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8/13/2019 Design of Compression Member BS
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Lecture 5. COMPRESSION MEMBERS
Content of lecture:
- Types and uses of compression members, loadings and cross-sections
- Axially loaded columns, general behavior, effective length
- Design procedure for axially loaded columns
- Columns under combined axial loads and bending moment
- Short and slender columns
5.1. Types and uses of compression memers
Compression members are one of the basic structural elements, and are described
by the terms columns, stanchions or struts, all of hich primarily resist axial
load!
Columns are vertical members supporting floors, roofs and cranes in buildings!
Though internal columns in buildings are essentially axially loaded and are
designed as such, most columns are sub"ected to axial load and moment! The term
strut is often used to describe other compression members such as those in
trusses, lattice girders or bracing! Some types of compression members are shon
in #ig! $!
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!i"ure 1. Types of compression memers
Compression members must resist buc%ling, so they tend to be stoc%y ith s&uare
sections! The tube is an ideal shape for such members! These are in contrast to the
slender and more compact tension members and deep beam sections!
'olled, compound and built-up sections are used for columns! (niversal columns
are used in buildings here axial load predominates, and universal beams are often
used to resist heavy moments that occur in columns in industrial buildings!
!i"ure #. Compression memer sections
Single angles, double angles, tees, channels and structural hollo sections are the
common sections used for struts in trusses, lattice girders and bracing!
Compression member sections are shon in #ig! )!
C$assification of cross sections.The pro"ecting flange of an *-shaped compression
member ill buc%le locally if it is too thin hile the rest of the member remains
straight! +ebs ill also buc%le under compressive stress from bending and from
shear! The reduction in compressive capacity should be obvious as the arped
portion of the member re"ects load and transfers it to other portions! To prevent
local buc%ling occurring, limiting outstand/thickness ratios for flanges and
depth/thickness ratios for ebs are given inBS .:Part $,Cl. /!.
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Column cross-sections are classified as follos in accordance ith their behavior
under load0
Class 1 - Plastic cross section! This can develop a plastic hinge ith
sufficient rotation capacity to permit redistribution of moments in the entire
structure! 1nly class $ sections can be used for plastic design!
Class 2 - Compact cross section! This can develop full plastic moment
capacity but local buc%ling prevents sufficient rotation at constant moment!
Class 3-Semi-compact cross section! The stress in the extreme fibers should
be limited to the yield stress because local buc%ling prevents development of the
full plastic moment!
Class 4- Slender cross section! 2remature local buc%ling occurs before yield
is reached!
#lat elements in a cross section are classified as0
- Internal elementssupported on both longitudinal edges3
- Outside elementsattached on one edge ith the other free!
4lements are generally of uniform thic%ness but, if tapered, the average thic%ness
is used! Compression members are classified asplastic, compact orsemi-compact
if they meet limiting proportions for flanges and ebs in axial compression given
in Tab. 5, BS .0Part *! #or rolled and elded column sections #ig! / shos
these proportions hich ere set out to prevent local buc%ling!
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Limitin" Proportions
E$ement Section Type P$astic
Section
Compa
ct
Section
Semi %
Compa
ct
Section
Outstand
e$ement of
compression
f$an"e
Ro$$ed &T
'e$ded &T
(.5).5
*.5(.5
151+
Interna$
e$ement of
compression
f$an"e
'e$ded &T #+ #5 #(
'e su,ect to
compression
t-rou"-out
Ro$$ed d&T
'e$ded d&T
#(+*
/#)5 & p y 0.5
!i"ure +. Limitin" proportions for ro$$ed and 2e$ded co$umn sections
Loads. Axial loading on columns in buildings is due to loads from roofs, floors
and alls transmitted to the column through beams and to self eight, #ig! 67a8!
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#loor beam reactions are eccentric to the column axis, as shon, and if the beam
arrangement or loading is asymmetrical, moments are transmitted to the column!
+ind loads on multi-storey buildings designed to the
a0
0
!i"ure 3. Loads and moments on compression memers
simple design method are usually ta%en to be applied at floor levels and to be
resisted by the bracing, and so do not cause moments! *n industrial buildings loads
from cranes and ind cause moments in columns, as shon in #ig! 67b8! *n this
case the ind is applied as a distributed load to the column through the sheeting
rails!
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5.# 4ia$$y $oaded compression memers
General considerations. Compression members may be classified by length! A
short column, post or pedestalfails by crushing or s&uashing, #ig! 7a8! Thes&uash
loadPy iin terms of thedesign strength is0
Py = pyA
+here3 A - area of cross section!
