A method for the redution of the cost of solution of large nonlinear structural equations was developed. Verification was made using the MARC-STRUC structure finite element program with test cases involving single and multiple degrees of freedom for static geometric nonlinearities. The method developed was designed to exist within the envelope of accuracy and convergence characteristic of the particular finite element methodology used

Jones, R. F., Jr.

Young, C. T., II

English

NASA Technical Reports Server

STUDY  OF  SOLUTION  PROCEDURES  FOR  NONLINEAR 
STRUCTURAL  EQUATIONS 
Cline T. Young I1 
Mechanical  and  Aerospace  Engineering 
Oklahoma  State  University 
Rembert F. Jones, Jr. 
David  Taylor  Naval  Ship  R & D Center 
SUMMARY 
A  method  for  the  reduction  of  the  cost  of  solution f large  nonlinear 
structural  equations  was  developed.  Verification  was  made  using  the  ITARC-STRUC 
structure  finite  element  program  with  test  cases  involving  single  and  multiple 
degrees-of-freedom  for  static  geometric  nonlinearities. The method  developed 
was  designed  to  exist  within  the  envelope  of  accuracy  and  convergence  charac- 
teristic  of  the  particular  finite  element  methodology  used. 
INTRODUCTION 
At  present  the  finite  element  codes  in  conjunction  with  the  large,  high- 
speed  computers  available  are  capable of producing  reasonable  solutions  to 
practically  all  static  problems  conceivable  in  structural  analysis.  In  addi- 
tion,  well-behaved  problems  such  as  those  involving  small  elastic  deformations 
are  solved  relatively  inexpensively  and  accurately.  Computational  difficulties 
do  not  arise  until  the  stiffness  of  the  structure  becomes  a  function  of  dis- 
placement  and/or  displacement  history. An opinion  widely  held  is  that  when 
this  does  occur  an  implicit  solution  scheme  is  necessary  for  accuracy. All 
implicit  schemes  require  an  iterative  solution  where  there  is  an  attempt  to 
reduce  some  error  term  to  zero  at  each  iteration.  Therefore,  a  nonlinear 
problem  is  more  expensive  to  solve  and  can  become  astronomically so depending 
upon  the  degree  of  nonlinearity  and  the  convergence  criteria  used. 
In the  solution of nonlinear  structural  equations  the  reformulation  of 
the  stiffness  matrix  is  a  first  order  contribution  to  the  cost. The  first 
logical  step in attempting  to  reduce  the  cost  would  be  to  seek  a  less  expen- 
sive  way  to  update  the  stiffness  matrix.  This  of  course  has  been  done  with 
some  success  and  i9.appare.ntl.y  still  being  researched.  Looking  at  only  the 
most  recent  developments  or  evaluations we  see  that  Mondkar  and  Powell [ ]
have  used  the  constant  alpha  technique  to  try  updating  the  stiffness  matrix 
for  the  modified  Newton-Raphson  approach.  Matthies  and  Strang [2,,3] 
have  taken  similar  approaches born from  a  paper  by  Dennis  and  More [ 4 ]  on 
Quasi-Newton  methods. The basic  premise  was  that  the  stiffness  matrix  could 
be  updated  without  going  through  the  full  process  of  reformulation  and  de- 
composition  or  inversion.  The  most  popular  approach  was  to  update  the  stiff- 
1 29 
https://ntrs.nasa.gov/search.jsp?R=19800024257 2020-03-21T15:59:08+00:00Z
ness  matrix by a mat r ix  of  rank  two. Th i s  is  known as t h e  BFGS (Broyden- 
Fletcher-Goldfarb-Shanno) u p d a t e .   C r i s f i e l d  [5] used a method similar t o  a 
BFGS update  of  rank  one.  A l l  of t hese  pape r s  show conclus ive  ev idence  of c o s t  
r e d u c t i o n  f o r  c e r t a i n  p r o b l e m s .  The las t  by C r i s f i e l d  is  c l o s e s t  i n  f o r m  t o  
