."(

Welded Continuo'IJ.s Frames end Thai.... C t... ompot".en s

Progress Report No o 9

!~TIC STRENGtd AND DEFLECTIONS OF CONTINUOUS B~AMS

by

C.,H .. Yang, _Kllud.-E, Knudseno Bruce Go Johnston and LJ-ynn So Beedle

with an appendix prepQ~3d by Walter lio Weiskopf

~lis work has been carried out as a part o~ aninvestigation spo~soredjointlyby the WeldingResearch Council and the Department of the Navywith funds furnished by the following:

American Institute of Steel ConstructionAmerican Iron and Steel InstituteCol~1mn Research Council (Advisory)Institute of Research, Lehigh UniversityOffice of Naval Research (Contract No o 39303)Bureau of ShipsBureau of Yards and Docks

Fritz Engineering LaboratoryDepartment of Civil Engineering and Mechanics

Lehigh UniversityBethlehem, Pennsylvania

24 November 1952

Fritz Laboratory Report Noo 205B .. 22

11/24/52 -1=

CONTENTS--------

10 0 Nomenclature

9 0 Referenoes

11 0 'lrables and Figures

12 0 Appendix by W.H o Weiskopf.

Page2·

3

8

11

1313131'121

21

23 .

26

28

30

31

34:

36

5,6,

Abstract

Introduction

Methods for Computation of Defleetions

(a) Numerical Integration(b) Mathematical IntegratiOJ(c) ¢ - Area Method(d) Simple Plastic Theory(e) Plastic Hinge Method

Comparison with Experimental Results

Influence of End Restraint

Influenoe of Load Distribution

Limitations

Summary

Acknowledgements

4"

..

205Bo22 11/24/52

ABS IfRAC T.-t _

=2=

While conventional structural design according to the theory ot

elasticity gives assurance against excessive deflections, the appli­

cation of design methods based on the ultimate carrying capacity of

a structure necessitates special consideration of critical deflections o

TIle simplicity of the so-called plastic design methods may be ove~•

shadowed by 'time-consuming deflection computations unless reliable

approximate approaches can be applied..

This paper demonstrates several methods for computing deflections

due to bending of mild steel beams of uniform cross section o The

effect of various simplifYing assumptions which greatly reduce the

numerical work involved is shot~ together with comparison with ex­

perimental results.. The influence on the deflections of various

degrees of end restraint and load dlstributi~n is computed..

The paper demonstrates for several loading conditions on con=

tlnuous beams the possible savings by using plastic design as against

conventional elastic design and suggests a specific design criteria

applicable to the examples giveno

205B,,22 11/24/52

I N T ROD U C T ION-------~----

",-

The prediction of deflecti,ons of mild steel beams strained beyond

the elastic limit is an important part of design methods based on the

ultimate carrying capacity.. This concept of structural behavior e~

phasizea collapse (or excessive deformation) rather than a maximum

unit stress as the limiting criterion on struotural usefulness o

Stl"'uotural'des1gn methods based on the theory of elasticityg

except in problems of elastic stability, define as tha limiting use­

ful load the load oausing initial yield at the most highly stressed

location in the struoture o The working load is taken as a certain

safe fraction of this yield load, and will leave the entire structure

well within the so-called elastic range.. The deflections in many

cases constitute no major design consideration and are readily es=

timated for those cases in which certain limits are imposed on their

magnitude 0

For statically determinate structures and in problems of sta~

ility the two concepts of structural carrying capacity yield basically

the same solution.. In the analysis of redundant structures g however 9

the differences are greato

Initial yield at some location in a redundant structure does

not render the st~lcture incapable of carrying additional load o Al­

though the deflections will increase at a faster rate above the

yield load than in the elastic range, danger of imminent collapse is.

obviated by the ability of mild steel to relieve the most highly

strained portions of the struaturso The strain-hardening character­

istics of structural steel further increases the margin between

initial yield and ultimatecollapse o A similar influence is ob­

served due to catenary stresses set up in beams which are not allowed

to move horizontally at su~portso Neglecting strain-hardening and

205B 0 22 11/24/52

catenary e~facts~ collapse occurs when a sufficient number of cross

sections have yielded to reduce the structure or a part of it to act

as a mechanism under any further increase o~ loado

As an illustration, consider a beron over three spans carrying

loads in the center span as shown in Figo lao For an amount of end

restraint o~ the center span and a load distribution which give larger

moments MA at the .supports than center moment MBs; Fig o lb g yield will

~irst take place at the supports (Flg o 10)0 Under ~urther increase'

of load the center moment MB will increase at a ~aster rate than in

the elastic range to relieve the yielding sections at the supports o

At the load PI these sections will have yield zones penetrating all

the way to the neutral axis and can carry no additional moment until

strain-hardening occurS o The center section, however p does not yieldI

until the load reaches P29 and loads Pp are required to produce yield

all through the center soctiono Except for strain harden'ing the

carrying capacity is now eY~austed~ and the center deflection, Fig o

ld~ inCreases rapidly under practically no increase in load!?,

According to this concept g the collapse load is defined as that

load at which a small load increment would cause a vel~ large increase

in deflection. However, this simple definition of the ultimate load

is not concerned with the magnitude of the de~lections at that load

but merely with their rate of change~ The magnitude or the deforma­

tion at this theoretical collapse loa~ may very w~ll be prohibltiva~

and the structure in reality rendered useless at a lower loa~ due to

excessive deflectionso In many cases, therefo~ew a design based on

the ultimate oarrying capaoity requires the computation of the de-

fleotions at the full loadu* If this deformation cannot be tolerated

~-----------~----------------------The full load is defined as working load tlmes the factor of safetyor load factor o

205B022 11/24/52

as a limiting criterion~ then the loading causing the maximum toler-

able deror~ation must be computed and substituted as ultimate loado

A constant load ractor ror each type of stru~ture leads to a uni­

f'ormsarety margin and thus to economy in design. In Progress Report

No o 3 those items covered by the load f'actor were discussedo It was

indicated that "A structure so designed will usually be loaded within

the elastia lim1t in the ran.ge of its working load" 0 However, since

the reserve stre:ngth \above __ the yleld load varies grently with the

amount or end restraint and the type or load distribution, an econ~

omical load ractor may under some conditions bring parts of' the

structure above the yield point under the working loado In these

cases a check on the defleetions under working load will be desirabls Q

The above discussion·point$ out the need of methods for eom-

puting beam deflactions be~o~d the elastic rangeo The simple linear

relationship between load and deflections is lost as soon as initial

yield is reached anyWhere in the beam (Figo ld)g consequentlys the

·calati1ation of the deflections becomes more involved.. It is the

purpose of' this paper to show the efrect of various slmplirying

assumptions outlined in Ref o 19 which can be made in order to reduce

the work involved in such calculations, and also t~~indicate the in~

fluence on deflections of' various end conditions and load distribution:

The underlying princ:'l.ples for predicting deflec·~ions·in t?-e. 12 20

plastic range were furnished by Nadal and Timoshenko 0 The lattere

with references to earlier work» outlines an accurate semi~graphical

procedure based on the actual stress-strain relationship using the

conjugate beam method and the concept of reduced modulus of elas...

tlcityo Van den BroeklO gives examples of' the calcQlation of beam

deflections based on an idealized stress-strain diagram g and gives

observed deflections of simple and continuous beams of steel and

various alloyso

11/24/52

13Roderick and Phillips . i:(1 a Ii terature survey on the carrying

capacity of simply supported mild steel beams, compare the deflections

according to various assumptions concerning the stress distribution

through the depth of the beam, and derive the slope-deflection equa­

tions in the case of symmetrical concentrated loads o

Roderlck15 , using Timoshenko ' s semi-graphical method g shows that

the simple plastic theo~y based on an idealized stress-strain dia­

gram determined from annealed specimens loads to a satisfactory pre­

diction of the ultimate load.. However, this theory does not predict

with sufficient accuracy the deflections of as-delivered specimens o.In Refo 14 the simple plastic theory is extended to take into ac-

count strain hardening.. Both these reports present results from

simply-supported, rectangular' beam tests ..17Symonds and Neal , by mea~s of the slope-deflection methodD

calculate the critical deflections according to the simple plastic

theory for the load under which yielding starts at the last hinge to

rorm (~2 in Fig.. ld) 0 The pl"edic tions are compared with tea t results

of Stuss! & Kollbrunner16 (1935) and Maier-Lelbnitz (1936).for

thl~ee-span beams of varying ratio between outer and center span

·lengths ll ~tra1n hardening and the spread or the plastic zones is

not considered.. These authors call for full scale tests of frames

and continuous beams made fram standard commercial sections using

normal fabrication procedures c

Such tests are in progress at Lehigh University as part of an

investigation of welded continuous frames and their components o

Results are reported for beamsl "5,, columns2 ,,6, corner connectic;ms49

and portal frames70 Ref o 3 in this aeries shows that plastic design

results in deflections which compare favorably with those resulting

fromslastlc design, and~t8_ a simple method of drawing the

205B o 22 11/24/52

load=deflectlon diagram for be~ns in the plastic rangeo In a dis-.,..

cussion of this paper Symonds19 suggests taking the deflection under

',I working load as "the de:fleation b"2' when the last hinge starts to

fo~. divided by the load factor o This approach, however g results

in defleotion estimates which may be several hundred percent too

largso

Although beams only are discussed in this paper, the 'resUl~-s

may be extended to portal frames with small axial loads and no side­

sway 0 Experimental deflections from tests of miniature frames under

various comblna.t:lons of V81"tlcal and side loads are given by Baker

e.nd Heyman9 C Hrenn.ikoff11 (and discus30rs) and ~Symonds17 offer

analytical solutions for the plastic deflections of frames o

. ,

"'!"

205B.22 11/24/52

LIM I TAT ION S_-..0 ..-" .. _

-8-

Several factors ir~luencing the deflections of beams in the

plastic Y"fu'1ge are not -included in the following treatmE!nt o The most

important of these factors aX'e b1"lefly discussed in thEl.following ..

The usual rolled sections are so proportloned as to prevent

local buckling of their elements in the elastic region c• When pal't

of a Cposs section has :y-leldedg ho'wevel", the resistance against local

buckling of the compression flmlge at that cross section is reducedo

Local instability therefore may cause greatly increased def'leetion..~

and collapse be.forethe beam is stressed very far lnto the plastic

range 0

Plastic design theories are based on the assumption that the

sections used have sufficient "r-otation capac! tyll to allow the pla.s­

tic hinge moment to be maintained through the required localized

angle of rotation necessary for- development -of hinge moments at all

other required locations 0 Pr-ema tlxrs lo~e.l buckling of:· cross sec tioD

elements .will destroy the l"otation capacity and thus invaJ.ldate the­

above basic assumptiono

Resi~ Stresses

AS-delivered beams show le.rge "locked-up" 01' res:'i.due.l strc-7Isses

due primarily to non··unifonn cool:lng at'ter hot··rolllng.. The proc:ess

of straightening or cambering afteI' rolling find also welding has a

similar although more localized effact e

wnen added to the stresses pr?dlWed by external loads the re­

sidual stresses lower the yield-load on a structure appreciably be­

low tha.t expected from simple tension tests50 A reduction or the

calculated initial yield load b-sr one-thir'd is commonly found o ThIs \

earlier yield causes deflections somewhat larger than predictedo

205B o 22 11/24/52 =9....

~,.

'.

Stress Concentrations. ..

Stre~s concentrations in the region of application of concen­

trated loads are generally not considered in the analysis g and raise

the actual peak stresses far above the computed nominal streases o

As a result g yield occv~s at lower loads than predicted and is more

extensive» thus causing a corresponding increase of actual deflections.

Shear Deformation

As is common procedure in calculating elastic deformations g the

effect of shear forces is neglected. However, yielding of the web

due to shear force may add appreciably to the beam deflections.

Axia~ Compressi!! Forces

Although only beams are treated in this paper the numerical ex­

amples simulate pin-ended portal frwnes without side-sway. If the

outer spans were rotated about the interior supports to a vertical

position, they would then form the legs of' the portal frame. In

such frames, however, all members carry compressive axial forces in

addition to moments, the effect of which is two-folq:

(1) For a certain moment the add:!. t;lon of a compressivo axial

force increases the curvature of the mentber', ·chus increasing 'fjha

def~ectiono This effect is minor for thrusts smaller than about

10% of the compressive yield load.

