WELDED COI~T Il\TUOUS FRAlviEiS & THEIR C01·:IPONENTS

by

Robert L. Ketter & Lynn So Beedle

This work has been carried out aspart of an investigation sponsor­$d . j bi11'cly by Jche Wel.ding Iteset\1~cJ1

Council, and the Navy Departmentwith funds furnished. by the fol­lowing:

American Institute of Steel ConstructionAnlerican Irorl and Steel I11stit1.l.te

Colunm Research COUI1Cil (Advisory)Irlstittlte of Research, Lehigh U'niversity

Office of Naval Research (Contract No. 39303)Bureau of Ships -

Bureau of Ya.r-ds and Docks

Fritz Engineering LaboratoryDepartmerlt of Civil Engineering e.11d Ivlechanics

Lehigh'UniversityBethlehem, Pennsylvania

~Tovember, 1952

Fritz Laboratory Report No. 205A.11

205A.ll

II o

III.

IV ()

'V Q

VI Q

VII.

Vlll tt

rrJTRODUCT ION

NOTES ON THE ANALYSIS

~ffiASUREMENT OF ROTATION

TESTS RESULTS AND DISCUSSION

a. Moment-Rotation Curves

b~ Rotation Capacity

CONCLTJSIOJ\TS

ACI\l\f O\,\JLI~DGI~1Vm~NTS

NO£i8~NCLATURE

RE~FI~REl\T CES

£aE...~~

1

2

4

5

11

16

18

18

19

!1PPEMDJ~

SAJYIPLE NU~1ERICAL DEF{IVATIO:f:\T OF 1v1-9 CURVIG 20

205'A.ll

If INTRODUCTIOl\T

In advance of a more detailed report being prepared on the sub~

ject of the load-deflec.tion and load-rotation behavior of WF' steel

columns, it is considered that current interest warrants preparation

of a short report discussing the moment~rotation problem. Not only

is this needed to further describe the basic behavior of caltoons loadr

ed with combined bending and thrust, but also to evaluate energy ab~

sorption of complete frames and to develop tentative proportions for

colunms intended f-or use in plas'tic design.

Performance of structures subjected to loads sUfficient to cal1se

plastic action is directly related to the ability of certain members

or connections to develop plastic hinges. Moreover, these members

must allow rotation through cOTIlparatively la~rge angles·' to en'sure .the

development of plastic hinges elsewhere o The pnecise requirement for

the tlrotation capacitytr of a particular structural cOYl1ponent is de...

pendent on such factors as (a) type of structure (degree of restraint)9

(b) type and ]~ocation of loads and (0) lengt11-depth ratio of members Q

An investigation is needed to establish a criterion of required ro~

tatio11 capacity talcing into account the apove mentioned factors 0 0118

such stUdy:(2) made for a contiUQous beam subjected to uniform load·~

j,ng, suggests that a member nlust be able to sustain a rotation of

eight times the predicted elastic value to ensure the necessary de(~

velopment of plas'tic hinges elsewhel'e in the strt1.cture.

This report will be primarily a presetitation of experimental re~

suIts of fl1J.l scaJ.. e column tests in which as ....delivered members of

various lengths have been tested under different combinations of end

bending moment and thrust. Four conditions of loading have been stud~

ied and are shown diagralnmatically in Fig. 1.

~_.~.-

p

Fig, 1

..2 ...

P

~,)MoiI

I Hd"I .

p

Each of these conditions will hereafter be referred to by its letter

designation. For example: condition t.tbtt would mean that the merflber

is held fixed (not allowed to rotate) a.t one end vlhile the other end

is sUbjected to an axial thrust, P, and a bending moment, Mo -

Tests have been carried out on two sizes of WF sections (8WF31

and 4WF13) t) These members viere tested in lengths of 8ft., 12ft~. and

1.6 ft. giving a ran.ge of L/r from 21 to llJL.

Table 1 surllmarizes the tes·t condi tiollS f'or each specimen. It

also serves to catalog four u'studies U that wiJ.. l be plon.tion,ed later"

For further reference, Table 2 shows on ~-Mo interaction curves the

load condition of,each test, test number, section, slenderness ratio

and rolling number.

II. l'JOTES ON TIlE Al~ALYS IS...........,....... * .....

The momen·c ....rotation (M...g) relationship depends on -the M....¢ curv~e~

since it is the integrated effect of the latter which determines the

load-deformation conditions. Therefore, of fundamental importance

is the determination of M-~ curves above the elastic limit o Such

Inoment...curvatul~e relationships mtlst include tIle variable P, and a sOllJ"_:

lution to this problem has been presented in a previous Lehigh rG~

port(6). To illustrate the effect of axial thrust, one figure from

that report is reproduced as Fig. 2.

205A.ll

p = 0

1.0It == o. 2Py- ___-----

8 VvF 31

__.__ ,..~.~_.._.... ~.~ t B.t i en~ aXIs

" = O. 8 ~y

,. = 0.6"------~-~y

P = O. 4- fit~_---------Jy-----

t:..--...:..----:;......~ ....---:.-...;....,.t_'---t- ....._L--L..---l.--'----L.--I-"--""""'""-.....-.J.0.5 1.0

MMy

Another varia-ble which lnaJT have a pronounced influence on the

moment-curvature relation is the presence of coolin.g residual stress~

This variable is to be discussed in a future report; however, one

figure has been included to indicate the degree of variation due to

such trlocked-intt stresses, Fig. 3 indicating the theoretical rela...

tionship. Curront work suggests that the effect of residual stress

becomes more pronounced as the ratio of axial thrust to Py is in­

creased. However, axial load has been relatively low in a majority

of the tests carried 011t thus far in the experimental investigation,

a condition corresponding to that preseIlt in portal frame structures

(p (0.15 Py ). For the 8WF31 members tested, Fig. 3 can then be con­

sidered as representative of the influence of this variable. Since

-the magnitude and distribution of rosidual s·tres;ses in rolled sec­

tions is undoubtedly a function of type and size of the section, such

varation may not be representative of all sections. (Fig. 3 is the

result of an analytical study based on measured residual stress dis­

tribution values.)

