required suitability for cold forming welding and galvanising The ratio of the specific minimum ultimate tensile strength fu to the specific minimum yield strength satisfies
The nominal (characteristic) values of the yield strength fyb and tensile strength fu for the specified steels are presented in table course notes
The basic material used for fabrication of the steel sections consists in flat sheet steel strips and the Romanian standards available are STAS 908-90 STAS 1945-90 STAS 9236-80 STAS 9150-80 STAS 10896-80 all of them being now harmonized with the european norms (SREN) Generally all these grades of steel will have the elongation at failure A ()gt20 Also supplementary measures will be adopted for the stripes of 02hellip8 mm thickness considering cold forming process and sensibility to brittle fracture
21yu ff
znomc ttt
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Examples of profiled sheeting and members (SREN 1993-1-3) a- basic elements b- structural elements suitable for axial loading c- structural elements suitable for bending
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Influence of cold forming (hammering)
The manufacturing process modify the mechanical properties of the profiles alteration of the stress-strain curve of the steel
Strain hardening provides an increase of the yield strength and sometimes of the ultimate strength that is important in the corners and still appreciable in the flanges while press braking lets these characteristics unchanged in these zones
ybuybya ffA
Cntff
2
902 in
i 090i
uybya fff 50
fyb fu - yield strength respectively ultimate tensile strength Nmm2 t - thickness of the steel plate A - gross area of the cross section (mm2) C =7 for cold rolling and 5 for other methods of cold forming n - number of folders at 900 having the internal radius rlt5t on the whole perimeter of the cross section for different angles from 900 the number n is determined with
wheren - relevant number of folders that increase the strength
-internal angle of the folder between 900 and 1350 for values under 900 we will use
and for values bigger then 1350 the folder will not be considered anymoreA superior limit value is imposed also for the average limit yield strength
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
DETERMINATION OF N ndashNUMBER OF RELEVANT CORNERS
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Etapele formarii la rece (Rhodes) 1992 ale unui profil
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Echipamente industriale de laminare (formare continua) a profilelor la rece
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
STRUCTURE OF COLD FORMED SHAPES-WALLS
Cold formed shapes are obtained from several walls The walls may be internal or external stiffened or un-sitffened according to their end (edge) conditions bull stiffened walls ndash that have their edges bound with another wall or with a folded end stiff enough as to prevent from its deformation in a direction perpendicular to the plane of the elementbull un-stiffened walls ndash that have one edge fee to displace (rotate) in a plane normal to the plane of the element
Stiffened walls of the cold formed shapes a)- external wall with end stiffener b) internal walls with intermediate stiffeners
Conditions are imposed for the stiffness of the walls
-for intermediary stiffeners
-for end stiffeners
4
2
411min 418
266000663 t
Rt
btI p
42411min 29
266000)(831 t
Rt
btI p
tRt
ata p 84
266000)(82 6 2
min
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Cold formed shapes with stiffened walls a)- intermediate stiffeners b)- with lip and clip (end stiffener)
OBSERVATION The end stiffeners of Ω and C shapes must respect also the condition aminge frac14 bp
from the total width (excluding the rounded corners) of the wall that is stiffened
Maximum bt ratios and modelling of the static behaviour
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Shapes with or without stiffeners their geometric characteristics and
mechanical behaviour according to EC 3 (SR EN 1993-1-13)
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
TIPURI DE RIGIDIZARI ALE PERETILOR PROFILELOR FORMATE LA RECE
Rigidizari intermediareRigidizari marginale a- cu rebord b ndash cu rebord intarit
Rigidizari intermediare longitudinale cu una sau mai multe cute
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
INFLUENTA ROTUNJIRILOR IN STABILIREA LATIMII bp
bullThe plane widths bp is measured from the midpoint of the cornerbull In the case when a cross section is made up from plane elements with sharp corners with rle5t and rbple015 rounding of corners is ignored bull All the sectional properties may be calculated based on an ideal section and the following approximations
