'1 <-~,

i' '~iI

,,I, ,--....,

},-/I

I(f)}1

a 0::;=)-/' II"3 ~c{ , ;:< q 0-'

INFLUENCE OF YI~ED ADJACENT ,SPANS,

O~ THE ROTATION CAPACITY OF B~

by

L ENG\NEER\NGDEPARTMENTNOEFERCI~~~ LABORATOR'{

FRITZ ENGILE\-\IGH UNIVERSIIY ,

'BE1HLEHENI, PENNSYLVANIA

A THESIS

Presented to the Graduate Faculty

of Lehigh Un~versity

in candidacy for the Degree of

Master of Science

JJ

June 1963

, I.,~.,j. •

. ~ ;~::..~.. ir •

,'--'.

,\ ~;.

iii ..

ACKNOWLEDGEMENTS

This work is a part of a project on r~elded Con-

tinuous Frames and Their Components" being carried out at

the Fritz Engineering Laboratory, Lehigh University, under,

the general direction of Dr. Lynn S. Beedle. Professor

William J. Eney is the Head of the Fritz Engineering Labora...

tory and of the Civil Engineering Department. The project

is sponsored jointly by the Welding Research Council and the

United States Navy Department under an agreement with the

Institute of Research of the Lehigh University with funds

furnished by the American Institute of Steel Construction,

Offioe of Naval Research, Bureau of Ships, and the Bureau of

Yards and Docks., The Column Research Council of the Engineer-

ing Foundation acts in an advisory capacity. Technical

guidance for this project is furnished by the Lehigh Project

Sub-Committee of the Structural Steel Committee, Welding

Research Council. Mr'li T. R•. Higgins is the chairman of the

Lehigh Project Sub-Committee. Valuable suggestions offered

by Mr. T. R. Biggins and the Sub-CoUlDittee are sincerely

apprec.1.ated.

The author i8 deeply, indebted to Dr. Theodore V."

Galambos, professor in charge of the thesis, for his con-

iv

t1nuecl aclvice t 8\l3Sestion anel help d.u'taa the p~epUat10a of

this thes18. Be also wishes to .press his grat1tude to .*.Anthony T. 'errara for h1e helpful acldee _ingthe pQ10cl

of development of the the.1s.

" Ct

The assistance, ex.tendecl by the wthor' 8 as.oclate.

working on the same research project i8 gratefully 8f:knowleclgecl.

He also wishes to express his gratitude to Mr~, Keno.th'R.

Harpel, Foreman ,of theh::Ltz Engineer1ng Laboratory, and bll1.?!

aS818tant& for their co-operat1on 11'1 prepuatiol1 of the.\~pe~i-

The author wishes to express his appr~iationto

Mr. R. Agliettl tor his a$s1stance in' performlng the .per1-: . ~

'ments ~cl help with varioue computations, The dr••tngs, w•• ,

,prepared bYMe~Sr8. 1\. s,opJto and, J. 'Sz:Llq,l 'aad ita. manusot1,t'

was typed by Miss Valerie A\lstin. Thei~ eo","operatlOn is

greatly' appreciate'l.,

-,

.>"1'~~

.~ "~,1 \:~1 .-/.1

':1.(

~

1<).'

1j:1.'i

-INTRODUCTION

2.1 Object;lve8

2.2 Teet Set-up

2.3 Instrumentation

2.4 Mat~tal Properties

3.1 Loac1..DefomaciQIl Behavior

3.2 Comparuonw1th' LB..Ser1es Teats

page

vii

1

6

6

8

S

·9

11

11

19

v

3.3 Stra1ae at Local Buck11o& 2.1

3.4 Compax'iaOil with ABC SpeclfJ.c4t1on 21

3.5 Relatlonahip betwe.Ver·t·i.cal andLateral DeforIDatlon 23

·3.6 Effective Length Concept 2S

s. 29

1. TAlLIS AND fIGURES

,qe

31

8•

• • ' r .~. ~,_;--.,.. -_.~. _._•.. "-

VITA 58

vii

ABSTRACT

Results of a series of test8 performed on lat~rally

braced wide-flange beams are presented inth1s report. The

main reason for these tests were: (1) 1:0 QtermiQe the influence

of a yielded adjacent span on the rotation capacity of the beams,

and (2) to compare the results of the b~s having yielded adja­

cent spans with that of the beams baving only one span yielded.

Test results have indicated that the beams with the

yielded adjacent spans have considerably less rotation capacity

than the beams with only one span yielded.

-~--....---- .--:'-:-~' .

1. INTRODUCTION

The problem of the lateral bracing is of considerable

importance in plastically designed Steel Structures. Simple

plastic theory assumes that at c~rtain pouts 1n a structure

plastic hinges will occur and these must be capable of 1'0-

tating tllrough finite angles in order that the structure may

become a mechanism. When I or WF ahape beams are subjectec:i

to bending about their strong axes, they have a tendency for:

lateral buckling. that is, buckling out of the pla~ bend­

ing. This lateral buckling gives r1se to excessive strain on

one edge of the compression flange thereby causing that edge

to buckle locally and thus terminating the us.wl life of the

beam.

The problem of lateral and local buckling can be

best understood with the aid of Fig. 1. This f1gltte is a plot

of IDOment versus end rotation for a simply supportec:i beam load.ed

as shown. The "adequate ll type of curve is obtained when the

beam not only delivers full Mp_ but a180 rotates sufficiently

to let a mechanism form. whereas the "inadequateJ1 type of

curve is obtained when the 'beam clelivers full Mp but fai18 to

,-,-' .~.,

--------

-2-

-' maiQ1;ain that with the increase in curvature.

.)

The problem of lateral bracing is essentially the

problem of postponing lateral and local buckling 80 that the

beam rotates sufficiently for the formation of a mechanism.

Both theoretical ancl experimental stuci1es (1.2) have been

carried out on the problem of lateral bracing in plastically

designed steel structures. The main objectives of this

reported(1.2) program were: (1) to find the optimum spacing

of the lateral bracing so as to get the "adequate" type of

curve, (see Fig. 1). and (2) to determine the required strength

and stiffness of the lateral bracing members~

Four tests on lOWF25 beams were performed to in-

vestigate the optimum spacing of the lateral bracing. Results~~of this series of tests have shown that when L/ry ~45 an

"adequate" type of an M-6 curve is obtained; by increasing

"the L/ry ratio, the M-e curve tends towards the "inadequate"

type.