\""--( P = 0)

\

1.0 ~..".--

~~/\~ -Neglecting Residual stress IP = O.~2P.)

I Including ~

/ Re.s j du a1St res s ~.....

I A'~:;A

Assumed Residual 5tr~ss Pattern

(8 WF 31)

0., 1.0

¢E

':y-

Fig. 3

Given the particular M..0 reJ~ation, moment-rotation curves can be

obtained by numerical integration. Newmark's method(?) is adapted

to a solution of problems of this type. An example is shown in Ap~

pendix A and in fig. 23 the theoretical curve (including the in-

fluence of residual stress) is compared with experimental results.

The experimental set-up, techniques of experimentation, etc.,

have been described in a previous report(l). In all tests, end ro-

tation measurements have been made with a level bar, illustrated in

Fig. 4.·

-5-

bubble

Fig. 4

This apparatus is connected to the base plate of the test specim~

as shown in Fig. 4(c). It is possibJ..e with differen'c brackets to

place level bars along the length of the member& Good reproducib­

ility has been obtained using this instrument,

L_JJ.Qmg!TI=]..9J1~~t1-9!l_.Qurv~s

The influence of four variables are presented in this report.

These are:

In anyone case, the other three variables are maintained constant o

Tl1.is j.s summarized in the folJ.. o,'!ing table which also shows the ,fig',r[':"~"

numbers applicable to each study.

STUDY

2

LOADINGCONDITION

SLENDERNESSRATiO

AXI ALTHRUST

The behavior of each test specinlen 1.3 shown in Figs. 5-15. In

each of these curves the end bending moment, Mo , has been plotted

against the end rotation, QA. The ordinate has been made non­

dimensional by dividing by MY, the theoretical yield moment for the

rolled shape in the absence of axial thrust. For each individual

curve, the computed yield and plastic monlents are shovln, due· rega..rd

being taken of the load condition and axial thrust. (These values

are detel-'mined from either reference lr or 5.)

Exeellent agreement between experimental and theoretical result2

was consistently observed in the elastic range 7 although the theo~

retical curves are not shown. (Fig. 23 is an example.)

By way of explaining the figures, the particular variable is

indicated in parenthesis after the test number in each case. Thus

in Study 1 (influence of loading condition), T~12 (c) means Test 12

and load condition !tOfte

It is apparent, of course, that in the early tests the defornJ

mation was not continued through a sufficient _angle change to give

205A.ll ~7~

a definite indication of the collapse pattern. In current tests,

every effort is made ~to determine the uunloading tt curve.

§l_TlQ¥-l.Llillg.-ginE Conq.i:tJon

As indicated in Table 2, three series of tests have been com­

pJ.eted or are under,~ay vlhich indicate the influence of this val'liable,)

In each seI'ies the ratio of axial thrus·t was held constant at

p =O.12~y.

~_ 5:_._U31l.F31--,-1:L~. .; 5.5...,.. PLRy:..:: 0.12.1 Elastically, a expected?

the stiffness increased in the order T-12, T~13, T~4 and T~14, the

corresporlding loading conditions being U eft, ltd", "btl and "au. Tests

T•.lt- and T-14 were not carried to their ultinlate load, although all

current and future te'sts are bej..ng contiIluecl to cOlnplete failure (t

T~12 carried an appreciably smaller end moment than did tests using

any of the other three loading conditions and the failure involved

lateral-torsional buckling. In ectch of 'the other three cases local

buckling was observed.

increased~ in the order tr c tt , tt'dtt a11d Bbu and there \vE:lS a correspond-­

ing improvement in rotation capacity. No condition "au tests have

been carrj4ed out for this slenderness ratio. Since these coltmms

were more slender than those of Fig. 5, the tendency toward lateral

buckling is more pronounced o ThiS supposition is supported since

both T..26 (ltett) and T..23 (ttd'~) failed due to lateral buckling. Test

T~24 CUbit ) had not reached the maximum loa~d when the test was stoppec j

however~ at that time there was noticeable lateral twisting of the

(~,:)111mn as well as a local wrinlcling of the compression flange. Bot11

~f these conditions could be observed by eye; however, it was im­

possible to determine which developed first.

205A .11 ..8..

F.i~L_Qt![[1.h.jJLr ~=..54J'.L4-=-~=2Q (lle lf) whieh failed

due to lateral buckling, had less rigidity and failed at a lower

load than T-17 (nbH ) 9 ,whicll. failed d1.le to local instability.

