\; -

l'

LATERAL AND LOCAL BUCKLING OF I :BBAMS IN PLASTIC BANGE

(A part from the PhoDo dissertation of Ching Hwm Yang)

/

IIII~'

I.,

55

(5) L.flTERAL AND LOCAL BUCKLING- OF I BEM~1S IN PLAsrrI(; RANGE

Shanley, in his paper on inelastic column theory. (10)

has proved that an ideal column will start to bend e,t a

.load equal to its tangent modulus load. According to the

average stress and strain· diagram of structural steel, the

"'slope of the curve in the plastic range before strain har-

dening is zero. That leaves the criterion of the tangent

modulus load ~~thout real meaning in the limiting case.

In columns loaded with a slight ecC'entrici ty (Dro-

. vided the- columns are made of a material which exhibits the

above-stated zero slope throught the plastic range), the

average compression stress on the column cross section can

never reach the yield point.

Ideally loaded columns made of perfectly plastic

material are actuallyunstablewhEln the average stress

reaches yield point. It may collapse at any instant when

the average stress reaches the yield point .regardless what9

length the column has.

. ,

F J\ r :;, ,".I G. / . ...Y'l)

. I

56

This statement can be demonstrated as follows:

Let Po be the load at which the average compression

stress on the column reaches the yield point

" Po = a CfYll

where "a " equals the total cross section area of the

column.

Suppose the ideal column is loaded with axial load Po'

and we make a virtual displacement along the column as the

dotted lines show aboveo Take a cross s~ctlon A,A. Let

the corresponding displacement at that section be Yo. Then

the section will be under ~otal axial load Po and a

moment IvI = Po "-yo. In order to balance both the mome nt and

the axia.l load, the stress distribution must be chane:;ed from

jk to hi. But the stress cannot be any higher the cry if

the material is perfect~y plastic. The stress distributed

in the shaded area in the. above Fig. 1s therefore impossibleo

The external moment will not be balanced. It is obvious

that the column will" collapse in bending at once.

This would lead to a conclusion that a member shoUld

never carry co~pressive stress to the yield point if it

is made of materials that have perfect plasticity after

yielding.

In simple plastic theory, sections in structural

members at which stresses exceed the yield point are assumed

-

57

to develop "plastic hinges". In the case of "I" sections

there is, from the above discussion, a possibility of

bucklin2, of the compression flange in the region of the

so-called tlplastic hinge". The buckling of the compression

flange would l~turally reduce the value of plastic hinge

moment. Therefore the' problem of instability of structural

steel members becomes very important in the simple pI Bstic

theory of structural design.

Stress and strain curves for ordinary tension or com-• I

pression tests do not give enough information for the

analysis of stability problems. in the plastic range. None

of the metals used in engineering structures are perfectly

plastic 2nd an ordinary stress and strain diagram usually

gives no information of the relation between stress and

strain rate in pIa sti c ranGe. The mechani sm of yielding of

the metal also affects the bucklinc ; strength of the com-

pres sian mera.ber.

Take the previous column; the moment at section Al A2

may be very STIlB.ll at the beginning. If this moment makes

the column bend, then the strain rate at slde .11 will

naturally be higher than at side A2- ~he stress at side

Al wil.l be raised by the higher strain rate that makes the stress

distribution over the cross section possible to balanoe the external

moment, ?nd the column wil] thaD bend and shorten at the same

time until the strain hardening range is reached. I

58

In a si.-rnple tension or compression test of a structur2.1

steel one will find that Luder's lines do not all appear at

onee when the yield point is reached. Yield lines'usually

are initiated in srnne places and then gradually spread over

the whole- spcc:i.meno· While the yield lines are progressing,

the region where the yield lines were initiated might~

have develo:?ed all its plastic strain and reached the strain

hardenin;:=; l'anse 10ca11 yo The specimen can not be considered

as perfectly plastic even though the portion of the stress

•and strain diagram is observed to be parallel to the abscissa.