A long orslender column fails by buc%ling, as shon in #ig! 7b8! The failure load
is less than the s&uash load and depends on the degree of slenderness! 9ost
practical columns fail by buc%ling! #or example, a universal column under axial
load fails in flexural buc%ling about the ea%er :-: axis, #ig! 7c8! The strength of
a column depends on its resistance to buc%ling! Thus the column of tubular section
in #ig! 7d8 ill carry a much higher load than the bar of the same cross-sectional
area!
This is easily demonstrated ith a sheet of A6 paper! 1pen or flat, the paper
cannot be stood on edge to carry its on eight3 but rolled into a tube it ill carry
a considerable load! The tubular section is the optimum column section having
e&ual resistance to buc%ling in all directions!
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a0 0 c0 d0
!i"ure 5. Be-a6ior of memers in aia$ compression
Initially straigt struts !"uler load#. Consider a pin-ended straight column! The
critical value of axial load P is found by e&uating disturbing and restoring
moments hen the strut has been
a) initia$$y strai"-t 0 strut 2it- initia$ c0 strut 2it- end
d0 co$umn
strut cur6ature eccentricity
section
!i"ure 7. Load cases for struts
given a small deflection y, as shon in #ig! ;7a8! The e&uilibrium e&uation is0
Pydyd!"y =))
pc8 7py>pc8 = 1 p4pc,
here0 py> design stress 7orpmax83
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p4- 4uler stress3
pc> limiting compressive stress, ,!.) 87 y!
y!
cpp
ppp
+
=
)
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!i"ure (. T-eoretica$ effecti6e $en"t-.
An alternative method is to determine the distance beteen points of contra flexure
in the deflected strut! These points may lie ithin the strut length or they may be
imaginary points on the extended elastic curve! The distance so defined is the
effective length! The theoretical effective lengths for standard cases are shon in
#ig! ! Fote that for the cantilever and say case the point of contra flexure is
outside the strut length!
"**ecti+e lengts !Cl. 4. .2#.The effective length is considered to be the actual
length of the member beteen points of restraint multiplied by a coefficient to
allo for effects such as stiffening due to end connections of the frame of hich
the member is a part! Appropriate values for the coefficients are given in Tab.)6
of the code and illustrated in #ig! /!)/7a8 and 7b8!
*n the case of angles, channels and T-sections, secondary bending effects induced
bG end connections can be ignored and pure axial loading assumed, provided that
the slenderness values are determined using Cls.6!5!$.!) to 6!5!$.! or Tab.)? in
the code!
*n the case of other cross-sections the slenderness should be evaluated using
effective lengths as indicated in #igures /!)/7a8 and 7b8! *n addition, Appendix D
of the code gives the appropriate coefficients to be used hen assessing the
effective lengths for columns in single-story buildings using simple construction!
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!i"ure *. Codes definition for effecti6e $en"t-
The coefficients given for determining effective lengths are generally greater than
those predicted by mathematical theory3 this is to allo for effects such as the
inability in practice to obtain full fixity!
5.+. Memers su,ect to comined compression and endin"
Column loads so far have been assumed to be concentric, i!e!, applied along the
axis of the column! This assumption is valid hen the load is applied uniformly
over the top of the column, or hen beams has having e&ual reactions frame into
the column opposite each other as ould be the case in #ig! $/a if the reactions of
beams A and ere the same, and those of beams C and D ere the same! *f,
hoever, beam ere omitted as shon in #ig! $/b, or if the reaction of as
considerably less than that of A in #ig! $/a, it is evident that the loads on the
column no longer ould be symmetrical and that the left column flange ould be
sub"ected to a greater unit stress than the right! This eccentric loading condition
occurs fre&uently in all columns of buildings, here a floor beam is supported on
the interior face ithout a corresponding load on the exterior face! 7#ig! $/c simply
illustrates onemethod of framing, hich may be used to lessen this eccentricity, or
even to balance the loads if the total reaction of theto spandrel
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a0 0 c0 d0
!i"ure 1+.
beams is nearly the same as that of the floor beam!8 These types of members are
sub"ected to bending moment in addition to axial load and termed Hbeam
columnsI! They are representing the general load case of an element in a structural
frame!