£lliI!l]§.j.~_:h~:ID~h.:hs__..§ ttl<;b:.t

Figs. 5, 6 and 7 indicate that stiffness of a beam~coluw1 is

dependent on the condition of loading. For those conditions tested,

stiffn.ess increased in t11e order tie", udu , ttbU and Hatt ; the first

being the weakest. Furthermore, the type of failure seems to be

markedly influenced by loading condition. In each case the mombor

with tIle less stiffness, (conditj.. on "cu loading), faj.led due to lat­

eral~torsional buckling, while the stiffer members showed a tendency

toward local. instability.

f33~VJ)X_g:~__SleI)fternJ?'§Lltill_io

Da.ta may be drawn from three series of 'tests to study this var~

iable. Two are· condition Itc" loadirlgs, wllile the third is condition

It b" •

Ej.1k.-§1-.l§YlE31-,-_901lg..i.tj.9.LL_~_9~y_1:. 0.12) Failure in each

case was due to lateral-torsional buckling. As expec~ed, stiffness

increased with decreasing length: (compare T-12, L/r : 55; T~16,

1,/r =41; and T-19, L/r = 27)~ The shortest menlber (T~19) had

greatest strength. In comparing rotation capacity, it would seem

that the longer members carried their load through a larger rotation?

however, such a comparison is misleading since the slender members

allowed greater rotations in the elastic range.

£}~&)--3:_..Q±~~.Q.illl<i.itionnb~y:Z-Q.,1.?)Relative stiffness

increased in the order T~9 (L/r =Ill), T-24 (L/r =84) and T-17

(L/r = 55). T..17 failed due to local buclcling; T-21.r was still

increasing in load when the test was stopped (rotation was over 8

times the elastic rotation) but there was noticeable lateral-torsional

deformation as well as a marked local wrinkling D T~9 failed due to

a pronounced lateral~torsionalmovement o Since T-24 was from a dif~

ferent rolling than T··9 and T-17, it is possible that a difference

in magnitude and distrlbution of residual stresses accounts for the

different behavior. Measurements of residual stresses in each of

these members are planned for the near future.

Fig. 1 0 C Q+WF13, Con..ditio~c" ,J:.L'Ey-= O.B). The slenderness

ratios in these two tests are 55 and 84. As would be expected, the

shorter, stiffer member ~ad the greater strength. Both members faileq

due to lateral~torsional buckling.

SYmm~~izing tho trends observed in study 2, stiffness in the inelastic

as well as the elastic range is influenced by the slenderness ratio~

As would be expected, the more stocky members were the more rigid.

One further observation that can be'made based on this study is, that

members tested under condi tj.on If en loading failed due to lateral­

torsional buckling regardless of slenderness ratio; whereas, the type

of faj.ll11'e for condition Itb" loading seerned to depend on the slender?l'-·

ness of the member.

§'~1W~-3L.~~..111e_ I~J:l111~Jl.QLQf_ Axj_aU!rr_\l§~

F~ig !l.wl1.t..._-lIDiF3~lli:li.t1on_.~lb:r4-_!i1'-=....2.2l Since T-lt (p :: Oc12Py.)

undoubtly wou~d have failed due to local instability had the test

been carried far enough, and since T-3 (p =O.lt9Py ) failed due to

lateral~,.torsionaJ.. bucl~ling ~ axial thrust plays an important role i,n

Q8termining the type of failure B These early tests give little in~

. formation on rotation capacity~

205A.ll

Fi~.K~.1..b.JlQDli.;it1-.Qp._.~tp~,'.,L/r,:: 111) T-7 (p = O.27Py ) \"a8

expected to carry less load than T-9 (p ~': O.lOPy). Such was the ob­

served condition. Failure in both cases was due to lateral-torsional

buclcling.

!11.f~_.llL~-i4W"."E13~Co1].Q1.tiQ;tL.~I= 551 Both of these members,

T-17 (p =O.12Py ) and T-2l (p =0450Py)' failed ~e to local insta­

bilj.ty. The "hump" in these curves seems to be characteristic of

those conditions in which failure was of the local instability type

(se(.~ also Fig. 11, T-4). Similar behavior has been observed in tho

continuous beam tests and has been discussed in Progress Report 5(3)0

U1timate moment~ carrying capacity of a beam--column is directly

related to the amount of axial thrust applied to the member. The same

correlation is not observed when considering the ability of each

member to plastically deform thru relatively large end angle changes~

Based on these few curves it would seem that increased axial thrust

does not have the pronounced detrimental effect on rotation capacity

that it has on moment carrying capacity.

STUDY 4: The Influence of Size of Cross~Section~~,..""~.~~&&;"~"'-~""_ ..;.:c::.~-~~~-.,~-.,-~---"-......".r.._~~~~ ..__~""""""'''-~~~,j..B;;;r.~~~·~~.:..JIC.P~~:r.:~~..~~'r"'"

A question frequently asked is, "Will the results of these tests

describe tho behavior of geometrically similar sections but of dif~

ferent size from those tested~tt Therefore, correlation between the

two sections tested under identical conditions of loading and slenf~

derness ratio is next examined~

Ej...g.,.J~..L....J.gJW...gj. t ~,-Id.r-.,:: .i2~n.y-=-.9.,L12.2. indicat es that

tll.o same general trends were observed~ in ea.ch test. The 8WF31 secticr.i.

yioldcd at a proportionately lower load, and this indicates a re~

latively greater magnitude of residual stress. Both tests indicated

205A.ll ....11...

the tthumptr or sUbsequent incrGase in moment strength ctfter a certain

amount of plastic straining~ Tl1is was mentioned in discussing Fig.

130 Each of the members failed due to l08al instability.

FJ~;.<,J~2.;_..i.Q..on<lL~J.91L H cl~..J-lr = 52',. )~L£y_';=--Q'p.12). For the con­

dition 11 en type of loading, failures were of -tile lateral-torsiol1al

buckling type. Here again the 8~inch section yielded first, but

there was no sUbsequent recovery of moment strength and consequently

the 8WF31 member carried less ultimate mOInent ·than -the comparison

test.