The co:nprGssion member can therefore be expected to have a

buckling strength of the tanE:ent modulus load in the plastic'

range where the tangent of the stress and strain curve is

chosen at the starting point of the strain hardening range.

E.

F/G~ IV 3/

HoVl does this yieldinL process affect the buckline;

strength of a compression member? It can be demonstrated

by the following analogi cal eX8l!Dle:

.-----.~-l\1\

I I\r-~·--Uy 1\

! i \ rr__________.l.L:.- __

CI. I. I II I, ;.. i: • I II I

, _r-'-J-'--L-~..L-J.t_t ~ ; '1 D

IIIi

!II ,,

A '-r-.-r-.,-,c--.-~,_,__._,.__r_'-,i' .B

n

A 59

(0.) (b)

FIG. IV 32

Suppose the compression member is slightly tapered on one

side, as shown in dotted lines in above Fig. IV.32(a).

Instead of having a uniform cross section throughout the

length, the stress distribution along the member will be as

shown in dotted lines Fig. IV.32(b). The root section

AB will reach the yield point first, and as stress increases

the yielding zones will progress to reach the top section

CD. If t'ne mechanical, properties of the material are homo-

geneous every section will be strain hardened as soon as

the stress exceeds o-yo During this progression of yield.ing

the strain lwrdening zone and elastic zone are separated by

only an inf'in5- te simally thin plane which has peJ'fe ct

plasticity. In this case the cO:':lpression member will,,'

however, have a buckJing strength at ~ast equal to the

tangent :nodulus load. The tangent of the stress and strain

diagram is selected at the point of the ~tartlng of the

strain hardening region as before.

60 .

To summarize the above discussion there may be two

extrer,le case s:

Fil~st, if the, rraterial is perfectly plastic and stressed

to the yield point, the compression member yIill be

and will bend no matter what the L Ir ratio of the

unstable

com-

pres~,ion me;~lber is. Secondly, if thE; ~)last:i.c flow in a

compr8ss1on Yi18mb3:L' is established plane by plane and all

the '018.1188 ,(.:etstrain hardened 3.8 ':)lastic flow oroc-resses~ . ~ G

the CO:',l131"'833ion menber YJill then have a blYJ.ckllng strenc;th

of tangent ~odulus load as defined above.

The practical case may lie between the above two

extremes. Apparently the tangent ~odulus load defined as

above" vJill be come the uppe:r lind t 'of the buckling s tren[.;th-

for the . .co~presslon member. For rectaneular sections the

t[;mgent :nodulus load can be calculated as follows:

t

FIG. IV33-------~----------

•

61

x 103E* = 0.636t

btG

I =:rn-~y = 37.5 kips!inch2

P = a cry = bt cryEtI it la

p = ~ 1.:3 •

d~ -dE:'-

~ = 3.75

L~ = 7.5 'for the oase of fixed end.

Compression coupons were tested at a ratio of ~ = 4. Test

conditions al~e simulated to the fixed end. No bending in

the plastic range is observed., Stress and strain diagr~l

showed a yield point in Fig. 35. More tests of very

carefully aligned.short compression members of various ~

ratios are needed to evaluate an effective value of the

tangent modulus at the starting point of the strain har-

dening region to predict the buckling strength of steel

structural mcmbers o

The lateral buckling problem of flexural members also

becomes vOl'y serious in the plastic rangeo Take a simply

\\

\

supperted I-beam under constant mornent and suppose that,both of the flanges are in the plastic range under constant

moment. . .~- - - ~ ~ - - - - - - ~ -- - - ~ ~ - - - - - - - - - - --* Et 1s the tangent at the starting point· of the strain

hardening portion of the stress-strain curve •

•

62

.( ,~----.-----.--.-- ..-..-. -'-"--=='~===:-'=~='::~l \,I I IIJ \:1-'-------.---~·-···-'---··-·-----··-.I I--- "-.--'-H .

