J. T. ODEN Finite Denections of a Nonlinearly Elastic BarProfessor of Engineering Mechanics,

Research Institute,University of Alabama,

Huntsville, Ala. Mem. A5ME

S. B. CHILDSAssociate Profenor,

Department of Mechanical Engineering,University of Hauston,

Houston, Texas

The problem of finite deflectioll s of a tlonlil/early elastic bar is investigated as an extensienof the classical theory of the elastica to itlelUde material nonlinearities. A mometlt-curt'ature relation in the form of a hyperbolic tangent law is itltroduced te simulate thatof a class of elastoplastic materials. The problem of finite deflections of a clamped-endbar subjected to atl axial force is given special aUetttrotl, atld llumerical solutWllS to theresultitlg system of tlot/litlear differential equatiO/ls are obtaimd. Tables of results forIlarieliS l'alues of the parameters defining the material are provided and solutiol/s arecompared tvith those of the classical theory of the dastica.

IntroductionIN Tilt.: two centuries t.hat have elapsed since Euler

first examined the problem of the elastica, problems involvingfinite deflections of thin bars have attracted the attention ofnumerous investigators. An accollnt of several applicllt.ions ofthe theory of the elllStica with references to most of the previouswork can be found in the monograph by Frisch-Fay (1(.1 In thepresent paper we investigate a problem of finite deflcctions of abar constructed of a nonlinearly elastic materilll charal'lerized bya moment-curvature law similar to that exhibited by a class ofelastoplastic matcrials.

Consider a straight prismatic bar of length [. which under~oesfinite deflections and rotations but lllllall strains. AS8uming thatplane cross sections remain plane during the deformlltion, thelocal bending moment tit a typical secl.ion cun be exprc!l.~ed as II

function of the chnnge-in-curvllture nt tho section:

M = P (~) (I)

M

¢'d9/ds

lIere 0 is the slope of the bar, 8 is the llrc length along the de-formed axis, and AI is the bending moment. The specific formof F( ) depends, of course, on the Illnterial properties of the bar.

In the following, we assume that the moment-curvature rela-tion for the bar under consideration is of the form

Fig. 1 Nonlinear moment-curvature relation

= dl)/cls and difTerent.illting (2) with respect to s, we arrive, onsimplifying, at the governing system of nonlinear diffcrenlinlequations

where Do and DI arc characteristic flexural rigidities, II>; indicatedin Fig. 1, 'Po is a Inllterial constant corresponding to a chllracteristicchange-in-curvntlll'c, and

(3)

dois = sin 0

dOds = <p (5)

This relation is of a form similar to the one-dimensional stress-strain relation proposed by Prager (2] for an elastoplastic ma-terialllnd used by Havner [3] in his annlysis of nonstrain-hnrden-ing materinls.

Cantilevered Bar

d<p -p sin 0ds - Do{l - (V,<p - PI)2/(Do - D,)DDlPo21

It is clenr thnt in the C88e ])0 = D" (5)2 and (5).l'cduce to

d20 P- + - sinO = 0ds2 Do

(tl )

which we recognize as the basic differentilll equation governingthe comparable problem in the classical theory of the elasticn [41.

To facilitate the solution of these equntions, we introduce thedimensionless quantities

where Pcr = 7('2Do/4£1 is the classical Euler buckling load. In-troducing (7). into (Ij) and augmenting the system with the ad-ditional condition that the load parameter A is independent of z,we arrive at the following set of four first-order nonlinear dif-ferential equations,

E = D,/Do(7)

p = <Pol,

'Y = 1/(1 - E)p2A = P/P ..

Z = 8/1.u = v/I,

!If = -/,1) (4)

where I) is the transverse deflection of the bar. Noting that sin 0

As a special case of (2), we consider a c1amped-end columnloaded axially by a force P as indicated in Fig. 2. In this case

I Numbers in brackets designate References at end of paper.Contributed by the Applied Mechanica Division for publication

(without presentation) in the JOURNAL OF APPLIED MECHANICS.Discussion of this paper should be addressed to the Editorial De-

partment, ASl\lE, United Engineering Center, 345 EllSt 47th Street,New York, N. Y. 10017, and will be llccepted until April 20, 1970.DiscWlSionreceived lifter the closing date 'will be retumed. J\Innu-script received at ASillE Applied l\lechanics Division, August 20,1968. PaperNo. 69-APM-T.