(2) ~ deflections grow larger the. additional moment of the

thrust multiplied by the deflection becomes appreulablsoFor one

of the portal frame tests in which a uniform 8WF40 section was

used. this effect added 11% to the maximum beam moment at the ulti­

mate loado The deflections are correspondingly increasedo

CatenarY Effect

Large deflections tend to shorten the distance between supports o

In beams with pinned supports axial tensile stresses develop and a

205B" 22 11/24/52

pal">t of the load is carried through this catenary effect with resulting'

smaller deflections than if carried by bending o

In portal frames as discussed above the knees are practically free

to approach one an.other» and ,the e.xie.l tension does not developo

~ummary £! Limitations

n16 integrated effect of local instability, resldualstresses g

stress concentrations, shear deformationG and thrust is to increase

the deflections and lower the ultimate carrying capacityo Hero 5

offers a more detailed discussion of these factors on the background

of. experimental evidence, which is smmnarized as follows:

(1) 'The calculated deflection at the predicted initial yield

load was reached at a 10 to 25% lower loado

(2) At the calculated initial yield load the deflections exoeeded

those predicted by 13 to 88~o

Common to all of the methods for calculating plastic deflections

discussed in this paper are the following basic assumptions o

(1) The longitudinal strain varies linearlly across the depth

of the beamo

(2) The stress-strain relationship in bending is identical to

that in simple tension testso

(3) The stress-strain relationship is the same in tension as in

....

(4) Under increasing load the ratios between the individual loads

are held consta.nt o

In the elastic thooX7 the l1.n~ar l"s;lation between banding

moment9.nd C'U:I:"ilaturo is expressed by

d l·1 *11 =. E! (1 )

\

The change in slope ot the deflected bewn between x =0 and

x ~ x equals the area under th~¢ ='curve between these po1nts~i\ '

<X.o- Ol ,-= J'(Jdx 12). c

In the example in F1go 2,0(,,, is the slope $.t the left -support

and is found by the "conjugate bae.:Oln method~

a:o m t JL¢> (L-x)d~ (3)'

The deflection Co at ,s. distance l. f'romthe sUPPO~t9 F1go 2, thtlln

-(4)

.' J.

()(t>l-j¢(l-x)~becoIlles . ., J?....

8 ::& focdx =. ~ \

which raay be :lnterp:r.ated as ~1lG moraent ot the area under' the J8 curve

batt~aen x ':::I ,10 and x = f. about the po int .01' ,8 9 includingth@l ef'f'@ct

of the slopG at the suppo:rt"

Above .the elastie; limit. the linear r~iationshlp Eqo (1) is lostg

I

but the sam.a method, as outlined above holds 11 the actual M;;.,J8,relation.~

ship ifJ used in place ot EG,o (1 )0 AAlsum.ing the same stI'aaa ....8t~s1n re,.,

lat1onsh5.p in banding as detem.1risd trom slmplotension' t68ts~ Figo 3 9

theM-tI d1s.g1"aIl'l may be 4erived fl'1l0m the stress...stl"ain diagram as 'out-o " ..... -\, ..' ,. •

lined in the appendix of llefo 10 In Figo 3 the measured stress-strain

curve is, closely approximated in the elastic range by

(5)

in the plastic range :for as=delivered steel having no distinct ,upper

yield point by

. rf v, 6':) (6)

and in the strain~hardentngrange by

= - • ~ _ _ ~ _ _ _ ~ • _ Q a _

4. See pgfl 35 £o%" "NomGnclaturen

205B .. 22 11/24/52 =12-

The latter part oJ.' the strail1#ohardenil1.g range will rarely be reached

due to failure by local buckling, and is of' little practical interest o

The M-¢ diagx-a.m evaluated f'x"om this stress-strain relationship

for an 8WF'40 sectlon is shown.in Ii'ig o 4 0 Using such a M=¢ relation­

ship in place of Eqo (1) the deflections above the elastic range may

be calculated by the procedure ex~mplified in Eqso (2) to (4)0 In

statically determ.:tnate bee.ms 110 diffIculties a.re encountered g but

for redundant bewns the non-linear M=¢ relationship necessitates a

t:e:tal-and-error p:r.·ocedl.1re to sa. t:lt~fy the boundary conditions <)

In the example of Fig o 5, (a symmetric, continuous beam with

thil'd-point loa.ds 111. the center span) ~ the "Jrie1d moment is exceeded

at supports and the ratio between the moments at support Mc and at

the centerline Me is no longer given by the elastic "analysis 0 WithLi

..~I

an assumed value of Mb the centerline moment is given by

ME ~ Mb~ PL2 (8)-"8

and the ~ = diagram can be plotted as shown using an 14....11 curve liRe

i~go 4. Due to sJIDIIletrJ', the rotation ot the beam axis at th1J (~)

as determined by integration of the ~ - diagram should be zero. This

bou.ndary condlt.ion will most likely not be satisfied by the first

trial assumption £01' Mbo The zel"()o.. lil1'(~ of the moment diagram 'must

then be shH'tf;H.1 1n the ppoper direc tion p the n.ew ¢ ." diagram plotted gI .

its areas cOlnpu1;ed a.nd s. second ehec.k on the resulting rotation at

the centerline.obtainedo The method is theoretically~atiafactorY

but the amount of nUmerical work involved is pl"ohibi tive for prac-

tical applications o

In p~lnclple the above approach is comraon to the various methods c

The methods differ only in the extent; to which simplifying assumptions

are made in order to reduce the amo~~t of work involved g

205B o 22 11/24/52 -13

I.:'

::

(a) Numerical !f.2!~gra~ .2f _~ ~tua! !:! .£urve

This method :1s outlined above and gives the e:2mct deflection in""

so-rar as the M-¢ diagram used is correct o In as-delivered specimens.

however. residual stresses and stress concentrations will cause earlier

yield and delay "the attainment of the predicted "plastlc hinga·moment"

as indicated in FigQ 40 In comparison with this effect the simplify­

ing assumptions employed in the following methods Introducerelatl~ely

small errors o

1he actual stress distribution across and along the beam as 1110

ustrated for a cantilever in Figo 6 (a) forms the basis for this

"exact" methodo

(b) MatEematic!11.!l..~f£ratlo£2!.. lslealized Mca~ Curve

This method isdu6 to MI'o W.Ro Wefskopf and is described by him

in the Appendix of this papero Mathematical integration of the M-~

curve is made possible by the idealization shown in Figo 7 0 The re­

sulting assumption of stress distributions in a beam is shown in Flgo

6 (b) 0

The effect of neglecting the short curved portion of the M-¢

diagram near the yield point (Figo7) 1s to slightly reduce the pre=

dieted a~rlectionao This error becoMes larger for sections having

la.rge relative fla.nge thic1:nessj) espocially if a considerable length

of the beam is under cons tant moment.. Neglecting the ,small tl"langles

in stress diagram 1 Flg~ 6 (b)~ also reduces the cQlculate~ deflec~

tions g but in significant amounts only for deep I~sections and rec~

tangular seetions o

(e) ...I!_-_A_rs_·a_ ~£C!

CoB. Yang21 suggested the further simplifioation of neglect-­

ing the spread of the plastic zones along the length ot the beam o

The equivalent assumption ·for the stress~strain and M-¢relationshlpsI .

205B4 22 11/24/52 -14c>

are shown 1n F1g~ 8 0 In a beam with a moment gradient as shown 1n

Fig o 6 (c) the l(lngth of' the pel"'fectly plastic region will conse­

quently be zeroo The cu~vatUl~e in the remaining two regions is re~

Iated to the momFQ.t by

fils ~

in the elastic ~egion up to Mp '

d lii5 M...B Z !)U _\____(, I

(9)

(10)

(11)

,.0;...,

in the strain hardening region ror moments exceeding Mp (F1So 8)0

In a region \11' constant moment ~ the curvatiurs will have a value

be tween the 11m!ts•

~ ~f6~ ~- BZ~ , -U I

and is detern",:'ned by the boundary cond1 tions"

The ef'fect of neglecting the spread of th~ plastic zone along

a member with .l moment gl'adient is to give somewhat sma.ller predicted

deflections~ This additional simplffying assumption thus adds to the

inaccunacin', of the prev;!,ous method, but tho (deviations, are small as

compared 'j:) the influencc:> of residual streBEl' and stress concentration

as ind1-c3~ed on the M-tl diagram" Flgo 4 0 These assumpti.ons consider=

ably sho"ten the numel~lca1 "'Jork» e.s is shown by the following ex=

ample :'ien compared with the Appendix.

G ),18 ider an 8WF40 b0~:.m ova r three spo.n.s7 ~, 14 0 , e.n.d 7' I wi tb.

conccr[;rated loads P at the thlrd··polnts of' the centrtl.l span (It'ig o

ga) 0 The maximum def'lect1on (which occurs at thecenterl1ne) will

be c/lculatede The material properties and the geometrical proper~

ties of the 8WF40 section as deter-mined by tests5 are shown 1n

From the elastic analysis of' the structure we have

Me'~ =Mb ~ ~ (12)

205B.22 11 /2;;,J'~'';JI _.( ......... ~15-

'.

where Me and Mb a1."e respectively the rilOh'1ents at centerline and supportQ

Elastic behavior 11 assumed until these mOnlents reach the :full plastic·

hinge value Mp (Figo 9b)g corresponding to the load

p ~. 6Me ~ = 6Mb ~ ~ ~ 53 0 5 kips (13)L L L

The curvature e.~ any point 1s proportional to the o~dlnate o£ the moment

diagram, Fig. 90 0 T.he beam axis at the center remains horizontal due

to symmetrYi I.nd the center deflection equals the moment. about the

S1lpport b 0' the sh~ded area of,,",'

ifc == 1 ~l l) L = ~ Q 2Mp LE!", 2~ i '2 -s--

the ¢."diagramp

o ,Lg_]' g.jj 19

~~t'"6'111 0 (14)

A £urthc,f' lncrease in the loads P will cause yield both at

.center and S f9POL'>t o Strain hardening will commence at th,e supportsI'

but the cen:.ar 'ioment will not :tncrease until the cur·vature is in""

creased ~ ~ 1!()5e £old( Table I. ) 0 Thus for loads 9 say

WY)kips (Fig- ~c )Me ~ 1497 ino ko

Mb ~ PL ~ Me ~ 1583 in. k o

"""'3

P gj 55

The points where the moment reaches ~ E 1491 ino kip on either

side of the supports are easily d$t~~1nad (Figo 90):

~ = 1497 Q 84 = 79 0 44 In om.m%2 =Mn =1497 ~ 27 0 22 1no, ~

... vv

Using EqsQ (9) to (11) or !Olga 8 the cOl"responding curvatm-s dia=

grams Fig 9 9d g can be plottedo The characteristic rotations are:

~p = ~ ~ 344 x lO-&rad/i~

918 l':l ~ "" BZ ;:: 3944 x 10-6 rad/in01-"'-

¢b = Mb =, BZ ~ 1583 ~ 1132 ~ 4870 x lO~~rad/in01 630 x 147"""" .

The curvature -0 in the center region is of a magnitude between

~p and¢s corresponding to the horizontal part of the M=¢ diagram g

Fig o 8 0 Its value must be dete~ined using the condition o~ zero

205B o 22 11/n /l, /,.:;:>t:; ~I <>.~

slope of' the beam axis at the centerline g ~05 0 0 In this example

only the al~ea H of the curvature diagram Flgo 9d depends on ~o~

which then in the following computation is determined to give @Cos 0 0

AD =1/2Bs "'"CD =1/2

'AREA OJ? 16= BOMER ARM MOMENT~ TO "a" ABOUT "a"

x 79 0 44x Oo000344~ = 0 0 0137 x 52 0 96 5 = 0 .. 7254 0 56x Oo0039441i11 = 0 .. 0180 x 8107~-@ "" 1~471

x 4., 56x 0 I) 000926ili = 0:;.;o~O;..;O;..;;2;;.:1~--=x;;,.-....,;;;8;.;;;2;.:;0..;;;4;.;;;8__S!__=_0~.. 1..7:-:3___'7 2 0 369

!;MOMENT ARMTO "bit

' ...

