Lecture 5. COMPRESSION MEMBERS
Content of lecture:
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Types ypes and and use usess of of com compr pres essi sion on membe embers rs,, loa loadi ding ngss and and cros crosss-se sect ctio ions ns
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Axia Axiall lly y load loaded ed colu colum mns, ns, gene genera rall beha behavi vior or,, effe effect ctiv ivee leng length th
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Design proc rocedure for axially loaded columns
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Colum olumns ns unde underr com combi bine ned d axi axial al loads oads and and ben bendi ding ng momen omentt
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Short and slender columns
5.1. 5.1.
Type Typess and and use usess of compre compressi ssion on mem meme ers rs
Compression members are one of the basic structural elements, and are described by the terms ‘columns’, ‘stanchions’ or ‘struts’, all of hich primarily pr imarily resist axial load!
Columns are vertical members supporting floors, roofs and cranes in buildings! Thou Though gh inte interna rnall colu column mnss in build buildin ings gs are are esse essent ntia iall lly y axia axiall lly y load loaded ed and 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! $!
!i"ure 1. Types of compression memers
Compression members must resist buc%ling, so they tend to be stoc%y ith s&uare sections! The tube is an ideal shape for such members! These are in contrast to the slender and more compact tension members and deep beam sections!
'olled, compound and built-up sections are used for columns! (niversal columns are used in buildings here axial load predominates, and universal beams are often used to resist heavy moments that occur in columns in industrial buildings!
!i"ure #. Compression memer sections
Single Single angles, angles, double angles, angles, tees, channels channels and structural hollo sections sections are the com common sect sectio ions ns used used for stru struts ts in truss russes es,, latt attice ice girde irders rs and and brac braciing! ng! Compression member sections are shon in #ig! )!
C$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 loca locall buc% buc%li ling ng occu occurr rrin ing, g, limi limiti ting ng outs outsta tand nd/t /thi hick ckne ness ss rati ratios os for for flan flange gess and and depth/thickness ratios for ebs are given in BS in BS .: Part $, Cl. $, Cl. /!. /!.
!i"ure 1. Types of compression memers
Compression members must resist buc%ling, so they tend to be stoc%y ith s&uare sections! The tube is an ideal shape for such members! These are in contrast to the slender and more compact tension members and deep beam sections!
'olled, compound and built-up sections are used for columns! (niversal columns are used in buildings here axial load predominates, and universal beams are often used to resist heavy moments that occur in columns in industrial buildings!
!i"ure #. Compression memer sections
Single Single angles, angles, double angles, angles, tees, channels channels and structural hollo sections sections are the com common sect sectio ions ns used used for stru struts ts in truss russes es,, latt attice ice girde irders rs and and brac braciing! ng! Compression member sections are shon in #ig! )!
C$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 loca locall buc% buc%li ling ng occu occurr rrin ing, g, limi limiti ting ng outs outsta tand nd/t /thi hick ckne ness ss rati ratios os for for flan flange gess and and depth/thickness ratios for ebs are given in BS in BS .: Part $, Cl. $, Cl. /!. /!.
Column cross-sections are classified as follos in accordance ith their behavior under load0
Class 1 - Plastic cross section! This This can can deve develo lop p a plas plasti ticc hing hingee ith ith
suffic sufficien ientt rotati rotation on capaci capacity ty to permit permit redist redistribu ributio tion n of moments moments in the entire entire structure! 1nly class $ sections can be used for plastic design!
Class 2 - Compac Compactt cross cross sectio section n! This This can can deve develo lop p full full plas plasti ticc mome moment nt
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 elements supported on both longitudinal edges3
-
Outside el elements attached 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 as plastic, as plastic, compact or semi-compact semi-compact if they meet limiting limiting proportions proportions for flanges flanges and ebs in axial compression compression given in Tab. 5 , BS .0 .0 Part *! #or rolled and elded column sections #ig! / shos these proportions hich ere set out to prevent local buc%ling!