FIG. J\/-.' .34-----

According to the first assumption that the yielded part of

the beam is perfectly plastic, it is obvious that the

lateral buckling strength of the beam will be ,equivalent to

a beam considering only the elastic part of the same be.ara

under the same moment. According to the assumption that

the yielded part of the beam will have a tangent modulus

strength, the lateral buckling strength of this beam can be

computed by regarding the beam as n~ving different moduli

in the elastic rErt and plgstic part. The actual lateral

buckling strength of s~c4 a beam is expected between the. above

two val~eso

The central· portion of the tested continuous beams bet·wee·n

the two loading points were all under constant moment. It

is natural that when both flanc;es of this portion enter the

plastic range~ the lateral buckling strength of the beam

will be greatly reduced.~

. I"evel bars were mounted perpendicular to the beam a.xis

to measure the rotatHm of the beam. Curves are shown in

Fig. 3Qo Par 8WF40section it is soen that lateral deformation

63

started bet~veen Vol = 45 kips to W = 50 kips. It was at this

regio~as observed from both t~e white wash and strain

gage~ that, the flanges went to the plastic range. Fig. 37

is a picture of beam B7 taken after the test, wluch shows

that the central span buckled laterally in tvvo half wave

Ie ngths.

Dial gages also were mounted near the supports between

the tension and compression flanges to measure the local

buckling in Beam B4 and B5 as shown ih Fig. 38. Beam B7

has thin..Yler flange thickness than all the rest of the

beams. The compression flange buckled very severely as

shown in FiG- 39 0

(6) SH'~;A!i- FAILURE IN EECTANGULAR BfAIWS OR IN VlEBS OF I-·

SECTIONS Dlffi TO TRANSVERSE LOAD

In the elastic beam theory, when a bending nlember is

under transverse load, maximum shear stress is developed at

the neutral axis. This shear stress is distributed as a

parabolic fun~tion across the section accol"'ding to the

theory. As the beam is loaded to the plastic Fange this

prediction of shear stress by the elastic theory is of

course no' lonGer valid. This shear stress, when its value

becmnes comparatively high, may initiate yielding in the

web of the beam earlier than that of' the normal stress in

the outer fibre of the flange_ The problem is discussed in-

the following three different cases:

(I)

(II)

Table' of Contents,

Synopsis

Introduction

Page

1

1

(III) Description of Experimental Program

(1) Test set-up

(2) Preparation of specimen

(3) Strain gages, deflection gages andlevel bars

(4) Test procedure

3

5

7

10

( IV) Presentation and Analysis of Test Resul ts· 12•

(1) Bending strength of I-sectionsunderpure moment 12

a) M~ relation of bending membors

b) Strain.distributj.on in plasticbending

(2) Initial yield strength of the con­tinuous beams

a) Residual stress

b) Stress concentration

12

17

18

18

24

(3) Ultimate strength of continuous beam~, 31

a) Strain hardening effect of struc­tural steel and the Ultimate strengthof continuous be~s 32

b) The influence of end restraint onthe Ultimate strength of continuousbeams 38

(4) Deflection of.beams in plastic range 41

a) Method of simple plastic theory

b) Method of numerical integration

42

46

0) Method of mathematical integrationfor I-sections 48

d) Simplified method of mathematicalintegration for I-sections 50

(5 )

(6)

Lateral and local buckling of I-beamsin plas ti c radge

Shear failure lin rectangular beams orin webs ofr-sections due to trans­verse load

55

63

(v)

(\11 )

Conclusi ons

References

73

74

(VrI) Appendixes

A., Coupon test ~esults 77

B. Residual stresses 82

c. Determination of initial yiel~ strengthin experimental load-deflection curves 85

(VIII) Nomenclature 87

... -... -..

Strain in inches per. inch

C COMPRESSION STRESS-STRAIN CURVE FOR AN 8WF40 BEAMFig. 35

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-0.01 -0.005 o 0.005 0.01 0.015 0.02

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1000

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Deflection in inches

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