Dlill =1/2 x 1056x 0 .. 000926 = = 0 0 0007 x 0052 lIfil = 0 0000E:l'!l 1 0 56 x 0 ..003944 m 0 0 0062 x 0 .. 78 :: .,,; 0 0 005F= =1/2 x 27 0 22 x 0 .. 000344 8 = 000047 x 10 0 63 :iii: "" 0 0 050G~ +1/2 x 27 0 22 x 0 0 000344 ~ + 000047 x 46 0 '93 := + 0 0221Hm + 28 0 00 x 0 0 001254 g + 0 0 0351 x 70,,00 :g + 2 0 457

~c ~ 0 So m 20 62 In

It is seen that the magnitude of the curvature of' the yielded

center portion of the beam g consistent with the condition ~ ~ O~ Is

round to be

The centerline deflection for P ~ 55 kips then is 6 ~ 2 0 62 ina ore

1/64 or the span lengtho It follows that strain hardening in the

centar portion corresponding to:,

~c @ 0 0 003944 rad/in

wIll not take plaoe while the defleotions are tolerable and often will

never be reached due to earlier oollapse caused by local'compression

f~ange buckling ..

Additional points on the load-defleotion curve may be obtained

in the same manner, but the relationship is very nearly linear above

the yield, point until strain hardening would commencse> ,The curve is

plotted in Fig o 10 1n comparison with predictions by numerical inte~

gration and the simple plastic theoryo The result from these three

methods raIl clQse together which indicate that the simplifications

of the theoretical M-¢ curve g Figo 4 9 are or small oonsequence o

205Bo22

Howaver2 all tlu"ea CUl"ves give smaller def.lections than those

obtained expe~1mentally,(5)~AS discussed earlier, the discrepancy

is dUG to the tact that the ~Leor~t1aa1 M-~ relationship derived

trom the st~ess""stra1n curve fail to give precise results for as..,

delivered membsrso

Muoh closer agreement ,with the experimental deflections are

obtained uhen there is no portion of the beam under constant moment

as shmin for'a cantlleve~ in Flgo 110 In such a case a large portion

of the peam is actually in tho tranmitton (ele.st;:i.cqplastic) range

which is influenced by, the assumptions D19..deo This t'igure, taken

fr~mRefo 19 9 shows very good ag~eement with tests ~xcept in the

early part of the plastic regiono At this load level the effect

of residual stresses and stress concentration mllst be expected to

'., be most pI"onouncado'

(d) The S1mPke Plastic Theo~I

The simple plastic theory neglects the str;ain hardening etfeat»

but considers t~e ~pread of t~e plasti~ zpnes along the msmber~ Figo

6 (d) ,shows the assumed stress dist~lbutlon in a cantilever· a.,ocord~

to this theory~ The calc~ation of dGflactiona i~ furthe~ s~nplifled

over the p~av1ous methods o

Using ag~in thIS example of Figo 9~ the loed and deflection at

and .(14) with My instead of Mp :inlti.a.1 yield are obtained by Eqse (IS).Py :Ii 8M,! :;; 47G 6 kips

L ', 2 .

Sy ~ ~ ~c= Og76

When' the full plastic hinge moment 1s reache4~

P ==~ =: 53 e 5 kips

and deflections inereaae without limit since no f~ther resistance

1s offered at the yielding sectlonso

Intermediato' po3.nts on the load';'de.f'l@ction ourve may be found,

~ bY' a nume~1ca.1 nl€d;;hod simUar to that used for the previous caso

exo'ept that the:) transfe1" :Crom elasttc to plast'.c behe.vionI- 1s grad..

ualas sbo1-m bY,the dotted lines1n Fig o 9 (d) 0 The reaul t1ng ot.trVe

is shown in F1go 10 and is hardly distinguishable in the early plastic

'reglon,f~om t~at determinod by th~ mor$ exact nume~lcal integration

methodo Figo 11 shows tho smue result fo~ a cmlt11everQ The neglect

ot st:z.-ain htwden:i.:ng ~ea1l1ts In So horizontal line at P !S Ppo

In order to demonstrate this method in ~eater detail the come

putatlon of, the center' deflection of" a. bull t·",in beam with thil-d...point

load~ng will be ShCrlno 'The rosulting load~daf1ect1onc~ve is given

in Figo 12 in comparison with G,q,8l?irnente.l values and. predictions by

other mathods o

Initial yield, takas place when the largest, elastic moment IJ at

the .f~ed ends, reaches Ioiy =,1334 inOlko (Table I) 0 T!1@ loads arG then

,p mJ J! ~ =: 35 0 8 kipsY,2L -----

" ' 2" ,Hmndb~@i: 'Voo:~!)' ': ' bY'= !'~ ~ ~~ 8 0 653 tm 2..~;>'..9 1116

Next, deta~1.ne 'tho point on 'the load<ndafiection Olll"ve COITes=

ponding to yielded .t"1anges at tho fixed ends (Figo ld) 8

rip "'l~H.? m 1497 1n"k~ (Tabla I),

l'Met a '8' P1L .." Mp := SSP1 ... 149'7

Her~~" the unknown value PI is det~rmil'!ed by 'the cOlldition of zero,

~esultant change in slopo f~am th~ rixed end to the beam cente~.

This condition is most easily sat~s~ied by a tria1=and~e~or pro­

cedureg asstn111ng a value for M{ 0 Afte:v, ~xceed1ng the yield load

(M~ =667 inoko) this moment is inc~eas1ng at a more rapid rata than

~~Fo ThUS, r1~ == 850 1n~ko -1s assumed in the first tlflla1 9 giving theJ

curvature...diag19am belowo The cu.rovatw:-, prC)duc:1ng yield tm-ough the

'flanges is f6p ' = _}1_. pJ"'tf' • 3504eiO~ rad/ino, h=2w cI '

2058.,22 11/24,,5 2

1497 ill.l<.."'- 1:;3'tin.k.

" "07 "'-""'-

" ~

.'

Assuming the area' (a) to be trapezoidal the total rotation from

support to centerl ine 121 a

~··area (a) I 00 5 0 (354+30"") 3.9 o 10=6 u 12900 10a6 rad.

¢ aroa (b) : 0.5 0 S07 0 31.8 e 10.6 ,. 4880 41 10.6 rad o

f6 al'ea (0) I - 0.5 ~ 195 " 20.3 .6 6l'adoo 10 -~1980010@

..61lka5469°10GG6 rat;lof6 area (d) Z .. 195 " 2800 o 10

-1370.10.6 l'~~= ~ 0

Thus the cente~l1ne moment is smsl1sIJ than as sumed above, end the

new value r.lt :: 750 inekr;> is tried" The .corresponding cordinates

are s~mm ulthout parentheses on the above pJ - d1agramo

f6 area (a)s 0.5 G (354+307) 4.0 & 10-6 =1320·iO~ rado

f6 al'ea (ell - 005 •'~ area (~)z ....

,~.

11 &Ite& (b): 0 0 5. 307 0 33.0· 10-6 =5110010-6 redo

172 0 18.7. lO~6=-1610019-6 rado

172 0 28.0 41 10-6 :Q4820 0 10·6 ~ado

30.10;;S:~;;d:

The 81"2;'or 1s V8l7 small, and. 3~

M<\: = 750 + (850-'750) "3io;:io~752 in.k., bY' :stralght....111'lG

ints1'polation, 1s taken as the final valus. Then3'1 .

PI == t' ·(I·iF + Met ) ~.(1497+?52) :3 i O,2 kips.

