lecture -0008 compression members

8/13/2019 Lecture -0008 Compression Members

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N.W.F.P. University ofEngineering and Technology

Peshawar

1

By: Prof Dr. Akhtar Naeem Khan

[email protected]

Lecture 08: Compression Members


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan 2

Compression Members

Structural elements that are subjected to axial compressive forces only are called columns.Columns are subjected to axial loads through

the centroid. The stress in the column cross-section can be

calculated as

where, f is assumed to be uniform over the

entire cross-section

A

=f


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan

Compression Members

This ideal state is never reached. The stress-state will be non-uniform due to!Accidental eccentricity of loading with

respect to the centroid

"ember out-of #straightness $croo%edness&,or

'esidual stresses in the member cross-section due to fabrication processes


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan !

Compression Members

Sometime they may carry bending moments as

well about one or both axis of cross-section.

The bending action may produce tension inpart

of the cross-section

(espite of tensile stresses or forces that mayproduce, columns are generally referred as)Compression "embers* because compression

stresses normally dominate their behavior.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan "

Compression Members

#n addition to mo$t common type of

compre$$ion mem%er$ &vertica' (em%er$

in $tructure)*compre$$ion may inc'ude the

Arch ri%$

+i,id frame mem%er$ inc'ined or otherwi$e

-ompre$$ion e'ement$ in tru$$e$


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Compression Members


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Compression Members


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Compression Members


9/91CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan

Compression Members



Compression Members



3'enderne$$ +atio4he 'on,er the co'umn* for the $ame 56$ection*

the ,reater %ecome$ it$ tendency to %uck'e and

$ma''er %ecome$ it$ 'oad carryin, capacity.

4he tendency of co'umn to %uck'e i$ u$ua''y

mea$ured %y it$ $'enderne$$ ratio

Compression Members Vs

Tension Members




Tension Members

7ffect of materia' #mperfection$ and 8'aw$

3'i,ht imperfection$ in ten$ion mem%er$ are can

%e $afe'y di$re,arded a$ they are of 'itt'e

con$e9uence.n the other hand $'i,ht defect$ in co'umn$ are

of ,reat $i,nificance.

A co'umn that i$ $'i,ht'y %ent at the time it i$ putin p'ace may have $i,nificant %endin, re$u'tin,

from the 'oad and initia' 'atera' def'ection.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan 1!


Tension Members

4en$ion in mem%er$ cau$e$ 'en,thenin,

of mem%er$.-ompre$$ion %e$ide compre$$ion force$

cau$e$ %uck'in, of mem%er.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan 1"


Tension Members

resence of holes in bolted connection reduce

+ross area in tension members.

resence of bolts also contribute in ta%ing loadAn Ag


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WHY column more critical

than tension member?

A co'umn i$ more critica' than a %eam

or ten$ion mem%er %ecau$e minorimperfection$ in materia'$ and

dimen$ion$ mean a ,reat dea'.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan 1/

WHY column more critical

than tension member?

4he %endin, of ten$ion mem%er$

pro%a%'y wi'' not %e $eriou$ a$ the

ten$i'e 'oad$ tend$ to $trai,hten tho$emem%er$* %ut %endin, of compre$$ion

mem%er$ i$ $eriou$ %ecau$e

compre$$ive 'oad$ wi'' tend to ma,nify

the %endin, in tho$e mem%er$.


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Compression Member Failure

4here are three %a$ic type$ of co'umn fai'ure$.

ne* a compre$$ive materia' fai'ure& very $hort

and fat).

4wo* a %uck'in, fai'ure*&very 'on, and $kinny).

4hree* a com%ination of %oth compre$$ive and

%uck'in, fai'ure$.&'en,th and width of a co'umn i$in %etween a $hort and fat and 'on, and $kinny

co'umn).


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4here are three %a$ic type$ of co'umn fai'ure$.

ne* a compre$$ive materia' fai'ure& very $hort

and fat).

4wo* a %uck'in, fai'ure*&very 'on, and $kinny).