Limitin" Proportions E$ement
Section Type
P$astic
Compa Semi %
Section
ct
Compa
Section ct Outstand
Ro$$ed &T
e$ement of compression
Section 15
(.5
*.5
).5
(.5
1+
#+
#5
#(
'e$ded &T
f$an"e Interna$
'e$ded &T
e$ement of compression f$an"e 'e su,ect to Ro$$ed d&T
#(
compression
+*
t-rou"-out
'e$ded d&T
/#)5 & p y 0 .5 !i"ure +. Limitin" proportions for ro$$ed and 2e$ded co$umn sections
Loads. Axial loading on columns in buildings is due to loads from roofs, floors
and alls transmitted to the column through beams and to self eight, #ig! 67a8!
#loor beam reactions are eccentric to the column axis, as shon, and if the beam arrangement or loading is asymmetrical, moments are transmitted to the column! +ind loads on multi-storey buildings designed to the
a0
0
!i"ure 3. Loads and moments on compression memers
simple design method are usually ta%en to be applied at floor levels and to be resisted by the bracing, and so do not cause moments! *n industrial buildings loads from cranes and ind cause moments in columns, as shon in #ig! 67b8! *n this case the ind is applied as a distributed load to the column through the sheeting rails!
5.#
4ia$$y $oaded compression memers
General considerations. Compression members may be classified by length! A
short column, post or pedestal f ails by crushing or s&uashing, #ig! 7a8! The s&uash load P y i in terms of the design strength is0
P y = p y A
+here3
A - area of cross section!
A long or slender column fails by buc%ling, as shon in #ig! 7b8! The failure load is less than the s&uash load and depends on the degree of slenderness! 9ost practical columns fail by buc%ling! #or example, a universal column under axial load fails in flexural buc%ling about the ea%er :-: axis, #ig! 7c8 ! The strength of a column depends on its resistance to buc%ling! Thus the column of tubular section in #ig! 7d8 ill carry a much higher load than the bar of the same cross-sectional area!
This is easily demonstrated ith a sheet of A6 paper! 1pen or flat, the paper cannot be stood on edge to carry its on eight3 but rolled into a tube it ill carry a considerable load! The tubular section is the optimum column section having e&ual resistance to buc%ling in all directions!
a0
0
c0
d0
!i"ure 5. Be-a6ior of memers in aia$ 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 )
!" y d y < d
)
Py
=−
This is solved to give the 4uler or loest critical load,
P !
) = π
!" y < #) !
*n terms of stress the e&uation may be rearranged as0 p ! = "
for r =
$
P ! $
) π =
!"
$#)e
! Substituting
and r )= "/$ gives0
p !
=
π ) !r ) )
#e
=
π ) ! )
)
#e < r
=
π ) !
π ) ! = ! ) λ ) 7 #e < r 8
here0 " y - moment of inertia about the minor axis :-:3 # - length of the strut3 P > axial load3 r y % &" y /$'(,)- radius of gyration for the minor axis :-:3 p ! % P ! /$ - 4uler critical stress3
λ % #/r y - slenderness ratio!
The slenderness * is the only variable affecting the critical stress! At the critical load the strut is in neutral e&uilibrium! The central deflection is not defined and may be of unlimited extent! The curve of 4uler stress against slenderness for a universal column section is shon in #ig! ?!
Strut +ith initial curature. *n practice, columns are generally not straight, and the effect of out ol straightness on strength is studied in this section! Consider a strut ith at initial curvature bent in a half sine ave, as shon in #ig! ;7b8! *f thc initial deflection at x from $ is y. and the strut deflects y further under load , the e&uilibrium e&uation is0
!" d ) y/d) %P&yy( ' here deflection y = sin ! $%&#
*f . is the initial deflection at the centre and δ is the additional deflection caused by P, then it can be shon by solving the e&uilibrium e&uation that0 δ =
δ . 7 P ! < P 8 −$
The maximum stress at the centre of the strut is given by0
pmax
2ut3
=
P P 7δ . $
+
−
δ 8
" y
, here h is shon in #ig! ;7d8!
p max % py - design strength pc% P/$ - average stress p4 % P ! /$ - 4uler stress " y % $ r y ) - moment of inertia about the 0-0 - axis $ - area of cross section r y - radius of gyration for the 0-0 - axis h - half the flange breath
Than the e&uation for maximum stress can be ritten as0 p y
=
pc
+
pc A$ −
δ . h 7 p ! < pc 8 − $ r y) $
@
δ . h = η 2ut r y) and rearrange to give0 7 p4 > pc8 7 py > pc8 = 1 p4 pc!