48 / MAR CHI 970 Transactions of the AS M E

y

L

1A

xFig. 2 Nonlinearly elasllc clamped-end column

dudz = sin 0

1.0

Q9

Qa

O~

0 Q6u~~ Q50

~, Q4~Q3

02

QI

00

£'1.0 (claulcal elastica)

£:0.75

10 20 30 40 50 60 70 80 90 100

a (degrees)Fig. 3 pip., versus end slope <l< 'or rJ>ol"7 0.40

d'"dz

(11" /ol)X sin 01 - 'Y(1I"XuI4 - E"'P (8) 1.0

£'1.0 (classIcal elastica)

Numerical SolutionFor convenience, we denote u = g" 0 = g., '" = g., and X = g.

and rewrite (8) in the vector form

where a is the end slope as indicnt.ed in Fig. 2. Thus the endslope is to be specified and the load required to maintain a givenslope characterized by the unknowu load parameter X is com-puted. A similar procedure is usually used in the classical prob-lem of the elastica [4].

£"0.75

£" 0.25

€' 0.50

0.3

0.8

0.6

0.7

0.9

o.~ 0.5~o

~~ 0.4

(9)o0°

"'(0)

0(1 )

ou(O)

0(0)

~ =0dz

Solutions arc Hubject to the uoulldllry conditions

q' = I(q)

where q = Igl, g" g., g,l, q' = dqldz, and

II = sin g.

(1T'/4)g, sin g,r. = -1 - 'Y(1I'\jIq,/4 - Eg.)'

(10)

(11)

0.2

0.1

oo 10 20 30 40 50 60 70 80 90 100

We wish to solve (10) in an iterative manner. Following Beil-man and Kalaba. [5) and Kalaba [61, we use the Newton-Kant~ro-vich technique t<l establish the linear equations relating the nthand (n + 1) iterations:

a (degrees)Fig. 4 p/p.t versus end slope a lor <Pol = 0.05

(13)

initial condition for t.he flrst iteration is easily estimated fromthe known solutions of the classical elastica.

Equation (12) is linear with varying coefficients. It is solvedas a superposition of the two particular solutions

subject kl the initial conditions

(14)'n + J.{p - qnl

fn + In{r - qui"p'

(12)

Here

J(gn) = ()f(q)J()q q=qn

The vector q. is the solution of (10) subject to the illitial cOlldi-tionf? that made (12) satisfy (9) OIl the nth iteration. Tl~e

Journal of Applied Mechanics MARCH 1970 / 49

1.6

1.4

1.2

1.0

0.::; 0.8

Cl0.6

0.4

0.2

1.6

1.4

1.2

1.0

0.::; 0.811\ '--- €·0.75

li: O·l~0.4€·0.25

0.2

oo 10 20 30 40 50 60 70 80 90 100

a (degrees)

oo 10 20 30 40 50 60 70 80 90 100

a (degrees)

Fig. 5 p/p.r versus end slope a lor f/>oL= 0.40 Fig. 6 p/P.r versus end slop' a lor .pol 0.05

Table 1 01/00 = 0.75

• 0' l' 2' C' 6' S' 10' 20' 40' 60' SO' 100'

