Associate Professor, S B Childs

J T Oden

CiteSeerX

J. T. ODEN 
Professor of Engineering Mechanics, 
Research Institute, 
University of A labama , 
Hunfsville, A la , Mem. ASME 
S. B. CHILDS 
Associate Professor, 
Department of Mechanical Engineering, 
University of Houston, 
Houston, Texas 
Finite Deflections of a Nonlinearly Elastic Bar 
The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension 
of the classical theory of the elastica to include material nonlinearilies. A moment-
curvature relation in the form of a hyperbolic tangent law is introduced to simulate that 
of a class of elastoplastic materials. The problem of finite deflections of a clamped-end 
bar subjected to an axial force is given special attention, and numerical solutions to the 
resulting system of nonlinear differential equations are obtained. Tables of results for 
various values of the parameters defining the material are provided o)td solutions are 
compared with those of the classical theory of the elastica. 
Introduction 
IN THIS two centuries that, have elapsed since Euler 
first examined the problem of the elastica, problems involving 
finite deflections of thin bars have attracted the attention of 
numerous investigators. An account of several applications of 
the theory of the elastica with references to most of the previous 
work can be found in the monograph by Frisch-Fay [ l ] . 1 In the 
present paper we investigate a problem of finite deflections of a 
bar constructed of a nonlinearly elastic material characterized by 
a moment-curvature law similar to that exhibited by a class of 
elastoplastic materials. 
Consider a straight prismatic bar of length L which undergoes 
finite deflections and rotations but small strains. Assuming that 
plane cross sections remain plane during the deformation, the 
local bending moment at a typical section can be expressed as a 
function of the change-in-curvature at the section: 
M = F (1) 
Here 6 is the slope of the bar, s is the arc length along the de-
formed axis, and A! is the bending moment. The specific form 
of F( ) depends, of course, on the material properties of the bar. 
In the following, we assume that the moment-curvature rela-
tion for the bar under consideration is of the form 
M = (Do - Ateo tanh (<p/<fo) + A v (2) 
where D0 and A are characteristic flextiral rigidities, as indicated 
in Fig. 1, <P(, is a material constant corresponding to a characteristic 
change-in-curvature, and 
dd 
Js (3) 
This relation is of a form similar to the one-dimensional stress-
strain relation proposed by Prager [2] for an elastoplastic ma-
terial and used by Havner [',>] in his analysis of nonstrain-harden-
ing materials. 
Cantilevered Bar 
As a special case of (2), we consider a clamped-end column 
loaded axially by a force P as indicated in Fig. 2. In this case 
M = -Pv (4) 
where v is the transverse deflection of the bar. Noting that sin 8 
1 Numbers in brackets designate References at end of paper. 
Contributed by the Applied Mechanics Division for publication 
(without presentation) in the JOURNAL OF APPLIED MECHANICS. 
Discussion of this paper should be addressed to the Editorial De-
partment, ASME, United Engineering Center, 345 East 47th Street, 