n- number of cornersm- number of flat widthsbi- length of the mid line of the flat widths
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE SECTIONS
Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limit of the steel The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections
The following assumptions are demonstrated to be inconsistentI The perfect planarity - the initial deformations of the sheets due to faults of fabrication must be between certain limits Still the real plane elements do have initial geometrical imperfections- initial deflection w0 which grows with the increase of loading Due to the effect of membrane behavior the ultimate strength of the sheet is bigger than the critical elastic force of buckling Ncr This reserve of strength clearly insures a post-critical behavior
Plate in compression conditions of supports and post-critical reserve
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
II Reduced deformations out of the plane of the plate ndash this assumption is normally available in the theory of linear buckling in elastic domain In reality the ultimate strength of the plate exceeds the critical stress the deformations being rather importantIII Axial loads - this assumption is impossible from the practical point of view the planarity of the plate being an ideal assumption
Measurement of residual stresses in a cold-rolled C shape a) ndash residual flower b)- slicing method c)- curvature method
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
IV Linear elastic behavior of the material ndash this condition is satisfied up to the yield limit Still due to residual stresses caused by rolling welding cutting etc in some fibers the plastic stresses are reached for applied stresses lower than fy
Local buckling in compression and bending of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
The two distinct stages in the post-critical domain of the behavior of a plate areElastic- uniformly distributed stresses under the critical forcePost-critic - below the critical force the plate is deformed more and more the stresses are not anymore uniform
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is ([Nmm2])
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall on the
boundary conditions on the ratio between the dimensions of this wall-non - stiffened walls kσ =0425
-stiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not happen
as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark leads
to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon
The cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids where the strains are smaller are able to take over stresses that may reach the design value of strength At the moment when the maximum strength value of the material fy is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where = 0) the deformations being very important
The width of the wall reaches its minimum value called the effective width beffFrom the point of view of the local buckling-the stiffened compressed elements (walls) are flat elements in compression with both edges parallel to
the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -the non-stiffened compressed elements (walls) are flat elements in compression which are stiffened
only at one edge parallel to the direction of the stress
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
BUCKLING OF THE THIN WALLS-WINTERrsquoS GRID MODEL
Local buckling in compression and bending
of the thin walled elements
Consecutive stages of stress distribution in stiffened compressed elements
Winterrsquos model (grid)
Effective width of a plate in compression
Stiffened walls-marginal and intermediate
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Buckling is reached for a critical value of the normal stress σc ge σcr where the critical stress is
determined with the known relationship
3
22
2
2
10190112
ppcr b
tk
b
tEk
The coefficient kdepends on the nature and the distribution of the stress on the width of the wall
on the boundary conditions on the ratio between the dimensions of this wallbullnon - stiffened walls kσ =0425
bullstiffened walls kσ=40 the supports are considered articulated
It is important to observe that in the case of a wall under compression in its plane the lost of strength capacity will not
happen as long as the longitudinal edges will remain rectilinear the limits of strength capacity are much increased for certain types of walls This remark
leads to the theory of effective width of the wallThe design concept the grid model proposed by Winter (1959) for the instability phenomenon The
cross section for these profiles is made up from flat elements (walls) with constant thickness inter-connected generating a grid
[Nmm2]