1'he COIT8sponding center deflection Sa equals the moment ot the

~-d1agr~ above about the cente~11n01

205B,,22

Ur.2J1A"JL~~.(a) 94 0 10eO

~~~~1~~'

820~7

Q!tlectlon InC)

0 0 008

0 0 352

0 0 101.

6809

3402

14eO

.~... '.:.~' .. 5110.10....u

-1610 0 10..6

(b)

(c)

(d)

",,0 0 055

. ,.,0 0 00''7,8, :II '0:34in&

Fr-om (P]" b,) the load...de.flec"cion CUI"Ve closely parallels that. ",' 3

or a simply SUPPolltsd beam~ .. $2:= 81 = ei6' lP2:rl121L ,

'.

.'

The daflec'1;1on1s then, .from the ctquatlon.. abov8,

c'Jj1. ,. 0, +~ (~l L~ .. 8, + ~.~.~. ¥:f ..1040.,34 oJ- 00148 0 ~o8oS53 ::: 0 0 71 in.

Finally. the caFFylng 'capaci"cy lSVG1"y nearly exhausted when yield

has penet~atad through the flanges also at the cente~ section which

oocurs, :roX' a load negligibly smal1al~ than ',.3~·1497 . .

Pp == 1:"2 Mp ~-,;rr- :;:1.5305 kips

The cUrvature between load points is now ~p = 3G4~lO&6 rad/in, while

the curvature at the fixed ends is larger and can be determined by

the continuity. condition of zelP-O I'sau.l tent change in slope from support

to beam centero From the ~, - diagram thuB determined the center deflect=" '.'

ion is found follOtd.ng the procedure that gavaSe a.bove $ At this poL-"t

the deflection, neglecting strain hs.1"dsn~g and ca:canary effects becomes

.J.ndeterr,ninate and the beam deflec·ts \-1ithout limit under constant loadsPp 0

..

205Bo22

(e) fl...~st~c Jl~~.ll~l.l2.~

Figo 6 (e) shows a stress distribution in which yielding 1s

assumed to be 1 L.vni ted to the cross section which 1"1.tDst: rea.ched initial

'. yleldo"\,

This assurnption diffex's only from the simplified method of

.'''' ~

mathematical integration, Flgo 6 ef! in,that strain hardening is neg...

lectedo ,The bem~ has elastic regions and localized plastic hinges

onlys 'and the rJI-$d relationship consis~s of the ,tt-TO stra:I.ght lines

~~ == EIfI in the elastic range and M =: I1p in the plastic rangeo

The member is assumed to behave ·elastically Lmtil a maximum

stressf.89 is reached (f is the shape factor)c' This is equiValent

to assuming that all of the material is concentrated in a 11..."16 at the

f'langso In the example Fig o 9, 'chs stress f.cS9 corresponds to a load

p= 6~p =5305 kips

and a deflectlonat .the centerline ot

8 == .~~ ,¥ft :: 0085 ina

(Eqso (13) and (14)} The sudden tOI~ation of a plastic hinge at that

load causes a sharp breale in the load-deflec·tion 'curvs,Figo 100

This simple approach was suggested in Reto3 g and, is 1l.lustrat~d

in Appendix C of that reference by the case oX a rixed-onded beam with

a concentrated load off centezoo This method ob'lrlates the necessity of

knowing' the r·1=$d ralationship abov0 the elastic :Nl.nge 9 and makes use ot

deflection formulas usually available in the design office and tabulated

in structural handbooks ..

Comparisons of·the predicted end experimental deflections fo~

a continuous beam and for a cantilever are given in Figso 10 and 11 s

respectively, which wereprevlously disoussedo

Flgo 12, shot~s a similar comp~lSon foI' an 8 WF 40 beam with

. fixed ends and thlrd~polnt loadlngo As in the case of the ,continuous

beams~ Fig~ ;LQp the di.fferences 1.-n ~p.epredictions :resulting f.'roll1 the

205B o 22 12/3/52

various meulods outlined above are smaller in the early plastic

range than the deviation f'rom the test results for the beam with

detail (a) (Fig o 12) at the supports ifhich was also used on the

tests in Figso 10 and 130 Detail (c) gives a stiffer beam with

deflections in close agl-eement with pradictionso The defl't>ctloru$. \

derived by mathematical integration are not shown~ but woufd fall

between the tt-ro upper curves o The plastic hinge method would give

a curve follO't·ring the simple plastic theo17 curve except for sharp

knees at the two curved regions o

Figo 139 finally~ compares computed and experlmanta15 detlections

for a simple beam under third~point loading o Strain hardening theoreti

cally does not begin until the cente~ defleotion reaches about 8 inches

and all methods give deflections falling on the curve ShOl~ for the

simple plastic theo~, the sha.rp knee of the plastic hinge method ex­

ceptedo In the tests yielding started at about one~third of. the cal~

oulated initial yield load" causL"'lg largeza deflections than those '

p:redlctedo

The comparison with tests~ Figu 108 13 9 indicates that hardly

any advantage 1s gained by using the 1110re refined and laborious

methods outlined above for predicting deflectionso The amount"of

wo:rk involved in a case like the continuous beam of Figo 9 by ·the

methods (a) to (e) iS g very roughly~ in the ratios 5: 4: 3: 2: 1 and

these ratios rapidly grow larger wi·~ more complicated problemso

The comparisons of Bligs Sl 10, 12:1 and 13 with experimental re­

sults are unfavorable due to ~~e large influence of residual stresses

over the length of the beams unde]! constant momento The agreement

in Figo ll·for the cantilever 1smora typical to!" most practical caseSo

205B,,22

•

.,

For ~nglllaering purposes the plastic hinge method p~obably

would be pre~erredo The othor methods belong in the ~esearch field

where greater accuracy is often r~qulred to explain experimental be­

havioro

The inclusion in the calculation of deflections of strain harden-

ineg ,although laborious, does not necessarily yield a truer answer

since earlier failure due to local buckling is probable in lnany caseso

In cases where strain hardening does occur but 1s neglected, the

estimates of both the ultimate load and the detlections will be on the

safe sideo

h 1:NFLUENCE OF END RESTRAI1!!

As a member' of a structural i'rama or part of acont1nuous beamg

a single b,eam span ms.y ha.ve any amount of end restraint betv.leen fully

fixed and simply supportedoTha distribution of the live load on the

span may also vary, and is usually sOlllewhere bE1/'(iueen un1to~ distribut­

ion over the whole span and concentrated center loadingo

In order to illustrate the effect of and restraint and load dis-

tributlon, a series of examples as shoi~ in Figo 14 is chosen and the

loadQdeflection curves computed by the simple plastic theo~o The end, '

restraint on the center span imposed by the OU1ier spans varies in these

e~amples from 10~ (:ruJ.l fixity) through 86 B 75 and 601' to the case of,

the simple spano

The frames simulated by the continuous beams are indicated in

Figo 140 Of ~has6o Frame (e) has been tested as such 0

Three loading conditions as shown in Fige 14 are 1nves~igated8

(1) A concentrated load 2P at cente~o

(2) Third-point concentrated loads Po

(3) Unifo~ load W= 2Po

An 8WF40 was used throughout these exampleso The section properties

are listed in Tnble Io

The load...def'lectlo:n ctwvas fo:J:.> tlHl fi'V'e degl'l>GSS of, end reErtraint;

are sho~m in Figo 15 for the case of a concant~ated load at the beam

cantero According to 'i:h~ s~.l11p10 pIa-otic thaol'79 the ultimata ce:rryco,

ing c,apacity ie 71~3 kips for any degreaof end ~est~Qint di~fe~9nt from. \ .

zero f, 0 In the latter casethe\ ul t:trnat~ load is reduced to one halt

of that valuo o Thus ti there appe[1:r's to be an illogical discontinuity in,

the value of the u1 tima.ta carr-ylng capacity Tt:'hen the end restraint is

increased from zsro o

Figo 15 shows g however9 that the ul'thaate load aocording to ithe

siinpla plo.sticthso17 should not be taken as a design criterion for

beams with maall end restraints o Very large detlectlons 9 which can­

not be toleratsd as a limit crite~lon for st~uctural usefulness9 will

take place before th:ta nul tlm~J;e .ioad" is t:1.tfe,~.n.edq Al so & local .flm-lgo

buckling '(·Jill l::l"}.l:.g ~:.bc'<.1t collapse bei'ore extreme dei'lsct1ons a~e reachedo

follOWing exa1"£l.ple Lrasums theSE; specifica~tiona fOl! plastic deslgn of continuou.9 beams 8 .-' .

Saf6 nJ:.O. ~cc,:Lc:r~r~1:;·j.~:.~ 1 .-:-r····~ -tis for the'deflection d th 17~ 1" n., ' ....,. g . un e1'" ,.8 _ :L, .. 0 •• Cr",d

C!o101'>ld.ng lead (',:J1 t :L">~,d by the load .fa~t~) must thera.fore be imp~sedorrl"s 1 0"" - ""'1' ,. . . ..~ ",- .• (.·.ct"'·GCi'.,.. ec·c:U~::l CUJ:!lves 1'010. the beams of Fig

o14 v11th. third....

point loading on. the center, sp;',n iJ.X"o· •snO't'll'n. in Fig o 16" and with tmii'or-m

1,oad on the center ap'.an in F.:tg o· 17'0' rm. ..ule diSCUssion above applies also

for thaseloading ensase For- small end restraints, the Q1timate load .

.will not be reached B...!."1.d the cori~espon.ding d61r:'le'c~ions~ -~ could not bo tol~

erated as the 'limit or struC~"~a1 u P 1II ....... · . seJa U ness o

Fo~ thepurposo of the.,

(a) Load Factor 1 loGSbased' em 110 :::: ~ (r'lp+M ) at the lasth

- _ ..~ . f.\ y-o" ?-nge to f"OIJ'J11 o This'·.PoS3-ibl1it;r Gtf

, suggested :tnan 'SaFl1ellll p:ape~o.;) ----,- ,

'( 1:» Maximum ~af2ectibn at full loadapb1tra~ily sel~ct;ad at

Beams t-d.th a concentrated load at t-l-}e cente~ and end restraint

varying froI'l 0 to 10~ of :ruity hs.va been analysed for these speol""

ficatlons o The resulting c~ying capacities are given in Table 1Io

1'010 end I'cstrnints la:ttger than 65 %" speoification (a) limits the·

working londo BelOto1 that value, specifioation (b) applies as is

shotm under "D1'1'lection at f'ull load" in Table IIo The eff'ect is to

aiva gradual reduotion inworklng load as the end restI'aint decreases

to zero (the simple beam) 0 The last tt-J'o columns of Table II afford a

. comparison with elastic design, theelastlc design working load being

tween as

!bUsoP1astlc design results in an increase in working load ~am

5% to 33% ove~ the elastic design. It should be noted, however, that

a p~t of this gain is due to the ~poice of a different basis for the

