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STATIC DESIGN OF LINDAB Z AND C BEAMS

DESIGN GUIDE

SECOND EDITION

By: Dr. Lszl Dunai

Sndor dny

LINDAB LTD. HUNGARY, 1998.

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Contents

1. Introduction...................................................................................................................3

1.1 The subject of this Design Guide............................................................................ 3

1.2 The Lindab Z and C profiles ............................................................................ 3

1.3 Applied Design Codes............................................................................................. 4

2. Geometrical and material properties .......................................................................... 5

2.1 Profile dimensions...................................................................................................5

2.2 Calculation of Section Properties............................................................................ 8

2.2.1 Design Thickness ...........................................................................................8

2.2.2 Interpretation of Effective Section Properties................................................8

2.3 Material Properties ..................................................................................................8

2.3.1 Steel Grade ..................................................................................................... 8

2.3.2 Interpretation of Design Strength................................................................... 9

3. Structural Detailing and Static model.......................................................................10

3.1 Structural detailing ................................................................................................ 10

3.2 Static Model .......................................................................................................... 13

4. Loads of Thin-Walled Beams.....................................................................................15

4.1 Permanent Loads ................................................................................................... 15

4.2 Variable Loads ...................................................................................................... 15

4.2.1 Live Loads....................................................................................................15

4.2.2 Meteorological Loads................................................................................... 16

4.2.3 Technological/Constructional Loads ........................................................... 18

4.3 Load Model ...........................................................................................................19

4.4 Critical Load Combinations .................................................................................. 20

5. Ultimate Limit States of thin-walled Beams ............................................................. 20

5.1 Bending Resistance ...............................................................................................21

5.1.1 Both Flanges Braced: Bending Resistance #1 .............................................21

5.1.2 The Tension Flange Free, the Compression Flange Braced: Bending Resistance#2........................................................................................................................... 22

5.1.3 The Compression Flange Braced, the Tension Flange Free: Bending Resistances#3 and #4 ...............................................................................................................22

5.1.4 Both Flanges Free: Bending Resistance #5.................................................. 235.1.5 Checking of the Bending Moments..............................................................23

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5.2 Shear Resistance of the Web................................................................................. 24

5.3 Web Crippling Resistance..................................................................................... 24

5.4 Resistance Against Combined Moment and Shear Force..................................... 25

5.5 Resistance Against Combined Moment and Concentrated Force......................... 25

5.6 Checking of Connections ......................................................................................25

5.6.1 Splices .......................................................................................................... 25

5.6.2 Cantilever Support of the Beam...................................................................26

6. Serviceability Limit States of Thin-Walled Beams ..Hiba! A knyvjelz nem ltezik.

7. Performing the Static Design By Calculation...........................................................27

7.1 Static Design Based on Design Tables.................................................................. 27

7.1.1 Composition of Design Tables..................................................................... 277.1.2 Using SpanLoad Design Tables .................................................................28

7.2 Static Design Based on Detailed Analysis ............................................................ 29

8. Examples of the Design of Lindab Thin-Walled Beams..........................................27

8.1 Checking of a Z Section Purlin.......................................................................... 33

8.1.1 Structural System #1 .................................................................................... 33


8.1.3 Structural System #3 .................................................................................... 368.2 Checking of a C and Z Section Wall Beam in a Lindab Industrial Type Building.....................................................................................................................................47



8.3 Checking of a C Section Floor Beam ................................................................ 50

9. Design Table of Lindab Z profiles ......................................................................... 30

10. Design Table of Lindab C Profiles.......................Hiba! A knyvjelz nem ltezik.

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1. Introduction1.1 The subject of this Design GuideThis is the second edition of the Design Guide published by Lindab Hungary Ltd. in 1996 un-der the same title. This Guide deals with the static design of thin-walled Lindab C and Zsection beams used for various different purposes, such as roof purlins, wall beams, or floor

beams. It explains the properties of thin-walled profiles significant from the point of view ofstatic design, shows the theoretical background of the dimensioning process, and as an at-tachment, contains useful tables and examples for practical design.

The composition of the second edition of this Guide is the identical to the first edition, and therecommended computational methods are the same. However, these design methods are nowapplied to a new, expanded profile library. The Guide explains how the new profiles may beused for larger spans, giving the appropriate design tables. It also includes a new structuralsystem not discussed by the first edition. The guide examines in a practical way the lateral

support conditions for the flanges of thin-walled profiles. Some of the examples are new orupdated. The example for the design of a roof purlin explains how the effects of differenttypes of snow accumulation may be considered during the static design.

1.2 The Lindab Z and C profilesThe dimension range of the new Lindab "Z" and "C" profiles has been modified: the range ofthe web height has increased from 100200 mm to 70350 mm, and the range of the platethickness has increased from 1.02.5 mm to 0.73.0 mm. The web/flange ratio has alsochanged in order to ensure a more effective load-bearing performance. For this reason, the 1st

and the 2nd editions give different results for old and new profiles with the same web height.The new "Z" and "C" profiles are included in Table 1.1 (the height in millimeters appears inthe name of the profile and tmarks the nominal plate thickness).

"Z" profiles "C" profiles

C-70 / t= 0.7, 1.0, 1.5

Z-100 / t=1.0, 1.2, 1.5, 2.0 C-100 / t= 0.7, 1.0, 1.2, 1.5, 2.0

Z-120 / t=1.0, 1.2, 1.5, 2.0, 2.5 C-120 / t= 0.7, 1.0, 1.2, 1.5, 2.0, 2.5

Z-150 / t=1.0, 1.2, 1.5, 2.0, 2.5 C-150 / t= 0.7, 1.0, 1.2, 1.5, 2.0, 2.5

Z-200 / t=1.0, 1.2, 1.5, 2.0, 2.5 C-200 / t= 1.0, 1.2, 1.5, 2.0, 2.5

Z-250 / t= 1.5, 2.0, 2.5, 3.0 C-250 / t= 1.5, 2.0, 2.5, 3.0

Z-300 / t= 1.5, 2.0, 2.5, 3.0 C-300 / t= 1.5, 2.0, 2.5, 3.0

Z-350 / t= 2.0, 2.5, 3.0 C-350 / t= 2.0, 2.5, 3.0

Table 1.1: Lindab "Z" and "C" profiles

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1.3 Applied design codesThe principles of the design method discussed in the 2nd edition are equivalent to those of the1st edition, as they had been reviewed and approved by the Hungarian Construction QualityControl Institute (MI). The loads and the serviceability limit state criteria are defined ac-

cording to the relevant Hungarian standards, while the ultimate limit state criteria are basedon the Swedish standard for thin-walled structures. References to the relevant Eurocodespecifications can also be found in this Guide.

[1]MSZ (Hungarian Standard) 15020-86: Static Design of Loadbearing Structures of Build-ings. General Regulations.

[2]MSZ (Hungarian Standard) 15021/1-86: Static Design of Loadbearing Structures ofBuildings. Loads of Buildings.

[3]MSZ (Hungarian Standard) 15021/1-86: Static Design of Loadbearing Structures ofBuildings. Stiffness requirements of Buildings.[4]MSZ (Hungarian Standard) 15021/1-86: Static Design of Steel Structures of Buildings.

Design Regulations.

[5]MSZ (Hungarian Standard) 15021/1-86: Static Design of Steel Structures of Buildings.Dimensioning Procedures.

[6]ME 04 18082: Design by Calculation, Joint Detailing and Control Analysis of Thin-walled Steel Structures.

[7]ENV 1993-1-1: 1992: Eurocode 3: Design of Steel Structures.Part 1-1: General Rules for Buildings.

[8]ENV 1993-1-1: 1996: Eurocode 3: Design of Steel Structures.Part 1-3: General rules Supplementary rules for cold formed thin gauge members andsheeting.

[9]StBK-5: Swedish Code for Light-Gauge Metal Structures, Swedish Institute of Steel Con-struction, March 1982.