S-ort co$umns
Sort column ea+ior.*n order to develop an expression that ill account for the
variation in stress over the column cross section due to the eccentric condition,
consider once more the short compression bloc%, this time ith a load P
eccentrically applied 7#ig! $/d8! The distance e is the eccentricity, and c is the
distance from the axis of the bloc% to the extreme fibers! The stress in any fiber, on
any cross section of the bloc%, such as - : may be considered to be the sum of
the average stress/$,and a stress caused by the momente.To the right of the
axis of the bloc%, i!e!, on the same side as , this moment causes a compressivestress on the section and to the left of the axis, a tensile stress! The unit stress at 0
is e&ual to the average stress/$,plus the extreme fiber stress4c/"caused by the
momentc. Substitutinge for4,the intensity of stress at 0and2 are expressed
by the formulas
*y= / % A ' /ec % I 0 * $= / % A / e c % I or * = / % A / e c % I
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in hichfis the unit stress at either edge of the section, depending on hether the
plus or minus sign is used, and " is the moment of inertia in the direction of the
eccentricity! The expressions are applicable to sections symmetrical about to
axes such as rectangles, *- and B- sections!
*n the investigation of eccentrically loaded columns, maximum compression is
usually the most critical, because seldom ill the tensile stress on the far edge of
the column due to the momentebe sufficient to counteract the direct compressive
stress/$. +here this does occur, it is of importance only hen the column is to
be spliced! *t is generally true in buildings that columns carry a direct axial load in
addition to any eccentric loads that may exist! +here such is the case, a more
convenient form of the expression is
* = / % A ' / e c % I
in hich /- is the total vertical load including the eccentric load, and
5-is the eccentric load alone!
Sort column *ailure.Consider vertical member having e&ual end moments4(,
deflecting to a shape shon in #ig! $6! 9aximum lateral deflection is6 mand the
moment at mid-height, is 4. +hen the axial load is applied to the already
deflected shape 7#ig! $6a8, there ill be an additional moment at mid-height e&ual
to6 m.This, in turn, causes
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/a0
/0
!i"ure 13. S-ort co$umn e-a6ior /a0 and fundamenta$ interaction cur6es /0
more lateral deflection, causing more moment, and so on! Conse&uently, the final
bending stress at mid-height of the column ill he the sum of the stresses caused
by each action, or
*= c % I ' / 5mc % I .
The additional stresses caused by 6m are very difficult to ascertain, often
re&uiring the complex mathematical processes %non as numerical integration!
Such procedures and accompanying formulas are unrealistic for routine design
application! Boever, some useful conclusions can be abstracted from the above-
described structural action! *t is seen that for a constant end condition such as thatshon in #ig! $6a 7e&ual end moments8, the lateral deflection ill depend upon the
slenderness ratio of the column ith respect to the direction of bending! A large
slenderness ratio permits a larger lateral deflection! The corresponding bending
stresses from the deflection ill increase ith increasing values of the axial load
P.
*n order to simplify the design procedure, a method based upon the application of
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the interaction formula is used! *t may be modified as necessary to agree ith
experimental test data!
The curves shon in #ig! $6b are typical of such test data for short columns
7straight lines, * = .8! Columns having e&ual end moments 4( ere tested to
determine hat additional axial load 2 could be applied before failure ould occur!
This as repeated for columns having different slenderness ratio, such ratios being
determined respective to the direction of the applied moment! These values, of
varying combinations of and 4, ere made dimensionless by dividing them
respectively by Pc > axial load causing yielding 7if it alone occurred8 and 4c -
bending moment causing yielding 7if it occurred in absence of an axial load8! The
similarity of the axial load ratios/Pcand the axial stress ratiosfa /pyshould be
apparent! The same similarity exists beteen the moment ratios 4( /4y and the
bending stress ratiosfb/py!
!i"ure 15. P$astic stress distriution in endin" aout 88 ais
#or short columns failure generally occurs hen the plastic capacity of the section
is reached! The plastic stress distribution for uniaxial bending is shon in #ig! $!
The moment capacity for plastic or compact sections in the absence of axial load is
given by0
= S py9 1.#6 py77see Section 7.8.) of BS )9)(: Part 8
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here0 S =plastic modulus for the relevant axis
6 = elastic modulus for the relevant axis!
The interaction curves for axial load and bending about the to principal axes
separately are shon in #ig! ?!$?7a8! Fote the effect of the limitation of bending
capacity for the :: axis! These curves are in terms of /Pcagainst4r/ 4cand
4ry/ 4cy, here0
= applied axial load
P % py$, the s&uash load
4r = reduced moment capacity about the axis in the presence
of axial load
4c % moment capacity about the axis in the absence of axial
load
4ry %reduced moment capacity about the :: axis in the presence
of axial load
4cy % moment capacity about the :: axis in the absence of axial
load!