The value of pc , the limiting strength at hich the maximum stress e&uals the design strength, can be found by solving this e&uation and η is the 2erry factor! This is redefined in terms of slenderness!
"ccentrically loaded struts. 9ost struts are eccentrically loaded, and the effect of
this on strut strength is examined here! A strut ith end eccentricities e is shon in #ig! ;7c8! *f y is deflection from the initially straight strut the e&uilibrium e&uation is0
"Id # y%d$ #= - P!e ' y#
This can be solved to give the secant formula for limiting stress! Theoretical studies and tests sho that the behavior of a strut ith end eccentricity is similar to that of one ith initial curvature! Thus the to cases can be combined ith the 2erry factor, ta%ing account of both imperfections!
(esidual stresses. As noted above, in general, practical struts are not straight and
the load is not applied concentrically! *n addition, rolled and elded strut sections have residual stresses, hich are loc%ed in hen the section cools unevenly!
A typical pattern of residual stress for a hot-rolled B-section is shon in #ig 5! *f the section is sub"ected to a uniform load the presence of these stresses causes yielding to occur first at the ends of the flanges! This reduces the flexural rigidity of the section, hich is no based on
a0
0 !i"ure ). Residua$ stresses
the elastic core, as shon in #ig! 57b8! The effect on buc%ling about the 0-0 axis is more severe than for the 2-2 axis! Theoretical studies and tests sho that the effect of residual stresses can be ta%en into account by ad"usting the 2erry factor η .
)esign strengts! An extensive column-testing programme has been carried out,
and this has shon that different design curves are re&uired for0 7$8
Different column sections3
7)8
The same section buc%ling about different axes3
7/8
Sections ith different thic%ness of metal!
#or example, B-sections have high residual compressive stresses at the ends of the flanges, and these affect the column strength if buc%ling ta%es place about the minor axis!
The total effect of the imperfections discussed above 7initial curvature, end eccentricity and residual stresses on strength8 are combined into the 2erry constant
η ! This is ad"usted to ma%e the e&uation for limiting stress pc a loer bound to the test results!
The constant η is defined by0 1 % .!..$a &* - * (8, here *. = .!) 73 ) !/py8.!
The value λ . , gives the limit to the plateau over hich the design strength p y controls the strut load! The 'obertson constant a is assigned different values to give the different design curves! #or B %sections buc%ling about the minor axis a has the value ! to give design curve 7c8 7Tab! )57c88! A strut table selection is given in Tab. ) in BS 6.0 Part $! #or example, for rolled and elded B %sections ith metal thic%nesses up to 6. mm the folloing design curves are used 7$8
uc%ling about the ma"or axis - curve 7b8 7Table )57b883
7)8
uc%ling about the minor axis :: - curve 7c8 7Table )57c88!
The compressive strength is given by the smaller root of the e&uation that as derived above for a strut ith initial curvature! This is0 7 p4 > pc8 7 py > pc8 = 1 p4 pc, here0
py > design stress 7or pmax83
p4 - 4uler stress3 pc > limiting compressive stress, φ = A p y
+
pc
=
p ! p y 7φ + φ )
−
p ! p y 8 .!,
7η +$8 p ! @ < )
The curves for 4uler stress p ! and limiting stress or compressive strength pc for a rolled B - section column buc%ling about the minor axis are shon in #ig! ?! *t can be noted that short struts fail at the design strength hile slender ones approach the 4uler critical stress! #or intermediate struts, the compressive strength is a loer bound to the test results, as noted above!
Compressive strengths for struts for curves a, b, c and d are given in Tables )57a-d8 in BS .0 Part $!
"**ecti+e lengts ! The actual length of a compression member on any plane is the
distance beteen effective positional or directional restraints in that plane! A positional restraint should be connected to a bracing system hich should be capable of resisting $E of the axial force in the restrained member 7 Cl ! 6!5!$8! The actual column is replaced by an e&uivalent pin-ended column of the same strength that has an effective length0 & " = ,&
here & - actual length3 , - effective length ratio 7to be determined from the end conditions8!