4hree* a com%ination of %oth compre$$ive and

%uck'in, fai'ure$.&'en,th and width of a co'umn i$in %etween a $hort and fat and 'on, and $kinny

co'umn).


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Flexural Buckling &a'$o ca''ed 7u'erBuck'in,)i$ the primary type of

%uck'in,.mem%er$ are $u%;ected to

%endin, or f'e5ure when they %ecome

un$ta%'e


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Local Buckling4hi$ occur$ when $omepart or part$ of 56$ection of a co'umn are

$o thin that they %uck'e 'oca''y in

compre$$ion %efore other mode$ of%uck'in, can occur


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Torsional Buckling 4he$e co'umn$ fai' %y

twi$tin,&tor$ion) or com%ined effect of

tor$iona' and f'e5ura' %uck'in,.


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Sections used or

Compression Member

#n theory numerou$ $hape$ can %e

u$ed for co'umn$ to re$i$t ,iven 'oad$.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan 2!

Sections used or

Compression Member


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Sections used or

Compression Member


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Sections used or

Compression Member


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Sections used or

Compression Member


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Column !uc"lin#

Buck'in,

7'a$tic Buck'in,

#ne'a$tic Buck'in,


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Column !uc"lin#

Buck'in, i$ a mode of fai'ure ,enera''y

re$u'tin, from $tructura' in$ta%i'ity due

to compre$$ive action on the $tructura'

mem%er or e'ement invo'ved.

75amp'e$ of common'y $een and u$ed

too'$ are provided.


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!uc"lin#75amp'e


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!uc"lin#75amp'e


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!uc"lin#75amp'e


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!uc"lin#75amp'e


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!uc"lin#

75amp'e &a) i$ temporary or e'a$tic

%uck'in,.

75amp'e &%*c*d) are e5amp'e$ of p'a$tic

%uck'in,.


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Column !uc"lin#3tee' co'umn %uck'in,


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Mechanism o !uc"lin#

=et u$ con$ider 8i, 1* 2* and $tudy themcarefu''y.

#n fi,1 $ome a5ia' 'oad P i$ app'ied to the co'umn.

4he co'umn i$ then ,iven a $ma'' def'ecion %y,ivin, a $ma'' force 8.

#f the fprce P i$ $uficient'y $ma''* when the force 8

i$ removed* the co'umn wi'' ,o %ack to it$ ori,ina'$trai,ht po$ition.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan /

Mechanism o !uc"lin#Fig 1


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4he co'umn wi'' ,o %ack to it$ ori,ina' $trai,ht

po$ition. >u$t a$ the %a'' return$ to the %ottom

of the container.?ravity tend$ to re$tore the %a'' to it$ ori,ina'

po$ition whi'e in co'umn$ e'a$ticity of co'umn

it$e'f act$ a$ a re$torin, force.

4hi$ action con$titute$ $ta%'e e9ui'i%rium.


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4he $ame procedure can %e repeatedwith increa$ed 'oad unti'' $ome critica'

va'ue i$ reached.


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Fig 2


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan !1


4he amount of def'ection depend$ on

amount of force 8.

4he co'umn can %e in e9ui'i%rium in an

infinite num%er of %ent po$ition.


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Fig 3


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4he e'a$tic re$torin, force wa$ not

enou,h to prevent $ma'' di$tur%ance

,rowin, into an e5ce$$ive'y 'ar,edef'ection.

Dependin, on ma,nitude of 'oad P*

co'umn either remain in %ent po$ition* orwi'' comp'ete'y co''ap$e or fracture.


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan !!


4hi$ type of %ehavior indicate$ that for a5ia''oad$ ,reater than Pcrthe $trai,ht po$ition of

co'umn i$ one of un$ta%'e e9ui'i%rium in that a

$ma'' di$tur%ance wi'' tend to ,row into ane5ce$$ive deformation.

Buck'in, i$ uni9ue from our other $tructura'e'ement$ con$ideration$ in that it re$u't$ from

$tate of un$ta%'e e9ui'i%rium.