The above comparisons indicate that the influence of size of

member is of little importance in the elastic' range. Further, these

.tests suggest that once yielding occurs, tl1en two members of dif~

ferent size will not behave the same unless the loading condition is

favorable or unless the cooling residual stresses are of nearly the

same magnitude.

~.ROTATION CAPAClI1

Mention was made in the Introduction of the importance of ro~

tation capacity, the ability of a member to deform thru a relatively

large angle change while sustaj_ning a mome,nt approximately equal to

the plastic hinge momont. Study of figures 5-15 reveals that few

of the columns exhibit a behavior that approaches the idealized

U plastic hinge U case. (:B"'or tIle idealized case rotation would contintle.

indefinitely at Mpc ).

The test curves previously examined fall into four general types

that are indicated diagramatically in Fig. 16 and in Table 30 CurVG

A might be considered as the basic type. Gurve B applios to a test

(T-14 is an example) ill vThich t~he loading was stopped before the

maximun It collapse" mon1ent vias reached. In ctlrve C the member con­

tinued to carry increased moment through a large angle change. Again

the test ~las stopped pl"ior to corl1plete "unloading". A member re ...

suIting in curve D actually has no rotation capacity since the max~

imum momont does not appr~aCh Mpc •

Rotation capacity, R, will be defined as the ratio of observed

end angle change, eA' to the theoretical end rotation at initial

yield, eye, or

R = eA/eyc

Fig. 16 indicates that non~dimensionaliz~ng the coordinates provides

an abscissa that is a dirGct measure of rotation capacity.

Because of the nature of the experimental curves, rotation cap~

acities will be compaI'od according to soveral It cases tl or criteria o

The trends will be examined as a function of slenderness ratio, ratio

of axial load 9 and loading condition.

1.0

TYPES 'OF CURVES

e IeI( lye

Fig~ 16

1.0

PROPERTIES OF CURVES

e IeIf ~ yc

205A.ll

The nomenclature used in this particular section is as follows:

MO " =Mc)men.t appli(~d at the and c:f a boctmr.-.-column

:: Predic·t8cl e.lastic Ij.nlit mOln8~1.~;~ tn~J.llcling the effect ofr~!~j.a.l t~J::rllst, If

= Observed end rotation corresponding to Moo

=PredictGd end rotation corresponding to Myc •

=beJ~:rC)L1SO j~n moment bela"'T that predicted at the elasticlinlit rotation o

- In.crease of Inaximum mome11t over predicted elElstic limitrotation.

= Increase in rota'tion over preo_ic'ted. elastic limit ro'"tation at the elastic limit moment.

=Rotation capacity at the maximum observed moment.

~ Rotation c~pacity at the elastic limit moment on theunloadi.Jlg curve.

Casez11l Incr8ment of rotation capacity at predicted

yield moment CaRl) and reduction of moment at the

elastic limit rotation (~Ml).

Such a comparison is primarily an examination of the tlinitial

yield" charactoristics of the member. The results are shown in Figls~

17 and 18. The variable here is slenderness ratio and test·results,

in general, indicate that the shorter the member the greater the

departure from predicted behavior o The maximum reduction in moment

is about 10% whereas the increase in rotation is more than 50%. It

is considered that the discrepancies should be the greatost for con~

ditiorl tt CU .loading and there is some agreerIlcnt here since the con~

d_.,1.tj OJ:J Hb tt tests sho'\V' less redllction.

205A .11 -14..

pase 2= Rotation capacity at the maXilTI11m moment

(R2 ) and increase in monlent above the elastic limit

value (AM2 ) ~

This case, which descl~ibes the ultimate strength of the various

beam"columns , is shown in figures 19 and 20. In Fig. 196M2 is

plotted against slenderness ratio. Fig. 20 is a plot of R2 versus

slenderness ratio. It is interesting to note that'those tests ex­

hibiting greatest strength and rotation capacity (as defined by R2)

failed by local instability; whereas, lateral-torsional buckling re­

sulted in weaker, less stiff, members, This is to be expected since

lateral buckling involves general yielding over a comparatively large

area whereas local instability is usually confined to a smaller

region that is often conducive to strain~hardening. As further work

is carried out, it may become possible to define the condition above

which local buckling will occur.

In no case does the rotation capacity approach the value of

eigh'c which was mentiOl1.ed in the Introdu.ction (page 1). In fact, a

rotation capacity of about two seems to be about all that most of

these columns can achieve at their maximum moment. The influence of

loading condition is pronounced and Fig, 20 suggests that condition

ub" and ttd" tests w01.11d provide nlore satisfactory performance in

columns of L/r less than 50.Fig. 21 was prepared to S110W the influence of axial thrust on

rotation capacity, and the latter function is plotted against the

ratio P/Py. Comparison was only possible in the case of loading

condj.~tion ttbu tests. Lat(~ral~torsional buckling \'las observed il1

the case of T9, T7, and T10 and R2 decreases gradually as the axial

205A.ll -15...

load increases. However, the trend indicated by tests T~17 and T~21

suggests that when the mode of failure changes to local buckling,

then the rotation capacity is actually improved by an increase of

axial thrust. This trend requires confirmation~

Ccg>JL_~: Rotation capaci"ty, R3 , at ·t11e elastic limit

(unloading curve).

The graph of Fig o 22 shows trends similiar to those observed

in Fig. 20. The rotation capacity measured at the elastic limit

(on the unloading curve) is plotted against slenderness ratio. An

increase in slonderness ratio is accompanied by a docrease in the

rotation capacity; however, the influence of this variable appears

to be less pronotmced than that of loading condition. The additional

tests planned for loading conditions "du , ttb tl and Ita" Shotlld provic1e

information covoring a more complete range of slenderness ratio.