[10] EN 10147: Specification for continuously hot-dip zinc coated structural steel sheet Technical delivery conditions.

[11] BS 5950: Part 5: 1987: Structural use of steelwork in building.Part 5: Code of practice for design in cold formed sections.

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2. Geometrical and material properties2.1 Profile dimensionsThe geometrical detailing of the Lindab Z and C profiles is illustrated on Figure 2.1 andtheir geometrical dimensions are given in Table 2.1.

Figure 2.1: Lindab "Z" and "C" profiles

Comments:

The geometrical dimensions are understood as the enclosing dimensions, The different size of the flanges enable the joint of the beams by slipping one member into

the next one.

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Z C

A[mm]

B1 B2 L t A

[mm]

B1 B2 L t

100 41 47 16.2 1.0 70 41 47 8.8 0.7

41 47 16.8 1.2 41 47 9.7 1.0

41 47 17.7 1.5 41 47 11.2 1.5

41 47 19.3 2.0 100 41 47 15.3 0.7

120 41 47 16.2 1.0 41 47 16.2 1.0

41 47 16.8 1.2 41 47 16.8 1.2

41 47 17.7 1.5 41 47 17.7 1.5

41 47 19.3 2.0 41 47 19.3 2.0

41 47 20.9 2.5 120 41 47 15.3 0.7

150 41 47 16.2 1.0 41 47 16.2 1.0

41 47 16.8 1.2 41 47 16.8 1.2

41 47 17.7 1.5 41 47 17.7 1.5

41 47 19.3 2.0 41 47 19.3 2.0

41 47 20.9 2.5 41 47 20.9 2.5

200 66 74 19.7 1.0 150 41 47 15.3 0.7

66 74 20.3 1.2 41 47 16.2 1.0

66 74 21.2 1.5 41 47 16.8 1.2

66 74 22.8 2.0 41 47 17.7 1.5

66 74 24.4 2.5 41 47 19.3 2.0

250 66 74 23.7 1.5 41 47 20.9 2.5

66 74 25.3 2.0 200 66 74 19.7 1.0

66 74 26.9 2.5 66 74 20.3 1.2

66 74 28.5 3.0 66 74 21.2 1.5

300 82 90 28.2 1.5 66 74 22.8 2.0

82 90 29.8 2.0 66 74 24.4 2.5

82 90 31.4 2.5 250 66 74 23.7 1.5

82 90 33.0 3.0 66 74 25.3 2.0

350 92 100 28.8 2.0 66 74 26.9 2.5

92 100 30.4 2.5 66 74 28.5 3.092 100 32.0 3.0 300 82 90 28.2 1.5

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82 90 29.8 2.0

82 90 31.4 2.5

82 90 33.0 3.0

350 92 100 28.8 2.092 100 30.4 2.5

92 100 32.0 3.0

Table 2.1: Geometrical data of Lindab "Z" and "C" profiles.

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2.2 Calculation of section properties2.2.1 Design thicknessFrom the point of view of static design, the thickness of the thin-walled Z and C profiles

is described by the dimensions defined below, according to Standard [9]:tn Nominal thickness: the thickness of the steel plate without any coating,

t Design thickness: the thickness of the steel plate to be used for design purposes,

tmin The minimum value of plate thickness: smaller thickness may not appear in a statis-tical sample.

According to Standard [9], design thickness is interpreted as:

ttmin=0 95.

(2.1)

2.2.2 Effective section propertiesThe section properties are computed according to the following principles:

Sectional geometry is defined by the plate midlines. Local plate buckling of the compression elements in thin-walled Z and C profiles is

allowed for by using the so-called "working" or effective" width and thickness. The fol-lowing steps are to be taken: computing the compressive stress, determining the effective plate width and thickness,

calculating the effective section properties.

In case of positive and negative bending moments, effective moments of inertia and sec-tion moduli can be calculated for the asymmetrical thin-walled Z and C profiles: Ieff

+,Ieff

-, Weff+, Weff

-. Since the flanges of the Lindab Z and C profiles differ only slightly,for practical purposes they may be considered to be the same size, and common section

properties may be calculated for the upper and lower flanges using an average flangewidth: Ieff, Weff.

2.3 Material properties2.3.1 Steel gradeThe Lindab Z and C profile beams are manufactured from sheets of the steel grade de-fined by standard [10]:

FeE 350G; Ry = 350 MPa, Rm = 420 MPa, E = 210,000 MPa.Ry is the characteristic value of the yield limit,Rm is the characteristic value of the ultimate tensile strength,Eis the modulus of elasticity.

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2.3.2 Design strengthThe design value of the yield strength, according to the general definition given by [9] is thefollowing:

H

y

mn

R

= (3.2)

mn m n= = 1 0. (3.3)

m - a partial safety factor accounting for the uncertainty of material quality,

n - a partial safety factor accounting for the intended purpose of the structural element.

Notes:

m = 1.0, according to [9]; (3.4)To obtain the value of this factor, the scatter of statistical variables (material- and geomet-rical properties) defining the tensile and compressive strength of the plates has been stud-ied, both individually and combined as loadbearing capacity values. Based on the resultsof the statistical analysis, the design values of thickness and yield strength were defined asthose belonging to the loadbearing capacity value with a 96% probability of exceedence.

n = 1.0, 1.1, 1.2, according to [9]; (3.5)The value of this factor depends upon the intended purpose of the given structural elementin the global structure. The relevant chapters of the Hungarian Standard [2, 3, 6] used inthisDesign Guide as a supplement, apply this safety factor in a different context and in a

different way, therefore its value in the calculations is considered to be a constant 1.0.

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3. Structural detailing and static model3.1 Structural detailingThin-walled beams usually have a so-called secondary loadbearing function, which meansthat they transfer the direct loads from roof and wall claddings and floor decks to the primaryloadbearing elements or the main beams. Roof purlins, wall beams and light-weight floor

beams are typical secondary loadbearing members.

The thin-walled beams are supported by primary loadbearing structures, therefore the detail-ing of the latter elements defines the span. The supports are usually devised so as to assure thesupport of the thin-walled beams in two directions, as shown on Figure 3.1. If the thin-walled

beam sits directly on top of the supporting member, then the bearing length of the beam is acharacteristic feature of the structural design as well. Thin-walled beams may be used as partof the global bracing system, and as such, they may have supplementary mechanical functions(e.g. supporting compression elements of the main beam.) The fastening of the thin-walled

beams must harmonize with this function also.

Figure 3.1: A typical support of thin-walled beams

The loadbearing elements of the cladding (e.g. profiled sheeting) are fixed directly to the thin-walled beams. The cladding according to its structural function may be connected to theZ and C beams in several different ways: to one of the flanges (to the upper- or the lowerflange, from the point of view of load direction), or to both flanges (e.g. when upper andlower sheetings are both used.) In this arrangement, besides its direct loadbearing function,the sheeting performs an auxiliary mechanical function as well, by laterally supporting theflanges of the thin-walled beams. The free flanges may be supported laterally by using staysor suspending bars (see Chapters 5 and 8 for details).

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The problem of the joints of the thin-walled beams can be solved from the point of view ofmanufacture, assembly, structural- and mechanical behavior. Simply supported beams, or in case of small spans continuous beams, can be constructed from a single unit, withoutany joints. In case of continuous beams made of multiple units, the asymmetrical flanges ofthe Z and C profiles allow one member to slip into the other, forming overlapping joints,as shown on Figure 3.2.

Figure 3.2: Overlapping joint of "Z" and "C" profiles

It is usually suitable to place the joints above the supports, by slipping one member into theother, or by inserting an connection element, according to Figure . This way, the greater bend-ing moment occurring at the supports of continuous beams is carried by the two profiles to-gether. This arrangement allows the increase of the loadbearing capacity and bending stiffnessof the outer spans, which are critical from the point of view of mid-span bending momentsand maximum deflections. This increase can be achieved in the following ways:

By using a thicker outer element of the same height as the base profile, by using a supplementary element of the same height as the base profile.The length of the overlapping joint is generally 0.2 L (whereL is the span); and0.3 L at the second support. The length of the supplementary element to be used in the outerspan is 0.8 L.