full load in the tHO design methods. On u strict comparative basis,

therefore, Oo5o(f-l)olO~ should be subtracted fI'om thos~ numb6I'sin

the last oolumn of Table II ~hlch ar$ determ~ed by Fult' giving a

gain of 0 - 2~ resulting .from plastic designo

Fig" 18 shot'1s the load-de.f1ectlon curves for these beams aoC:Ord­

ing to the plastio hinge method, and in Figo 19 are g1ven the s1niilm.o

curves tal! the case ot un1.fol"Dlily distributed 108do Tho upper limits

tor allowable working loads according to the plastic and elastio

analyses outlined above are indioated in the ti~es. The itJorking

loads reaultirig f:C'om "G1).c91aat;1~ snaly~is are eve~h.N \'lith1n 'theelasT;J.c

limlto ' This a point about which ~larc has bean considerable discussiono

The area betvleen tho lines showing the elastic and plllstlc working

loads indicates the economy resulting ~om the 1attero

A possi_le additional specification,

( c) Maximum deflection at working load* ,would decrease the t-Torking loa.ds as indicated 1..1'1 FieSt> 18 & 190 Since this

.;,.\

19

.al1Q ,'~:L$;£:tic dl':;tl~;::'ti~l1 ~:.C:~~l" \~\;);':II.d.r,-g l{'lld on1yo Roweve!', togo oUb.$Jl ~

of $tructure~8 like fraIDa~ 3ubja~t0d to 81d~ lo~ds~ it may be ne~e$e~·

~'k,""y to reve:rt to themo:re basic plastic design specification·· (b) a~

bovso

205Bo22 11/24/52

60 I~FLUENCE OE' LOAD DISTRIBUTIOb

A 'comparison between Figso 15~ 169 and 17 reveals how the dls~

tribution of a total load 2P influences the lo8d~deflectlops reletion=

ship for cortinuous beamso In the three figures the load is concen~

treted at the center D divided between the third=polnts D and uniformly

distributed, respectivelyo

Concentrating the load neer the beem center lowers the ultimate

carrying capacity and increases the deflectionso As en example ~ryme

chsracteri~tic values from Figso 15-17 are given in Table III

for the beam wi th 75% end res train t (LlIL2= j) 0 Twlos the ul time. te

concentrated center loed can be cBx'ried as uniformly distributed over

the ~xUe ~l~ and causes a deflection of only 2/3 of that of the con=

centra ted loado

For a certain load in the elastic rangeD say 2P ~ 50kD the center

deflection is increased from 0029 ino to 0050 ino by changing the load

distribution from uniformly distributed to full concentration at the

center polnto

205B,,22 11/24/52

SUMMARY--------

..,28-

(1) The plastic desi3~ of continuous beams and frames requires

the calculation of critical deflections at the ultimate:ioadD

and often at the working load alliij,t1J 0

(2) Several methods for predicting the load."deflectloln relation­

ship of continuous beams are outlined o These methods ere

b~sed on the moment."curvature relationship derived from the

measured stress~strain relationship. and differ only in the

number of assumptions made to simplify the computation"

(3) The simplest approachg the plastic hinge methodg is most

suitable for engineering Qpplicationso The additioIlal work

involved in the theoretically more "exact" methods is not

justified by the accuracy gained except for research p~~~pose8~

(4) Tests of as~dellvered beams with welded details at supports

and load points give larger deflections than predictedo The

discrepancy probably is due to residuel stresses and stress

concentratlonso Beams with regions of constant moment show

the largest discrepanciea (Flgso 10, 12" 13) D while s canti,;",

lever gives good agreement with the predicted deflections

(5) When the end restraint is small the ultimate load p~edicted

by the simple plastic theory is accompanied with excessive

deflections (Figs 0 1St) 169 17) 0 The simple' plastic theory

must therefore be combined with a limitation on the deflec~

tions at the ultimate loado

205Bo22 1J./24/52

(6) A comparison of allowable load and resulting deflections

of continuoue beems analysed by elastic end plastic theory

(Fig~o 18 and 19) shows en increase in working load by the

latter method of 0 to 29%0 depending upon the end restraint

and load distribution,>

(7) The effect of the load distribution may be demo~strated by

the fact that the ultimate load of a uniformly loaded beam

1s twice that which can be applied concentrated at the cantero

The larger capacity obtained by distributing the load 1s

accompanied by reduced deflections ( Table III )c

•

This report is part of a current study of WELD~D

COHTIlIUOUS FRAr~BS AnD TIlliIR :COdPOlr.c;ITTS being conducted at

Lehigh University in the Fr~tz'Engineer1ngLaboratoryo This

research is sponsored jo1ntl'Y by the Department of -the navy

and the trfelding Research Council th.rough the Lehigh Project

'Subcommit"tee of 'its structural steel Committeeo

The authors wish to', acknowledge the assistance of

W~lliam. Spraragen, Director of the trfelding Research CouncUJ

LaMotte Grover, ~hai~aan of the structural Steel Committee}

and To Ro Higgins, Chairman' ot the Lehigh Project Subcommit~

teeo The t-lork is performed ~der the general guidance of

Protessor William Jo Eney, Di»ector of Fritz Engineering

Laboxaatoryo

The valuable contrU)ut:ton contained in the Appendix

by Wo fI o \velsJoo,~~bi Consultin$' Engineer and l-lember or the

Lehigh Project Subcommittee, ~s greatly appre~iatedo

205B 0 22 11/24/52

9 • RE1<~RENCES

-31-

Reports in the Lehigh Serle~ on Welded Continuous Frames and

Their Components

1. Luxion~ W.W., and Johnston, B.G.Progress Report No. 1PLASTIC BEHAVIOR OF WIDE FLANGE BEAMSWelding Journal, l'Tovember 1948, po 5388

20 Beedle, L.S •• Ready, JoR., and Johnston, BoG.Pr~'eess Report No. 2TESTS OF COLUMNS UNDER COMBINED THRUST AND MOMENTSESA Proceedings, Vol. VIII, No.1, December 1950, po109

3. Yang, C.H., Beedle, L.S., and Johnston, B.GoProgress Report No. 3PLASTIC DESIGN AND THE DEFORMATION OF STRUCTURESWelding Journal, July 1951, po 3488

40 Topractsog1ou, A.Ao, Beedle, LoS., and Johnston, BoGoProgress Report No. 4CONNECTIONS FOR WELDED CONTINUOUS PORTAL FRAMESPart I - Test Results and Requirements ror ConnectionsWelding Journal, July 1951, p. 3598Pa~t II - Theoretical Analysis or st~aight KneesWelding Journal, August 1951, po 397sPart III - Discussion of Test Results and, ConclusionsWelding Journal, November 1952" po 5438

50 Yang, C.H.. Beedle, L.S., and Johnston, B.GoProgress Report No. 5RESIDUAL STRESS' AND THE YIELD STRENGTH OF STEEL BEAMSWelding Journal, April 1952, p. 2058

6. Ketter, R.L., Beedle, L.S., and Johnston, B.G~

Progress Report No.6COLUMN STRENGTH UNDER COMBINED BENmNG AND THRUS'rPublication scheduled in Welding Journal

7. Ruzek, J.M., Knudsen, K.E., John~ton, E.Re, and Beedle, L.S.Progress Report No.7.WELDED PORTAL FRAMES TESTED TO COLLAPSEPublication scheduled in Proceedings, SESA

8. Johnston, B.Go J Yang, C.'H., and Beedle, L.S.Progress Report No. 8AN EVALUATION OF PLASTIC ANALYSIS AS APPLIED TOSTRUCTURAL DESIGNMimeographed September a, 1952

205B.22 11/24/52

othel'» Reports....... .ne......-ul'..

. -32-

9 0 Baker, JoF., m1d Heyman,JeTESTS ON HINIATURE PORTAL FRAlmSB1~1tish t'lelding Research AsS{) eiation.Research Report Nco FEe 1/17, August 1949

100 Van den B~oeksJ.AoTHEORY OF LIHIT DESIGNJohn \vlley & Sons, Inc .. , New Yorlr, 1948

110 Hrenn!koff, AoTHEORY OF INELASTIC BENDING· \vITH RBFEREUCE TO LII'lIT DESIGNTranso ASCE. Vol. 113, 1948, po 213-268

12 0 Nadal, AoPLASTICITYHcGI'al-1 HUl Book Coo, Inoo, USl"1 York 1931

130 Roderick, JoWo and Phillips, IoH$THE CARRYING CAPACITY OF MILD STEEL BEA}~S

British Welding Rese~h Association,· .Rese~ch Report Noo FE. 1/14, Ftabruary, 1949

140 Roderick, JQWO and Heyman, J.EXTENSION OF THE SD1PLE PLASTIC THEORY TO TAKE ACCOUNTOF THE STRAIN HARDEIHNG RAlilGEBritish Welding Research Association,Research Report Noo FEe 1/15, March, 1949

15 0 ft.rJt!~~'i!:k- ;'l" 11 y ,

THE ~ELATIONSnIP BETWEEN LOAD AltD DEFLECTION FOR APARTIAT~Y PLASTIC ROLLED STEELJ01~T

British Welding Research ASsociation,Research Report Noo F~o 1/24j; March 1951

16 0 St~ss1,F. and Kollbrunne~~C.F•. BEITRAG ZUIJI TRAGLASTVERFAHREN

Bautechnik9 Vo1~ 13, 1935, po 264

1'7 0 S:vznonds, PoS o and Neat~ .. BeG"THE INTERPRETATION OF FAILURE LOADS IN THE PLA~TIC THEORY

OF CONTINUOUS J3EAliIS AND FRll.l1ESJourn.al' of the· Aeronau"li1cal Sciences,Vole 19, noo 1 9 January 1952, po 15-22

. .

180 Symonds, PeSo .A REVIEW OF }mTHODS FOR TfIE PLASTIC ANALYSIS OF RIGID FRM1ES

OF DUCTILE METAL .Brown University, All-So/SS, May 1950

19 0 Symonds. PeSoWELDED CONTINUOUS FRANES: PLASTIC DESIGN MID THE DEFORMATION

OF STRUCTURESDiscussion ot Hero 3Welding JOUI'nal, Vol. XVII, No o 1, 1952, P e 33s..36s

205B022 11/24/52

',11 200 T1moshanko j SaTHEORY OF ELASTIC STABILITYMCGraw Hill Book Co., Ine~, N~YoCoD 1 0 Edit, 1936,po450 51STRE1IGTH OF MATERIALS V61 It II'D. Van Nos·t~and Company, Inc." lI.YoCo, 2 0 Edo• 1941, p. 363-379

210 Yang, CoH.THE PLASTIC BEHAVIOR OF CONTINUOUS BEAMSPhoDo Dissertation, Lehigh Un1versity~ 1951

205B.22 11/24/52

10, NOMENCL!TURE

A c~oss section area

b inange't.,1dth

B strain hardening or:r-sat3 f·1=BZ + CI~

C strain hardening modulus

E modulus ot elasticity

l' shape factor ih depth 01' section

I moment 01' inertia. 01' o~oss section

s section modulus 21*Ii

t 1feb thickness

w flange thickness

x dIstance along undefo~ed beam axis

Yo distanoe tromnautral axis to nearest fibre in yield

"18 distance from neutral uIa·to nearest fibre in strain

haI'den1ng

Z plastic modulus ot cross seetioD

..

ZE

Zp

dJoyc

$60..

86F

Ow

plastic modulus of elastic part of cross section

plastio modulus or plastic part of cross section

unit no~al stress

lowe~ yield point stress

unit normal strain

curvature of bent member

8lo~~ of deflected beam axis

deflection

def1ect1o~ at full load

deflection at working load

205Bo22 11/24/52