'lloL- .40

PIPel 1.000 1.000 .999 .995 .990 .9S3 .974 .92S •S57 .S25 .809 .800

x./L 1.000 1.000 I. 000 .999 .997 .995 .992 .969 .878 .734 .550 .337

Y./L .000 .011 .022 .0..4 .067 .089 .111 .222 .429 .601 .727 .797

Cl'oL- .20

P/Pcl 1.000 .999 .995 .9S3 .965 .945 .925 .857 .807 .789 .7S1 .776

x./L 1.000 1.000 1.000 .999 .997 .995 .992 .969 .S7S .736 .553 .341

Y./L .000 .011 .022 .045 .067 .089 .112 .223 .428 .599 .724 .795

CPoL = .10

PIPel 1.000 .995 .9S3 .945 .907 .S7S .S57 .S06 .779 .770 .766 .763

x./L 1.000 1.000 1.000 .999 .997 .995 .992 .969 .S79 .73S .556 .345

Ya/L .000 .011 .022 .045 .067 .090 .113 .222 .426 .596 .722 .793

cp.L •. 05

PIPel 1.000 .983 .945 .S7S .S41 .S20 .S06 .779 .765 .760 .758 .757

xa/L 1.000 1.000 1.000 .999 .997 .995 .992 .969 .S80 .740 .557 .347

Ya/L .000 .011 .022 .045 .06S .090 .112 .221 .424 .595 .721 .792

so that the (n + l)th solution of (12) is then

q. = (1 - (3)p + {3r

Here

and we have set

(3=

ResultsSolutions were obtained using the procedure described in the

previous section for IX = 1,2,4,6, 8, 10, 20, 40, ... , 100 deg nudfor various values of E and p. After a few iterations, (3 ap-proaches zero and the process is considered to have convcrged.In the cases considered, if iteration.~ were continued until (3 <10-', O(l) was < 10-1 in all instances. Integrations werc alsopcrformed for the case in which E = 1, P = ex> ('Y - 0), and re-sults wcre obtained which agreed with thosc of the classicalelastica.

For purposes of discussion it i8 convenient to compare themoment-curvature relation in Fig. 1 (lind equation (2» with thatof an elastoplastic material ill which no unloading is permitted.Then the parameter IpoL may be regarded as measure of the(18)

(17)

(15)

(16)

1

0IX

r(O) = 0

0.98>"(0)

p,(I)

rIO) - p.O)

>". + O.02{3X.

oIX

oX(.)

p(O)

50 / MAR CHI 970 Transactions or the AS M E

Table 2 DdDo = 0.50

, O· l' 2' 4' 6' 8' la' 20' 40' 60' BO· lOa'

ll'oL = .40'

PIPe1 1.000 .999 .996 .986 .969 .947 .920 .758 .544 .457 .413 .3SS

xa/L 1.000 1.000 .000 .999 .997 .995 .992 .96S .865 .703 .499 .269

Ya/L .000 .011 .022 .045 .061 .090 .112 .230 .45S .641 .766 .S26

ll'oL = .20

P(Pel 1.000 .996 .986 .947 .S90 .S24 .75S .543 .407 .360 .337 .324

"aiL 1.000 1.000 .000 .999 .991 .995 .992 .965 .864 .710 .515 .293

Ya/L .000 .011 .022 .045 .068 .092 .116 .239 .457 .631 .752 .S14

'PoL· .10

PIPe 1 1.000 .966 .947 .S24 .698 .606 .543 .406 .333 .307 .295 .289

xa/L 1.000 1.000 1. 000 .999 .997 .995 .991 .965 .869 .721 .531 .315

VaiL .000 .011 .022 .046 .07l .096 .121 .23S .447 .618 .739 .805

«'PeL II .OS

PIPe I 1.000 .947 .S24 .606 .499 .442 .406 .332 .293 .260 .273 .270

xa/L 1.000 1.000 1.000 .999 .997 .994 .991 .966 .873 .729 .543 .331

VaiL .000 .011 .023 .04S .073 .097 .120 .233 .436 .607 .731 .799

Table 3 DdDo = 0.25

Q 0' l' 2' 4' 6' 8' 10' 20' 40' 60' 80' lOa'

'PoL· .40

PIPel l.000 .999 .998 .991 .980 .965 .94S .S46 .708 .646 .615 .597

"aiL l.000 l.000 l.000 .999 .997 .995 .992 .969 .873 .724 .533 .315

VaiL .000 .011 .022 .044 .067 .089 .112 .225 .439 .615 .740 .S06

Cl'oL= .20

PIPel 1.000 .99S .991 .965 .928 .S87 .846 .707 .611 .577 .560 .551

xa/L 1.000 1.000 1.000 .999 .997 .995 .992 .96S .874 .728 .542 .327

Ya/L .000 .011 .022 .045 .067 .090 .114 .229 .437 .609 .732 .800

(PoL •. 10

PIPe 1 1. 000 .991 .965 .887 .S09 .750 .707 .610 .557 .539 .530 .526

xa/L 1. 000 1.000 1.000 .999 .997 .995 .992 .966 .S76 .733 .549 .337

VaiL .000 .011 .022 .045 .069 .092 .115 .22E .432 .602 .727 .796

'PoL = .05

PIPe1 1. 000 .965 .SS7 .749 .616 .636 .He .557 .529 .520 .516 .513

xa/L 1.000 1.000 1.000 .999 .997 .995 .992 .96E .87S .737 .554 .342

Ya/L .000 .011 .023 .046 .069 .092 .115 .22. .42S .59S .723 .794

relative "elastic zone" of the material lind E = Dt/Do is an illdica-tion of the relntive stiffness of the material after yielding. Thus,for a given Do, the smaller the value of cpoL the lIlore narrow theelastic portion of the M-<p curve characterizing the mnterial.Likewise, the smaller the value of E the softer the material afterinitial yielding. A value of E of unity, for example, might bem;ed Coran clastic-perfectly plastic material.