Two things are yet needed to establish the sUitability, of plasti0

design techniques:

(a) how much rotation capacity is required in order that all

plastic hinges be developed in building frames, and

(b) is the trend. suggested by loading condition n cit tests

borne out for the other conditions of loading.

Four of ·the seven tests whose reslllts are plotted in Fig. 22 hav~e a

rotation capacity of about 2, the slendernGss ratio ranging from 40

to 84. Adlnittedly condition "c" is the worst caso, but it is

heartening that the few results available for the other loading con~~

ditions show improved behavior in this respect. If the roquired

rotation capacity for typical structl1res approaches the value of

205A.ll -16..

eight, mentioned earlier, then these tests indicate that rotation

capacity may be more serious as a limitation to plastic analysis

and design than previously anticipated. A rotation capacity of 8.0

is kno,~ to be a severe requirement since it was developed for the

case of fixed-ended beams. In actual framos, the columns themsolvos

supply less than complete restraint, and i.n the plastic range they

will also contribute to inelastic rotation. T~lS a required rotation

capacity of 4.0 might not be unrealistic for It cOlnplete restraint tl

and this value would be further reduced since every column allows

some flexibility.

V (I CON Q L-1L S ION S

Based. on the few tosts thus far carI'iecl ··Ollt, the following ab...

servations are made:

1. The stiffness of a beam~column is dependent on the condition

of loading. In increasing order of stiffness the loading

conditions investigated are "e", tid", Itb" and ttau. (Figts o

5....7)

2. Failures, have been of two types: lateral.torsional buckling

and local wrinkling of the rlange elements. For all members

tested under condition "e" loading, failure was of the

lateral--torsional. type rega1'dles s of slender11ess; hO'\'Jever,

for the other loading conditions there is a tendency for

the stiff, short members to buckle locally. (Fig.'s 8~lO

and Table 3)

3. Axial thrust deC11 eases the ultimate moment carryi.ng capac::.ty

of a beam-column but does not" necessarily decrease its ro(.I"'

tation capacity. Further tests are needed to establish

205A,11

definite trends. (Fig's. 11~13)

4. The influence of size of member is of little importance

in the elastic range; however, when yielding occurs, two

members of different size will not behave the same unless

the loading condition is favorable or unless the cooling

residual stresses are of nearly the·same magnitude. (Fig's.

5. There seems to be a limit or ultimate carrying capacity above

which local buckling is the type of failure and below which

lateral~tqrsional instability takes place. Further ana­

lytical work is needed to establish this transition zone.

(Fig. 19 and Table 3)

6. Increased slenderness ratio in general decreases both the

moment value and rotation capacity. However, in many cases

the variable of loading conditions has a more pronounced

effect, (Figts. 17-22)

7. Only a few of the members had a rotation capacity greate'r/'.

than 4.0, and a conunon value ,vas about 2.0. If the re..

qUired rotation capacity for typical structures approaches

the value 8.0 (the ttidealtt requirement for a fixod...ended

beam, loaded at the third~points) then rotation capacity

may be a more important factor thanprevio1.1s1y anticipated.

(Fig. 22 and Table 3)0

-18-

YL__~_AS;I\N01J~~~MIG1I1S

TIle work is being carl~ied Ollt unc1er the direction of the Lehigh

Project Subconnnittee of the Strllctural -S·teal Committee-, Welding

Research Council, at Fritz Engineering Laboratory of which Professor

William J o Eney is Director.

r

811

e'yc

cry

Young's modulus of elasticity

Total length of a beam-column

Moment

Moment applied at the end of a beam-column

Moment at which yield point is reached in flexure

Predicted elastic limit moment including the effectof axial thru..s·t

Plastic hinge moment, modified to include the effectof axial compression

Decrease in moment below that predicted at the elasticlimit rotation

Increase of luaxj..lTIUln 1110ment over predic'ted elastic lirnitrota~tion

Concentric axial load

Axial load corresponding to yield point stress acrossentire section

Radius of gyration

Increase in rotation over predicted elastic limit ro~

tation at the elastic limit moment

Rotation capacity at the maximum observed moment

Rotation capacity at the elastic limit moment on theu1110ading curVG

Observed 811(:1 rotation corresponding IVlo

Prodj.ctocl cnd ro·tation, corresponding to l/Iyc

Lower yield point stress

Curvature at a section (the reciprocal of the radiusof curvature)

205A o ll

Y.~~Jl_~~~~~FE11EN~

Reports in the Lehigh Series:,

-19-

1. pr..Q.g~£L,B&ILort !'I..<h=k UTests of Colunll1s under CornbinedThurst a11d MOlnent tt , by Lynn S. Beedle 9 Joseph A. Ready,and Bruce Go Johnston, Proceedings of the Society forExperimental Stress Analysis ~ Vol. VIII, No.1, -1950 •

2. P~2E,r,~§LJ3.53J)...9-r__LEo~.f=<~.:..~,Par.LI,t!Connections for 1tJeldedContinuous Portal Frames-Test Results and Requirementsfor Connections", by A.A.Topractsoglou, Lynn S. Beedleand Bruce Ge Joln1ston, Welding Journal Research Sup.,Jll1y 1951 0

3 c. Pt9.E!.§.:ss~BQ.nos~-..~o .~..2, ttResidual S,tress alld the Yieldstrength of Steel Beamstr , by Ching Huan Yang, Lynn SeBeedle and Bruce G. Johnston, Welding Journal ResearchSup. 9 April 1952.

