Design of Compression Member BS

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

Design of Compression Member BS

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