New York, N. Y. 10017, and will be accepted until April 20, 1070. 
Discussion received after the closing date will be returned. Manu-
script received at ASME Applied Mechanics Division, August 20, 
1908. Paper No. 00-AI'M-T. 
Fig. 1 Nonlinear moment-curvature relation 
= dv/ds and differentiating (2) with respect to s, we arrive, on 
simplifying, at the governing system of nonlinear differential 
equations 
'"' = sin 6 
ds 
dd 
- = i? yi>) 
dip —P sin 8 
ds = Dojr-'(D^^Pvy/(D^^D7)D^ 
It is clear that in the case Do = D\, ('•>)•> and (5)3 reduce to 
cP6 P 
h - sin 0 = 0 (6) 
ds'- Do 
which we recognize as the basic differential equation governing 
the comparable problem in the classical theory of the elastica [4|. 
To facilitate the solution of these equations, we introduce the 
dimensionless quantities 
u = v/L s/L e — Di/Do p = <p;iL 
ip = L<p X = P/Pm 7 = 1/(1 - e)p'
2 
(7) 
where Pc r = 7r
2A/4/>2 is the classical Euler buckling load. In-
troducing (7) into (5) and augmenting the system with the ad-
ditional condition that the load parameter X is independent of z, 
we arrive at the following set of four first-order nonlinear dif-
ferential equations, 
48 / M A R C H 1970 Transactions of the ASME Copyright © 1970 by ASME
Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/30/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Fig. 2 NonMnearly elastic c lamped-end column 
du 
- = sin 
as 
df 
dz 
d\ 
(7T2/4)X sin 8 
1 — 7(7r'-\w/4 — txj/'f 
(8) 
= 0 
re-
solutions are subject to I ho boundary conditions 
-(,(()) = 0 ^(0) = 0 
0(0) = a 8(1) = 0° 
(!>) 
where a is the end slope as indicated in Fig. 2. Thus the end 
slope is to be specified and the load required to maintain a given 
slope characterized by the unknown load parameter X is com-
puted. A similar procedure is usually used in the classical prob-
lem of theelastica [4], 
Numerical Solution 
For convenience, we denote u = qt, 8 = q-., \p = q3, and X = f/i 
and rewrite (8) in the vector form 
q' = f(q) 
where q = {ql: q-,, q:l, qt\, q' = </q/<b, and 
/i = sin q-i 
(7r2/4)(/4 sin q-. 
f3 = 
1 — yiir-qiqi/4: — eq:i)
2 
h = ?, 
h = 0 
(10) 
( I D 
We wish to solve (10) in an iterative manner. Following Bell-
man and Kalaba [5] and Kalaba [61, we use the Newton-Kantoro-
vich technique to establish the linear equations relating the nth 
and (it -\- 1) iterations: 
Here 
q'„+ i = f(q„) + J ( q „ ) { q „ + l - q „ ! 
&f(q)" 
J (9.) = dq 
(12) 
(13) 
q=q» 
The vector q„ is the solution of (10) subject to the initial condi-
tions that made (12) satisfy (9) on the nth iteration. The 
a (degrees) 
Fig. 3 P/Pci versus end slope a- for <j>{tL = 0.40 
^=f.O (classical elastica) 
10 20 30 40 50 60 70 80 90 100 
a (degrees) 
Fig. 4 P/P0i versus end slope a for r/>ol = 0.05 
initial condition for the first iteration is easily estimated from 
the known solutions of the classical elastica. 
Equation (12) is linear with varying coefficients. I t is solved 
as a superposition of the two particular solutions 
p' = fn +
 J,.{p - q« 
' ' = f» + J«tr - q»i 
subject to the initial conditions 
(14) 
Journal of Applied Mechanics MARCH 1 9 7 0 / 49 
Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/30/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