4f) :er.QEr§.-.q§,_ RG.P_Q~r~--liQ.,, __~_~!).. ttColU~lnn strongth Under CombinedBending and Thrust tf , by Robort L. Ketter, Lynn S. Beedleand Bruce Go Johns'ton, vJeld.:ing Jor!rnal Research Sup.,DeC81nbor 1952.

5tt Pr-9Eress cB~t L, It IntoIl(1ction Curves f'or Columns tt byRober~t L. I(etter, and~ Lynn S. Be,':)dle , Fritz Labora·toryReport No. 205A.4, April 1951~

6. PrQRr__ess~_B§1I!...9rt.".-l:,ttThe I\1oment--Cur\lature Relation For WFColulnns lt by Robert L. Ketter and LJTnn So Boedle, FritzLaboratory Report No. 205A.IO, September 1952.

Additional References:

7. Ne\·nnarlc, N·. M., ttl\T1,11n(1rical Procedure for COll1puti11g De­flections? Moments? and Buckling LoadS",. Transactionsof the A.S~C,Ea, Vole 108, 1943, pge llbl--1234 o

APPENDI~ A: Numerical Detormination of Moment-Rotation Curves

OUTLINE of' the general method of solution (using the numerical procoSfof Ne~nark(7») is as follows:

1. Assume a deflection curve.

2. Determine the moment acting at equally spaced pointsalong the member.

3. From the M~~ diagram (for example Fig. 3, page 4) readvalue of ¢ corresponding to the moment values at eachof these points along the column.

4. Numorically integrate these 0 values, and· obtain a de­flection curve.

5. If this new deflection curve does not check that orig­inally assumed go thru steps 2,3 and 4 against usingthe ne\v defJ..ection curve.

6·. When t118 defl.ection curve is sUfficientJ.y accura'ce­correc·t slope values to the end of the collU1In (usingconventional finite difforence equations).

An example using this procedure follows.

Determine the value of end rotation for the 8WF31, 12 ft. columnloaded as shown in the following sketch.

P = O.12P, -= 41.1-.9k

No := l050"J:~

-21--

]MUltiPlication ~Factor

._------_.j-----~--,._.-'. I

Total Moment

---r--:;JIt"--.----- --..-~--- ..~-.,----

A6corr = -(7¢L + 6¢o -cPR)

24

A-= -(,1.771:+ 1.740 - 0.317)

24

Iherefore,

[1.119 + o. I33] A. IT IEy

0.0302 inches/inch

TEST ! LO'ADI NGNUMBER COND IT f ON

SECTIONp

py

L--r STUDY ILoading

Conditien

STUDY 2Slenderness

Ratl.

STUDY 3Ax i a'

Th ru s t

STUDY 4Size "fMember

.\--...~--_._.~.~ f---' ,_......."__ ' .~~.o._ __ -__-...-__~_......_.__•__... _ _ •__~-_:-...-."-_~-

i=~~=-:~~=~~~=---=~~it=~l~;---=~~t~~!-~~~==.~~~~·~=.-~.~_=~t:-:-~:..t~-_-~it-~_~;~]I T-7 b 4WFI3 a.V I" Fig. 12 I .I'---'->~---·-.. --~-..--,-._ -.-_. t-._ ..--~ ,~._--, -, ..- ---~-- --._.- .-••_~- .•. -.-..----...-• .-.--~.--- -- _4 ",.,••••• - ,~•••• _ .•• , •• - ~---~.-_.--t--~ _.,., .. ----.-..-.--,T-9 b 4WFI3 0.10 III Fig. 9 Fig. 12 I'I--"-~'·_'_'_ --'-','- c_.· _.'. __ .• ,....-f---. --- ..---.----.>-*-,-~- --- ~-_ ..- ..__._. ,.._-,_.--..~ '_'__'''~'''''_'~_''_.''_'b. •__._,.,.--...---.__.•'--- ...._~--~_ ... ,v-._·_"'_"_· ~

T-IO b 4WFI3 0.51 J II Fig. 12l-~"'--'---"-'~'- f--~----------""-"- .-._._~ ._ ....~._.. - --~----- :----.-~_.. .'-~-- - ....---""--------- --~-, -.--,-..... '.-'- _ ......_-_._-_.-.--~~ ..' i

T-12 c 8WF31 0.12 55 Fig. 5 Fig. 8 I-~.-- •...-...-- _.-••- ..............--._..- ...._.,.•••- -.-_.__' • ~,_....~ ....._-~ .... ,~r__"r--- ._.'-"'-_...-.-- .., .. - • ,-_....~... ~ .. _.. -.....~_. -_.-" ,.~_.'..••' _._ .......~ ~- .__ ...-~- ~.·.-P._- ~-- - .' ....."-_----, '-"~ ._~

T-13 d 8WF31 O. 12 55 Fig. 5~_. ._ ...... __~_~ ~~ __• __ ~~ ••8_ ,__ "._._ ._..__.......'._,_, ~~'" u·__ ..... __._.,__ +,.~ .. _ ......._.,._ • __ .~.__ . J~ ..._._. "",.'_'_'_._'.' _. __••• ~.••_ .•. .. _._••• ,

j T-14 a 8WF31 0.12 55 Fig. 5

r.-.---.----~- 1---._-_ _,...... -...-...--,~_._--- ..~'"._--_ ~ -'--.--'-.-. 1-- ----.-..--•• ----- --_.--_._~-----~*---_.----._ -._ .._,_.