The above mentioned assembly units have the following length according to Figure 3.3:

Z in the outer spans: 1.2 L + cantileverin the inner spans: 1.2 Lsupplementary element: 0.8 L

C in the outer spans: 1.1 L + cantileverin the inner spans: 1.0 Lconnection element: min. 0.2 L + 150 mm (standard length: 1,600 mm)

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Figure 3.3: Overlapping system of "Z" and "C" beams

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3.2 Static modelThe static model of the thin-walled beams based on their primary structural function may be simply supported, or continuous girders. In a general case, the span length, bendingrigidity and the loads may vary along the beam. Mechanical behavior is considerably influ-

enced by the lateral support of the flanges. Depending on the type of support, different situa-tions may occur, from the relatively simple case of plane bending to the combination of biax-ial bending and torsion. Therefore, the design of a thin-walled girder with arbitrary geometryand loading is a very complex process.

Considering the requirements of practical design, the static model can be simplified to allowfast computations for typical situations, based on the following assumptions:

In case of continuous beams, the spans are equal (a span is the distance of the center linesof the supports).

The load model is an equally distributed total load. Six typical girder models are defined (see Figure 3.4):

1. Simply supported beam,2. Continuous beam with three supports, without any joints, or with joints using connecting

elements,

Figure 3.4: Typical static models

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3. Continuous beam with four or more supports, without any joints, or with joints using con-necting elements; possibly employing stronger (thicker) profiles in the outer spans,

4. Continuous beam with three supports, with an overlapping joint at the middle support,5. Continuous beam with four or more supports, with overlapping joints at the inner sup-

ports, and equal- or higher strength profiles in the outer spans,6. Continuous beam with four or more supports, with overlapping joints at the inner sup-

ports, and supplementary elements in the outer spans.

The bracing action of the cladding and the transfer of the loads is taken into consideration inthe presentDesign Guide by making the following assumptions:

The loadbearing structures and the connections of the claddings are rigid and strongenough to laterally support the flanges of the Z and C beams.

The lateral supports of the flanges may be assumed to be continuous. The claddings are fastened to the beams having sufficient resistance to transfer sucking

loads.

Three basic support and load-transfer situations are defined, as shown on Figure 3.5:1. Both flanges are braced,2. The tension flange is free, the compression flange is braced,3. The compression flange is free, the tension flange is braced,

a) Pushing load,b) Sucking load.

Figure 3.5: Lateral bracing of the flanges

The Guide provides design tables for quick dimensioning of the typical structures shownabove. In case of static models other than those defined above, detailed static analysis must be

performed, with the help of loadbearing capacity data given by the Guide. Even then, how-ever, the design tables may be used effectively for preliminary calculations.

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4. Loads of thin-walled beams4.1 Permanent loadsThe self-weight of the thin-walled beam (qg kN/m), and all other loads and effects acting

permanently and constantly on the beam must be considered as permanent loads. The charac-teristic value of the permanent loads from the self-weight of the elements supported by thethin-walled beam (pa), and the partial safety factor () that belongs to the design value can besummarized according to Standard [2] as follows:

Self-weight of the thin-walled beam:qg is the self-weight of the beam [kN/m],g = 1.1, if the direction of the self-weight coincides with the studied effect (e.g. in case ofsnow load), or g = 0.8, if the direction of the self-weight is opposite of the direction ofthe studied effect (e.g. wind-sucking.)

Cladding layers:pr is the self-weight of the given layer [kN/m2], calculated from its average air-dry bulkdensity, that must be reduced to the thin-walled beam, qr[kN/m].

1. Concrete, and reinforced concrete structures, masonry, metal and wooden structures,r= 1.1, or 0.8,

2. Pre-fabricated light-concrete structures, thermal and acoustic insulation,r= 1.2, or 0.7,

3. Light-concrete slabs, plaster, leveling and smoothing layers,r= 1.3, or 0.7,

Concentrated permanent loads:The concentrated loads G [kN] occurring in case of thin-walled beams used inroof, wall and floor structures (e.g. fanglight, column), must be assumed to act in the

place and arrangement specified by the plans. Interconnected partition walls, supportedon each story by the floor structure, with a thickness not more than 10 cms without plas-ter, may be regarded as a distributed load on the co-acting section of the supporting floor,and reduced on the beam accordingly.

The permanent load acting on a thin-walled beam that supports a roof or floor structure con-sisting ofn layers is calculated as follows:

characteristic value: q q qa,a g r,= += ii

n

1

(4.1)

design value: q q qa,sz g g r, r,= += i ii

n

1

(4.2)

4.2 Variable loads4.2.1 Live loadsTable 2 of Standard [2] contains the characteristic values of live loads (p h [kN/m2] and Ph[kN]), carried by floors comprising of thin-walled beams. The characteristic values of dy-namic live loads must be multiplied by a dynamic factor (), unless a detailed dynamic calcu-

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lation is carried out (see Table 3 of Standard [2].) Partial safety factors of live loads are de-fined by Standard [2] as follows:

h = 1.2 in case of concentrated live loads (Ph) and live loads distributed along lines (qh).

h = 1.4 if ph < 2.0 [kN/m2],

h = 1.3 if 2.0 ph < 5.0,

h = 1.2 if 5.0 ph.

4.2.2 Meteorological loadsSnow load:

According to Standard [2], the characteristic value of snow load on a roof surface with an an-gle 30 to the horizontal plane, reduced to the horizontal projection of the roof area,

measured in [kN/m2

], is the following:At an altitude of M 300 meters above sea level:

ps = 0 8. (4.3)

At an altitude of M > 300 meters above sea level:

pM

s = +

0 8300

1000 2. . (4.4)

If the angle of the roof is 60, snow load is not assumed to act, if 30 < < 60, the char-acteristic value is obtained by linear interpolation.

Snow loads are usually uniformly distributed (in case of 20 roof angles, or flat archedroofs, when arch height/span 1/8.) However, if roof shape or the position of several con-necting roof planes results in possible snow accumulation on some parts of the roof, then this

possibility must be reflected in the calculations (see locally increased snow load in AppendixF1 of Standard [2].)

The partial safety factor of snow loads for thin-walled beams is defined by Standard [2] asfollows:

s = 1.4, (1.0) if ga/ps 1.0,

s = 1.75, (1.25) if ga/ps 0.4,

for intermediate ga/ps ratio values, partial safety factors are determined by linear interpolation.The numbers in parentheses may be used for temporary buildings (with maximum 5 years ofdesign life.)

Wind load:

According to [2], the characteristic value of wind load is calculated as follows:

pw = c w0, (4.5)

c is the pressure coefficient,

w0 is the dynamic pressure of the wind.

Dynamic pressure of the wind at a height ofh meters from the ground, for buildings not tallerthan 100 meters, standing in an open area, can be calculated in [kN/m2] as follows:

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wh

o =

0 7 10

0 32

..

(4.6)

If the neighborhood of the building is an urban or industrial area, with evenly distributedbuildings higher than 10 meters:

wh

o =

0455 10

0 44

..

(4.7)

For a building of constant width, standing in an open area, an average value may be usedalong the entire height:

wh

o =

0603 10

0 32

..

(4.8)

The lowered average value to be used in a built-up area:

w ho =

0373 10

0 44.

.(4.9)

Standard [2] gives wind pressure coefficients for different types of buildings. Closed, orpartly open buildings where no more than 30% of the surface area is open or can beopened are typical when designing roof purlins or wall beams. According to Standard [2],the pressure coefficients for the outer plane surfaces of these buildings can be summarized asfollows:

For lateral walls, on the windward side: c = +0.8 (+ wind pressure, wind sucking), For lateral walls, on the lee side:

c3 = 0.4, if h/l 2, c3 = 0.6, if h/l 3,where h is the height of the lateral wall, and l is the width of the building parallel to thedirection of the wind; intermediate c3 values can be obtained by linear interpolation,

For lateral walls parallel to the direction of the wind: c4 = 0.4, On a plane roof surface, the values ofc1 and c2on the windward and the leeward sides

must be determined according to the roof angle, assuming two different possibilities ofwind load; Case 1 is illustrated on Figure 4.1, Case 2 is illustrated on Figure 4.2.