O 10 20 30 40 50 60 70 80 90 100 
a (degrees) 
Fig. 5 P/PCT versus end slope a for 0cX — 0.40 
O 10 20 30 40 50 60 70 80 90 100 
a (degrees) 
Fig. 6 P/Pcr versus end slope a for <j>0l = 0.05 
Table 1 D,/D(, = 0.75 
a 
p / p c l 
x a / L 
y a / L 
p / p c l 
x a A 
Y a / L 
p / p c l 
X a / L 
y a /
L 
p / p c i 
X a /L 
Ya/L 
0° 
1 .000 
1.000 
. 000 
1.000 
1.000 
. 000 
1.000 
1.000 
.000 
1.000 
1.000 
. 000 
1° 
1 .000 
1 .000 
. 0 1 1 
. 9 9 9 
1 .000 
. 0 1 1 
.995 
1.000 
. 0 1 1 
. 983 
1 .000 
. 0 1 1 
2° 
.999 
1.000 
.022 
.995 
1.000 
. 022 
. 983 
1.000 
.022 
. 94 5 
1.000 
.022 
4 ° 
. 995 
.999 
. 0-.-4 
.983 
. 999 
.045 
.94 5 
.999 
.045 
.878 
. 999 
.04 5 
6 " 8 ° 1 0 ° 
cp„L = . 4 0 
. 9 9 0 
.997 
.067 
. 983 
.995 
.089 
.974 
. 992 
. 1 1 1 
cp0L = . 2 0 
. 965 
.997 
.067 
.945 
.995 
.089 
.925 
.992 
. 112 
CP„L = -10 
.907 
.997 
.067 
.87 8 
.995 
. 090 
.857 
. 992 
. 1 1 3 
cp0L = . 0 5 
. 8 4 1 
.997 
. 0 6 8 
.820 
.995 
.090 
.806 
. 992 
.112 
2 0 ° 
. 9 2 5 
.969 
.222 
.857 
. 969 
. 223 
.806 
. 969 
.222 
.779 
.969 
. 2 2 1 
4 0 ° 
. 857 
. 8 7 8 
. 429 
.807 
. 8 7 8 
.428 
. 779 
. 8 7 9 
.426 
.765 
. 880 
.424 
6 0 ° 
. 8 2 5 
.7 34 
. 6 0 1 
. 789 
.7 36 
. 599 
. 770 
. 7 3 8 
.596 
. 760 
. 7 4 0 
. 595 
8 0 ° 
. 809 
. 550 
.727 
. 7 8 1 
. 553 
.724 
.766 
.556 
.722 
.7 58 
.557 
. 7 2 1 
100° 
. 800 
. 337 
.797 
. 776 
. 341 
.795 
. 763 
. 345 
. 7 9 3 
.757 
. 347 
.792 
p(0) = 
0 
a 
jo 
r(0) = 
0 
! a 
0 
0.9SX(,o | 
so that the (n + l) lh solution of (12) is then 
q„ = (1 - /3)p + jSr 
Here 
p,> 
0 
/>,"> 
r,u> _ p,o> 
and we have set 
X(„+i, = X„ + 0.02/3X,, 
;lo) 
(16) 
(17) 
(IS) 
Results 
Solutions were obtained using the procedure described in the 
previous section tor a = 1, 2, 4, 0, 8, 10, 20, 40, . . ., 100 (leg and 
for various values of e and p. After a few iterations, /3 ap-
proaches zero and the process is considered to have converged. 
In the cases considered, if iterations were continued until /3 < 
t()"6, 9(1) was < 10~8 in all instances. Integrations were also 
performed for the case in which e = 1, p = m (y —»• 0), and re-
sults were obtained which agreed with those of the classical 
elastica. 
For purposes of discussion it is convenient to compare the 
moment-curvature relation in Fig. 1 (and equation (2)) with that. 
of an elastoplastic material in which no unloading is permitted. 
Then the parameter (ftlL may be regarded as measure of the 
50 / M A R C H 1970 Transactions of the ASME 
Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/30/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
Table 2 D,/D0 = 0.50 
a 
p / p c l 
y-a/ i-
Ya/L 
p / -°G 1 
>:-./L 
Va/'L 
5 / f c l 
x a / L 
y a /
L 
p / p c l 
x a / L 
y a / L 
0 ° 
1.000 
1.000 
.000 
1.000 
1.000 
.000 
1 .000 
1 .000 
.000 
1 .000 
1.000 
.000 
1 " 
. 999 
1.000 
.011 
.996 
1 .000 
. ' I I I 
. 986 
1.000 
. 0 1 1 
.947 
1.000 
. 0 1 1 
2 ° 
. 996 
1. 000 
.022 
.986 
1.000 
. 0 2 2 
.947 
1.000 
.022 
.824 
1.000 
. 0 2 3 
4° 
. 986 
.999 
.04 5 
.94 7 
. 999 
.04 5 
.824 
. 999 
.046 
.606 
. 999 
. 0 4 8 
6 ° 8 ° 1 0 ° 
¥ 0 L = . 4 0 
.96 9 
. 997 
.06 7 
.94 7 
. 995 
. 090 
. 920 
.992 
. 112 
cp„ L = . 20 