Having experimentally determined the spacing of the

lateral bracing in a beam with elastic adjacent spans, the

next problem was to determine the strength and stiffness of

the bracing members themselves. Several tests were performed (2)

'. "~'. " - ."

which were similar in all respects to those described above(l)

except that the strengths and stiffnesses of the bracing

members were varied. The results of these tests havd indi-

catec1 that the M-a curves were of the "adequate" type f,or

the lateral bracing details used. Furthermore, it was shown

tbat(2) the main function of the lateral bracing is to keep

the compression flange of the beam in pos1tio~and any kind

of _lateral bracing which is capable of doing the above function

is also adequate. Experiments were also performed to investi-

gate the influence of such variables as beam size, method of

beam to purlin attachment (bolted or welded), half stiffeners. ,

in the vicinity of plastic hinges, and bracing on one side

only.

In the previously performed tests(l,2) the span

under inve.tigatio~ was always subjected to a uniform bending

moment and the adjacent spans were elastic. Fig. 2b show~

the schematic view of ,the test set-up, beam section, and

the location of the lateral, bracing for theSe tests. For

a more detailed study of ,the probl_ of lateral bracing,\: .

four additional tests (G-9to G~12) on 10B15 beams were per­

formed. Fig. 2& shows ,the schematic view of the beam seetion,

moment diagram, loading points ~4t1d'tq.e position of the lateral

._ ..---,;--'1', ...... - : •

~I\'

bracing for these tests. The only difference between these

tests and the p1:evious ones was that in this 8e1:ies there "ere

five beam segments with the central three segments under uni...

form moment, whereas in the previous ones, only one sepent

was under uniform moment. The test set-up, data it instrument.-

tion and the test procedure were essentially the same a8 i.11

the previous tests.

The main objectives of this series of tests can be

summarized as: (1) to investigate the influence of the

yielded adjacent spans on the rotation capacity of the beam,

and (2) to compare the results of the beams having yielded

adjacent spans with that having an elastic adjacent span.

The mechanism of failure of the beam haVing three

yielded spans is essentially the same as the beam having only

one yielded span. It can best be described by the M-e rela­

tionship (Fig. 1). The beam first behaves elastically up to

about 0.8 Mp, tg.e M-e relationship becoming non-linear after

that point due to the pres.ence of the residual stresses.

When the whole section has yielded, the beam co~pletely loses. .

its rigidity in the weak. axisdi1i'ection. The beam buckles

laterally at this point. Furtber lateral deformation and

..'" -" ...... -.--..~...-,

, -5-

transverse bending is accompanied by increased strain on

the compressive edge of the compression flange. When the

strain at the compressive edge becomes large, the flange

buckles locally, thereby bringing an end to the useful life

of the beam. This mechanism of failure is the same for bothI

(''y'

types or bracing, that is, for the beams having three seg-

ments yielded, or 'for the beams with only one segment yielded.

But there is one basic difference: the beam having elastic

adjacent spans obtains elastic restraint against failure

~n1ereas the beam having yielded adjacent spans does not

get any restraint. Hence, the beam with yielded adjacent

spans should exhibit less rotation capacity than the beam

Hith elastic adjacent span for the same L/ry ratio. This

fact ha~ been very clearly demonstrated by the results of

these tests.

The failure mechanism as described above looks

very simple but a mathematical formulation of the problem\ •.I!.';'1" l-. ~

is extremely complicat~d duti to the numbers of variables

involved (see Ref. 6, page 62). However, an attempt to

find a relationship between the deformations in the plane

of loading and in a plane perpendicular to that has been,

made here.

': ...... ~.

2. DESCRIPTION OF THE EXPERIMENTAL PROGRAM

2.1 OBJECTIVES

Four tests on 10B15 beams were performed in this

experimental program on the lateral bracing requirements of

plastically designed beams. Fig. 24 shows a schematic diagram

of the beam, the loading condition, and the position of the

lateral bracing. Loads were applied at the ends of the beam

(except in test G-ll, where loads were applied at B and E)

by means of two hydraulic jacks connected in parallel to

deliver equal loads. In all the tests, the beam specimens

were divided into five equal spans by six sets of lateral

bracing at A, 3, C, D, E, and F. Web stiffeners were used

at all of these six points. Two one inch diameter rods of

high strength steel t-Jere used to support the beams at Band

E. This way of loading produced constant moment in the

three central beam segments (B to E). The central segment;

(C D) is termed Lor and segments BC and DF are called Ladj.

These tests collectively will be termed as G-series tests.

The bottom part of Fig. 2 shows the schematic diagram

of the beam, loading conditiQns and position of the lateral

bracing for tests LB-10, LB-ll, LB-1S, andLB-16. These

tests have been reported in detail in reference (1), and

,--- ---~

':

I:I:;1-/I~~ ~

will be termed in this report as La-Serles t.sts. The loads

1

\'.-/

were applied at the ends causing uniform moment in the central

sepent of the beaau. The lateral bracing usec:l was the same as

that in the O-Series teats. It has been shown (1 ,2) on the

basis of these La-Series tests and others(2) that the post­

buckling strength furnished by the experimental specimens was

mainly due to lateral restraint provided by the adjacent

elastic spans. The Q-Series tests (0-9 to 0..12) were per'"

formed to verify the above statement, that is, the influence

of the yielded adjacent spans on the rotation capacity of the

beams, and to get a better understanding of the behavior of

such beams under loading. It can be seen from Fig. 2 that

the LB-Series and the O-Series of tests were exactly the same

in all respects except that the latter bad the three central

beam segments under uniform moment which yielded simultaneously

after full Mp was reached and hence the adjacent spans did not

provide any elastic restraint.

Table 1 lists some of the major test variables such

as beam section, length of critical and adjacent spans, and

the purpose of these tests. The beam section used was a

10815 rolled WF shape and the values of L/ry ratio were 30,

35, 40 and 45. For convenience, this 1nfo:t'mation is a180

.:".;"""'-:::''--:~'-.'

Ii

, ..-/

-8-

given for the corresponding LB-Series tests.

2.2 TEST .SET-UP

Figs. 3 and 4 show different views of the test set-

up. The set-up was essentially the same as that reported

in references (1) and (2) except that in this case, there

wert! five beam segments instead. of three. The lateral brac-

ing was the same as that used 1n the LB-Series tests. Fig. 5

shows the beam and the test set-up at the end of the test.

2.3 INSTRUMENTATION

The instrumentation also was essentially the same

as that in references (1) and (2). In all the four tests,

deflections i~ the vertical and lateral directions, and curva-

ture measurements in both the elastic and the inelastic range

were recorded. In the elastic range increments of load and

in the inelastic range increment of vertical deflection were

used in recording measurements.