t h e  method developed here. 
A s e c o n d  l o g i c a l  s t e p  is  t o  r e d u c e  t h e  number of i t e r a t i o n s  r e q u i r e d  t o  
s a t i s f y  t h e  c o n v e r g e n c e  cri teria.  This  can  be  done  by  determining  an  estimated 
displacement  as a c c u r a t e l y  as p o s s i b l e .  The development  which  follows shows 
how t o  do t h i s .  A l l  f o r c e s  and loads  are of   an   incrementa l   na ture .  
DEVELOPMENT 
I n  g e n e r a l ,  t h e  i t e r a t i v e  m e t h o d s  of s o l u t i o n  u s i n g  t h e  s t i f f n e s s  f o r m u l a -  
t i o n  w i l l  by some l o g i c a l  means c a l c u l a t e  a gene ra l i zed  d i sp lacemen t  fo r  a 
g iven   gene ra l i zed   l oad .   Re tu rn ing   t hen   t o   t he   e l emen ta l   l eve l ,   t he   e l emen ta l  
s t i f f n e s s  matrices are a l t e r e d  t o  r e f l e c t  t h i s  c h a n g e  i n  s h a p e  and t h e  t o t a l  
r e s i s t a n c e  of t h e  s t r u c t u r e  t o  t h e  a p p l i e d  l o a d  i s  d e t e r m i n e d .   I f   t h e   s t r u c -  
t u r e  i s  to  be  cons ide red  in  equ i l ib r ium,  the  app l i ed  load  mus t  be  exac t ly  
balanced by t h e  r e s u l t i n g  r e s i s t a n c e  of t h e  s t r u c t u r e .  Any imbalance i s  termed 
a r e s i d u a l  f o r c e  and  must  be  considered as a n  e r r o r .  An a t t e m p t  i s  made t o  
r e d u c e  t h i s  e r r o r  by a l t e r i n g  t h e  e s t i m a t e d  d i s p l a c e m e n t .  The r a t e  of conver- 
gence depends on the manner  of  es t imated displacement  select ion.  
The v a s t  m a j o r i t y  of i m p l i c i t  s c h e m e s  a v a i l a b l e  u t i l i z e  o n l y  t h e  m o s t  
r ecen t   r e s idua l   and   t he reby   i gnore   any   poss ib l e   t r end   de t e rmina t ion .   Fe l ippa  
[ 6 ]  recognized  th i s  and  proposed  a v i a b l e  method f o r  d e t e r m i n i n g  t h e  d i s p l a c e -  
ment t h a t  would y i e l d  t h e  least r e s i d u a l  w i t h i n  s p e c i f i e d  l i m i t a t i o n s .  T h i s  
approach  requi red  the  de te rmina t ion  of  a w e i g h t i n g  m a t r i x  t h a t  w a s  dependent 
upon the  e lements   chosen  and  the  appl ied  loads.  The  development i n  t h i s  pape r  
is independent of t h e  p h y s i c a l  c h a r a c t e r i s t i c s  of the  e lements .  
A key element i n  t h e  s u c c e s s  of the approach developed i s  t h e  f i n i t e  
element  method  used. A s  ment ioned  before  the  MARC-STRUC s t ruc tu re  p rogram was 
u s e d  b u t  t h e  v a r i a t i o n a l  f o r m u l a t i o n  of t h e  s t r u c t u r a l  e q u a t i o n s  w a s  performed 
a c c o r d i n g  t o  t h e  method  of  Jones [7 ] .  It i s  most  important  to  have  the  most 
a c c u r a t e  method p o s s i b l e  f o r  t h e  d e t e r m i n a t i o n  of t h e  r e s i d u a l s .  
Cons ide r ing  the  so lu t ion  fo rm,  in  F igu re  1 a graph of f o r c e  v e r s u s  d i s -  
placement is  shown.  The c u r v e   r e p r e s e n t s   t h e   c a l c u l a t e d   r e s i s t a n c e  of t h e  
s t r u c t u r e .  The o r i g i n a l  s t i f f n e s s  m a t r i x ,  Roy a s s u m e s   l i n e a r l y  e las t ic  de fo r -  
mat ion  and  y ie lds  the  d isp lacement ,  uo ,  and t h e  r e s i d u a l ,  Ro, f o r  t h e  a p p l i e d  
load , F. The displacement  , uo , and r e s i d u a l ,  Ro ,-are t h e n  u s e d  i n  a Quasi- 
Newton f a s h i o n  t o  u p d a t e  t h e  s t i f f n e s s  m a t r i x  t o  K 1  and a new disp lacement ,  i1, 
and  consequently a new r e s i d u a l ,  R i ,  are ca lcu la ted .   Highly   accura te   answers  