-onc'u$ion$


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan !"


Buck'in, of 'on, co'umn$ i$ not cau$ed %yfai'ure of materia' of which co'umn i$

compo$ed %ut %y determination of what wa$$ta%'e $tate of e9ui'i%rium to an un$ta%'e one.

-onc'u$ion$


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Mechanism o !uc"lin#-onc'u$ion$

Compression member


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan !/

Compression member

!uc"lin#

Buck'in, occur$ when a $trai,ht* homo,eneou$*centra''y 'oaded co'umn $u%;ected to a5ia'

compre$$ion $udden'y under,oe$ %endin,.

Buck'in, i$ identified a$ a fai'ure 'imit6$tate forco'umn$. Pcr

Pcr

P

P

(a) (b)Pcr

Pcr

P

P

P

P

(a) (b)

Compression member


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Compression member

!uc"lin#

4he va'ue of P at which a $trai,ht

co'umn %ecome$ un$ta%'e i$ ca''ed the

-ritica' =oad.

hen co'umn %end$ at critica' 'oad* it i$

$aid to have %uck'ed.

4herefore* critica' 'oad i$ a'$o ca''ed the

%uck'in, 'oad.

$lastic !uc"lin# o


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$lastic !uc"lin# o

Columns

4he critica' %uck'in, 'oad Pcrfor co'umn$

i$ theoretica''y ,iven %y

4endency of compre$$ion mem%er$ to

%uck'in, i$ ,overned %y =r

$lastic !uc"lin# o


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$lastic !uc"lin# o

Columns

The intersection point , of the two curves represents the

maximum theoretical value of slenderness of a column

compressed to the yield strength. This maximum slenderness

$sometimes called uler slenderness&

$lastic !uc"lin# o


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan "1

$lastic !uc"lin# o

Columns

%nelastic !uc"lin# o


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Columns

#n e'a$tic %uck'in,* it wa$ a$$umed that a co'umnmade of a meta' who$e $tre$$6$train curve i$'inear unti' a yie'd p'ateau reached.

8or a co'umn with intermediate 'en,th* when%uck'in, occur$ after the $tre$$ in the co'umne5ceed$ the proportiona' 'imit of the co'umnmateria' and %efore the $tre$$ reache$ the

u'timate $tren,th. 4hi$ kind of $ituation i$ ca''edinelastic buckling.



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Columns

4an,ent6(odu'u$ 4heory



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan "!


Columns

7n,e$$er$ -onc'u$ion wa$ cha''en,ed with the

%a$i$ that %uck'in, %e,in$ with no increa$e in

'oad.

The tangent-modulus theory oversimplifiesthe inelastic buckling by using only onetangent modulus. In reality, the tangent

modulus depends on the stress, which is afunction of the bending moment that varieswith the displacement w.

4an,ent6(odu'u$ 4heory: Draw%ack$



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan ""


Columns

4an,ent6(odu'u$ 4heory: Draw%ack$

The tangent-modulus theory tendsto underestimate the strength of

the column, since it uses thetangent modulus once the stresson the concave side exceeds the

proportional limit while the convexside is still below the elastic limit.



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Columns

+educed (odu'u$ 4heory7n,e$$er pre$ented a $econd $o'ution

to the ine'a$tic6%uck'in,* in which the

%endin, $tiffne$$ of the 56$ection i$e5pre$$ed in term$ of dou%'e modu'u$

Erto compensate for the

underestimation given by thetangent-modulus theory.

http://www.efunda.com/formulae/solid_mechanics/columns/inelastic.cfmhttp://www.efunda.com/formulae/solid_mechanics/columns/inelastic.cfm


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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan "/


Columns

+educed (odu'u$ 4heory

8or a co'umn with rectan,u'ar cro$$

$ection* the reduced modu'u$ i$ defined %y:

4he corre$pondin, critica' $tre$$ i$*



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Columns

+educed (odu'u$ 4heory: Draw%ack$

The reduced-modulus theory tends

to overestimate the strength of thecolumn, since it is based on stiffnessreversal on the convex side of thecolumn.