Vertical deflections were measured by means of a

surveyor's level and a l/lOO...in .. ·travellill8 scale at pre..

viously punch-marked points at, the center of the top flange

of the beam. Ames Dials, were also used to measure vertical

deflections at the center of, Lor and Ladj. Lateral deflee..

',,-,-:-- ..

-9-

tions of both the compression and the tension flange were

recorded by mean. of a transit fixed in a vertical plane

and a l/lOO~in. travelling scale.

Curvature was obtained. by means of electrical strain

gage readings and also by rotation gages. Fig. 6 shows the

location of strain gages for all the tests except that in

test 0-12 where strain gages were also attached in spans

Ladj.. Curvature was calculated by the formula

j E: TJ + (tiLd (1)

Rotation gages can be seen in Fig. 5. A 29-in.

level bar was mounted on a plate with the help of a knife

edge at one end and a "utical serew at the other end .. An

Ames Dial was connected to the level bar and the pl~te was

welded to the lug at the support point of the beam. The

amount of rotation at support points was lleasured by meane

of the dial, and since there was theoretically a constant

moment across the three beam segments. the curvature was ob...

tained by dividing the change in, slope by the length.

2.4 MATERIAL PB.OPERTIES

The test specimens were rolled 10115 section of

\' .,~- - ~.'.

1),\I:}{;

Ji

"I!

I:1

I'~

ASTM A-7 structural steel. Nine standard tensile CQupon tests

were performed, au from the flanges and tlu:ee from the web and

their values are listed in Table (2). The geometrical dimen-

sions of the section were also measured and MP was calculated

by the formula

Mp &II bt (ii-t) 6 yf + ~ (d-2t)2 6 yw (2)

The values of Mp so obtained are also listed in Table (2).

; -'.,-.... : .: ~..~:. . .. ,

('~

-11-

3. DISCUSSION OF THE EXPER.IMEN'lAL RESULTS

3.1 LOAD-DEFORMATION BEHAVIOR

The types of load-deformation relationships which

define the behavior of beams failing by a combination of

lateral and local buckling are: (1) defo~tion' 1n the verti­

cal plane wbich is also the plane of the web and loading,

(2) deforQ8tion iQ the lateral plane which is perpendicular

to the p~ane of the web, and (3) local buckling. Appropr~te

terms have been used t:o non-d.1mensionalize the load-deforma-

tion behavior such that comparison between teats is made easier.

Moments have been divided by Hp, vertical deflections at the

9!fyL2 '. S!!yL2. '. .' 'center of LeI' and Ladj b;r 8 r, I and 8 E I' respectively and.

curvature (,) by ~1!:. In Fig. 7 vertical defl.ctlons at the

center of LeI' and Lad~ at the limitation of yielding have been~L2' 5MyL2

derived as aT I and 8& I respectively and hence these terQUJ

have been used as non-dimens.!onalizing parameters. Similarly J

curvature at initiation of yield.ing bas been shown to be20'y(S)ell -.

3.1.1 Moment Versus Vertical Deformation.Relationship

Fig. 8 shows a typical non""dimenaiona1 moment-veraus­

vertical deformation relationship. This eurve'essentiallYLcon­

sists of four parts: (1) elastic range, (2) inelastic range,

-12-

(3) 'plastic hinge plateau, and (4) unl.oad1.l'lg ;r:ange. In the

elastic range, the ~ent-ver8us-verticardeformation relation-

ship is linear. The elastic range continu.e8 up to about

0.8 Mp (Point L) and after this point M-v relationship becomes

non-linear due to the presence of residual stresses in the

section. When v/vy approaches about 2, the whole cross­

section of the beam is plastified and tip is attaineci as per

f9rmula (2). 'rests baveshown that the point of flattening

out the M-v curve (point M) is also the point at which the

compression flange of the beam buckles laterally. This

phenomenon i8 observed visually as well as recognized by a

big difference between the strain readings on the compresston

flange tips at the center of Ler. The point of the lateral.

buckling has been marked by @ 00 the M-v curve.

After lateral buckling, the be~ can not sustain

anyaddieional moment and deformationcontinl1es to increase

at' the same moment. LOcal buckling occurs in the compression

flange of the beam after it goes through sufficient deformation

depending upon the spacing of the lateral bracing. This

point has been marked by @ in M-veurVe ud is observed

ViSUAlly. In the unloading range. the beam is unstable, and

deformation increases with decreas~ in moment capacity • Hence ,

-;--

:)]

IiII

--- -- -- _._---

point (j) has been c:lefined as the end of useful life of the

beam.

Figure 9 is a plot of M/Mp versus vlvy , tJlt/Jy and

9/9y at the center of Lcr for tests 0-9 to 0-12. Curvature

and angle of rotation values used here are those obtained by

the rotation gages. A comparison between the curvature as ob-

tained by the rotation gages will be made with the curvature

obtained from 5&...4 gages. All of these curves follow the same

pattern as that described above and their values up to local

buckling are also the same. Hence. any of these three para­

meters (v/vy. 'Ifly • 9/9y) may be used as a measure of the ro­

tation capacity of the beam. In this report, the parameter v/vy

bas been used to describe the performance of the beam. After

local buckling. the curvature seems to increase more than the

vertical deformation as can be seen 10 Fig. ;&J ~ By the time

local buckling takes place. the beam not only buckles laterally

but also goes through sufficient twisting thereby causing the

observed vertical deflection less than the true one.

Figure 10 shows a plot of M/Mp versus vlvy at the center

of Lcr for all the four tests side by side to compare the results.

For convenience. these results bave been tabulated in Table 3.

... -." ....:, ..: :"~ -.' .. ~--~. '~r ~. "_ -'" '"".-~"",".'<:-". .,.:.,~._,- ..... -.. ",. "--,-_'';' (. -, ,,-,1_;_ ....•••--.----.

-14-

From this table it can be Seen that lateral buckling occurred

at v/vy values of about 2.3, 1.3, 1.7, and 1.6 for tests 0-12,

0-10, 0-9 and 0-11 with L/t:y ratios of 30, 35, 40 and 45

respectively. The variation in v/vy values at the initiation

of lateral buckling did not seem to follow any definite trend

with the variation in L/ry value. The minimum and the DlaxiDNID

value of v/vy at initiation of lateral buckling was 1.3 and 2.3)

respectively, giving an average value of 1.8.