110 TABLE§ M!E .-F...IG....,U..RE.......S

205B 22 11/24/52

TABLE I Q PROPERTIES OF 8WF40 SECTION 5.,....,.. .Matel'ia1 Pro;g~r~:tes W!!suredl

YOung's Modulus or Elasticity

Yle1dPoint (lowe~)

Strain Hardening Off-set (Fig~ 8)

Strain Hardening Modulus (Figo 8)

E =2906 x 106 pai

6r 3'7?60 psi

B =28560 psi

C == 630 psi

Geometrical P:roperties of 8WF40 Seotion (Maasuredl

h =8032 !no

t a 0 0 370 1n~

A I: 11 0 66 ina

S = 35034 1ziSZ = 39065 1rt5

..

I =14'7 0 0 in4..

f • 1 0 122

b D 8 0 06 ino

W a 0 0 552 1n~

Cross Seotion Area

Flange tr/idth

Flange Thickness .

Depth

Web Thiokness

Seotion Modulus (strong axis)

Plastio 140duluB (st~ong axis)

Moment of Inertia (strong axis)

Shape Factor

Section Properties of 8WF40 sect;on (derived from above! ,

Yield 140ment My • 1334 in~ kipso

Plastic Hinge Moment . Mp := 1497 !no kipso

CurvatUl'a at Initial Yield ~y =~ ::I 307 x lO...arad/~ ,

91p = to~y ;: 344 x 1006 rad/in

CurvatUre at Which Strain ~s =; ~p .. BZ = 3944 x 10cu6rad/inof '" .

rJf.ft~ Q 9 0 653 iho

~-rrr- .. 90710· inc

205~o22 11/24/52

TABLE II o HORKUTG LOADS AnD THEORETICAL DEFLECTIONS

OF CONTINUOUS ~1S DESIGNED BY ELASTIC ~

PLASTIC THEORIES

!Zp " _ Svf=40

"* 4Leo ::=.)41 ..~ L, ..1

END PLASTIC ELASTIC INCREASE IN WORK-R65TRAIWl' DESIGN DESIGN ING LOAD (Plastic

over ela~t1c)

Working Def'lectlon Deflection Working, .

load at Working at tull load2P load load

~ i; kips x L2 X,La kips %

100 0 40.6 1/730 1/410 36.6 5

90 0017 42 0 0 1/540 1/250 3400 17"

80 Oe38 42 0 0 1/440 1/170 3200 24" , ,

70 0 0 64 42 0 0 1/370 1/120 2906 30

65 0.61 42.0 1/340 1/100 28.5 32"

60 1.00 40.8 1/330 1/100 27 0 5 33

50 1050 ~6.1 l/~~O '1/100 25~'1 29.. ..

40 2,,25 3201 jl~-~,)o ' 1/100 2400 25

30 3050 ;!9.1 ... #:'\'""0 1/100 22,8 22~~J~.~ ~.~ .

20 6.00 26.4 1/330 1/100 21.5 19

10 13.00 23.'2 1/340 1/100 2003 13

0 00 20 0 4 1/360 1/100 19.2 ' 6

205B 22 11/24/52

TABLE III o INFLUENCE OF THE DISTRIBUTION OF THE LOAD ON THE

STRENGTH AND DEFLECTION OF A CONTINUOUS 8WF40

BEAl-i \1fITH 75% END RESTRAI~TT AND 148 SPAN

LOJW ULTD-iATE -DEFLECTIOU--:-UEFLECTIOU AT 50DISTRIBUTION ~ ' OAPACITY' ULTIMATE LOAD KIPS TOTAL LOAD

- , -

Concentrated 7103k 1 0 21 ino 0 0 50 inat Center

Th.1Fd-Po1ntlOOo9kConcentrated 0 0 85 ino I 0 0 40 ino

Loads-

.. ..

Unitornll"! 142~6k 0081 ino 0 029 inoDistributed1-

-2053 22 11/24/52 40

F,G. I. MOMEKT5 AND OEFLECT(ON~ OF CONTI NUOU5 BERt"I6

I B iI I <rI \-

p. \ P

t_~_.-;';:--2..2C­flI

I

i\

',.' I~~---i--+-...I.-_;.J....~- --...,( b)

(a.)

•

(c) I­ZuJ1:oi:

Po

LO~D P

(d)a.

CENTER. DEFLE.CT/ON

205B 22 11/24/52

M

p

...--

4l

F/6.. 2. LOfto ~ DfFLE.CTlON RE.LATlON,5H-' P IN THe. ELASTIC RAI"t6-E.

60 r---i;-'-:-;--+-,-~:---+-:---i---,:---t--r-..,.I .I I 5tr1itl Haroen; !9 Ral1~ ,.

50 ···· ··· l ···..l ·····.. t·· ·..·~r ..·e.·:tf~~~ .. ·:..~ -1....1"'-- .,--_ .

_ 40 P~~!j~~n9~ ).... .. \ ·· i·: .. ···.. · l· ·~ ..· ·· t· · 1 .~ :..--' j ! ! ill

*' 30 ·'f--l-l--\---·lt··t···\·····~ ,0 EJo~tic Rrrlgc -1 ..·..· ·..·,· · 1 ·..·..1· 1·..·.. '1'· .

~ 'E ' , : 1 l' I j

fO ·~·· ·~~··· ..·:· ·!..··· ····.j. .. ·····..····..··t· ······..1· ··· ..· i..····· ·· l..·· ·· ..····I···..·..·· , .! ! j i I I i l~ ~ i t ~ 1-_.1;,.;_....J.~--,.L,,~-~

o 0 5 10 /5 2f) eli 30 3S 4tJ ItS soSTRfllH ... 10 3

F,c,.. 3. MEASUR.ED AND IOEflL.ll.E.O 'StRESS - STR.A/N Dlli6RAM

z--

co

........".~ ..... ...... ,~ ."

...

M

F!6. 5, MOf'rlENT AND RorftTtON DtA6RAr-t5 FOR A PLASTfGALL Y

BENT. BEBM

"." 1

I"H\TlYi:NATICAl

It\TEbRAr'OH :;-;_~'r=-="-=_.~=~~=--::=-~- --;'".._~,. to "__~

(b)

@

f

@

I4>- ~REfI

® 0 @ 0-) ®

\L£z:pcf'j % J~" "r-u::rHOP (C)

f. !!J :Ai d!j f'~3

I ~ CD .@ ®SlT1PL.E Pi.A5'r1(. p ---,

Cd) 'cP cf J'rftFOR.Y

Ii:.. ~ 69 °9G 0 @

PU\5T1c "kNOt(e) c-P 1_._..-

ttETA()J)

p E 6'9 f: 6",

Fi&.6. 5TRE5S DI5TRI8lJTION5 IN BEAMS flBOvE. raE. ELASTIC RAri6Ei~5GUf1ED . iN TttE. ME.Tt'tOD5 DI5ClJSSED

-----------------,p~tiG Roflge

M .. Hp -(~)~4-t

. -..-~ .5train tbrdel?irg Range

__ -- ,.., ~ ez. ... Cl4'

-------

Elo5ti" Rttl1ge.~ M=E.ltp "

---_._--

...... .J

CURVATURE. c:P

FI6.7. IOE.ALIZEor-I-d> WRY-E. FOR' t{fiTHEMflTfCRL

INTEGRftTIQN M~r~()D

.,

'~'r,r;n trj,).~: ',,, ~ ..I .t__ 1 l. ~,{ "I

50 I--~-'--------;---;---r--~r-----,I ;; i I \

4 f~9.,~1~;~;~~J: .. i··~l.. ···, j5traifl f!(Irdem"g ROfl9t- ;

i 1 " _~ d +8 + C€.i l ...!.: "" .. , .... "1" r.... ." .. \. .. 1" r !.. IEhsfic. l1a'fige! ! i i j

~ .... -.jd.cell················· t··· ···+··+···········r············· ...

!~ 1:1

.·····-1············1··· ··1 j ····t·······:····r ···1

o 10 20 '0 403TRA:n :It 10:3

200°1 t l ! i IPlastic ~ange I· I !

.~ 1500 I ...,."'",..·..1..M·;=-tJf-~ ..*'· !. -' i ·:·it·f(i/n....i~;d~n;(J~;1~ei i! i !i - ~'1fS. 8Z tut/> 1 J... i! i l" • : •

ii, 'j 't ; i1000 · · ~· · l·"· · ··· ~· .. ,, .. · · ); , " j ,,, .

~ . Efd5+i, Ra~nP 1 j 1 i 1% i ,'j- i i ! l !IIJ ~M=E.ltA Iii'; 1

~ 5: ····_-r-···r-··t·······+·············!·I···I····o 2 4 ,6 8

CUl~VATln:;, Rt.d1ans X 10 Per Inch

FIG.8. JTH~S..UTnAIU AN~} Leltt RU.ATIOl~JIIIl\3 ASdUL.:,J;IDIN fd A1tF...A I·1ETITOD

,,.

'.'

205B 22 11/24/52

(a.)

(b)

(C)

(dJ

"45

LUI'"va rure5for p. 55 It.

F/6.9. MOMENT AMD CuRVATUR.E D,AGRAMS FOR CoNTJNUOU~ BfAt'1

(<1" ~Re.A M E.Tl'tOO)

..

120 ~----,-.--------....-----~-----,-c------;------:------:-~i, j

: . ~,.. ~.... --J:--., ..:..=::...:'.:." -"-r-... ---,J---,k~:_--~j-_·-"------l-=:-=J,=--=--~----=-:--1------ ..CaIc.tJlot,d Imh I Y,d ~ ~ i' i

_---Pi Pi ,C 8w ~o.15: is 'A" 6

L 7' ~~~/4-_'_~ 7' .1

--<>-- .:x.perimento!!J ' i ! ----Predic.ted, NUlYlfr;£Of lofeqartioo I

I i I :: ~eD~~~~~ Io 1L----+-----+----~----::!----.:.:.::.:::---:-~:...--7---'..,-+..5;::-----74

o OS 1.0 I.S 2:0 2.5 3.0 v .0

20

100

...0..<\l

Cl 60§

C.ENTER OEFI..ECTION, INCHt6

F16. 10. loAD - DEFLEC.TION RELATIONSHIP OF A CoNTlN()O()S8r:~M

( .'

0.4- 0.8 I.e. ':6 2.0 c.4 2.B 3.2

fMo DEFLEc.nON, INCHES

Fro. II. lOA D ... OI:FLEGT tON RELATIONSHIP CANTI LEo" ER"9

OF' R .

oo

2J) .-----..:---",.......;.--r----~_:__---__:_-------'---------- .....

J

' ! . 'Ii b \ I __ -.L------~.I I . --ft="--.- ._--=-.~i-----~=:-- .==----. - --;:::..':=-' ,

Ii.. ..~ ";"~'-, - --_. - ~.=-- 3, ---~:t==: ---- --r- -- . iy:;. :-_7: ~::z;;:; -=-=::--~ = ----:-:7-4-- .. . ~.. . . . . . : .......~..

l¥'" .--+ - -j ..... ··1·· ...... · "',' .' ... '" Ii *'J ... .,..........-:-. '1"" ••. ,. '" ! 1 i

16 ·_··_..·--_·_···--·i·-.._·····....·_·_- ..·¥:.------+·--·---·---t--·-..·..·-·---·---t·-···---·-----..--i---·-·-----·-·..-·-..-~--_·_-_·--··-·· ....! //! I I j ji! V § I . j iI ! i I J! iI i I T I II I I i l

....................................- ..._.!............... .. ..·..·..··-....··......·l.._·--·....··-·-·-·..•..·....·..·~·-····· . ·..·i..·....·....·..·-··-......·-..·--·--~·-·-·_ ....·_··-..-..··....-·-I .! iii

! t, ~ ~

I !! I I; i 1 ~ ~. ! I - . - . - KIIHE.R"AL. II'4TU,RllTIOM 1 1.: : 1

j ..l!! i I.'f'.-.- -_._-.- -- _ _.•._ i........... I"1at1ER."~1. INT£6RRTION, +__._._._ _ ~ ,_ _. _._._.j 1 i CJsjnn Centro' flCtll>1 M-t C.II....e i j. . I .." . .! ! ! HA11t!.t1ATIGftl. II'lTf6Rf1TIOK ! jI I! i i

- - - - -' <p-ARF-A MF.THOC

___ • '. _r" SIMPLE Pl.~rl(. THEOR.V

- - Ejl.pERJI"1Er\T/TL

! I I I

'03.5E;S

IM-CHES

I:!J z. ()CENTER DEFLECTION ')

/.'0

120 r------.,..-------:------:-----~--------....__---_,_---___.! I ! .. --i- .~i , ... ....,.L. ... --- I. __ "" --- =i ~ ... ---- i ~·o" ......-, ii ~·--'··"i .-- .' -~. . i i ~

I f ....···1 e ~ ! i~. - ..... ----~ ._.:..i i- , :-

I /'/,w ~ . i ~ ~ i i . i

IOO---'--l·rA~I-------l-------·-t----".----.-----i:·-y~--+---·-·I---~-,--,+----- ..·,·I,-----l----·--l·--,·'I

. 11 i ! ! i ~ ·l~ ~ i I Iiii 1 ! I I ! I_+ _ ~ _....... .. . }... . ····..· ·..······..· l..··· ·· · ··..··..· ·'I..·..·..···..·· ·· · •· •..····r.._· _..·..·..·..·.._ ·t ·..·· ··· · ·.. ··· ······ ..ill ~ ;:

I i I Jup/#f DefrJi', ! , ii . i p P i

40 - L.. .J............. . 1. ~~) t !T=~' I_~-l_! 8 w 4b .._..~_ .._.._._•._._..-

I I Ile) l:;}l? !. 14' .1 1,1

! . I! I i I

20 ..- ··_....·..·.._..·..·....·r..··--....···..··_...._....-r-_··__........·..-T··.. -()- ~:~~~~a~umeriCdI Intet}ration _..., ..._-_....._--

i I ~ - ." - .. f/i-nrt:!4 f1itliOd.· II ! i II 6imple Pla5 ftc.. l1JeD(9 i

! iii,'·I : !

80

~~

....60a..

<\J

. .. ,i

,

- 01

Ii

2.0

, i~ Ji i. i· I! t!

i .l··· .~.;:.:-».--.+ - - --1- - - +- - - -+ ~'I ·c -/ .; i = ~

50 C;iwiQW~Id-f.;-7- [-iO- - -:t- -- -i-i--

~ ! i! ... __-+i L . P PJ.

-j'. .- i -r k + 14'=-=; 8 W'40 -c-----.,f::"",::.:·..--.·--·...·.·....·..--·--·

i.;. I::. II. ~~.~ .~

- --j.-- ----..- ~---- - -i"-'.--Ei)-~.... E"perimen tal

I I ! Pr-edicted, SimpfG Pla~tic D7~rg ,

iii" ,. P/a~t;, Hinge Hethod:.i.:o !- -i.' ..,i-' -"- .;...- ~-----"-----~----_ ___:"

o as 1.0 1.5 20 2'-5 3.0 3.5 4..0

60

--\ (-~

'\....

. r '

CENrER DE FLEJ::..rlON , INCHES

Fi6.t3. LOAD-DEFLECTION. RElATIDN5HIP, SIMPlV SUPPOR.TED BeRYl

205B 22 11/24/52 .

~_. -.;",. ..

BEA~: .EXAMPLE SIPJ.1JLATED FRAME END RESTRAINT OFCENTER SPAN

(a)

~100 ~

(b) l i -,~.[ =11lizs i.F4 86 %

~~ 16&' J.s~ .'iTf'

( c)! t t

.[ 1,( K A A 15 %\.K+ 168" .. ~ 84" ~.

( d) 4 k A.,

k ~ :& 60 %~

16&" + 1{>.8" "I. 166 " t

"I

( e) sJ &1- f.,Im tI. rtr o %

~168'

~

-60

CENTRAL SPAN LOADINGS :

'------------- :....- ,.--_-...J

FlGo 140 ILLUSTnATIVE ;EXAU>LES OF EFFECT ON MAXtlL1J1! BEAM.

DEFLECTIONS OF END RESTRAINT AND LOAD. DISTRIBTifTION

"0 ,f '

60

,~----

i.1

f-----1----ji ---.. -'.!ji

Ii

...L -L._ _ .~ ~

"----+-~-+---+---+---!l.-~---.,

!' !~~~'\:P=~ Varl/l'(}r.J E(}d Restrain f !. '--1-'.'-'i- , ;; t ~) frrJ~ 100 % ({";ced ended) I! i"" 8:~ .1 to 0% (6impltj wfpDrmJ) 1