Figs. 3 and 4 indicate the variation of the nondimensionalloadPI POl, where Pol is the end force predicted by the classical theoryof the elastica, versus the end slope a for values of IpuLof 0.40and 0.05, respectively. For the rather wide elastic zone casetypified by <PoL= 0.40, it is seen that a reduction in the load re-quired to produce an end slope of 100 deg is over 60 percent forthe realistic case in which E ~ 0.25. For most mild steel bars,however, the value of cpoL ranges between 0.02-0.10. For 'PoL= 0.05, it is seen that only 27 percent of the load predicted by theclassical theory of the elastica is required to produce an a of 100deg. Other values of PIP.I versus a for various valucs of 'PoL

Journal of Applied Mechanics

and E are given in Tables 1-3.For infiniteRimal changes in curvature, e(iuation~ (5) reduce to

the familial' differential equation governing the stability of pris-matic columns. Thus the bar under consideration buckles at.the classical critical lond p" = 1f'Do/4L', where Do i:s the in-stantaneous flexural rigidity at cp = O. According to the classicalela:stica theory, an increase in axial load is required to increase thedeformation (end slope) of the bar so that the stiffness of the bal'increases with an increase ill end slope. A quite different be-havior is encountered in the present case, as is indicated by theplot of PIP., versus a given in Fig. 5. For cpoL = 0.40 it is seenthat a sudden reduction in load occurs immediately after thecolumn buckles. In cases in which the material is extremely stitTnfter "yieldillg" (that is, in those cases in which E = Dt/Do ~lJ.5 (say», the column stiffness apparently becomes zero at a"" 40 deg find the curve acquires a positive slope as a becomeslarger. At this point the behavior approaches asymptoticallythat predicted by the classical elastica for a bar with flexural

MARCH 1970 / 51

\I

rigidity D,. For /poL = 0.05 and E = 0.25, a caso not too dif-ferent from many mild steel columns, there is a sharp decrease inload after buckling and the slope remains negative for a largeportion of the range of values of a considered, Fig. 6. TIlls, ofcourse, is in agreement with exporimental data available on thepostbuckJing behavior of elastoplastic columns, e.g., (7]. It isnoted, however, that as long as E > 0, there is always a value of afor which the slope of the P/ Pct versus a curve will begin to in-crease and to approach asymptotically the curve correspondingto the postbuekling behavior predicted by classical elastica theoryfor an elastic column with flexural rigidity Dt.

References1 Frisch-Fay, R., Flexible Bars, Butterworth and Co. Ltd., Lon-

don, 1962.

2 Prager, W., "On Isotropic Materials With Continuous Transi-tion From Elastic to Plastic State," Proceedings, Fifth I nternatwnalCongrell8 of Applied Mechanics, 1939,pp. 234-237.

3 Havner, K. S., "On the Formulation and Iterative Solution ofSmall Strain Plasticity Problems," Quarterly of Applied MathemcUic8.Jan. 1966,pp.323-325.

4 Timoshenko, S. P., and Gere, J. Jl.I., Theory of Elalfii.c Stability,2nded., McGraw-Hili, NewYol'k, 1901,pp. 76-82.

5 Bellman, R. E .. and Kalaba, R. E., Quasilinearization and Non-linear Boundary- Value ProbtemlJ, American Elsevier Publishing Co.,New York, 1965.

6 Kalaba, R. E., "On Nonlinear Differential Equations. theMaximum Operation and Monotonic Convergence," Journal ofMathematiclJ and Mechanics, Vol. 8, 1959,pp. 519-574.

7 Horne, M. R., lind Merchant, W., The Stability of Frames,Pergamon Press, Oxford, 1965,p. 26.

\I

52 / MAR CHI 970

Reprinted from the Marcil 1970JOlmtal of Applied Mechanics

Transactions of the AS M E