In the post critical stage (post buckling strength) the central grid do not work anymore while the extreme grids
where the strains are smaller are able to take over stresses that may reach the design value of strength
At the moment when the maximum strength value of the material R is reached in the extreme zones a bigger portion in the internal part of the wall already isnrsquot working anymore (where the
deformations being very importantThe width of the wall reaches its minimum value called the effective width beff
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
CLASSIFICATION OF THE WALLS OF A COLD FORMED SECTION
From the point of view of the local buckling-stiffened compressed elements (walls) -flat elements in compression with both edges parallel to the direction of stress which are stiffened by web elements flanges or edge stiffeners of sufficient rigidity -non-stiffened compressed elements (walls) -flat elements in compression which are stiffened only at one edge parallel to the direction of the stressConsidering that in the situation of buckling in elastic of a wall having its effective width beff the stress σcreff reaches the maximum stress in the plate in post-critical domain that is σmax = fy Then the former relationship becomes
22
2
2
112
eff
pcr
effeffcr b
b
b
tEk
From this relationship it results that the effective width of the wall depends on the ratio σcrσmax
max cr
peff bb
In buckling stage the averaged stress on the whole width of the wall is σu the equivalence between the stresses will impose the following equation
upeffupyeff bbbfb max
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Von Karman determined the following relationship for the effective wall
max
2
2
2 1
112
ppeff b
tEkbb
In the case of the plate articulated all around and uniformly compressed kσ = 40 max
91E
tbeff
in order to simplify the further design specifications EC3 uses the following relationships relative slenderness (of the plate) referred to bp
kt
bf
p
cr
yp
428
bull reduction factor y
u
p
eff
fb
b
bull influence of the elastic limityf
240
11
andp
The slenderness of a wall λp is the ratio between the flat width of the wall bp and its thickness t
It results that
Winter proposed a semi - empirical relationship derived from that of von Karmanrsquos that takes into account the imperfections
maxmax
4150191
E
tb
Etb
peff
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
This is used by EC3 in the design of the strength of very slender sections
The following annotations are used
6730p 1
6730p
pp 220
11
SpecificationsThe effective width of a flat wall in compression andor in bending is determined considering the relative slenderness referred to the width of the flat wall bp and also the limit of yield strength fyb In order to identify the way the cross section of a wall is working we have to compare the effective slenderness with the limit slenderness
The recommended values of the maximum slenderness (limit slenderness) for different types of cold-formed sections are presented in table 1 The common experience and the tests in laboratory impose these values
The limit slenderness is defined as the ratio between the width and the thickness of the wall in the case when the normal stresses are uniformly distributed on the whole cross section and equal with the design strength of the material The values of the limit slenderness depend on the kind of the wall and the grade of the steel The presence of the imperfections reduces the theoretical values of these limits over which buckling may occur anytime see table
For
For
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
DIFFERENCIES BETWEEN BUCKLING OF IDEAL SLENDER MEMBER HOT ROLLED SECTION MEMBERS AND COLD FORMED SECTIONS
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
STRESSES DEVELOPED IN THE WALLS OF A COLD FORMED SECTION SUBJECTED TO BUCKLING
Thin walled C section in compression
Buckling ranges depending on slenderness
EUROCODE buckling curves
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
BUCKLING MODES OF COLD FORMED SECTIONS
Local Flexural+ Flexural Torsional
interactive buckling
a) Flexural + Flexural-Torsional interactive buckling
b) b)- Local+Flexural Torsional interactive buckling
Simple buckling modes for a C column (thin walled section)
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
DISTORTION OF OPENED SECTIONES WITH THIN WALLS
Combined modes of buckling leading to distortion of thin walled opened sections (wavelength is considered as buckling length in the case of a column)
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Moduri simple de flambaj ale unui profil C fara rebordF-flambaj prin incovoiere generala FT- flambaj prin
incovoiere-rasucire Lw Lf ndash distorsiunea inimii si respectiv a talpilor
Moduri de flambaj prin distorsiune