The partial safety factor of the wind load is usually w = 1.2 when designing thin-walledbeams. In case of temporary buildings (whose design life is maximum 5 years), a partial

safety factor ofw = 1.0 may be used.

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Figure : Wind load, case 1 Obtaining the pressure coefficients

Figure 4.1: Wind load, case 2 Obtaining the pressure coefficients

4.2.3 Technological/constructional loadsDuring construction, the loads acting on the floors must be considered to take their worst pos-sible values. A minimum uniformly distributed load of ge = 1.0 kN/m

2 must be assumed to actas technological load, or if more unfavorable two concentrated P e = 1.0 kN concen-

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trated forces 1.0 meter apart, distributed on 10 x 10 cm surfaces. Partial safety factors for thetechnological loads are calculated as explained above for the live loads.

4.3 Load modelThe load model is a reduction of the standard loads defined in the preceding chapters onto thestatic model of the thin-walled beam. The load model is determined in two steps:

1. The reduction of distributed surface loads (p) to distributed loads acting along lines (q),based on the static model of the cladding. For practical purposes, sufficient precision canbe achieved by assuming a simply supported static model for the cladding:q = p bg (4.10)bg is the distance of the thin-walled beams.

2. The reduction of the linear loads (q), to the static model of the thin-walled beam (calcula-tion of normal (qn) and transversal (qt) linear loads):

In case of permanent loads:qa,n = qa cos (4.11)qa,t =qa sin (4.12)where is the roof angle.

Figure 4.2: Reduction of permanent loads

In case of snow load:qs,n = qs cos

2 (4.13)qs,t =qs cos sin (4.14)

Figure 4.3: Reduction of snow load

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In case of wind load:qw,n = qw (4.15)qw,t = 0 (4.16)

Figure 4.4: Reduction of wind load

4.4 Critical load combinationsThe characteristic value of the critical load combination forn variable loads is the following:

q q q qa a,a e, e, e,= + +

=1

2

i i

i

n

(4.17)

qa,a is the characteristic value of the permanent load,

qe,1 is the characteristic value of the critical (most unfavorable) variable load,

qe,i is the characteristic value of i-th variable load,

e,i is the simultaneity factor of the i-th variable load:

e = 0.6 for meteorological loads,e = 0.8 for the live load of floors, if at least 50% of its characteristic value is quasi-

permanent.

The design value of the critical load combination forn variable loads is the following:

q q q qsz a,sz e,1 e, e, e, e,= + +=1

2

i i i

i

n

(4.18)

qa,sz is the design value of the permanent load,

e,1 is the partial safety factor of the critical (most unfavorable) variable load,

e,i is the partial safety factor of the i-th variable load.

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5. Ultimate limit state analysis of thin-walled beams5.1 Bending resistanceBending resistance of thin-walled beams depends mainly on the lateral bracing of the flanges.Based on their longitudinal distribution, the supports may be continuous (e.g. when profiledsheeting is fastened frequently to the flange) ofpartial (e.g. when using stays or suspending

bars). Based on their rigidity, the supports may be rigid orelastic.

When performing the ultimate limit state (ULS) assessment, this Guide assumes that theflanges are braced laterally by continuous and rigid supports according to the following pos-sibilities:

1.both flanges are braced (Figure 3.5/1),2. the tension flange is free, the compression flange is braced (Figure 3.5/2),3.

the compression flange is free, the tension flange is braced, and a gravitational load (com-pression) acts upon the tension flange (Figure 3.5/3a),

4. the compression flange is free, the tension flange is braced, and wind sucking (tension)acts upon the tension flange (Figure 3.5/3b).

Next, the above mentioned possibilities will be discussed in detail. The ULS assessment offree or partially braced flanges is reviewed generally in this chapter, suggestions for practicaldesign can be found in Chapter 8.

5.1.1 Both flanges braced: bending resistance MH,1Failure is determined by the local plate buckling resistance of the compression flange and thesections of the web in compression. (For less slender plates, failure may be determined by theresistance of cross-section.) As a result of biaxial bending and torsion, the forces illustratedon Figure 5.1 act at the supports of Z or C section beams.

Figure 5.1: Both flanges braced support reactions

Local plate buckling resistance of thin-walled members is usually computed based on the ef-fective or working plate width. Depending on the compressive stress in the flange, thecrippled portions are eliminated and working stripes are assumed. The bending resistance

based on the ULS loadbearing capacity of the above described effective cross-section will be:

M wH,1 eff H= (5.1)

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H is the ultimate tensile strength of the plate material,

weff is the section modulus of the effective cross-section.

5.1.2 The tension flange free, the compression flange braced: bending resistance MH,2For the compression flange, failure is determined by the local plate buckling resistance de-scribed in the previous part. For the tension flange, failure by combined flexure and torsion isdetermined by the resistance of cross-section. The resistance of the flange in tension is usu-ally higher than the resistance of the compression flange, therefore the bending resistance isthe following:

M MH,1 H,2= (5.2)

5.1.3 The compression flange braced, the tension flange braced: bending resistancesMH,3 and MH,4

Failure is determined by the lateral-torsional buckling resistance, or the local plate bucklingresistance of the compression flange. Lateral-torsional buckling of the compression flangeresults in the deformation of the thin-webbed profile. The part of the profile in compressionmay be analyzed as a beam element laterally supported by a continuous spring, as shown onFigure 5.2, (flange rigidity analysis). Spring stiffness is different if gravitational (compres-sive) or wind sucking (tensile) loads are transferred from the braced tension flange. Accord-ing to this, different bending resistances may be computed for compressive loads (MH,3), andtensile loads (MH,4). The general formula for the calculation of bending resistance is the fol-lowing:

M wH,3/4 eff ,ny kH,3/4= (5.3)

weff,ny is the section modulus of the effective cross-section for the compressionflange,

kH,3/4 is the lateral-torsional buckling strength for compressive or tensile loads, de-pending on the bedding and stiffness characteristics of the equivalent compres-sion member.

Figure 5.2: A model for the lateral-torsional buckling of "Z" profiles

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5.1.4 Compression flange partially braced, tension flange continuously braced: bendingresistance MH,r

Failure is determined by the lateral-torsional buckling resistance and web buckling resistanceof the compression flange, as described in the previous chapter. Between two lateral point

supports, this occurrence can be analyzed with the model explained in Chapter 5.1.3. Besidesthe nature of the loading, the distance between the point supports and the structural detailingof the supports are also important features of bending resistance computation. The generalformula for the bending resistance is the following:

M wH r eff ny kH r , , ,= (5.4)

weff,ny is the is the section modulus of the effective cross-section for the compressionflange,

kH,r is the lateral-torsional buckling strength for compressive or tensile loads, de-pending on the bedding and stiffness characteristics of the imaginary equiva-

lent compression member and the structural detailing of the supports.

5.1.5 Both flanges free: bending resistance MH,sFailure is determined by the lateral-torsional buckling resistance of the member in flexure, orthe local plate buckling resistance of the plate fields in compression. In practice, this situationmay be critical during assembly. Bending resistance is computed according to the followingformula:

M = wH,s eff,ny kH,s (5.5)

weff,ny is the is the section modulus of the effective cross-section for the compressionflange,

kH,s is the lateral-torsional buckling strength, as a function of the lateral-torsionalbuckling slenderness of the laterally unsupported beam.

5.1.6 Checking of the Bending MomentsBending resistance must be checked in the critical cross-sections of the static model accord-ing to the following formula:

M MM H,i (5.6)MM is the critical bending moment calculated from the design values of the load,

MH,i is the bending resistance depending on the current support- and load characteristics.