The amount of the plastic hinge plateau depends upon

the spacing of the lateral bracing(l). Local buckling occurred

at v/vy of about 6, 3.8, and 3.7 for tests 0-12, 0-10 and 0-9

with L/ry ratio of 30, 35 and 40, respectively. Hence, it is

seen that greater the value of L/ry ratio, the smaller the length

of the plastic hinge plateau. For test 0-11 with L/ry ratio of

45, unloading atarts at the initiation of lateral buckling and

points @ and ® coincide, thus furnishing no rotation capa-

city to the beam.

It has been stated earlier that the onset of local

buckling would be considet:ed as the termination of the useful

iife of the beam. But in actual tests, the termination of the

useful life of the beam was not spontaneous but a gradual process

~ after local buckling. Even in test 0-11, where no plastic hinge

~" .' .I'" .

i;

plateau was formed, the unloading phenomenon was a slow and

gradual process.

In Fig. 9 it was observed that the values of v/Vy

and 0/0y at the center of LeI' of the beam were almost the same

up to the point of local buckling ®. The next point is to

cheek whether the values of v/vy obtained at the center of Ler

were equal to that obtained at the center of Lac1j as the three

beam segments were under uniform moment. Figure 11 is a plot

of v/Vy at the center of LeI' and an average value of v/Vy of

both adjacent spans, Ladj. They almost overlap eaeh other

,'\ showing that the middle three beam segments were uncleI' unifoB

moment and consequently plastified to the same extent under a

particular loading in the plastic .range.

The values of curvature used so far were those obtained

by rotation gages. Curvature was also calculated by strain gage

reaclings by formula (1). The location of the strain gages may

be seen in Fig. 6. The average value of the curvature was

calculated from the three sets of strain gages located at the

center of Ler in the middle of the flanges and at 2-in. on

either side of that point. Figure 12 is a plot of M/Mp versus

_/0y as obtained from the rotation gages and the strain gages("

~. to compare their values. These curves are essentially the same

.. ' -0'--'" "--, .' .,,~.-._.~..'.c..........-,'.~__.~__-,----,-,-,--,---,_-,------,"~_~-...;..'~_-_.......... ..........

\

i\

,.: t

-16-

up ~o the point of lateral buckling. For test Q-10, the

curvatures at local buckling obtained by both methods were

the sam.. but wiele variation in their values were observed

for the other tests. The difference was small i.n the beginning

but grew bigger with the increase in deformation, Furthermore.

curvatures at the initiation of local buckling from the rota­

tion gages were always smaller than or equal to that obtained

from the strain gage readings.

The higher values of curvature as obtained from electrical

strain gages may be attributed to the yielding pattern of the

structural steel (3,4) it The mechanism of yielding in structural

steel is discontinuous, t~ing place in slip-bands by a sudden

jump of strain ~rom yield strain (EX) to strain at strain barden­

ing (co). According to thistbeory. during yielding, part of

the material remains elastic whereas others reach strain hardening.

After each part of the material bas reached the strain bardening

range. it again becomes homogenous but direction dependent.

Further. during this yielding process; some of ,the slip-bands

may actually be formed under the strain gages. Still another, ,

problem with the strain gages is that the gages themselves may

be capable of going through sufficient elongatiQnbut the

cementing material between strain gages and the beam may give

way. Due to these reason8. 8tra1rs ,as recorded by strain gages

-17-

do not necessarily represent true strains.

So far, discussions were confined of the test results

in the vertical plane only. Now an attempt at presenting and

evaluating test data in the lateral plane will be made.

3.1.2 MOment Versus Lateral Deformation Relationship

Figure 13 is a plot of the deflected shape of the

compression flange of the beam for tests G-9, 0-10 and G-12.

Similar curve was obtained for test a-ll also. It has been

discussed earlier that rolled WF or I Section beams buckle later­

ally at v/vy of about 2 and M .~. The three central beam

segments were under uniform moment, bence lateral buckling could

have occurred in anyone of them, but the segments adjacent to

Ladj were elastic, offering restraint to the lateral deformation

of Ladj. Hence, the lateral deformation could be initi.ated only

in Ler which di.d not get any elastic reStraint from Lad.j. In

three out of the four tests performed (0-9'to 0-12), exactly

this happened: the initiation of lateral buckling took place in­

the cGl1Ipression flange of the Lcr segment. In test G...IO~ lateral

buckling started. in the span Ladj. This may be clue to· initial

crookedness or variation.1n the sectional properties in that

beam segment.

Before the initiation of lateral buckling. both eage.

of the cOJDpt:ession flange of Ler wereunc1et: uniforQl compressive

strain aue to bending. Once lateral buckling started. the eclass

of the compression flange were under different kinds of strain.

The compressive edge of the compression fhinge was subjected to

combined compressive strains due to bending and also due to

lateral deflection of the flange. whereas the tensile edge of

the compression flange was under the action of compressive strain\

due to bending and tensile strain due to lateral buckling. With

the increase in vei'tical deformation. lateral deformation al.o

/--- increased causing excessive compressive strain on the comp:;;es&ion

edge and reversal of strain on the tensile edge of the cOIDp:;;ession

flange of the b8alD. This phenomenon can be seen in Fig. 14 which !

is a plot of a typic.l strain distribution oithe compression

flange at the center line of Letr for test 0-'. The location of

the stt:ain gages may be eeen in Fig. 6. Similar curves were

obtained from other tests.

Local buckling. which has been stated ae the point of

termination of the useful 11fe of the beam occurs due to exceasive

compressive strain on the compressive edge of the compression

flange. After this point. the beem ia in ustable equilibrium,

'-, and Wlloading starts with the increase in aeformatiora".

L;

-19-

Figures 15 and 16 are plots of M/~ versus fJ and

Ue/L for all the four tests. ~ is defined in rig. 16 a. the

angle of twist and VeIL is defined as the. lateral deformation

parameter for the beam. Table 3 gives the values of 13 and

Ue/L for different tests at local buckling.

3.2 OOMPARISON WITS LB-SERIES TESTS-

test data and their discus.ionfor tests 0-9 to 0-12

have been presented in the previous sections. A comparison of

these results with the results of the LB-Series tests, which

have elastic adjacent spans, 1s given in Figs. 17 to 19.

3.~1 Comparison of Vertical Deformation

Figure 17 is a plot of moment versus vertical deforma­

tion relationship for the two types of tests. Co~arlaon is

made betweell beams haviq the same L/ry t:atio, that is, teats

0-9, 0-10 and a-ll are compared with the corresponding tests

LB-15, LB-ll and LB-10 with L/ry equal to 40, 35 and 45

respectively. Test 0-12 can not be compared with any other

test as no tests Were performed in the LB-Series with Llry • 30.

Table 3 lists v/vy values for both types of test8 at

the start of lateral and local buckling at the center of Ler.