!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!
!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 one method of framing, hich may be used to lessen this eccentricity, or even to balance the loads if the total reaction of the to spandrel
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 moment e. To the right of the axis of the bloc%, i!e!, on the same side as , this moment causes a compressive stress 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 stress 4c/" caused by the moment c . Substituting e for 4, the intensity of stress at 0 and 2 are expressed by the formulas
* y = / % A ' /ec % I 0
* $ = / % A / e c % I
or
* = / % A / e c % I
in hich f is 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 moment e be 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 moments 4 ( ,
deflecting to a shape shon in #ig! $6! 9aximum lateral deflection is 6
m
and 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 to 6 m. This, in turn, causes
/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 ' / 5 mc % 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 abovedescribed structural action! *t is seen that for a constant end condition such as that shon 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
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 P c > axial load causing yielding 7if it alone occurred8 and 4 c bending moment causing yielding 7if it occurred in absence of an axial load8! The similarity of the axial load ratios /P c and the axial stress ratios f a /p y should be apparent! The same similarity exists beteen the moment ratios 4 ( /4 y and the bending stress ratios f b /p y!
!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 p y 9 1.# 6 p y 7 7see Section 7.8.) of BS )9)(: Part 8
S = plastic modulus for the relevant axis
here0
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 /P c against 4 r / 4 c and 4 ry / 4 cy , here0
= applied axial load
P
% p y $, the s&uash load
4 r
= reduced moment capacity about the axis in the presence
of axial load 4 c
% moment capacity about the axis in the absence of axial
4 ry
%reduced moment capacity about the :: axis in the presence
load
of axial load 4 cy
% moment capacity about the :: axis in the absence of axial
load!
Jalues for 4 r and 4 ry are calculated using e&uations for reduced plastic modulus given in the ;uide to BS .0 Part $0 $?,
/%P c ' $ % c$ 1 and /%P c ' y % cy 1
here
$ % applied moment about the axis y % applied moment about the :: axis!
This simplification gives a conservative design! 2lastic and compact B sections
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 :-: axes 4 and 4ry, 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
/%P c ' $ % 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
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 is no 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
Case c - A slender column sub"ected to axial load and uniaxial axial bending
about the ma"or axis 8-8 . 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 9-9 planes 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
: = : 5; 1 :
Fe terms used are0
:
/ % P c0 r$ % c$ 0 ry % cy
/ % P c0 a$ % c$ 0 ay % cy
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!
$! 4 ay; the moment capacity about the :: axis, is not sub"ected to the restriction $!) p y = y . )! At Lero axial load, slenderness does not affect the bending strength of an B section about the 0-0 axis! /! At high values of slenderness the buc%ling resistance moment 4 b about the - axis controls the moment capacity for bending about that axis! 6! As the slenderness increases, the failure curves in the /P c , :-:-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!?!/ of BS .: 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%!
*n each case to procedures are given! These are a simplified approach and a more exact one!
7$8 Loca$ capacity c-ec
/%A g p y ' $ % c$ ' y % cy 9 1.
+here3
- applied axial load3 $ g - gross cross-sectional area 4 - applied moment about the ma"or axis 2-2 4 c - moment capacity about the ma"or axis 2-2 in the absence of axial load 4 y - applied moment about the minor axis 0-0 4 cy - moment capacity about the minor axis 0-0 in the absence of axial load!
More ri"orous ana$ysis given 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$ ' m y % y 9 1.
here0
9x > maximum buc%ling moment about the - axis in the presence of axial load and e&uals the lesser of0
c$ ;! 1- / % P c$ # % ! 1 ' .5/ % P c$ #<7 or
+here3
! 1 / % P c y#
4 ay > maximum buc%ling moment about the :: axis in the
presence of axial load and e&uals
cy ;! 1- / % P cy # % ! 1 ' .5/ % P c y#<7
here0
P c and P cy > compression resistances about the ma"or and minor
axes respectively!
7)8 O6era$$ uc<$in" c-ec< ! The simplified interaction relationship to be satisfied is given in Clause 7.>.?.?. of the Code! This is0
/%A g pc ' m $ % ' m y % p y 6 y 9 1.