Notes: The bending resistances to be used in the practical design of Lindab Z and C sec-tions (MH,1, MH,2, MH,3,MH,4) can be found in the Design tables of Chapters 10 and 11 (forC sections, only MH,1 and MH,2 bending resistances are specified). For free or partially

braced members, practical recommendations can be found in Chapter 8.

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5.2 Shear resistance of the webShear failure of the web usually occurs as a loss of stability: shear buckling (for less slender

plates, failure may be determined by the resistance of cross-section.) Shear resistance in theplane of the web is obtained according to the following expression:

T b tH g H*= (5.7)

bg is the theoretical length of the web (the distance between the flange connections),

t is the design thickness of the plate,

H* is the shear buckling strength, depending on the ultimate tensile strength of the mate-

rial and the plate slenderness of the web.

Shear resistance must be checked in the critical cross-sections of the static model according tothe following expression:

T TM H (5.8)TM is the critical shear force calculated from the design values of the load.

5.3 Web Crippling ResistanceWeb crippling is a typical failure of thin-walled beams, due to direct compressive loads usually support reactions. In case of thin-walled beams without web stiffening, web cripplingresistance may be calculated as follows:

( )( ) ( )( )F t E r t b tH 2 H= + +015 1 01 05 0 02 2 4 902

. . / . . / . / (5.9)

E is the modulus of elasticity,

H is the ultimate tensile strength of the material,

b is the bearing length of the concentrated force,

t is the design thickness of the plate,

r is the inner bend radius of the sheet (3 mms for Lindab beams),

is the angle between the web and the loaded flange.

For outer supports, if the bearing length is less than 1.5 times the height of the profile, webcrippling resistance must be reduced by half.

Web crippling resistance must be checked in the critical cross-sections of the static model ac-cording to the following formula:

F FM H (5.10)

FM is the critical concentrated force calculated from the design values of the loads.

Note: Shear- and web crippling resistances of Lindab Z and C sections can be found inthe design tables in Chapters 10 and 11.

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5.4 Interaction of bending moment and shear forceM

M

T

TM

H

M

H+ 1 3. (5.11)

MM and TM are the critical internal forces and moments calculated from the design valuesof the loads, considering the simultaneous action of M and T,

MH and TH are the bending- and shear resistances.

5.5 Interaction of bending moment and concentrated forceM

Mif

F

FM

H

M

H

1 0 0 25. . (5.12)

M

M

F

Fif

F

FM

H

M

H

M

H

+ 0 64 116 0 25 1 0. . . . (5.13)

MM and FM are the critical internal moments and concentrated forces calculated from thedesign values of the loads, considering the simultaneous action of M and F,

MH and FH are the bending- and the web crippling resistances.

5.6 Checking of connections5.6.1 SplicesIn case of structural systems described in Chapter 3, where continuous beams are formed by

constructing joints at the supports, the overlapping members, or the members and the connec-tion elements, are bolted together as illustrated on Figure 5.3.

Figure 5.3: Detailing of bolted connections

Single-shear bolted connections must be checked for critical shear forces. In case of the staticmodels described in thisDesign Guide, the following shear forces may be considered for thedesign of connections:

Connection #1 and #2 of two-span beams:T q Lk,M sz= 0 57. (5.14)

qsz is the design value of the critical distributed load intensity,

L is the span.

Connection #1, #2 and #3 of continuous beams with four or more supports:

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T q Lk,M sz= 0 44. (5.15)

Connection #4 of continuous beams with four or more supports:

T q Lk,M sz= 0 2. (5.16)

5.6.2 Cantilever support of the beamThe typical detailing of the support of thin-walled beams is described in Chapter 3 (See Fig-ure 3.1). Bolted and welded connections of the support are checked for the design value of thecritical support reaction:

Beam supporting cantilever connection: single-shear bolted connection, checked for thesupport reaction in the plane of the web,

Cross section of the supporting cantilever: checked for the support reactions in the planeof the web and perpendicular to the cantilever,

Supporting cantilever main beam connection: The connection is welded around by con-tinuous fillet weld, checked for support reactions in the plane of the web and perpendicu-lar to the cantilever.

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6. Serviceability limit state analysis of thin-walled beamsThe serviceability limit state of thin-walled beams in terms of rigidity can be defined by beamdeflection. Maximum deflections due to the characteristic values of the loads are limited by

the stiffness requirements of the relevant standards, according to the following formula:e eM H (6.1)

eM is the maximum beam deflection due to the characteristic value of the loads,

eH is the deflection limit according to the appropriate stiffness requirement.

The following assumptions were made for the calculation of deflections when preparing thedesign tables of this Guide:

When calculating deflections, the bending rigidity of the thin-walled beams may be com-puted from the moment of inertia of the gross cross-section, i.e. without the deduction of

fastener holes. Internal forces are computed by assuming an unvarying cross-section along the beam (lar-

ger section moduli of overlapping joints or stronger profiles in the outer spans are ignoredas simplification on the safe side);

The maximum deflection of a certain span is obtained from the moment of inertia in thatspan.

Stiffness requirements pertaining to maximum beam deflections are summarized below, basedon the relevant Hungarian and Eurocode Standards:

eH = L/200 (6.2)generally for roof- and floor beams, according to [3] and [7],

eH = L/300 (6.3)if the beam is part of the global bracing system, according to [4], (note: the pertainingEurocode 3 prestandard [8] requires the analysis of the beam for eccentric normal forceloading.)

eH = L/150 (6.4)in case of low requirement levels, according to [3].

7. Performing the static design by calculation

7.1 Dimensioning based on design tables7.1.1 Composition of design tablesTable 1: Section- and loadbearing properties of Lindab Z and C profiles

Cross-section dimensions, Material properties, self-weight data: qg, moment of inertia of the gross cross-section for the SLS analysis: I,

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moment of inertia of the effective cross section: Ieff, Bending-, shear-, and concentrated force resistances for the ULS analysis:

MH,1, MH,2, MH,3, MH,4, TH, FH, (calculated using the bearing length typical for the givenbeam),

Table 2: Span load table for Lindab thin-walled beams

This table specifies the loadbearing capacity of the given static model, for the Ultimate- or theServiceability Limit States.

Input data of the table:

The type and plate thickness of the Lindab thin-walled profile, The static model (simply supported, or continuous with three, four, or more than four sup-

ports, without joints, or with overlapping joints, with or without supplementary elements,assuming uniformly distributed loads, see figure 3.4),

Span (each span assumed equal), Lateral bracing (both flanges continuously braced; the loaded flange continuously braced,

assuming compressive or tensile loads).

Results of the table:

1. Loadbearing capacity based on the ULS criterion (qH,t) both flanges braced,2. Loadbearing capacity based on the ULS criterion (qH,t) loaded flange braced, compres-

sive load (only for Z profiles),

3. Loadbearing capacity based on the ULS criterion (qH,t) loaded flange braced, suckingload (only for Z profiles),

4. Loadbearing capacity based on the SLS criterion (qH,h) L/200 deflection limit,5. Loadbearing capacity based on the SLS criterion (qH,h) L/300 deflection limit.

Notes:

a) The tables have been compiled using elastic theory for the calculation of internal forcesand moments.

b) The ULS loadbearing capacity has been computed considering all possible modes of fail-ure; therefore the values given by the table do not belong to a specific mode of failure of a

certain structural element. Since web crippling doesnt occur under usual support condi-tions, it is not included among the possible modes of failure (see Chapters 3 and 5).

c) Since the SLS loadbearing capacity is a linear function of the deflection limit, the loadsbelonging to deflection limits not mentioned above can be simply obtained (For instance,the load value for the L/150 limit is twice the value for the L/300 limit.)

7.1.2 Using the spanload design tables1. Definition of a static model based on the given structural detailing:

simply supported, or continuous beam with three, four, (or more than four) supports, withconstant span lengths, without joints, or with overlapping joints at the supports, with lat-eral bracing of one flange or both flanges, assuming uniformly distributed loads.