For 1;e,t a-10, v/vyat local buckling was about 3~.8, and for. tElst

.,"

·1\;

(

,1ij,

J"

1

-20-

LB-ll, it was 13.1, shawing a reduc~ion in ratatian capacity of'

about 701 due ta yieldec1 ac1jac,ent spans. Simil.rly for tests

0-9 and LB-1S, v/vy was about 3.6 and 13 respectively, 1noi­

eating the rec1uction in rotation capacity of about 1ot-. These

results have been shown in Fig .. 22 which is a pl~t of v/vy

against L/ry at local buckling and at one load before local

buckling.

3.1.2 .CompaJ,"isonof tatfu:al Deformation

Figure 18 is a plot of M/Mp V8l:sua P for the LB-Series

and G-Series tests. The value of P for tests G-9 and La-15 at

local bl.lckling was 0.047 anc1 0.190 respectively. Similarly ~

for tests a-10 and LB-ll was respectively equal to 0.035 and

0.115. These number,S clearly indicate that the angle of twist

of the a-Series tests wa, in the neighborhood of 30 ~ 40~ of

LB-S..ie. tests at the 'onset of local buckling. The value of

~ at the onset of local buckling for LB-Series and a-Series are

tabulate~ in Table .3. ~n the same table, the value of Uc/L

at local buckling for both series of tests are also tabulated.

From t~s table and from Fig. 19 which 18 a plot of M/Mp ver$US

Ue/L it CaD bea.en that the local buckling takes place at a

lesser value of VeIL for the e-Series tests than the La-Seri••

tests. Local buckling takes place at a Uc/L of 0.015 and 0.026

I /,

-2.1-

for tests G-10 and LB-ll respectively. Similar results were

obtained for the other tests.

3.~ STRAINS AT LOCAL BUCKLING

Table 4 gives the value of strains at observed local

buckling and at one load before that for all the four tests.

The strain at the initiation of local buckling lies somewhere

between these two values. These values are compared with that

predicted by Haaijer and Thurlimann (4). According to them,

local buckling should occur in 10815 section (b/t a 2.00/0.269

:.:. 7.•45) at a strain of 21 x 103 in/in. wllich is equivalent to

21,000ti'/in. The strains at local buckling as recorded in

G-Series tests were varying from 8,000 ti'/in. to 15,000 ~'/in.

Hence, the observed strains are lower than the predicted ones

indicating that the predictions(4) are not valid in beams haVing

yielded adjacent spanS.

3.4 COMPARISON WITB AlSO SPECIFICATION

Section 2.8 of AISC Specification states that, ''The

laterally unsupported distance lcr, from such braced hinge

locations to the nearest adjacent point on the frame similarly

bra~ed, need not be less than that given. by the formula

lcr == (60 - 40 M/Mp)ry

•.•",_._.~_•• J._.'. ~ . ~--- --~- .'

-22-

nor less than 35ry •

Referring to Table 3 again. it is seen that the

v/vy value at local buckling for tests G-10 and La-lt having

L/ry ratio of 35 is about 3.8 and 13.7 respectively. Reference

2 has shown that a value of v/vy of 13 is more than adequate

for beams. Hence. the AISC Specification seems conservative

where situations are similar to those of the LB-Series tests,

that is. only one segment is yielded. In cases where situations

are closer to G-series tests, th~t is, the three continuous

segments of the beam are yielded, the rotation capacity is

reduced to 30% to that of LB-S~ies tests. This may not sound

conservative at all compared to the LB-Series tests, but things

are not that bad when other things are considered.

At this point it should be realized that all hinges

are not formed Simultaneously in any structure but usually one

at a time. The first formed hinge is required to rotate IDOst

but the last formed hinge ,the,lea8~. Secondly, this is a rather

very unusual and severe case that, a beam woulc:l bend in a slngle

curvature for such a long length ,(35ry x 3 • l05ry). But 1f a,

situation arises where the beam bends into, a single curvature

for such a long length and the h~nges a~e amongst the first to

, . ~ -,." .

,.-," .

',;{

'._~':::'-~'-.' .------ , •••• ~~-"'':'' '.':;:' ...: I""-t-

------_.- '.;

/,

-23-

be formed, the spacing of the lateral bracing have to be made

closer than 35ry • Due to the limited number of tests performed

on the beams with yielded adjacent spans and the lack of

theDretical work. One solution of the problem seems to be to'

make the spacing of lateral bracing equal to l8ry (5), requiring

full rotation to the point of strain hardening.

3.5 RELATIONSHIP BETWEEN VERTICAL AND LATERAL DEFORMATION

Figure 20 shows a schematic view of the compression

flange before and after lateral buckling of the beam.

Strain on the outside face of compression flange = 0:Total shortening of the compression flange == @d L

2

Total shortening of the compression flange after

lateral buckling == bS = (~v - 0vL) ~ (3)

Assuming that all the compression flange shortening is

taking place due to bending after lateral buckling,L

6S == 1/2f0 ,(U' )2dx

Assuming a sine curve as the deflected shape of the

beam

. .. "

. , .,-....--,'"" .

u == Ua Sin i) xL

..•---......-..~ ....... , .- .-....-.-...,.,- ~.~:--:-.

u' Uo TI K= - COS.u...::.L L .

U 2 . 22 UK(U' ) o 11

COS., =L2 L

. L

'S = U02

U 2 J Cos ~2L2 L

ai

j Vo2 -rr!.:~S •I 4L (4)

Equating equations (3) and (4)

Uo21f24L

Uo 1 V" 4,'_- == - 2d (t)v - "';'t)L 11

using the relationship 0y = 2~y , Equation 5 becomes

(5)

(6)

Equation (6) gives a relationship between the lateral

deflection at the center of Lcr' curvature in the vertical plane

and the material properties. Figure 21 is a plot of Uo/L versus

~v/~ for test 0-12 as obtained experimentally and also by the

equation 6. Similar curves were obtained for other tests also.

In this curve it is seen that the plot of equation (6) gives a

'r· .• -~._, __ ..,,-----:-- __ ~

-25-

considerably higher value of DelL than the experimentally

obtained value. Equation (6) is derived on the assumption that

all the compression flange shortening after the lateral buckling

is taking place due to lateral bending. This assumption does not

seem to be very close to the actual behavior of the beam because

in the actual beam the compression flange shortening after

lateral buckling is due to the combined actions of bending in

the lateral direction, and axial distortion of the compression

flange. Unless the influence of both of these two factors on

the shortening of the compression flange can be separately

determined, it does not se~ easy to establish a relationship

between the curvature in the vertical plane and deformation in

the lateral direction.