. 890 
. 997 
. 0 6 8 
.824 
. 905 
.092 
.758 
.992 
.116 
cp0 L = . 1 0 
. 6 9 8 
.997 
. 0 7 1 
. 606 
.995 
.096 
. 5 4 3 
. 9 9 1 
. 1 2 1 
tp0L = . 0 5 
. 4 9 9 
.997 
. 0 7 3 
.442 
.994 
.097 
.406 
. 9 9 1 
. 120 
2 0 ° 
. 7 5 8 
. 9 6 8 
. 230 
.54 3 
. 965 
.239 
.406 
.96 5 
. 2 3 8 
. 332 
. 966 
. 233 
4 0 ° 
. 544 
.865 
. 4 5 8 
.407 
. 864 
.457 
. 333 
.869 
.447 
. 293 
.873 
. 4 3 8 
6 0 ° 
.457 
. 7 0 3 
. 6 4 1 
. 360 
.710 
. 6 3 1 
.307 
. 7 2 1 
. 6 1 8 
. 2 8 0 
. 729 
.607 
8 0 ° 
. 413 
. 499 
.766 
. 3 37 
.515 
.752 
.295 
. 5 3 1 
.739 
. 273 
. 543 
. 7 3 1 
100° 
. 3 8 8 
. 269 
. 826 
.324 
. 293 
.814 
. 289 
. 315 
.805 
. 270 
. 3 3 1 
. 799 
Table 3 D,/D0 = 0.25 
a 
p / p c l 
*a /L 
y a /
L 
p / p c l 
* a / L 
y a /
L 
p / p c l 
X a /L 
y a /
L 
p / p c l 
*aA 
y a A 
0 ° 
1 .000 
1 .000 
.000 
1 .000 
1 .000 
.000 
1 .000 
1 .000 
.000 
1.000 
1.000 
.000 
1 ° 
. 999 
i .ooo 
. 0 1 1 
. 9 9 8 
1 .000 
. 0 1 1 
. 9 9 1 
1 .000 
. 0 1 1 
.965 
1 .000 
. 0 1 1 
2 ° 
. 998 
1 .000 
.022 
. 9 9 1 
1.000 
.022 
.965 
1 .000 
.022 
.887 
1.000 
.023 
4 ° 
. 9 9 1 
. 999 
.044 
.965 
. 999 
.045 
.887 
. 999 
.045 
. 749 
. 999 
.046 
6° 8° 1 0 ° 
if0L = . 4 0 
.980 
.997 
.067 
.965 
.995 
. 089 
. 9 4 8 
.992 
.112 
tp0L = . 2 0 
. 9 2 8 
.997 
.067 
.887 
.995 
. 0 9 0 
. 846 
.992 
. 1 1 4 
cp„L = . 1 0 
. 8 0 9 
.997 
. 069 
. 750 
.995 
.092 
.7 07 
. 992 
. 115 
¥ 0 L = . 0 5 
.676 
.997 
. 069 
.636 
. 9 9 5 
.092 
. 610 
.992 
. 115 
2 0 ° 
.846 
. 969 
. 2 2 5 
.707 
. 9 6 8 
. 2 2 9 
. 610 
. 9 6 8 
. 228 
.557 
. 9 6 8 
. 2 2 5 
4 0 ° 
. 7 0 8 
. 873 
. 439 
. 6 1 1 
.874 
.437 
.557 
.876 
. 432 
. 529 
.87 8 
. 4 2 8 
6 0 ° 
.646 
.724 
.615 
.577 
. 7 2 8 
. 6 0 9 
.539 
. 733 
.602 
. 520 
.737 
. 5 9 8 
8 0 ° 
. 615 
. 533 
. 740 
. 560 
.542 
. 7 3 2 
. 530 
.549 
.727 
.516 
. 554 
.723 
100° 
. 597 
. 315 
.806 
. 551 
. 327 
. 800 
.526 
. 337 
. 7 96 
. 513 
. 342 
.7 94 
relative "elastic zone" of the material and e = D\ 'D0 is an indica-
tion of the relative stiffness of the material after yielding. Thus, 
for a given I)tl, the smaller the value of <fuL the more narrow the 
elastic portion of the M-tf curve characterizing the material. 
Likewise, the smaller the value of e the softer the material after 
initial yielding. A value of e of unity, for example, might be 
used for an elastic-perfectly plastic material. 
Figs. :> and 4 indicate the variation of the nondimensional load 
/•" J'e\, where P^\ is the end force predicted by the classical theory 
of the elastica, versus the end slope a for values of ip0L of 0-40 
and O.Oo, respectively. For the rather wide elastic zone case 
typified by <fuL = 0.40, it is seen that a reduction in the load re-
quired to produce an end slope of 10(1 (leg is over 60 percent for 
the realistic case in which e < 0.25. For most mild steel bars, 
however, the value of if„L ranges between 0.02-0.10. For <f(,L 
= 0.05, it is seen that only 27 percent of the load predicted by the 
classical theory of the elastica is required to produce an a of 100 
deg. Other values of P Pc\ versus a for various values of <ft>L 
and t are given in Tables i -o . 
For infinitesimal changes in curvature, equations (5) reduce to 