Jalues for4r and4ryare calculated using e&uations for reduced plastic modulus
given in the ;uide toBS.0Part $0 $?,
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sub"ected to axial load and biaxial bending are found to give a convex failure
surface, as shon in #ig! ?!$?7a8! At any point $ on the surface the combination of
axial load and moments about the - and :-: axes4 and4ry, respectively, that
the section can support can be read off!
A plane dran through the terminal points of the surface gives a linear interaction
expression0
/%Pc' $%c$' y% cy= 1
This results in a conservative design!
S$ender co$umns
Slender columns ea+ior.The behavior of slender columns can be classified into
the folloing cases0
!i"ure 17. S$ender co$umn su,ected to aia$ $oad and moment
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Case a- A slender column sub"ected to axial load and uniaxial bending about
the ma"or axis -! *f the column is supported laterally against buc%ling about the
minor axis :-: out of the plane of bending, the column fails by buc%ling about the
- axis! This is not a common case 7see #ig! ?!$57a88! At lo axial loads or if the
column is not very slender a plastic hinge forms at the end or point of maximum
moment
Case - A slender column sub"ected to axial load and uniaxial bending about
the minor axis :-:! The column does not re&uire lateral support and there isno
buc%ling out of the plane of bending! The column fails by buc%ling about the :-:
axis! At very lo axial loads it ill reach the bending capacity for :-: axis 7see
#ig! ?!$57b88!
a0 0
!i"ure 15. !ai$ure surfaces /a0 and contours /0 for s$ender co$umns
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Case c- A slender column sub"ected to axial load and uniaxial axial bending
about the ma"or axis8-8. This time the column has no lateral support! The column
fails due to a combination of column buc%ling about the :-: axis and lateral
torsional buc%ling here the column section tists as ell as deflecting in the 8-8
and 9-9planes 7see #ig! ?!$57c88!
Case d> A slender column sub"ected to axial load and biaxial bending! The
column has no lateral support! The failure is the same as in previous case above
but minor axis buc%ling ill usually have the greatest effect! This is the general
loading case!
/ailure o* slender columns! +ith slender columns, buc%ling effects must be ta%en
into account! These are minor axis buc%ling from axial load and lateral torsional
buc%ling from moments applied about the ma"or axis! The effect of moment
gradient must also be considered!
All the imperfections, initial curvature, eccentricity of application of load and
residual stresses affect the behavior! The HendI conditions have to be ta%en into
account in estimating the effective length! Theoretical solutions have been derived
and compared ith test results! #ailure surfaces for B-section columns plotted
from the more exact approach given in the code are shon in #ig! $7a8 for various
values of slenderness! #ailure contours are shon in #ig! $7b8! These represent
loer bounds to exact behavior! The failure surfaces are presented in the folloing
terms0
Slenderness0 : = : / % Pc0 r$% c$ 0 ry% cy
: 5; 1 : / % Pc0 a$% c$ 0 ay% cy
Fe terms used are0
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a$ - maximum buc%ling moment about the - axis in the
presence of axial load
ay - maximum buc%ling moment about the :-: axis in the
presence of axial load!
The folloing points are to be noted!
$! 4ay; the moment capacity about the :: axis, is not sub"ected to the
restriction $!)py=y.
)! At Lero axial load, slenderness does not affect the bending strength of an B
section about the 0-0axis!
/! At high values of slenderness the buc%ling resistance moment 4babout the
- axis controls the moment capacity for bending about that axis!
6! As the slenderness increases, the failure curves in the/Pc, :-:-axis plane
change from convex to concave, shoing the increasing dominance of minor
axis buc%ling!
! #or design purposes the results are presented in the form of an interaction
expressions and this is discussed in the next section!
Code desi"n procedure
The code design procedure for compression members ith moments is set out in
Cl.6!?!/ ofBS.: Part$! This re&uires the folloing to chec%s to be carried
out0
7$8 Kocal capacity chec%3 and
7)8 1verall buc%ling chec%!
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*n each case to procedures are given! These are a simplified approach and a more
exact one!
7$8 Loca$ capacity c-ec
greatest bending moment and axial load! This is usually at the end, but it
could be ithin the column height if lateral loads are also applied! The
capacity is controlled by yielding or local buc%ling! The interaction
relationship for semi-compact and slender cross sections and the simplified
approach for compact cross sections given in Cl. 6!?!/!) of theCodeis0
/%Agpy' $%c$' y% cy9 1.