I i j 1 Io ~::=...--~-~-...L-__l___~!_:__-+!---;+I;:;----f;--J~--7iQ--:;o 0.2 0,4 ~6 0,'8 lo 1.2. {4 1.'6 ',8 ,,".0

80 --------:---~---,---r------'!-:---__r--_r_--_r-~

eo...

CENTER DEFLECTION, INGHE5

F;6./5. INFWENGE. OF END RE.5TRHIN.T ON BEA~IO.EFLECTIOr{5

..... ._-

.'

"

, j

Nog

I

',8

._~ .._ - - - .~

I

I ;I ~: :

1,6

Vorqifltj EoctRestroi()t' from100% to 0 % ....r....·..·..·_··......·....·....··

~~.........~~==-_+""':::;,.....s;;:. J,~!Lcl.. 51J~@_--t'71' """""'---~..

IiI

I-r--------+.-.--..-..~_.-.-.-.--.-- .

~ ~~ ~! ~i ~r-------·..t-......·····......··-·-··l--··..·..···-·- ... , .

0.6 0:8 I~o 1,2 1.4

CENTER. DEFLECTION, fNCHES

0:4o.c.oo

/Zl> r----.----;----:--;-----r----r-------y---r--r---r--,

80U)~

~

cL(\.1

Qa:3-l':3:....~

Fib. 16. INfWEliCE OF END REGTRRINT ()N BE~M _DE FLEC nONS.

.. .'..

120

140

o O~---L_--+------+-..;.;......-~--~---*----;~---;":;;-----;,-";,8;---;;Z. '.IOZ 0.6 0:8 1.0 I.Z .1.4 '.6

CENTER.. O€FLfGTlON ') , INGttE5

'60 '---"'---~---;------r-----r-, ----;-----r-----r-----r---,j;

~

I+-

~)a.:~

....~

0rJ:C'-l 40

Frtr','i7. f'I'\tdiEN~E" OF . £1iD 'RE6TRAIN.T, OM.. BE"AfVJ DEFLeCTiONS-. .,'. , "'-,---..,..,-.---.;...~--- '------

roomotvro

"

/-81.<;'/.2.

."

/,00.2

'.

o

10

80 ;------;----------~----:------;-':,.;.:----~---:-------....,-

.l.;!_--~...__"..-__;--'-'-'--'---~-~-. \ 1

o

4-0 ·· ········_········· ·1..· --~.=. l~_.._.. ) ·····+..··..· ·..··· · ·t· · _· · ··..· l ···· · ,O'~..: · t· ·..··· _+ ···..· ··.._... on I I

70

60

"

CC;VT~Q. D~FLbCTlON, lNcl·n;s

FI,G, i8. 'INFlUbNCb OF 6.lVD Qt:STOA\NT O~ [Sw\M DbFLbCT\OlJ6

'.<.

i

-t--t-~----~-----, ,

I 1

. O~ I,

I 1 8 L.1 I i sjOO

.__.._.~.....t---'''''-'9-::--·-·f--..·--_·t···_..·_·...·..·····..··...··..·-t..._····..·...···_·_..··-t·.._·······......·············...·.! VO'!fiYl!J E.nd R~tJtmint ! 1

• i frem 100 'Ill ((il'-d elida} i iPLFl';)flc. 01::51" N I to b'II (,simpllJ 4UPPOrtedJ. I I

. ~ limltUl b~ SF" L/r~ 'vi:: pL !!-t·__·_··~;_ ....·..·_·r···_··_·.._--_·_.._.._·r··,~\ ···T--··....__··..·......-1··--·_··········....__..g.,. -= r360 1 ! 8yF4() , i 1

• I I i L 14' !!I I ! I" I!j t. Ii! !; I i I ~I I ' i i",.--r------r---l---T-l-------i---r-----r-----·

().e d. 4- 0.6· 0:8 1.0 ,:2. it,. 1.'6 f8 e~o

CE.I'lTE.R. DEFLE:CnON INCHE.S

o

160.------;----~---:---,-I --"---'---~---r---r----,

!I,I

100

120

80f/)a.-~

-~ 60a~Q....

. fl6'. 19. IMFLUENC.E. of END REStR.AINT ON BEAM DE.FL~nON5

..

205B o 22

.. ,.:. .J;':

11/24/52 .

APPENDIX

FLroctTRE OF I SECTIONS ABOVE THF. ELASTIC RANGE,

Walter H. Weiskopf

205B.22 11/24/52 67

Int~od'Uct10!1

For I-sections in flexure in the elastic range thed'C) tv(

familiar expression d~z ~EJ[ applies, and 1t~ integrations pro-

v1.de 'the slope~ the deflection cUI've~ and the soiutions of in­

TlUmberoble problems. In this appendix corresponding expressions

,are developed for I-sections in' the plost,.c end strain hardening

ranges and methods of integrating them are 1nd1cated.

Fig. 1 shows the elastic, plastio and strain hard~ning

~anges of a spectmen in tension~ In flexure, however~ the situa­

tion is more complioated. As load on e beam 1s inoreased~ some ot

the fibres d1stant from the neutral ex18 end vJhere the bending

moment is great enter the plastic range. There are then two por­

tions to the beam. one fully elastio and one in whicll the fibres

~ far from the n~utral axis are plastic. As the load is fu~ther

increased. some of the,fibres enter the strain-harden1ng range and

there are then ·three portions o~ beam~ one still elDstic. one in

wh1ch some of the fibres are plastic, ond one in wh1ch Borne of the

extreme f1bres are in the stra1n-ha~lening range. The plast1c and

'strain-hardening sections will be analysed separately.

The Plastic, Range

The stress-strain diagram is idealized as shown in Figo2.

A~p~1ed,to an I-section in flexure the strain and stress diagrams

are shown in Figs. ~a and ~bo Fl"Om Figo 38 it 1s seen that the ro-

tat10n per unit length of beam 1s

(1)

~~,.

<-~-~

11/24/52 58

Taking moments about the neutral axis in Fig. 3b there is obtained

(2)

In this expression IE is the moment of inertia and ~ is the stati­

cal moment of the elastic portion of the cross s~ction and Z is the

statical moment of the entire cross section.

'~hen the value of y 1s such that thee1aetic portion at

the cross sectlon is entirely in the web)equat1on (2) becomes

M = q, lz--¥)In equation (3) t 1s the web thickness.

Eliminating y from equations (1) and (3)

¢ = ..0;. @,t ~. E (3(djZ- M (4)

In equation (4) the expression o;z 1s the li~iting value

of the bending moment in wh1ch all of the fibres are stressed to the

yield point as shown in Fig. 300 This value, sometimes called the

plast1c hinge value, is designated Mpo

Then equation (4) becomes

ria ..Qi../ Q;t 'Y' - E 3(MP-M) (5)

Th1s is the tundamental relation of M to'~ for the plastic range.~

In Fig .. , (4) -this relation is' plotted for an 8 WF 40. FromAorigin.

to poiI1..t,l the beam is elastic. From point 1 to po1nt 2, y is 1n

the flange orfll1et. To the right of point 2, y is in the web

and equr:r~ion (5) applieii. The portion 1 to 2 1s a small portion

of theusetul range. To simplify the mathematics equation (5) will

11/24/52

be extended to the left until it meets an extension of the elastic

range at point 4G ' The value of the resisting moment at point 4 9

Mi , can be found by treating equation (5) simultaneously with.the\Vl'

elastic expression ~.-= Eo!. ~

3 .1..:<, 3 03~1;:. - 3MrM~ -t- I 0; t = (.6)

. M1 can be found b~~ solving ~'Quation (5) 0 It 1s a function. .

of the crOBS section an~ .~ only and can be tabtaated for all beam

sizeso .Tho elastic. range will then be. ·taken from zero moment to'Mi. .

and the plastic range from ~:~1 'to Upe

Integratins the M...~ Curve!) (Pl'a'st1c Ran..E.!) • I'

\.

As'is u~uall'1 ass~ed in the tlleory of flexu:re ~~z. is

tsk~n equ~l ~o ¢ and equation (5) becomes

This expres~10n must be integrated along the length of the, beam 1n

the piastic range 0 For this integration M is a function of Jt.. and

therema~nder of the right hand'side of equation (7) is constant o

The integrations have been perfor.med for three cases: 'Fig. (5),

the bending moment!) M, 1s constant; Figo (6) M follow,s, fl _2!trs1gb.t

line v(lriation (as bet~een concentrated loads); Fig. ('7) M follows"

a parabolic diagram (as for a uniform load) 0 In ,these' ~~presslons,

.Q. is the length of th'e plastic portion which is genezoslly only a

smell part ot the total length of the be~o

The e~press1ons first eiva the slope.~ , and··the deflec...d~ "

tlon, ~ , at any point in the lengtho . The constants of' ',,In,tegrat1on

have been determined so as to g1ve values in' terms of thG.bending

moments at the ends and the slopeQG(L~ and def1ection,~~~~ at thel

left endo . Nut (making?' =;.) the slope, o(R' and deflection, 'jR ,at

·205B.22 11/24/52

the right end are obtalnedo Finally express10ns for tX-R and 'jfl. ere

g1ven for the spec1al case, ML. =M p the t 1s the full plast-1c range.

The Strs1ne harden1ng Range

Above the plast1c rengethe stress-stra1n curve aga1n

r1ses as shown in Fig. 10 It is assumed that the d1agram follows

a stra1ght line in the strainohardening range given by the equat10n

d=C£-rB (8)

C and B are constants of the material, C belng analagous to E ln the

)elastic rangeo For the steel 8ho,~ in Flg. 1 p C equals 667 k1ps per

square lnch end B equals 28 klps per square incho F1g. 8 repI"Qsents

the straln and stress dlagrams for anI p'eam~ From the figure

:is -= oj-B xb.. (9)·o'-B 2-

When the dls·tance, 'j'H 1s entirely 1nthe web_ the reslsting moment

of the crosss8ct1on is

M=BZ. 1- (o'-B) 41" .L (O)-B)t: \l~ (10). h '3 J

In the r1ght hand rn~be~ of equation (10) the f1rst ter.m covers the

rectan.gle 0, 1, 4, 3: the second term the triangle 1, 5, 4 and the

th$rd term tbetrlang1e 1, 2, 6. Computations have shown this last

term to be very sma110 If it 1s neglected equation (10) becomes

M: BZ. + (d-B)&h

From F1g. 8 .

¢ =~ = Z(d-B)h 'he

SUbst1tuting d-:-B from equatIon (11) into (12)

(11)

(12)

,)

¢ =d2~_ = M-6ld~2. Ie

(13)

205B.22 11/2/1/52

a

In Fig. 9 the rel~tlon of M to ¢ for an 8 WF 40 1s again shown. This

is sim1lar to Fig. 4~ 'but extended 1nto the strain-hardening range.•

The solid line is the theoretical curve ond the dotted line is equa-

tion. (13).

I.t has. been found thnt failure takes place before the stress

gets very far into the strain~hardening range. Using the approxtma~

tion of equation (13) therefore appears well justified.

In integrating equation (13) M is a function of'X- and the

re~1ning sy.tilbols are constants. The 'integration~are so simple that

they can be perforrfl8d for each case without difficulty and will not,"

be given here~

Numerioal Example

To i11uetrate the method i twill be applied to a bean11/i th

f1xed ends and loads at the third points. Thie is ~est No o B2 and

is shown 1n Fig. 10 0 The fOllow1ngvalues ~re used:

B: 28 kstC=667 ksiIJ:: 168 in

8WF 40Ia146.3 1n4s. 35.5 in2.z- 39.8 in3t.~365 1h

From equation (6)

Solving this

0) 1s taken as 37~6 ka1which is an average v&lueof the test Coupon ResUlts,

M)/ • !S.5 x 31.6 :. 13'35 'I k.

Mp • 3"1. ca" 37.6 =- 15CO Ilk

3Ml - 3)( 1500 tvl~'-t ~.34 x 37.~')(. 3bS' '=' 0

moment Eb-.~

Then Z.;L :: MJ,.

In the elastic range the f'i.xed!~d moment 1s ZPL and the e,enter,

pi= ~ M.1. -=-2.L'ix 1433 = ~8.4"2)( 168

This 1s the l1mit1ng·toad fo~ the elast1c rangeo For

greater loads the ends of the beam enter the plastic rangeo The

205B o 22 11/24/52

bending moment diegrem for this condition is shown in Figo lOb.

The plastic section extends from the left and to the point where