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Columns

+educed (odu'u$ 4heory: Draw%ack$

The reduced-modulus theory

oversimplifies the inelastic bucklingby using only one tangent modulus.In reality, the tangent modulusdepends on the stress which is afunction of the bending moment thatvaries with the displacement w.



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Columns

3han'ey$ 4heoryThe critical load of inelastic buckling is in

fact a function of the transversedisplacement w

Practically there are manufacturingdefects in mass production and geometricinaccuracies in assembly.

This is the reason why many designformulas are based on the overly-conservative tangent-modulus theory.



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Columns

3han'ey$ 4heory

Factors eectin# !uc"lin#


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1. 7nd -onnection$

2. 7ccentricity of 'oad$-rookedne$$

. +e$idua' $tre$$e$



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1. 7nd -onnection$


+otation of end$ of co'umn$ in %ui'din,frame$ i$ u$ua''y 'imited %y %eam$

connectin, to them.



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1. 7nd -onnection$: 7ffective 'en,th


KL i$ ca''ed effective lengthof co'umn and K

effective length factor.



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1. 7nd -onnection$: 7ffective 'en,th


A column with fixed ends can support four times as much load asa column with pinned ends

This benefit decrease with decreasing L/r until Fcr finally

becomes virtually independent of K



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2. 7ffect of initia' crookedne$$


4he initia' out6of6$trai,htne$$ i$ a'$o termed Cinitia'crookedne$$C or Cinitia' curvatureC.

#t cau$e$ a $econdary %endin, moment a$ $oon

a$ any compre$$ion 'oad i$ app'ied* which in turn'ead$ to further %endin, def'ection and a ,rowth inthe amp'itude of the 'ever arm of the e5terna' endcompre$$ion force$.



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A $ta%'e def'ected $hape i$ po$$i%'e a$

'on, a$ the e5terna' moment* i.e. theproduct of the 'oad and the 'atera'def'ection* doe$ not e5ceed the interna'moment re$i$tance of any $ection.



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hen $trai,ht co'umn %uck'e$* it

a$$ume$ a $ta%'e* %ent e9ui'i%rium* %ut

with $'i,ht'y 'ar,er 'oad.#n -rooked co'umn def'ection increa$e$

from %e,innin, of 'oadin, and co'umn i$

in un$ta%'e condition when it reache$ toma5imum 'oad.



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. 7ffect of +e$idua' 3tre$$e$


#n ten$ion mem%er$ +e$idua' $tre$$e$ cau$e$the $ection to yie'd at a $tre$$ 'ower than the

yie'd point of the materia'.A$ a re$u't* the e'on,ation for a ,iven 'oad i$

,reater than wou'd %e ca'cu'ated form e'a$ticpropertie$.



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan /1



-omp'ete yie'din, of 56$ection did not occur unti'app'ied $train e9ua'$ the yie'd $train of %a$e

materia'.4he re$idua' $tre$$e$ doe$ not affect the 'oad

corre$pondin, to fu'' yie'd of 56$ection.



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#f the ma5imum $tre$$ n

reache$ the yie'd $tre$$ fy*

yie'din, %e,in$ to occur in

the cro$$6$ection. 4heeffective area a%'e to re$i$t

the a5ia' 'oad i$* therefore*

reduced.



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan /!



7ffect of re$idua' $tre$$e$ in cau$in,weak a5i$ %uck'in, at 'oad$ $ma''er

than tho$e for $tron, a5i$ %uck'in,.4hi$ $u,,e$t two co'umn formu'a$

for the $tee' W.