3.6' EFFECTIVE LENGTH CONCEPT

Mr. Johnston in the discussion(7) of reference (1) has\

estimated the effective length co-efficient of the LB-Series

tests as 0.6. The effective length coefficient is defined to

be the distance between the points of inflection divided by the

unsupported length. If 0.6 is the effective length coefficient

for a critical length of 45ry (1), then this is really indicative

of an effective L/ry of approximately 27 for beams with a yielded

.~ adjscent span.

'. ' . .:. .. " , '-..

------ --

-26....

The four tests (G-9 to G-12) performed at the Fritz

Engineering Laboratory, Lehigh University, seem to justify the

views expressed by Mr. Johnston. In these tests, the ends of

the central segment (Lcr) are hinged and hence the effective

length coefficient is equal to L. For test G-12, the rotation

at the initiation of local buckling is 6.02, whereas for test

LB-ll, it is 8.21. Based on these results, it seems justified

to brace the beam laterally at a L/ry ratio of 25 to get a

plastic hinge plateau of a sufficient length.

- -----

-27-

4. SUMMARY AND CONCLUSIONS

On the basis of the tests described in this report.

a number of conclusions may be drawn. Strictly speaking, these

conclusions are valid only for the tests performed, but when an

actual situation is similar to that in the test, these con-..

elusions may be used with advantage.

1. The rotation capacity of the beam at local buckling

with the three yielded spans was only about 301 to that of the

beam with only one segment yielded. Because of the limited'

numbers of tests performed, the percentage reduction in the

rotation capacity may not be conclusive, but it shows very

clearly the reduction in the rotation capacity due to yielded

adjacent spans.

2. The angle of twist (or lateral deformation of the

compression flange) of the beam with the three yielded spans

was also in the neighborhood of 30 - 40% of the beam with only

one span yielded.

3. The present AISC Specifications governing the lateral

bracing requirement in plastically designed beams is conserva~ive

for the beam with only one span yielded but may not be adequate,

in the beam having three y!elded spans and the hinges being

.;.:; ~:': ..~_., ,'.' '. ~'.

-28-

amongst the first to be formed. However, this type of case

is very rare in actual structures.

4. The problem of predicting strains at local buckling

is still unsolved. Previous theory as proposed by Haaijer, G.,

and Thurlimann, B.(4) does not predict strains at local buckling

in the beam with the yielded adjacent spans. This may be due to

the fact that the above theory does not take into consideration

the influence of the elastic or the inelastic adjacent spans.

5. An attempt to find a theoretical relationship between

the deformations in the vertical and the lateral plane has as

yet been unsuccessful.

The problem of the lateral bracing can not be considered

as solved unless the strains at local buckling can be predicted

with sufficient accuracy, and a relationship between the deforma-

tions in the mutually perpendicular directions are established.

Unfortunately, both of these problems have not been solved but

there is now abundant test data available to check any theore-

tical solution proposed. Besides this, these tests have re-

vealed the actual behavior of the beam under loading in the

plastic range which should be of a great help to a designer.

'.1..

M

z

Lcr

Ladj

d

b

t

( )A

( )B

( )A,R

=:

==

I::

=:

c:

=:

=:

=:

=:

=

=:

=:

1CI

=:

••,' t" ','

-29-

5 • NOMENCLATURE

Bending moment

Full plastic moment == 6 y .z

Plastic modulus

Length of'span under investigation

Length of adjacent span

Overall depth of the section

Flange width

Flange thickness

Web thickness

Radius of gyration about the y-y axis

Curvature

Curvature at the start of yielding

Apgle of rotation.'!

Angle of rotation at the start of yielding

Yield stress

Modulus of elasticity

Strain in the top flange of the beam

Strain in the bottom flange of the beam

( ) at lateral buckling

( ) at local buckling

( ) at lateral buckling from rotation gages

-30- ,.<

,'----!

( )B,R = ( ) at local buckling from rotation gages

( )A,G = ( ) at lateral buckling from strain gages

( )B,G = ( ) at local buckling from strain gages

--_.-~-'-.";-~ .

----- ------

,I. ~

l<!

6. REFERENCES

G. C. Lee and T. V. GalambosPOST-BUCKLING SYUNGTB OJ' WIDE-FLAIGE BEAMS.ASCI Proeeeclinga raper 3059, Vol. 88. DO. ,Februuy 1962.'

, '--/

2. G. C. Lee. A. T. FerI'ua anel T. V. Gal_boaLATERAL BBACXlG lmQUIREMENTS OF PLASTICALLY DESIGttEDBEAMS, FI:'1t8 EDaineer1q {..a.Dorato1'7Report No. 2058.6.

3. A. Nac1a1THEORY OF now AND FRACfUU OF SOLDS,McGraw-8111, New York, 1950.

4. G. Baa1jet' and B~ Tb.urlimannINELASTIC BUCKLING IN STEEL, Proceec11nga of theASCE Paper No. 1581, April 1958.

Lynn S. BeedlePLASTIC DESIGN OF STEEL FRAMESJohn Wiley &Sona, Inc. New York, 1958.

6. G. C. LeeINELASTIC LAtERAL INSTABiLITY OF BEAMS AND THEIRBRACING REQUIREMENTS, Pb.D. D18sertation t Lehigh

.University, OCtobet: 1960.

7. Bruce G. JohnstonDISCUSSION or npOST-ijUCKLING STRENGTH WIDE­FLANGE BEAMS" BY G.' C. LEE AND T. V. GALAMiOS.ASCEP1:ocee41nga Paper 30S', Vo1~ 88. r.b. 1962,..ASCE Jow:na1. Vol. 88, EM4. fart 1. August 1962.

/ "-,

7. TABLES AND FIGURES

-32-

--- ---...... ~- ..:' ~ , , ",

Test Beam Critical Adj ISpan Remarks INo Section Span(LclLadj I

G-9 IOBI5 40ry 40ry. Effect of

G-IOII

35ry 35ry,Inelastic

G-II II45ry 45ry I

Adjacent Span.

G-12. II30ry 30ry

LB-IO IOVF25 45ry 45ryEffect of the

LB-IIII 35ry . 35ry

length of

LB-15II

40ry 40ryL cr

. LB-16 II50ry 50ry

-32-a

10 B 15 :

10 VF25

b1t =14.85

bIt . =13.4

d/w =43.5

d/w =40.0

TABLE 1. 0UTLINE OF TEST PROGRAM,

TEST NO. SECTIONCTYT CTYW Mp M?