+here3
m - e&uivalent uniform moment factor from Table $? in the
code 4 b - buc%ling resistance moment capacity about the ma"or axis - = y - elastic modulus of section for the minor axis :-:
The value for 4 b is given in Cl. 6!/!5!5 of the Code as0
= p S $ =
here0
pb > bending stress determined from Tab. $ for values of * and
. * @ slenderness # ! / r y. @ torsional index = A / T 7approximately8 or can be calculated from the formula in Appendix or from the published table in the Muide to BS .0 Part $!
A more exact approach is also given in the code! This uses the convex failure surfaces discussed above!
Eamp$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
$!
Kocal buc%ling capacity chec%!
#rom Tab! ; find design strength, py = )5 F
#rom Tab! 50
for flanges - b/T = ?!$; N ?! 7plastic8 for eb - d/t % $;.!? < ? = ).!$ N / 7plastic8
9oment capacity for plastic or compact sections in the absence of the axial load0 4 cx = S x py = ;? x )5 x $. -/ = $;!) %F m > > $!) = x py = $!) x $. x )5 x $. -/ = $;?!6 %F m, so 1O!
*nteraction expression gives0 ??. ×$. / ;;!6 × )5, ×$.
/
−
+
/, $,;!)
=
.!6? + .!))
=
.! 5
The section is satisfactory ith respect of local capacity!
)!
1verall buc%ling chec%
$!. , so 1O!
4ffective length, Tab! )6!
#4 = .!? x 6.. = /?) mm3 Slenderness * = #4
= /?)<$!; = 56!$
#rom Tab!) select design curve and from Tab! )57c8 for * =56!$ and py =)5 F
#rom Tab! 7for the support conditions of the beam column that is laterally restrained against torsion, but partially free to rotate in plane8 the effective length #4 = .!? x 6.. = /?) mm3 The ratio of end moments = 4 min < 4 max =$)
E =
" cf " cf
" tf
+
= .!3 *< = 56!$<$!? = 6!; P D = .!?/)
4&uivalent slenderness, *KT = n u D * = $ x .!?6? x .!?/) x 56!$ = )!$
) #rom Tab! $$ the bending strength, p b = )/)!5 F
moment capacity, 4 b = S x p b = )/)!5 x ;? x $. -/ = $/)!$%F m! #rom interaction diagram0
- $ g
+
m4 4 b
=
??. ;;!6 ×$5) !?
+
.!;-5 × /, $/)
=
.!-,
$ !. ,
so it’s 1O!
The section is also satisfactory ith respect of overall buc%ling!
5.3. Co$umn ases
A$ially loaded sla ase plates ! Columns hich are assumed to be nominally
pinned at their bases are provided ith a slab base comprising a single plate fillet elded to the end of the column and bolted to the foundation ith four holding don 7B!D!8 bolts! The base plate, elds and bolts must be of ade&uate siLe, stiffness and strength to transfer the axial compressive force and shear at the support ithout exceeding the bearing strength of the bedding material and
a0
0 !i"ure 11.
concrete base, as shon in #ig! $$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
t
=
A
)!,+
p yp
7a )
−
.!/b ) 8@.!,
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 distribution p y - the design strength of the plate 7Cl. ?.. or Tab. ;8, but not greater than )5. F
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 e N $<; A p, the resultant load lies ithin the middle third of the base length and pressure occurs over the hole of the base! Than, P % F/$ Q 4/= R bearing strength of the concrete = .!6 f cu, here the area of the base $ = bA p and elastic modulus, =%B p A p<; 7 or - gives pmax and pmin pressure respectively8! Consider $ mm ide strip hich acts as a cantilever from the face of the column! This approach gives a conservative design for the thic%ness! ending in to direction is not ta%en into account! ase pressure at section -,
px-x = pmin 7 A p - b8< A p@7 pmax - pmin8 = pmax - 7b< A p87 pmax - pmin8!
The trapeLoidal pressure diagram loadin the strip is divided into to triangles! The moment at - is0
4 x-x = 7 px-x x b x b
4 = py = x = py x t )<;
#or eld connection find $ and = from section properties table! Axial stress f c = F/$ and the bending stress f bt = 7 4 x $./8< = x!