the bending moment equals M~ 0 The coordinate of this point is. ,

~L. Treating the center section of the beam and the'el&s~ic

portion of the left section (whe::."'e ~ i8 greater than,. %LJa'il exo

pression for the slope cen be found by the usual elastlc theory.

Thls is

(14)

The fixed end moment MF1S of course not the same ss when

the beam ls entirely in the elastlc range. In equatlon (14) it 18

. unknowno

The slope where the elastlc portlon meets the plastlc

portlon ls found by puttlng 'Jt...;.., for ~ 1n equatlon (14) 0

~ "- ~...L l-MF" (:t..i.-b) + pi~- '~:)J (15)d~ EI Z \ '1 Z

The slope at this polnt worklng from the left end of

the beam through the plastic portion can be found by puttlng the

proper values ln equation (38)0 o(~ equals zero since the beam 18

fixed at the end.

For MR put M,L and for ML. put MF • For J. put~.:..

Then

O<R ::. ~/d,t "Ix ';l:... [/Mp- ~IL' -IMp- M;]E 3 MF ~tv1;,. (16)

Obvlously ~~ J.- of equation (15) must equal "'It of equ9-!' tlon (16). Equatlng glves expression tn whlch )::...t and MF arean

unknown 0 Fl'QID Pig 0 10

;t.;,. ::.~f- MiP (17)

205B 0 22 11/24/52

Substituting thi~ in. 'equation .(15) and (16) equating the

values, and simplifying there 1s obtained

fixed

,<I'f

tion is

is that which makes the fixed end moment. MF. equal to Mr. T1ii8

can be obtaineq. by making Mr equal to Mp in equation (18) and"

solving tor P. The resulting equation is a quadratic and its solu-

p. ~~P t..-tJ~+¥ t 2Io;l¥(M~~M"'Y (19)

For the numerical example in Fig. 10 equation (19) gJye~

p-= +1.0" and ''1:;;. -: '.63//. This is the greatest value P can he)is

before the fibres near the end or the beam, e~te,r the stra1.n..,harden­

ing range. -For this value of P the moment at the centerot the,

beam 115 800" ~ well wi thin the elastic range~,

For values of P grenter than 11.0 k the strain...hardening

range 1s entered and the condit10n is that shown in Fig. lac. The

bending moment at any point in the end th\i;l~ is of course

From the figure the following relations can be obtained:

~1..-::::~.p

~P 'C .lli..:.M.,P

~L ":f:k. p ~ Mp-M.c:.V>

(20)

(21)

(22)

(23)

.....,I.

The slope at any point on the elastic portion of the end third of

205B 0 22 11/24/52

(26) .

(28)

.

.'

the beam is again given by equation (14). ~e slope when ~ equals

-;kL can be found by p,uttlng ;:e...:. for ~ in this equet1on. Then

using the value of ~L as g1ven,in equation (21):

~ .i.. = -L [ M; - f\1t - Mf-L 1"..EJ.:.1 (24)d%- EI. 2P 2.. q ]

For the plast1c section equation (40) app11es. o(~ 1s g1ven by

equation (24) and .MR, is M,t 0 ;.. is Mp -ML g1ven by equat10np ,

(23).(I(~ '" - 2~ p~ I ,Jl1p-ML' + I rM~-ML _ Md. + PL·' (25)

e. 3 p EIt 2 P :2 "J JIn the s'tx-a1n-harc1en1ng sectlon, from equation (20)

4 ..~. trl-6~ "".L [M F - P;t. -8Z]d~~ c.t c1 .

Integratlng

~ :::..l-.[MF;l::.. - ~ - BZ~ .... k]~ CI .. z (27)

From the ~ond,.tlon that k -- 0 when ;2:."0 1t 1s seen that the constantd"r-

of 1nte~at1on. K, equals zero.

Putting %p fo'r ~ 1n equation (2'7) g1ves the. slope where

the J)ls$"tlc and strll1n-hardenlng sect10ns meet lt At th1s po1nt.

from (27 land (22 )

~=' [M~-M~-26Z(MF-Mp~d~ 2CIP J

Tone slopes as g1ven by equations (28) and (25) must be the same •

Equating them gives an expression that can be solved forN1 Fe It ls

(~L .L)~ (~_\ c ... E. n+'Cl

205B.22 11/24/52

..~This is a quadratic ond vJjlues of MF can be computed

without difficultyo F1g. 11 shows the end ond center moments for

Test Specimen B2 as measured. The plotting extends through the

elastlc. plastio and stra1n~hsrden1ng rangeso A curve of values S8

g1ven by the theory here presented is shown Marked "Theoretical".

Values were computed up to s .loodlng \vhlch mokes tJ.?;e center moment•enter the plastic range. Beyond this point the endmoment;s· woUld

turn upward ond the ~enter moment would flatten out. '

Bendlng About the Weak Ax181

For I sect10ns subject to flexure about the wea~ axls

all of the equations for the plastic range (Figso 5, 6~ and 7) oan

be used, if for the web thickness, t, twice the flang~ tPickne88

is used. These equations can also be used for reQtangulsr sections

1f the width of the beam ls used instead of the web thickness, ·t.

The straln-hardening equations do not apply a8 ~ccurately to the weak

ax18 or to rectangular shapes. As a matter of fact~ however, failure

about the weak ax1s or In rectangular shapes Is apt to take place

beto8:go1ng far Into the stralnc:>hardenlng range if at ell, so that

for these cases ~p could be considered the ult1mate bending

momentc!

wr-----r-----r-----r----~---_...------...;....;....-..----~..----..,----~---....,,-----....~

IiIi

I,!

!!

~~::t:;;;~~::t::~:: J !I ~

.......__ -... _._ ---.-i._ _·__ ~ _ ..I :, ,! ~

'! ;i ~

I I!. :: i fI . i i· I i...-r-..--._..f·---·~-+-_·_··_-t-·_·_ .._---;--··__..--1·_·_-_···-.__.i-- __._ .I . ! . i . I· I 1

I ! . i 1 'I' I I !r/'I I I j iii 1

j ._..__L_ .. L__J J__.J .__..L ..L_..__I. ._L.._J_. .!lO .... I I I I' iii ! i !I I' :,. I ' ,

i j I. i j i '1 i ! jIi! I i ~ . I ! j

... ~-j- ~-~-.- .-..-.---.~.----~~.--.---t--1.--- ..-J~--- ..-L-.T---L._.-~---l--.--.-! I! i!: I j!I I I IS~40 I I I I

.. -i ----r----~--.--~-~--~-+---~-----ll-·--r·-----I--~.-I Ii! ·1\ I I I I !i I ! j !!. !!Iii , 1 i I' i 1! ! ! I l j. I ~oll...- ..Il- -':-~_ _:_+-----'-_--~--='~~~_--~--_~-__~__-~__------=-'~ 0.005 ~ '""- Sh-a.in

.~

Fig. 1 TENSION STRESS - STRAIN CURVES

205B ~22 11/24/52

Fig.L

0::....

.

- -

t

• r

----1+----- -- -If--- 4--_ -~---

.'

r I

4. STRAIN b. STRES~ c. STRESS

, ,

.'

..

l~r-------~'-----'''''';'''----• j ~! il ,i i i~ i - tI .i 1- l

i ~,~ i :~.I r···· -

I . I ! ! , .... I Mp ·-'313 .! --,... ------~-~r------'---l-"------:-·~"-IK;r;;II1i:--,--l----"-c--'"--+'"---C--'-",-----J----,-

I.Mi.. =1253 I 1<1>:::'3~54-)(,ol .1 . I 'a'

I¢ Sl2..'I><I<51 ' ~~--- a.' I . I I .I.Z-OO ··········..·······..·····..........:·........···......t· ..·........·····..·..···..·..·:........··..·..·..·..··t........·......··..........·....i'·.... ~ ·....·i....·..·......·..·-....·..·..........·..·....·..·....···I-·..:..·....·_..·-..-....:...........·....-....·f..........--'-.·..----....---~-- ..··..·..·..·

I 1- ~ ! M" =,17.2. I. . i . !i : 'I"'! ! ,.I i .1 Ii ..""\ I iI I 1""- ;os2.ttl. ~ 10 I I I= i ! iIi

I... -,,,'"--~--~"'"'"'-"+-'---'~'--"-"-'-+--'-",:-,,--,:-,,:,:~,-+,--'"--'":-:--~,-~-,,-+-,,-:_-:,-,;""-'"----",I",-,--;,;·,'"'-,-""---,~,-,-'"i i ! i' I ;

.~ ,I I L I I"'.... I j ! i . I i·....,;)~ . 1 ., ~ , i i.~··:· .~ ...:.±.: ,.·..·· ..··..·..1..················ · ·..·····:·:·· ··{-·:· · ····:·..· ·· ·..··..·:· ·t·· ·..· ·..·..·· ·..·· ·,·..· ·:..·..·I..···..·: · ·: ~..· l·· :···:· ··..:..·;.:·;..·.' ·..·..·: t ~..: ~..

V ! 'j ~ i i !~ i ! Ii i 1

· ~~._---"..f--- -------j-----~,-c-----I------------I~"i~M)-----I-~-----~,,----[,,"---'E I i I : i r

~ ~-" '~:--~c-----71--;-~-,------:"-j-,-----,,---J~~-~~~~~-"-J------.------1,,-. . i I" • I! i : . ! i

. ! ! II . ! . ! I

.! '.. ' i ! I· 1 j! ! i I' j If i i . I !f iii i f

• 209---- ----··---t-:---:.:~.:.---~--'"--------I---~---,,--c-f-,,-------f,,-'"------~--i-",··,"','i ' I :- l i

iii I I !I j Iii i

oO;----~--~;--------;~:-:-._------_~~~------.........~I_-,;,.-,---~l:---- -ilLI_-

'.

¢ \n Radians ..~ 10-+

....,

11/24/52

..

•

·$lope D(~

def'te.ction 'Y~ ...-~....

Fig- 5

(az)

(3+)

~- ...

0(.L. l'

';it- Jl.. ~ Y

. .'lope

clt,flection

•

•

(35)

(31)

(41)

'.~

205B.22 1~/24/52

PARA60L.1C MOMENT DIAGRAM

,hear ' \,{ '):. .

slope 0<.",deflection ":I... """'---.....:;.;;...--..........c

v\. ,- M,-M", _ Wi..1 2.

VR ::. .M IS - M... ... JJJ.!J. .2

71

~ =.sZ..J'St'I . (4,2)d~ £. 3 'Mf-M"'-V"'''--~ft£·

a =G.-~£ [Sin-' w -i. =t v.... _ sin-' ,Vis:' "'] + o(.L. (+3).d~ E?>W {zW(Mr-M...)+V~I.' .p.W(t¥1p-ML.)t-V...:4

'Y :: at PCf;t.' (W:it. -rV~ sin-l W%-ry" .. - W;;I;+\lt. 9in' ~k 1E ..3W W /ZW(Mp-ML.)+V...a.' W /Z.W(Mp-:ML +vt'

+f'~M') - ?<.~ -l.~;; -flM};MJ j-r o<..~1'- + 'Yc (44)

,,' Fo... full len9th (x~.Jl)

0(.11 '" Sf/~t'[ :;in-'/~W{Mr"'!M_)+vt- 9In-'/.z.W(M~M_)+v.:l..-o<.... (i5)

'J - 0; /za:.t '~ :;in' VB. _.YL. sir'-' y- .R - E. SW [w J2W(Mp-M...)-+-V!:' W . I~W(Mf'-M ...)-+-V:'·

+llMy;M"Bi' _/4(M'M_2}oCcJ. + 'j~ (46)

~Fi-'J. 7

205B.22 11/24/52 72

.0(,. R::. 0; l2:K£[S\n- l ..Yn. _~] + (1(. '- (+7)

E [3W "~2-

. :y R :: OJ ~[.':1.B.. sin-''y&' - 1T"VR + [?(MP-M;;)'] + o<.~ Sl +)jl. (+8)E .,--m w Vt,. 2W J w

Fig, 7

. ".

i

20SBo-22

. d.

11/24/52

. £,:1

b.5TRA\N

, .,.. 51 jj

I

i~i I7

~ .. 't77'7 \j,I

-f'T- - -1 4

6

~

I,

II

e. STRESS

:

:

:

!

Ii

Ro.dianSin

1700 ..~_.~-..--- ..

\4QQ I-----(.;;~---·;;...'""'!"--'---------r----.-,;.-----"'T----.:...;..---.:...--,.----------..,..-----------,---,..-~-..:...:.----.,

I !,;" ,I /7

i I ,/~~II :{ ~(,~t- I /~Gl~I :,1 ~ iI! / II I / / l

---t-------t----------t----------t---- -----~-+---,----.----+---------! ! ! P\o.s-nc. i S+~aJn- J1t:l.\>"dgnt ~

j i i ! B 1l A. 8 ks\ ; i

! Ii! I I~ i j 1 C~2..1+ kosi! 1

\100 1.---II1o------~~------~-------_;i:--------~------__=~------~"_--o:__---_'o 10 YJ 0 40 50 60 70

F'9· 9

~

D-o-~.,-5.5t 1%00.-

....C

"£0

~

?,C5Bn22 11/24/52

p

'15

p I

, ,}tr-------~5~~~----16-S-I'-----'t'----..;.::2-8-"--....-jl---

.FixedEnd

d .

..

I %l.

b.

. I

M<t:-r--------f--'-

Ela,stic

I~

-;

S+r4lin--har -f"-~~~f---.=::":':::::':":"':'=-----------_'-:"":_---

205Bo22. 11/24/52aV!€A"T

76

~oo

I Fi9·11 Load-Moment:"CUrVeSI Te5t 12...!

·· ·· ···..·· ..······_·..······..;f··········..···..·..· : f··..·..···..··..···········..·· ··_..·r- ··· ···..· ··_·-····· ·r~ : If,"I : I I

I I II I°0~~---.......;,O-----.z.....O-----3;...O----~...i-O----~;O~----~60

Loa.d WI l~ip5

r~.,fr

--.:1' t ~-.~,

ATE DUE

"'.