the familiar differential equation governing the stability of pris-
matic columns. Thus the bar under consideration buckles at 
the classical critical load }'„ = TT'2/AI/4/>'2, where A> is the in-
stantaneous flexural rigidity at <p = 0. According to the classical 
elastica theory, an increase in axial load is required to increase the 
deformation (end slope) of the bar so that the stiffness of the bar 
increases with an increase in end slope. A quite different be-
havior is encountered in the present case, as is indicated by the 
plot of P/P,;r versus a given in Fig. 5. For <̂<>L = 0.40 it is seen 
that a sudden reduction in load occurs immediately after the 
column buckles. In ca.ses in which the material is extremely stiff 
after "yielding" (that is, in those ca.ses in which e = Di/IK > 
0.5 (.say)), the column stiffness apparently becomes zero at a 
~ 40 deg and the curve acquires a positive slope as a becomes 
larger. At this point the behavior approaches asymptotically 
that predicted by the classical elastica for a bar with flexural 
Journal of Applied Mechanics MARCH 1 9 7 0 / 51 
Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/30/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
rigidity A - For <paL = 0.05 and e = 0.25, a case not too dif-
ferent from many mild steel columns, there is a sharp decrease in 
load after buckling and the slope remains negative for a large 
portion of the range of values of a considered, Fig. 6. This, of 
course, is in agreement with experimental data available on the 
postbuckling behavior of elastoplastic columns, e.g., [7]. I t is 
noted, however, that as long as e > 0, there is always a value of a 
for which the slope of the P/PN versus a curve will begin to in-
crease and to approach asymptotically the curve corresponding 
to the postbuckling behavior predicted by classical elastica theory 
for an elastic column with flexural rigidity A . 
References 
1 Frisch-Fay, K., 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 International 
Congress of Applied Mechanics, 1939, pp. 234-237. 
3 Havner, K. S., "On the Formulation and Iterative Solution of 
Small Strain Plasticity Problems," Quarterly of Applied Mathematics, 
Jan. 1966, pp. 323-325. 
4 Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, 
2nd ed., McGraw-Hill, New York, 1961, pp. 76-82. 
5 Bellman, R. E., and Kalaba, R. E., Quasilincarization and Non-
linear Boundary-Value Problems, American Elsevier Publishing Co., 
New York, 1965. 
6 Kalaba, R. E., "On Nonlinear Differential Equations, the 
Maximum Operation and Monotonic Convergence," Journal of 
Mathematics and Mechanics, Vol. 8, 1959, pp. 519-574. 
7 Home, M. R., and Merchant, W., The Stability of Frames, 
Pergamon Press, Oxford, 1965, p. 26. 
52 / M A R C H 1970 Transactions of the ASME 
Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 06/30/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use


Finite Deflections of a Nonlinearly Elastic Bar

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=075B9BAFA4B9308F37C7D2D7F6FAC10D?doi=10.1.1.1047.9849&rep=rep1&type=pdf

Finite Deflections of a Nonlinearly Elastic Bar

Abstract

Similar works

Full text

Available Versions

CiteSeerX