T-16 c 8WF31 0.12 4r Fig. 8~ -<'7_~ " .••••__ .~w_..~...~ ., .. _._ .......... ·.r.~·~ __·..~ __.•_.._. .. ....._ ...... ..-_,__.. _~_~._ ...__....., ..... __• •.-._........_-.. . . . _

t T-17 b 4WF13 0.12 55 Fig. 7 Fig. 9 Fig. 13 Fig~ 14... _-...--.... ........ ...~~~.......".A .... _ ....... .---.... ~_ ~. __ ' ..1 ..... _"#o__~_ ..... _ ..... ~ ... -.. ..... ' .... ~- ...... --...-..~_....._.. _ .... _ .... '..-. ...~..II".._ ,.--__ ... _I ............ r ...... 4~ l .. _........ _-..". ....._- .......... __ ................._~ ...... _ .-...-.....~----.- ...... -¥ ............... - __.....""'-...-:.- ..'1.1Io ............ t

T-19 c 8VvF31 0.12 27 Fig. 8 ".-._....- ~ __........_r~.~.----... .._ -" .---.~.~-----~ ....-.- -.----.-~--~---,-..--_..~ .1--- ....---------........--.- --~,_..----._-----------' --- .._.1

T-20 c 4WFI3 0.12 55 Fig. 7 Fig. 10

Fig. 15T-2f-.-.---i----.----............-- -._-...->.---.-~-. ---------- -_.,--- I--.........----.~ io------------- -.-.--.--+--..----

I

b 4WFI3 0.50 55 Fig, 13Ir----,---+------.-- .-------.--.--.-..... _,, 1-_~ __...4__ -------------.-_•...,••- "'"---,--...--r---__ ~-'__,_o,_...._

T-23 d 4WFI3 o. 12 84 Fig. 6-.-._.--__..__..•_.~._f- ..,~~ ">_-__.........__~.,_~"~.,-f--- ..----.-..- ..-- _.__ ._~_._.--.~~.,~---...-...._ ...._.- ...._,..- ""'~--"'-'-'.

T-24 b 4WFf3 0.12 84- Fig. 6 Fig. 9_________._,,-~--~- ..,---.,..-.- ~- ._~.~ ,_.~-.-......~. _._--....._- .~._~_._-- ......·..-------·-·.....·····...··- ......---·--~to_,-·-· ..~------··I

T-26 c 4WFI3 0.12 84 Fig. 6 Fig. 10I~====!:t====~======±===..~~__..~__..........."":""_~~.~~~~.-~~~ ...~~.~..._~.... ~._..~._,~~~~~~=:===.J.

TAB L E 1..1.

T-l9 (8- J

-~-= 27r

205A.ll

~::: 4 ~

~ ,T-16 (5- 2)

''',~.\rl'01

~OJ

1= 55 T-3(A

~r

T-14( A-3) T-It( A- 2) \ T-12( A-l) T-l)( A- 2)

.~= 55~, ~~

"T-17(A-2) ~

t Y.)rl

..~~ J::. =84 '.r

T-24-(B-l) r-26 (B-1) T-23(e-l)

205A.ll

y c

.1h~~Q--U~~h ATI 0 tL:::2:K.:::._.-~-,X~[.) -~_~P~J_l1_~_N ~~~~~~_.Ii_~7~..U-f2

t---i---~-' ..----~2_--- _"",_?-l

1.0 1 -t7.t ~~tt1·'\nax.

( ; ] I

n. 6t'l~" 1 II __ ..+__ .__-R~_...__ ...------·::-i

M::; i

PR-geERIJ~~~: OF <;UJ:L~~~

1.0 B A......--~"III ' .... ::: Reye

0.181 ).10 0.04-4 4.93 5.35

o 0 0.3;8 2.07.-_.- .------- --._- ---*-_ ... ~~- -1'

.~~~ '~~o~- ~::;. ~~±~j

Locat Buckl i'ngA550.504WFI3T-21 br------ ~-----,.-._~- .._.-......._..._-~ .----I T--23 d 4WF130.12 84· A Ben.ing+Twist!~_ .. ",._~_ ._~ . .~ ._._ ._. .._----...-.~ _~ ------_._---.._1' ..__·- ~-_._•..__- __~,

! T-24 b 4WFf3 O. 12 8~1- C 8ttM"'tiJng, Twist

i-- -T=26-j- _.... --;--- _h ~4WF·IL3'-O~I;··t'-'0:" ~··---·A--:t:~d ..;~~~C~;-;:~~ ...-b-~----- --~--- .:.-.-_ --.J=c=- =--- -.-:--..=:.--=:-:.:--------

* L~st recorded value - '(Test stopped prior,to unload'ing)

205A,.11

I ~

0.02

8 WF 3.1

Lr = 55

p = O. 12Py

-,-,·I__~_-l-_-L j ,.!.....__--t-__--L -,L-_0.03 O.O~ 0.05 0.06 0.07 0.08 0.09 O.JO

0A (inche~/inch)

Fig. 5

M•

My

1.0

4 WF J3

L - 8? -- -4

P = I. 12Py

0.10

eA

(Incheslinch)

Fig. 6

0.15 0.20

205A.l1

~M

y

1.0

0.5 --:I-4- WF ~

1- = 55rp = O. 12Py

0.01 0.02 0.03 O.O~ 0.0, 0.06 0.07 0.08

eA ( 1nch esJ inc h )

Fig. 7

1.0

0.01

8 WF 31p =O. 12PY

0.08

L-...r-­0.09

--=1f--. ~ WF 1312."'?J' .....T.d P - o. I 2Py

0.0;

"'---__,'-"-'----.lL_.........--._l,_-L._-L.-_l---'~._, __1__t_~L__l-..L._L-l._

O. 10 o. J..5

1.0

eA (inches/inch)