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2. Specifying the characteristic and design values of the critical load: qa, qsz.3. Obtaining the loadbearing capacity of the given model and beam using the span load

table: qH,t and qH,h.

4. Checking:qsz qH,t (7.1)qa qH,h (7.2)

5. Evaluating the results, and carrying out modifications if necessary.7.2 Dimensioning based on detailed analysisIf the static model corresponding to the actual structural arrangement does not satisfy the re-quirements described in this Guide, then the Lindab design tables cannot be used directly forstatic design. Such differences may result from varying spans, non-uniform loads, or othertypes of lateral support. In this case, it is advisable to use the spanload tables together with

one of the given static models that best reflects the current situation for preliminary design,then perform the detailed static analysis using the geometrical and loadbearing capacity data

provided by Table 1.

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8. Lateral support

8.1

Continuous supports

The design method explained earlier presumes one of the static models described in Chapter3, and the resulting mechanical behavior explained in Chapter 5. It was assumed during themodeling process, that the loads acting perpendicular to the plane of bending (the plane of theweb) were carried by continuous lateral supports of the flanges.

Based on practical and experimental experience, this assumption is justified in case of suffi-ciently rigid and strong metal sheeting properly fixed to the thin-walled girders. Employingthe specified fastening methods, Lindab profiled sheetings provide continuous lateral supportfor the flanges or webs of the Lindab Z and C profiles. The bearing of the out of planeloads must be checked by computing the resistance of the fastener components and the an-

choring structure.

8.2 Partial supportsBesides continuous supports, it may be necessary to employ partial lateral supports at specificlocations along the beam, to perform the following functions:

1. To brace the thin-walled girder against lateral-torsional buckling during assembly, and toensure the designed geometry of the structure,

2. To bear the loads acting perpendicular to the plane of bending (the plane of the web),3. To brace the free compression flange (nut supported by profiled sheeting) against lateral-

torsional buckling.

Notes:

In case of the first function, the partial supports are usually temporary; they must be useddepending on the span and the (roof) angle, as specified by practical experience and rec-ommendations of the standards [11].

The second function becomes necessary, if the loads acting perpendicular to the plate ofthe web cannot be borne by the continuous lateral support alone; in this case the suspen-sion or the support of the thin-walled beams must be checked for the load component act-

ing perpendicular to the web. In case of the third function, the lateral-torsional buckling slenderness may be decreased

by laterally supporting the free compression flange at certain locations, thus increasingthe bending resistance. Therefore, the bending resistance using partial support will behigher than MH,3 or MH,4, and employing sufficiently frequent supports, it may reach thevalue of MH,1.

Naturally, partial supports designed for a specific purpose (e.g. assembly), may be takeninto consideration for other functions as well (e.g. bracing of a compression flange in the

permanent stage).

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8.3 Structural detailing of partial supportsAccording to their function, partial supports may be:

Suspension members bearing forces acting perpendicular to the web plane (Figure 8.1), suspension members or stays preventing lateral displacement of free compression flanges(Figure 8.2), supporting members preventing the displacement and angular twist of the profiles (Figure

8.3).

The partial supports must be arranged and dimensioned according to their function. Duringthe design process, the problem of transferring and bearing the forces in the partial supportsmust be solved. This is possible by connecting two profiles across the roof crest (Figure 8.3),suspending a profile from the roof crest (with inclined bars, to a main girder joint), or attach-ing it to the molding.

Figure 8.1: Lateral support suspending member

Figure 8.2: Lateral bracing suspension member/stay bracing the free flange

Figure 8.3: Lateral bracing supporting member bracing the entire profile connect-

ing webs across the crest

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8.4 Recommendations for applicationThis chapter gives recommendations for one of the functions of partial supports described

previously, namely for roof purlins during assembly. The proposals summed up in Table 8.1are based on practical experience and Standard specifications [11]. Assuming the usual as-

sembly methods, the table gives the number of lateral supports needed for given roof angles,purlin spans and girder heights: 0 no lateral support is necessary, 1 one support in themiddle of the span, 2 two supports in the thirds of the span, 3 three supports in the fourths.The recommendations assume that the distance between the purlins is not more than 2 meters.The values given in the table are informative, possibly varying with different assembly meth-ods. If the table does not recommend any supports during assembly, the webs must still beconnected across the roof crust before attaching the profiled sheeting, as described in the pre-vious chapter. In case of roof angles over 22, lateral supports must be checked by staticanalysis. If the supports used during assembly remain in the structure permanently, they may

be taken into consideration for other functions discussed earlier, with the following restric-tions:

(1)the partial supports recommended for assembly are not taken into consideration by thedesign tables included in this Guide when computing loadbearing capacities,

(2)if the transfer of loads acting perpendicular to the web plane is not secured, then the ar-rangement and dimensions of the supports must be checked by static calculations.

Note: Lindab provides expert consultations in the above mentioned questions.

Roof angle Profile height Span

5 m 6 m 7.5 m 9 m 12 m

< 5 100 - 350 0 0 1 2 2

5 - 10 100 - 200 0 0 1 2 2

250 - 350 0 1 1 2 2

10 - 16 100 - 200 0 0 1 2 2

250 - 350 1 1 1 2 3

16 - 22 100 - 350 1 1 1 2 3

Table 8.1: Using lateral supports

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9. Examples for the design of Lindab thin-walled beams

9.1

Design of a Z section purlin /1

The problem: design of a Z section purlin in a free-standing Lindab industrial building.

9.1.1 Structural System #1Structural arrangement:

The distance between main girders: Lf = 6.00 m, The distance between purlins: Lsz = 1.80 m, Overlapping joints are placed at the supports, The beams are braced only along their upper flanges.

Static model of the purlin:

The structural arrangement according to static model #5, described in Chapter 3.2: con-tinuous beam with four or more supports, with overlapping joints over the inner supports,and same-strength or stronger profiles in the outer spans.

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Loads:

Loads transferred from the roof panels, based on the geometry of the industrial building:Pushing type load:

characteristic value: 2mkN

tIa 869.0q =

design value: 2mkN

tIsz 445.1q = Sucking type load:

characteristic value: 2mkN

tIa 209.0q =

design value: 2mkN

tIIsz 270.0q =

Self-weight of the beam:Assuming a Lindab Z 200 beam with 1.50 mm thick walls

mkg

g 4.43g = mkN

ggn 0.042cos9.81gq ==

partial safety factors: g,1 = 1.1, and g,2 = 0.8.

Critical load combinations:

I. self-weight of the beam + pushing type load:- characteristic value: sztIagnIa Lqqq += m

kNIa 606.1q =

- design value: sztIszgng,1Isz Lqqq += mkN

Isz 647.2q =

II. self-weight of the beam + sucking type load:- characteristic value: sztIIagnIIa Lqqq += m

kNIIa 334.0q =

- design value: sztIIszgng,2Isz Lqqq += mkN

IIsz 453.0q =

Checking by the design tables:

The table used is: Z 200, static model #5, the loaded flange of the beam is braced laterally.

Load combination I (pushing type load):

Ultimate Limit State analysis:

- span: L = 6.00 m

- wall thickness:

inner spans: 1.50 mm

in outer spans: 2.00 mm

-Loadbearing capacity: mkN

IszmkN

H 647.2q4.66q =>= Passed.

Serviceability Limit State Analysis:

- deflection limit: eL

H = 300

- loadbearing capacity: mkN

IamkN

H 606.1q2.12q =>= Passed.

Load combination II (pushing type load):Ultimate Limit State Analysis:

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- span: L = 6.00 m

- wall thickness:

in inner spans: 1.50 mm

in outer spans: 2.00 mm


IIszmkN

H 453.0q38.2q =>= Passed.



H = 300


IIamkN

H 334.0q12.2q =>= Passed.

Therefore, Lindab Z 200 profile beams of 2.00 mm and 1.50 mm wall thickness are used inthe outer spans and the inner spans respectively.