Forme de flambaj forte critice de flambaj si rezistente la flambaj n functie de lungimea elementului
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
The effective width and effective area of the walls in buckling
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
THE EFFECTIVE WIDTH AND EFFECTIVE AREA OF THE WALLS IN BUCKLING
bull Von Karmanrsquos theory mentions that the maximum stress in the wall σmax systematically reaches the elastic limit fy so a pattern of the determination of the effective widths comprises
bull Determination of the stress ratio ψ that shows the distribution of the stresses in the wall considered with its effective width (tab 2 and 3) For doubly supported elements the stress ratio may be based on the properties of the gross cross section
bull Considering the supports (internal wall or end wall as cantilever) and again the value of ψ the buckling coefficient is determined kσ
bull Relative slenderness is determinedbull Reduction factor ρ is determinedbull The effective width is calculated with the help of tables 2 and 3 Specificationbull In the case when the initial stress applied to the wall is small enough the amplified stress due to the lost
of the efficacy σmax may reach a value much lower than the elastic limit fy It is rational in this case to determine the effective width on the basis of the compression stress and not based on the limit of elasticity For that the parameter ε is computed by replacing fy with σcom as a first approximation of the σmax value
bull A new altered value of σmax is determined for the effective width based on the reiteration of the method and starting from the determination of the relative slenderness of the wall A procedure of convergence for the stress σmax until is reaches the recommended values is based on calculation of the relative slenderness of the effective wall and then using this value in the expression of ε
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
FLAMBAJUL PRIN DISTORSIUNE
Flambaj prin distorsiune a unui profil Z supus la compresiune (a) si la incovoiere (b)
a) Modelul de flambaj prin distorsiune utilizat de SR EN 1993-1-32006 b) rigidizare de capat pe fundatia elastica reprezentata de un resort c) modelul utilizat pentru determinarea coeficientului de rigiditate
Aria eficace a unei rigidizari de capat dupa SR EN 1993-1-32006
Perete nerigidizat (in consola)
Tensiunea critica elastica de flambaj la un element zvelt comprimat prins intr-o fundatie elastica avand coeficientul de pat K este determinata dupa Timoshenko si Gere in 1961
Coeficientul de flambaj prin distorsiune
unde zveltetea relativa pentru flambajul distorsional este
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
ELEMENTS WITHOUT STIFFENERS (PLANE ELEMENTS)I step
The reduction factor for the determination of the effective widths according to tab 4 for doubly supported or 5 for singly supported elements shall be obtained as we have already seen The value of relative slenderness is determined with
where
σcom ndash effective stress of compression on the extremities of the wall σ1 determined with respect to the effective area of the transversal section and multiplied with the safety factor γM1
kσ ndash buckling coefficient according to tab 4 and 5
II step
The design for the limit state of serviceability σ1-fy The value of the reduction factor ρ is determined with the relative slenderness obtained as in the I step where σcom = σ1 γM1 and the effective stress calculated is σ1 lt fyγM1
The following relationships are used
For we take ρ=1
For we take
After determining the values
III step
In tables 6 and 7 the geometrical width of the flat wall is bp In the case of the lateral webs without intermediate stiffeners (the folders of the sheeting) the annotation sw is equivalent with bp
kEt
bcomp
p 0521
6730pd01
60180
2201
pu
pdpu
pd
pd
kEt
bcomp
pd 0521
kE
f
t
b yppu
0521
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
DETERMINAREA LATIMII EFICACE A PERETILOR COMPRIMATI IN FUNCTIE DE TIPUL LOR SI DE DISTRIBUTIA DE TENSIUNI
Perete nerigidizat Perete rigidizat
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
ELEMENTS WITH EDGE OR INTERMEDIATE STIFFENERS
Determination of the stiffness (elastic restraint) of the spring determined by the
presence of edge stiffeners for C and Z sections real and equivalent system
Annotations for edge stiffeners for
a)- lipped channelb) with lip and clip
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
The design is based on the assumption that the stiffener works as a beam on elastic foundation represented by a spring stiffness depending on the bending stiffness of adjacent parts of plane elements and on the boundary conditions of the element