Tests carried on W shapes

Structural Stability Research

Council (SSRC)proposed a single

forula to siplify the deign

procedure



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan /"

. 7ffect of +e$idua' 3tre$$e$: 33+- 8ormu'a


!esign procedure be siplified by using Parabolabeginning "ith a #erte$ at Fcr%Fy"here &'r and

terinating at Fcr%Fy'2 "here it intersects and tangent to

uler yperbola*



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-om%ined 7ffect of -rookedne$$ E +e$idua' 3tre$$e$


An initia' out6of6$trai,htne$$ eo* produce$ a

%endin, moment ,ivin, a ma5imum %endin,

$tre$$ $B

#f $ma5i$ ,reater than the yie'd $tre$$ the fina'

di$tri%ution wi'' %e part p'a$tic and part of the

mem%er wi'' have yie'ded in compre$$ion.



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CE-409: Lecture 08 Prof. Dr Akhtar Naeem Khan //

-om%ined 7ffect of -rookedne$$ E +e$idua' 3tre$$e$


Code &e'uirements


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A3D 8ormu'a

Code &e'uirements

=Cc

r

KL

rKL

rKL

E

Ccr

KL

Cc

rKL

Cc

rKL

CcFy

Fa

rKL

22

2

3

2

1+,---

23

12

.

1

.

3

3

/

2

11

Code &e'uirements


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=+8D 3pecification$

Code e'u e e ts

4he de$i,n $tren,th of co'umn$ for the f'e5ura'%uck'in, 'imit $tate i$ e9ua' to

cP

n

here* cF .0" &+e$i$tance factor for compre$$ion mem%er$)

PnF A

,8

cr

8or cG 1." 8

crF 8

y

8or cH 1." 8crF 8y

c/01.2

c

133.2

-

4

r

56 y

Code &e'uirements


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=+8D 3pecification$

'

c

,-

4

r

56 y

4cr74y

8.2

8.0

2.9: 4cr

, 4y

c

133.2

4cr

, 4y( )c/01.2

c

,-

4

r

56 y

c

,-

4

r

56 y

4cr74y

8.2

8.0

2.9: 4cr

, 4y

c

133.24cr

, 4y

c

133.2

4cr

, 4y( )c/01.2 4cr, 4y( )c/01.2

Local !uc"lin#


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#

;f the column section is made of thin$slender& plate elements, then failure canoccur due to local buckling of theflanges or the webs in compression well

before the calculated buc%ling strength

of the whole member is reached.

hen thin p'ate$ are u$ed to carrycompre$$ive $tre$$e$ they areparticu'ar'y $u$cepti%'e to %uck'in,

a%out their weak a5i$ due $ma''moment of #nertia.

Local !uc"lin#


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#

Local !uc"lin#


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#

Flange Buckling

Laterally buckled beams


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Local !uc"lin#


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#

;f local buckling of the individual plate elementsoccurs, then the column may not be able to develop its

buc%ling strength.

Therefore, the local buc%ling limit state must beprevented from controlling the column strength.

Local !uc"lin#


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#

5ocal buc%ling depends on the slenderness $width-to-thic%ness b/t ratio& of the plate element and the yieldstress $4y& of the material.

ach plate element must be stoc%y enough, i.e., have ab/t ratio that prevents local buc%ling from governingthe column strength.

Local !uc"lin#


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#

4he critica' $tre$$ for rectan,u'ar p'ate$ with

variou$ type$ of ed,e $upport$* and with'oad$ in the p'ane of the p'ate di$tri%uted

a'on, the ed,e$ in variou$ way$ i$ ,iven %y

0% Constant depends on


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#

The coefficient k has a minimum value of 4 for

a/b=1!" etc.The error in using k=4 #ecreases $ith increasing a/b

an# for a/b= 1% or more it is extremel& small.

Local !uc"lin#


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#

-ritica' $tre$$e$ for p'ate %uck'in, can %eeva'uated %y determination of e9uiva'ent

$'enderne$$ ratio for which a co'umn wi''

%uck'e at $ame $tre$$* u$in,

Local !uc"lin#


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#

The A;SC specification provides the

slenderness $b7t& limits that the individual plate

elements must satisfy so that local bucklingdoes not control.

Consult table >-> of +aylord


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Than"s

lecture -0008 compression members

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