The same as in Chapter 9.1.1, but the joints are not overlapping.


The structural arrangement according to static model #3, described in Chapter 3.2: con-tinuous beam with four or more supports, without any joints, or with joints using connect-ing elements.

Loads:

Identical to the loads given in Example 9.1.1.




Ultimate Limit State Analysis:

- span: L = 6.00 m

- wall thickness:

in all spans: 2.00 mm


IszmkN

H 647.2q03.3q =>= Passed.

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H = 300


IamkN

H 606.1q2.12q =>= Passed.

Load combination II (sucking type load):


- span: L = 6.00 m

- wall thickness:



IIszmkN

H 453.0q38.2q =>= Passed.



H = 300


IIamkN

H 334.0q12.2q =>= Passed.

Therefore, Lindab Z 200 profiles of 2.00 mm wall thickness are used.


Identical to the one given in Example 9.1.1, but the purlin is a series of two-span girderswith overlapping joints at the middle supports.


The structural arrangement according to static model #4, described in Chapter 3.2: con-tinuous beam with three supports, with an overlapping joint at the middle support.

Loads:

As in Example 9.1.1,


The table used is: Z 200, static model #4, the loaded flange is braced laterally.


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- span: L = 6.00 m

- wall thickness:



IszmkN

H 647.2q85.2q =>= Passed.



H = 300


IamkN

H 606.1q2.06q =>= Passed.

Load combination II (sucking type load):


- span: L = 6.00 m- wall thickness:



IIszmkN

H 453.0q97.1q =>= Passed.



H = 300


IIam

kN

H 334.0q06.2q =>= Passed.

Therefore, Lindab Z 200 profiles of 1.50 mm wall thickness are used.

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9.2 Design of a "Z" section purlin /2The problem: design of a Z section purlin in a Lindab industrial building built next to anexisting building.

The dimensions of the building are the same as in Example 9.1, The position of the studied industrial building and the existing building:

Note: Snow may accumulate on the roof of the studied industrial building, along the wall ofthe adjoining taller building; this example shows how this problem may be addressed during

purlin design.Structural arrangement:

Same as in Chapter 9.1.1, but the distance between the purlins is not constant; the locally in-creased snow load is taken into consideration by changing the purlin distance. The position ofthe two outer purlins is unchanged (200 mm from the molding and 163 mm from the roofcrest), the position of the inner purlins is obtained according to the variable intensity of theload and the loadbearing capacity of the purlins.

The static model of the purlin:

The structural arrangement according to static model #5, described in Chapter 3.2: con-tinuous beam with four or more supports, with overlapping joints over the inner supports,and same-strength or stronger profiles in the outer spans.

Loads:

Only ULS analysis is carried out, and only for pushing type loads. The cladding is assumed to transfer its load to the purlin as a simply supported structure.

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Based on the geometry of the industrial building, the load transferred from the claddingcan be obtained assuming snow accumulation:

Pushing type load:at the trough between the two buildings:

design value:2m

kN

tvsz326.5q =

at the roof crest:

design value: 2mkN

ttsz 445.1q =

Design of the purlins:

According to the change in load intensity, the design of the purlins may be carried out in twodifferent ways for optimum performance:

1. Following the changing load intensity by modifying the purlin distances,2.placing the purlins at equal distances, but modifying the profile thickness according to the

changing load intensity.

From the practical point of view, the first solution is the simpler one. In this case, the positionof the purlins can be obtained by iteration (trial and error) or by more precise methods. In thisexample, a computational technique is shown to obtain purlin positions, ensuring uniform

purlin distribution.

The loadbearing capacity of a purlin according to chapter 9.1.1:ULS analysis: m

kNH 4.66q =

the maximum load that may be transferred from the roof panels to a purlin is computed bysubtracting the self-weight of the beam from the loadbearing capacity:

mkN

gng,1HHsz 4.6120.0421.14.66qqq ===

Introducing to characterize the change in load intensity:S

qq ttsztvsz

= ,

where S = 9.363 m is the length of the main girder of the frame.

3mkN0.415 =

Studying one specific purlin, the following data must be known: the position of the previous purlin: Le is the distance between the two purlins,

the load intensity at the edge of the previous purlins loading area: qe ][ 2mkN .

Let the distance between the current purlin and the next purlin be Lk. The surface load intensity in the middle of the current purlins loading area is:

4

LLqq ekeszf

+= ][ 2m

kN

Based on this value, the distributed load acting along the midline of the purlin is:

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2

LLqq ekszfszv

+= ][ m

kN

This value may not be higher than the loadbearing capacity minus self weight (qHsz) As a result, a quadratic equation is attained, with Lkas the only variable. Solving the equa-

tion, the lower of the two solutions will give the maximum distance of the next purlin:

eHsz

2

eeK L

8q

2q

2qL

=

Note: the above explained procedure calculates the distance between two neighboring purlinsbased on their optimum performance, inevitably resulting in alternating values (longer/shorterdistances follow each other). In the shown example, the following purlin distances are ob-tained with this method (rounded values): 0.2m, 1.395m, 0.546m, 1.584m, 0.804m, 1.968m,1.468m, 3.666, and because of geometrical limits, 1.235m, 0.163m..

In practice, the outer purlins always have a lower load/resistance ratio. For example, in caseof a uniformly distributed load and equal purlin distances, the outer purlins only carry 50% oftheir loadbearing capacity. As an analogy: gradually changing purlin distances following thelinear change of the load may be obtained by choosing a load/resistance ratio of approx. 50%for the outer purlins. Therefore, the distance between the first and the second purlin must becalculated by the following formula:

4q

2q

2qL Hsz

2

iiK

= ,

where qi is the load intensity at the first purlin ( ][ 2mkN ). Using this method, the following purlin

distances may be obtained: 0.2m, 0.895m, 1.002m, 1.071m, 1.238m, 1.412m, 1.797m,2.631m, and because of geometrical limits, 1.585m, 0.163m.

As an example, the position of the second and third purlins is computed here:

Computation of the first purlin distance (the location of the second purlin)given: 0.2mLe =

2mkN

etvszi 5.243Lqq ==

result: 0.895mLk =

Computation of the second purlin distance (the location of the third purlin)given from the previous result: m895.0Le =

2mkNk

etvsze 058.52

LLqq =

+=

result: 1.002mLk =

Applying the method repeatedly, the above given values may be obtained.

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Since this procedure is sensitive for numerical rounding, it is advisable to round the valuesafter performing the entire calculation.

In this case, a possible purlin arrangement is the following:

0.2m, 0.8m, 0.9m, 1.0m, 1.15m, 1.35m, 1.65m, 2.298, and because of geometrical limits,

2.15m, 0.163m.

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9.3 Design of a "Z" section purlin /3The problem: design of a Z section purlin in a Lindab industrial building built next to anexisting building.

The dimensions of the building are the same as in Example 9.1.1, The position of the studied industrial building and the existing building:

Note: Snow may accumulate on the roof of the studied industrial building, along the wall ofthe adjoining taller building; this example shows how this problem may be addressed duringpurlin design.

Structural arrangement:

Same as in Chapter 9.1.1, but auxiliary elements are employed.

The static model of the purlin:

Continuous beam, with overlapping joints over the inner supports, and stronger profileswhen needed (in the outer spans, or because of snow accumulation).

Loads:

Design is carried out only for pushing type loads. Based on the geometry of the industrial building, the load transferred from the cladding

can be obtained assuming snow accumulation:

Loads transferred from the roof panels, based on the geometry of the industrial building:Pushing type load:characteristic value: m

kN1a 609.1q = m

kN2a 27.4q =

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design value: mkN

1sz 649.2q = mkN

2sz 306.7q =

Critical internal forces and support reactions

The bending rigidity conditions of the girder must be known for internal force computation.

For this reason, Z 200 profiles are assumed with the following thicknesses:- Span 1: 2.00 mm- Span 2: 1.50 mm- Span 3: 1.50 mm- Span 4: 1.50 mm- Span 5: 2.00 mm- Span 6: 2.00 mm + 2.00 mm.