1.0 (Eending l' TuJist)

4 VIF 13

~) p::: O. 12Py,I

~_~ Ll__-L__...-...-_l-__-J-- J.- L-- , -----,J-

0.01 0.02 0.0) O.Olt 0.06 0.08

eA (inches-inchJ

Fig o 10

205A.l1

1.0

0·°3

0A (Inches! inch)

Fig. 11

8 WF 31L/ r = 55

1. 0

o ,

0.01 0.02 0.03 0.04- 0.05 0.06

4 WF 13

L/ r = I' f

0.0.7 O. 08 0.10

eA (inches! inch}

Fig. 12

2051l.1l

1.0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

8A

(inches/inch)

Fig. 13

4 WF 13L/ r = 55

0.09 0.10

2051t .11

1.0

~T-17 (4WFI3) (Local InstabilitlJ) ~~'\ ..~~~.:~,~./.~=ti--P '~-'"

~~~' ~

M r,>';:/'. \"

YC,' \ '\I "---- T-4 (SvvF31) (Local Instability)

/~- y

L/ r = 55

p :::: O. 12Py~~~

(Benrlin(: f TIJ)isiJ

L/ r = 55

\J. ~ p = O. 12Py

''1'../

Li

O.5~­

f

t-Lj

J.-\;

-<-~---":_--'--'-_J__L.-l__L-':'-1--a1 L __.L.__J ._l__I__..1.-1-1_.-.\0--,..,.a-,,!_-L-11.0 2.0 3,,0 4.0

Fmg. 15

205A.ll

0.10

0.05

10 20 3° ltO 60 70 80 90 100 110

O. 7

O.12PyI PM 1Mo yc

60 70 80 T-24 100 11050LtO

T-17 (4 fJ )

I I

30

l I20

\!·~O T- f9i"

I ~"C"I, \!

.~ I\\T-16

1\,! \1 \

! \; ~

I .~

GWF31 Sections -+-----.~~ : 1-----4VIF13 feet ionei \ if \1; IIb1'\ . "b T- 12 (8 tI )

\~\ !\\'bT-4 (4 11 )

10

0.6

0·5

L\R, 0.4-

0·3

0.2

0.1

Lr

Fig. 18

205A.ll

p = O. 12Py

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

e /eIf lye <

. ~IT-13 (8""') (Local Instabiltty)lid'!? '

;

,tt) r~ ~;,b T- 4 (8 f1) (Local Instabi l- i tIl)

?~ I

I~_~!-17 (A.If) (Local Instability)!--",--.,....I .,~~

, '-"'-~~'~:t.~::::..____ Itb If~~

~~~~"-~~~ (DenriinrJ "J-~~ m · 't)~~. J.uJZ,9,

~. :

~

(IJending f TVJist)

LO 30 ,-10 ~ 60

Lr

Fig. 19

10

j

~T~O (4 ft)

1~~~lf'lf ;

II ~~~ I

~.:.'-L~,~ T-26

I (Bendtn~ t Thitst)

8VJF31 Section -------I ' ~-- <-l:VJF1~3 Section!

I(Bending + T_ 19 4,0 T- I 2 (8") (Bendin~'fi Twist)

TIJ)ist) O=.~CI1 T-11~~ IL-._--L_---l_~....JI =:::rr:_·___l':':_.---!.' -l-i__~_!_I__~_.....\.___---1..-_~_-'---_--L-

70 80 90 JOO r 10

0.1

0.2

5.0 M 1Ma yc

p= O. f2Py

4.0e /e

A yc

"el" \~~,~ IT-13 (8 t1 )" ~~

fib II '~? T_ 4 (8")\

IiI! T-17 (4")~-'=========.==- lIb II

3.0 -

2.0

l.a 8WF-31 Sect ion ---1I

Iltd 'II-j T-23T-20 (4") i

~l -==-=~" I~T-26

~...'~·oT-t2 (8") I

. I~ 4WF13 Section

I;I

T-9.='-={)-t

40

-~r

fa 70 [30 90 100 110

Fig. 20

205A.ll

4.0

21

4WFI3 Section

-:(--

/8.18P.. yc

1T-24(I astl reco rded reading 8.9)

I\

I;I

R.3 I~---_._--' _.-.->~

--------..~,._-~-~ .. - ..._'.....1.

M 1Mo yc

T~+·17 (4")"=.I =-=- "b"

IT-20 (·1ff ) I

li '=-~~0'4_T-26

~T-16 ii~~ I! T·-12 (8") 1

i I ;! I I,

!!.i; _ I I \ _ i~ i r-4\'lF13 Sectifn

! I I! !i ji I

i

~T-13(Oll) lIdll

8\MF31 SectiO!l1.0

3·0

2,0

4.0 -

5.0

:I Ii I , 1 I I I I..

10 2J 30 40 50 60 70 80 90 lOa 110

+kr

Fig. 22

205A.ll

p

•dua\ st resS\ t

.\on Neg\ECting Res \ _L_SOU _._--s--- -.::;;_-:I~~;:ding f€sidual Stress,..,- ,,-'

/ """,,"I //

/ ,,// ..........-.nMy c II' 7 u--- --------:----

II

I , ---"Twistlng of specimen first observed by eye.Lateral-Torslcnal ~I ,,,'Displacement first I'

indicatt'<d ty lateral 9deflection gages. ~I

1.0

0.5

~J1(j

P

8 WF 31

L = 41r

o -0--- Expe ri men,t a I Resu Its

(Test···T-16)

0.01 0.02 0.03 0.04 0,05

Fig. 23