The determination of the spring stiffness is illustrated in figure 6 for intermediate and edge stiffeners respectively where Cs = 1fs and Cr = 1fr
The significance of the terms are
f- the deflection of the stiffener due to a force equal with 1
fs and fr ndash are taken as in the figure
For the rotational stiffness in the supports Cө Cө1 and Cө2 the effects of other stiffeners are considered if there is the case for any element that forms the cross section in compression
For an edge stiffener the deflection fy is determined with the relationship
where
In the case of C and Z
In the case of a intermediary stiffener Cθ1 and Cθ2 will be conservatory annulled deflection being thus determined - reduction factor χ due to buckling is determined considering the relative slenderness of the wall and imperfection coefficient according to the table of classification the cross sections by buckling curves (curve a 0)
- σ crs ndash the critical stress in the ideal plate (without imperfections)
3
22112
3 tE
bbf ppy
C
bp
1
3
2
21
22
21 112
)(3 Etbb
bbf
scr
ybf
222
201501
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Edge stiffeners
In order to determine the effective widths that split into several sections a stiffened wall the general method applied follows 7 successive steps it also may be simplified in a restrained form by imposing initial conditions Both methods may be developed iteratively General method
An initial effective area of the edge stiffener is determined based on the fact that it will act as an element infinite rigidly supported and subjected to a stress
The reduction factor χ will be determined for this stiffener but this time the elastic spring will be considered
The reduction factor will be improved by iteration The initial values of the effective widths bef1 and bef2 are obtained from the indications in the table considering that the wall is an intermediary one
The initial values of the effective widths cef si def are obtain as it follows
Lip
in the relationship ρ and are prior determined and the values of the local buckling coefficient kσ is determined as it follows
For kσ=05
- for
1
M
ybEdcom
f
cpef bc
350 p
cp
b
b
60350 p
cp
b
b3
2
35083050
p
cp
bb
k
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Lip and clip as for an intermediary wall and then as for an edge element
Sectional characterist ics of the effective section of the stiffener will be
- Area - Moment of inert ia with respect to the neutral axis of the effective section
Critical stress of the edge stif fener is determined with K=1f
Effective reduced area of the stif fener
The reduced thickness of the wall used for the effective area
Reducing factor χ for edge stif fener is determined based on the value of σcrs obtained priory but its value may be improved via iteration method if χlt1 through a value of the factor ρ after determining the compression stress
as to have
Iteration stops when χn+1 χn but χn+1lt χn cpef bc dpef bd
efefes dcbtA 2
s
sscr A
KEI2
sEdcom
Mybsreds Af
AA
1
s
redsred A
Att
1
M
ybEdcom
f
predp
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Simplified method
If the following condition is satisfied by the wall of which the width is bp
where - h - depth of the web adjacent to the wall on the opposite edge to the end stiffener- As- effective area of the edge stiffener that is
determined for an even distributed stress with be2 cef def determined according to the general method and
- r =031
Effective area of the stiffener is obtained with
but and χ=05
Effective cross section characteristics are determined based on the reduced thickness of the wall tsred In the case when χ=1 and r=486 the stiffener plays the role of a support for the adjacent wall
32
2 51
t
b
E
f
b
hAI pyb
psrs
efefes dcbtA 2
1
M
ybEdcom
f
predp
Edcom
Mybsreds
fAA
1
sreds AA
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
ETAPIZAREA CALCULULUI SECTIUNII EFICACE A UNEI TALPI RIGIDIZATE MARGINAL
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Determinarea coeficientului de rigiditate k si a coeficientului de reducere dupa SR EN 1993-1-32006
Pereti rigidizati
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Strength Verifications of the Cold Formed Thin Gauge Members(Considering the Buckling of the Walls)
1 Members in Tension RdtEd NN
NEd ndashtensile force due to design load
NtRd ndash the minimum of )(
0
RdtM
ya FAf
RdtF - net section resistance depending on fastener type according to chapter 4 Joints and Connections
2 Members in Compression RdcEd NN
NEd ndash compressive force due to design load
1
M
effyRdc
AfN
The shift of the neutral axis eN is the cause of an additional bending moment NEdEd eNM
3 Members in Bending RdcEd MM
MEd ndash the bending moment due to the design load
McRd ndash the bending moment resistance of the section 1M
effy Wf
Recommended