. J.k s i ksi in - kip In - KiP

G-9 lOB 15 42.1 50.0 747 640

~ -- G -10 II II II736

II

G -II II~ II II770 659

- G -12 II II II748 644

LB-IO 10 W"25 35.22 38.80 1046 918

LB-IIII II II

1040 915

LB-15II II II

1036 918

LB-16II II II

1040 915

TABLE 2. MATERIAL PROPERTIES . I

·wWI

TEST NO. ( V/VY)A ( V~Y)B (¢/¢Y)R,A (¢I¢)R,B (¢/¢)G,A (¢I¢Y)G, B (,B )B (~)LB

-") VG-9 1.65 3.63 I. 71 3.42 1.82 5.01 47.5x10 . 0.0169

3.76 1.52 3.72 1.52 3.78-3

0.015 ~vG-IO 1.32 35.x 10-

G-II 1.58 * 1.62 * 1.46 '* '* *'-)

0.023 t..-V

G-12 2.31 6.05 2.22 5.5 1.921 8.38 51.3 x 10

-'1LB-IO 1.54 8.21 '* *. '* '* 2.49 12.53 123.4x10 0.0236

~~

LB-II 1.8 13.72 '* '* * '* 3.52 14.69 114.8x10 0.0265

..3.1f4

- ,)LB-15 -*="'* ~ 12.95 * * * * ·12.6 189.6x10 ·0.0385

~·S4-

., ..

LB-16 1.65 '* '* * * * 3.72 * * '*'* Local buckling did not occur

'* * Rotation gages were not used

TABLE 3. TEST RESULTSI

w.+,-I

Strains at Observed Local Buckling Strains at one Load Above ObservedTEST NO. in Micro- inch/in. Local Buckling in Micro-inch/in.

Average C A B Average C A B

G~9 8,160 935- '14" 385 5,255 " 1,280 .10,085,..

".

G-IO 5,055 6,585 7,010 '·3,507 3,245 2,850

G-12 15,320 2,010 33,090 12,470 2,470 25,005

LB-IO 14,085 1,390 '* 12,575 1,760 28,265

LB-II 20,135 * '* 17,895 * *

LB-15 16,090 3,240 30,145 11,815 3,350 22,385

* Strain gage went out of range

TABLE 4. TEST RESULTS

t of LcrCo mp. Flange

I . i

w .!VII

jOdeQUote

-I-

--LocalBuckling

MOMENT

ROTATION 8

FIG. 1 TYPICAL M-9 RELATIONSHIPS

IW0'I

Jack Rod' Rod

-37-

Jack

;

A B C 0 E

d MP IMP IMP ~A B C 0 E F

Lateral Bracings at A, 8, C, ·0, E, and F

a) SCHEHATIC DIAGRAM FOR TESTS G-9 TO G-l2..

F

Jack Rod Rod Jack

~-,-M_p__.~.A L 8 L C L 0

1- ~l- cr -1-. ~

Lateral 8racings at Section· A, B, C and 0

b) SCHEMATIC DIAGRAM FOR TESTS LB-10, LB-ll, LB-15 andLB-16.

FIG. 2 SCHE}~TIC DIAGRAM OF TEST SET-UP

-38-

2

2

BJsuppor~Column

3

FRONT VIEW

Line

3_ :rrSupporting Girde~I I - 1=I

i irir:;: =T(or ~l'of--""'"

I'

Beam " II I

Supporting I III

Frame~--I

tiI I

Vertical Column ofIIII

, Support--. Supporting IIII

Frame- II

10 Loading Jack- ,l.J,l...l(_'I

V 10 B-;;~, 11'I 15 '1 :111 11

I, I' _IJ.UIII " Tl-r II Test Specimen~ Lateral ,. I

I'II

BracingsII I.. IJ I

I I ~w., " I'I. L--

! I I . ! I -rrI,

i,

III I i !I

r 'J' ~ '(" '

~Center Lab. Floor

- -=--=-~~----=--=-----------------=

"'--- Lateral Bracings---

SECTION I-I

FIG. 3 SCHEMATIC SKETCH OF TEST SET-UP

~Pin

--...- Sapporting

~~!:L Girder

:~Bracing

~'T"'.Fr~

Test Beam --f+--ll*o-I:I

- ._...._. .,,_~ __ .• ~__~__~.. _0_- ..~:.,. . _. ,. :.....- .. __ .. _.. ,...~_ ".,

T \"Supporting IfIIII

GirderII

:~I II

. , ]1I~

Cross Bea~"U,

r V JackI, --

I~

,I rcolumn

I'II c::

- yRoller

II.L..--J

~

I'0 0 I.--TestI' Laterol~ 0 ~Z;

}6Beam II

,I Support 0 0

IIII I

., ,I"

Ii Stiffener,I I v-Base ,I

1/ Beam II,Ii I 1/ ~I I :". '

. ,

\\ \ \ '1 \ '1\ ,'\\ "" '1'" "I'" 'I. 1

(c) SECTION I-I (d) SECTION 2 - 2

FIG. 4.

. IW\DI

PIG. S TEST 0-9 AND 0-10

-40-

":"41-

E-J

E-,

c - i

B -- IA 2"

12"1

A -J IB-----J

C ------'

.~ Ladj .1_ L cr a~ Ladj .1_

D

ID.-J

1-

i--, I'

n I I- r

I I I I II I I I II I I J I

I

(a) Sections where Strain Gages are Attached

b/2

b/2I- -I -I

m'3 • Eft ""'

---I l- -1l- II,~

.Yrv~ "$IJI h_~

II- - - I I

Section BB Section AA, CC, DD, and EE

(a) Location of Strain Gages

FIG. 6 ARRANGEMENT OF STRAIN GAGES

o

-42-

P

t.~

L

P

t

F

L1-eIl

A

·1... -1- --I.. ..I ...Ladj =L· Lcr= L Ladj =L L

Moment. Diagram when P = PyIk

B C ~G D E....------:-t~

O.5k O.5 k

2! I { I . 3 ~ _ 9 My L:

8G =EI 2" (3 L) My ( 14 L ~ - 8 EI

FIk

B tH C' D E~---'·A~~!-.-----:-

. T~-L.·. T'2.35. .. °35

2

8H =~I{t(3L)My( 265 Ll}= ~ ~¢

A

FIG. 7 DEFLECTIONS AT THE CENTER OF Lcr AND Ladj

1.0

0.8

MM 0.6

p

0.4

0.2

A B~ost Buckling Strength of 'IF Beams.~

, M t Plastic hinge Plateau N\.... Inelastic range

--- Elastic range

~ Unloading range

p

® Lateral Buckling

® Local Buckling

i:

v/vy

FIG. 8 TYPICAL MOMENT-VERSUS-VERTICAL DEFORMATION RELATIONSHIP

o 2 4 6 8 10 12 I.p.I..VI

® Latera I Buckling

® Local Buckling

{ G-12 I

0.6 <-

~0.8MpI G- 9 I

. 0.6

MM . 0.8

p

-0 ®1.0 1.0 ~t

M M- -Mp 0.8 Mp 0.8I G-II I ( 6-10 I· 818 c

y

¢Iep -Vy

® ® u/u 0..... y,1.0 1.0

a 0.2

"Ivy ,

4.0

epl 81iIe/>' ey y

FIG. 9

a 2.0

M-9-0-v RELATIONSHIP

4.0

\. \y' 4J",y

6.0 8.0

1~~I.