+here3 - applied axial load3 $g- gross cross-sectional area
4-applied moment about the ma"or axis2-2
4c- moment capacity about the ma"or axis2-2 in the absence
of axial load
4y- applied moment about the minor axis 0-0
4cy- moment capacity about the minor axis 0-0 in the absence
of axial load!
More ri"orous ana$ysisgiven is used to produce an alternative e&uation, hich
ill generally produce a more economic design!
This is based on the convex failure surface discussed above! The folloing
relationship must be satisfied0
m$%a$' my% y9 1.
here0 9x> maximum buc%ling moment about the - axis in the
presence of axial load and e&uals the lesser of0
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c$;!1- / % Pc$# % !1' .5/ % Pc$# maximum buc%ling moment about the :: axis in the
presence of axial load and e&uals
cy;!1- / % Pcy# % !1' .5/ % Pcy# compression resistances about the ma"or and minor
axes respectively!
7)8 O6era$$ uc bending stress determined from Tab. $ for values of* and
.
* @slenderness#!/ ry.
@ torsional index =A / T7approximately8 or can be calculated
from the formula in Appendix or from the publishedtable in the Muide toBS .0Part$!
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A more exact approach is also given in the code! This uses the convex failure
surfaces discussed above!
Eamp$e.
A braced column 6! m long is sub"ected to a factored end loads and moments
about the x-x axis! The column is held in position but only partially restrained in
direction at the ends! Chec% that a )./x)./ (C) in Mrade 6/ is ade&uate!
Solution
$! Kocal buc%ling capacity chec%!
#rom Tab! ; find design strength,py= )5 F $!)=xpy= $!) x $. x )5 x $.-/= $;?!6 %F m, so 1O!
*nteraction expression gives0
.!$5!.))!.6?!.)!$,;
/,
$.)5,6!;;
$.??./
/
=+=+
, so 1O!
The section is satisfactory ith respect of local capacity!
)! 1verall buc%ling chec%
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4ffective length, Tab! )6! #4= .!? x 6.. = /?) mm3 Slenderness*=#4
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A$ially loaded sla ase plates! Columns hich are assumed to be nominally
pinned at their bases are provided ith a slab base comprising a single plate fillet
elded to the end of the column and bolted to the foundation ith four holding
don 7B!D!8 bolts! The base plate, elds and bolts must be of ade&uate siLe,
stiffness and strength to transfer the axial compressive force and shear at the
support ithout exceeding the bearing strength of the bedding material and
a0 0
!i"ure 11.
concrete base, as shon in #ig! $$a! Clause 7.?.8.8 of S .02art $ gives the
folloing empirical formula for determining the minimum thic%ness of a
rectangular base plate supporting a concentrically loaded column0
,!.)) 8@/!.7,!)
A bap
+t
yp
=
here0
a - the greater pro"ection of the plate beyond the column as shon in #ig! $$b
b - the lesser pro"ection of the plate beyond the column as shon in #ig!$$b
+- the pressure on the underside of the plate assuming a uniform distributionpy- the design strength of the plate 7Cl. ?.. or Tab. ;8, but not greater than
)5. F
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Column ases.Column base transmit axial loads, horiLontal loads and moments
from the steel column to the concrete foundation! The main function of column
base is to distribute the loads safely to the ea%er material!
The main types of column bases are, #ig! $)0
$! Slab base. Depending on relative value of/4 to cases occur0
a8 The pressure over the hole base3
b8 The pressure over part of the base and tension in the holding don
bolts!
Cl.6!$/!$ assumes that the nominal bearing pressure under base-plate is distributed
linearly, so elastic analysis is used in design! The middle third rule applies, and if
the eccentricity eN $
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)! ;usset base. Mussets support the base plate against bending and this is hy
a thinner plate can be used than ith the slab base! The gussets are sub"ected to
bending from upard pressure under the base as shon in #ig! $)b! The top edge
of the gusset is compressed and has to be chec%ed for buc%ling! To ensure that this
ill not occur use limiting proportions from Tab! 5 for semi-compact section! To
re&uirements must be satisfied0
$! Musset beteen elds to the column flange0A G )?t
)! 1utstand of gusset from column or base plate0 S G $/t.Bere =
7)5thic%ness of the gusset plate3pyg> design strength of
gusset plate!
/! Pocket base. *n this type of base the column is grouted into a poc%et in the
concrete foundation, #ig! $)c! The axial load is resisted by direct bearing and bond
beteen the steel and concrete! The moment is resisted by compression forces in
the concrete acting on the flanges of the column! The forces act on both faces of
the flanges of a universal beam