- the result of the static computation considering the change in bending rigidity andthe effect of overlapping joints is the following:

Bending moment: [kNm], support reaction: [kN]

Shear forces:

at support #2: kN85.9TM2 =





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Checking

Ultimate Limit State Analysis

-Bending resistances of the members are superposed at the overlapping joints andthe auxiliary elements.

- In case of the usual structural arrangement, web crippling does not occur, there-fore it may be ignored.

-at support #2:

bending resistance:

- compression flange free, tension flange braced (MH,3)

11.43kNmM18.18kNm66.1152.6M 2H =>=+= Passed.

Sheer resistance:

kN85.9TkN34.5838.4196.16T 2H =>=+= Passed.Bending moment + shear force:

1.3798.058.34

9.85

18.18

11.43

T

T

M

M

H

2

H

2 =+= Passed.

Shear resistance:

kN75.7TkN92.3396.1696.16T 3H =>=+= Passed.

Bending moment + shear force:

1.3823.033.92

7.75

13.04

7.76

T

T

M

M

H

3

H

3 =+= Passed.

Shear resistance:

kN14.8TkN92.3396.1696.16T 4H =>=+= Passed.


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1.3924.033.92

8.14

13.04

8.92

T

T

M

M

H

4

H

4 =+= Passed.

Shear resistance:

kN31.8TkN34.5838.4196.16T 5H =>=+= Passed.


1.3772.058.34

8.31

18.18

8.21

T

T

M

M

H

5

H

5 =+= Passed.

Shear resistance:

kN60.20TkN76.8238.4138.41T 6H =>=+= Passed.

Bending moment + shear force:1.3108.1

82.76

20.60

23.32

20.03

T

T

M

M

H

6

H

6 =+=++ Passed.

- Between supports #3 and #4:

bending resistance:

- compression flange braced, tension flange ( H,2M )

kNm58.3MkNm69.7M 4-3H =>=++ Passed.

- Between supports #1 and #2:

bending resistance:

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- compression flange braced, tension flange ( H,2M )

kNm89.6MkNm43.13M 2-1H =>=++ Passed.

- Those sections of the girder must be checked as well, where an overlapping jointor an auxiliary element ends.

- in the last span, at the end of the auxiliary element:

bending resistance:

- compression flange braced, tension flange free (MH,3)

kNm58.8MkNm66.11M MH =>= Passed.

Shear resistance:

kN54.17TkN38.41T MH =>= Passed.

Bending moment + Shear force:

1.316.141.38

17.54

11.66

8.58

T

T

M

M

H

M

H

M

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9.4 Checking of a C and Z section wall beam in a Lindab industrial buildingThe problem: design of a C and Z section wall beam in a Lindab industrial type build-ing.


The detailing of the Lindab industrial type building is the same as the one that appears inExamples 8.1 and 8.2 of theDesign Guide Static Design of Lindab profiled sheeting,

The distance of the main girders: Lf = 5.40 m, The distance of the wall beams: Lsz = 2.30 m, The Lindab C 200 profile beam elements have no joints. The beams are braced along both flanges (inner and outer profiled sheeting).Static model of the wall beam:

The structural arrangement according to static model #1, described in Chapter 3.2 is: sim-ply supported beam.

Loads:

Identical to those given in the profiled sheeting problem:wind pressure: pw

kNm1

0 505 2= .wind sucking: pw2

kNm2

= 0 203.

partial safety factor: w = 12.

Load model:

Reduction of surface loads to linear loads:The load transferred to one beam: q qLg g= L 2.30mg =

profiled sheeting: q p Lw1 w1 g= qw1kNm= 1212.

insulation: q p Lw2 w2 g= qw2kNm= 0 605.

Reduction to a load perpendicular to the static model: the wind load is perpendicular tothe static model, therefore it does not have to be reduced.

Critical load combinations:

I. Wind pressure: q qIa w1= q Ia kNm= 1212. - characteristic value: q qIsz w w1= q Isz

kNm= 1.454

- design value:

II. Wind sucking:- characteristic value: q qIIa w2= q IIa kNm= 0 605.- design value: q qIIsz w w2= q IIsz

kNm= 0 726.

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The table used is: C 200, static model #1, both flanges of the beam are braced laterally.



- span: L = 5.40 m

- wall thickness: 1.50 mm


IszmkN

H 454.1q12.2q =>= Passed.



200H=


Iam

kN

H 212.1q70.1q =>= Passed.Load combination II (sucking type load):



IszmkN

H 726.0q12.2q =>= Passed.



IamkN

H 605.0q70.1q =>= Passed.

Therefore, Lindab C 200 profile beams with 1.50 mm thick walls are used.


As an alternative to Example 9.4.1, using Lindab Z 200 beams, Overlapping joints of the wall beams are placed above the supports, The beams are braced along their outer flange (with outer profiled sheeting only).

Loads:

Identical to those defined in Example 9.4.1Checking by the design tables:




- span: L = 5.40 m- wall thickness: 1.20 mm

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-Loadbearing capacity: q qHkNm Isz

kNm= > =2 65 1. .454 Passed.



200H=


IamkN

H 212.1q62.2q =>= Passed.

Load combination II (sucking type load): Not critical.

Therefore, Lindab Z 200 profile beams with 1.20 mm thick walls are used.

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9.5 Checking of a C section floor beamThe problem: design of a C section light floor beam.

Structural arrangement:

In the studied light floor structure, C section beams support Lindab LTP 20 profiledsheeting that carries a light-concrete leveling layer and cladding; a 2 cm thick plaster ceil-ing is connected to the beams from below.

The distance of the beams: Lg = 0.80 m The distance of the walls supporting the beams: Lfal = 4.20 mStatic model of the floor beam:

Static model #1 defined in Part 3.2 of the Design Guide: simply supported beam, bothflanges braced.

Loads:

The loads transferred from the roof panel: (LTP 20/0.4 mm, 5 cm light concrete, cladding,live load assuming a family home not elaborated)characteristic value: q Ia

kNm2

= 2 284.

design value: qIsz

kN

m2= 3072.

Plaster ceiling:density: gk

kg

m800 3=

thickness: v cmgk = 2

self-weight: g 9.81vgk gk gk = g 0.157gkkNm2

=

partial safety factors: b,1 = 1 2. or b,2 = 0 7.

Self-weight of the beam: assuming a C200 beam with 1.50 mm wall thicknessg 3.51g

kgm= q 9.81g 0.034g g

kNm= =

safety factors: g, .1 1 1= or g, .2 0 8= Load model:

Reduction of surface loads to linear loads:q qLg g= L mg = 0 80.

The load transferred to one beam:- load transferred from the floor panel:

characteristic value: q q LgIa Ia g= qgIakNm= 1827.

design value: q q LgIsz Isz g= qgIszkNm= 2.458

- load transferred from the plaster ceiling:

q g bgk gk sz= q 0.126gk kNm=

Critical load combination:

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Load transferred from floor panel + plaster ceiling + self-weight of the beam

characteristic value: q q q qIIa gIa gk g= + + q IIa kNm= 1987. design value: q q q qIIsz gIsz gk,1 gk g,1 g= + + q IIsz kNm= 2 646. Checking by the design tables:

The table used is: C 200, static model #1, the both flanges braced.


- span: L = 4.20 m

- wall thickness: 1.50 mm


IIszmkN

H 646.2q51.3q =>= Passed.



200H=


IIamkN

H 987.1q64.3q =>= Passed.

Web crippling analysis:

Note: The Design Tables do not check for web crippling, since in case of typical support ar-rangements, this mode of failure is not critical; in this detailing, the C profile beam sits di-rectly on the wall, therefore a supplementary analysis is necessary:

- Support reactions: F 0.5q LM IIsz= F 5.556kNM =

- Web crippling resistance: F kN FHszls M= >7 36. Passed.

Therefore, Lindab C 200 profile beams with 1.50 mm thick walls are used.

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