, .. j

1.0r®

I G - 12 II G-IO I

0.8 G-9

Test No Lfry

G-9 40

M-0.6 G-IO 35

- G-II 45MpG-12 30

0.4

0.2

OL--__--L --L. --l- .L-__-----L__

2 4 6 8 10I+:--

InI

FIG. 10 MOMENT-V RSUS-VERTICAL DEFORMATION CURVES

1.0

0.8

1.0

0.8

0 v/vy at t- of Lcr0.6 0.6

M6, ("IVy) Avg. at t of Ladj

Mp

0.4 0.4

0.2 0.2

o 2 4 6 a 2 4 6

v/vy

FIG. 11 M-v CURVES CONTINUED

®

. L@

I G .... 9 I © Strain gage

@ Rototion gage

1.0

I G-IO I

1.0

0.8

M aMp

1.0

0.8

I G-II I

1.0

l© 08

IG-12 I

86·42

___-I-__---L ...L--__---L._.:----ll .

10o642o.cA .

7epy

FIG. 12 MOMENT-VERSUS-CURVATURE RELATIONSHIP

I~'-II

- ~I[. I"[

30r. 'y 30ry 30ry 30ry 30ry

G-12

35ry 35ry 35ry 35ry

G-IO

G- 9 [~:I[ I"

BRACING POI NTS __---,--...,..-...::Y

FIG. 13 DEFLECTED SHAPES OF THE COMPRESSION FLANGEI

.J::'­~I

~LANGE TIP --r- STRAIN(Micro - inch/inch)--e:-TEST G-9

(TENSION) 4000 o 4000 . 8000 (Compression)

CENTRE OF FLANGE

o 4000 3000 12000 16000 (Compression)

. LFLANGE TIP --- STRAIN (Micro - inch/inch)~I~\0I

FIG. 14 STRAIN DISTRIBUTION OF COMPRESSION FLANGE AT ~ OF Lcr

1.0

0.8 L 1 G-IO I

Uc .,

M0.6

tJMp

0.4 VI I.--...j

II

"~UT

0.2

0 0.02 0.04 0.06 0:08 0.10 0.12 0.14I

Ln0

/3 (radians) I

FIG. 15 M-13 CURVES

-51-

IG-±e1l~

IG ~1-(~----'91M

.. Mp 0.75·

Compression Flange

0.5- •

1.0

0.75

Tension Flange

..0.5 l

o 0:0125 0.025 0.0375 0.050. ,

U'LFIG. 16 MOMENT-VERSUS-LATERAL DEFO&~TION RELATIONSHIP

® r® ,®'1.0

0.9

LILS-III0.8 rz.1.0

\®:M

0.9

['LB-151-Mp tl G- 9 I0.8

j®1.0 r®0.9

L ILS-IO!0.8

0.7 .

. I ! I I I I I

·0 2 4 5 8 10 12 . 14 16

"IVyI

V1tvI

FIG. 17 M-v RELATIONSHIP

.' G- 9 I~-----..;,----..!....----o

®

II G -II I

®

1.0

1.0

1.0

0.8

0.8

MMp

0.8

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16I

,8 (radians)V1w

/I

. FIG. 18 M-~ RELATIONSHIP

-54-

® ® '@1.0 I~ • t

ILB-III

IG-IO IM

0.75Mp

Compression Flange

0.5

1.0

M 0.75Mp

0.5

ILB -III

IG -10 I

Tension Flange

o 0.01 0.02

u

L

0.03 0.04

FIG. 19 MOMENT-VERSUS-LATERAL DEFOIU1ATION RELATIONSHIP

"

-55-

A BCD EF

t---~---~i- ---t----t----I1_ L _~Ladj=L_~ L cr =L ..1_Ladj=L .. 14 L ~.

Compression Flange of Beam before Lateral Buckling

I.. Lcr ..IL cr after Lateral Buckling

Ev

~. d/21

Strain distri but ion at t of L cr

1-L

~Ix ...p

-C~~;[v , Va ~... p

~

Compression Flange ( Lcr )

FIG. 20 SCHEMATIC VIEW OF COMPRESSION FLANGE

FIG. 21· VERTICAL AND LATERAL DEFORMATION RELATIONSHIP

IVI0'0I

Experimental curve

654

epv

epyCURVATURE

32o

0.02

0.04

u·oL

-57-

14

12

10

6

4

® v/v at observed local bucklingy

® V/v at one load before observedy

local buc kling

G - Series tests

LB - Series tests

o 10 20 30 40 50

L/ry

FIG. 22 LENGTH-VERSUS-VERTICAL DEFORMATIONS RELATIONSHIP

8- VITA

The author waa bom 1n 'atua t Bihar. Iac11a on

Rev_ber 19. 1935, the secoad chl.14 of Sri Brausadeo Nal'ai.n

and Sat _ Kaua1l1a Devi.

Be obta1necl a cS1ploma La Civil Engineering from the

Tichut School of Eagine_1na. B1bu. 10 Jal'lUArJ 1"6 and vox-keel

foX' the Govemment of Bihar fJ:'OlD March 1956 to J'w\e 1959. He

joloed Rovud Onlv..aity :Ln W••b1ngtOD D.O _ 10 Sept_her 19S9

.an4 received Ms Bachelor of Science c1epee 10 eivil lagia."­

1ng 1ft June 1961.

He ... awuctecl a Reaeuch AlalataDtabJ.p 10 Civil

Ens1lleer1oa at hitz EDsiDeeI'lna Laborat0J:1. Leb1p Unl••a1t7.

BeChl.... PeDDS11vaola and bea- atucU.•• foX' • Mast_' 8

Dep-•• there 1a Sept_bar. 1961. The author baa be. aaaocu.ted.

with the r ••eucb coacen1Da lateral bl'ac1Da requ1.l'emeat8 in

plaat1a deaign.