may r e s u l t ,  b u t  t h e y  are c l e a r l y  e x p e n s i v e  t o  o b t a i n .  
- - 
- - 
- 
The e x t r a p o l a t i o n  method presented i n  t h i s  p a p e r  i s  c l e a r l y  e x e m p l i f i e d  
by t h e  t r i a n g l e ,  ACE, shown i n  F i g u r e  2. The  method  used w a s  i d e n t i c a l  t o  t h e  
d i r e c t  i t e r a t i o n  method  (shown i n  F i g u r e  1 e a r l i e r )  up t o  and through the 
130 
c a l c u l a t i o n  of  uo, R, and  ul, R1. It w a s  t he  de t e rmina t ion  o f  t he  new esti- 
mated  displacement, 9 ,  t h a t  w a s  p e r f o r m e d  d i f f e r e n t l y .  I n  a one  dimensional 
s e n s e  t h e  r e s i d u a l s  Roy K 1  and the dis tance between them, a, were u s e d  t o  
c a l c u l a t e  a scalar, w, t h a t  p r e d i c t e d  t h e  d i s p l a c e m e n t  a t  which equi l ibr ium 
supposed ly  ex i s t ed  unde r  the  load ,  F. Of c o u r s e ,  t h i s  w a s  n o t  t h e  e q u i l i b r i u m  
p o s i t i o n  and a new r e s i d u a l ,  _R2,-was determined. The r e s i d u a l s ,  El, E2 , and 
the dis tance between them, wd - d , were t h e n  u s e d  t o  p r e d i c t  a new equ i l ib r ium 
p o s i t i o n .  The p rocess   con t inued   un t i l   conve rgence  w a s  s a t i s f i e d .  
- -  "
- 
- 
- 
A major  d i f f i cu l ty  encoun te red  w a s  t he  de t e rmina t ion  of t h e  scalar, W. 
I n  a one dimensional  case it w a s  easy enough to  see t h a t  
However, s i n c e  i n  g e n e r a l  t h e  v e c t o r s  Eo, El and d are h e t e r o g e n e o u s  i n  t h e i r  
components '  uni ts ,  a d i v i s i o n  as mentioned above i s  impossible  even when u s i n g  
vec to r   l eng ths .  The s o l u t i o n  t o  t h i s  d i f f i c u l t y  came about  by c o n s i d e r i n g  t h e  
u n i t s  of  work. I n  f a c t ,  t h i s  e x t r a p o l a t i o n  p r o c e s s  may be   symbol ica l ly   thought  
of as minimizing the work  done  by t h e  r e s i d u a l s .  I n  t h i s  l i g h t  i t  w a s  t hen  
dec ided  tha t  equa t ing  the  area of t h e  t r a p e z o i d ,  ABDE, p l u s  t h e  area of t h e  
t r i a n g l e ,  BCD, t o  t h e  area of t h e  t r i a n g l e ,  ACE, would r e s u l t  i n  a n  e q u a t i o n  
with  only  one unknown. Simplifying  and  rearranging,   the   fol lowing w a s  
ob ta ined  . 
- - 
Ro d 
w =  - 
(Eo-R1) d 
A t  t h i s  p o i n t  i t  w a s  decided to  implement  the theory and test  f o r  a one degree- 
of-freedom case and fol low that  with a more  complex case .  
VERIFICATION 
I n  a n  a t t e m p t  t o  d e t e r m i n e  t h e  v a l i d i t y  of the  a forement ioned  ex t rapola-  
t i o n  method i t  w a s  de t e rmined  tha t  a one dimensional buckling problem would be 
a p p r o p r i a t e  as a f i r s t  t es t  case. The bar-spring  problem of Jones  w a s  
s e l e c t e d .  
Bar-Spring Problem 
I n  F i g u r e  3 the dimensions used on the problem may c l e a r l y  b e  s e e n .  The 
l e n g t h  of t h e  s p r i n g  w a s  unimportant  as long as n o n l i n e a r  e f f e c t s  d i d  n o t  e n t e r  
t h e  c a l c u l a t i o n s  f o r  t h e  s p r i n g ' s  d e f l e c t i o n .  The b a r  w a s  modeled s o  as t o  
a l low only  a change i n  l e n g t h  and no bending deformation, hence the absence of 
an E 1  t e r m .  A load  w a s  a p p l i e d  a t  t h e  end  of t h e  b a r  and s p r i n g  i n  t h e  d i r e c -  
t i o n  of de fo rma t ion  to  r ende r  the  p rob lem one  of a pu re ly  s ing le  d imens iona l  
1 31 
case. The buck l ing   l oad  was a t  2.7 kg. ( 6  l b . )  w i t h  t h e  r e s u l t s  t a b u l a r i z e d  
i n  T a b l e  I. The exact deformat ion  w a s  c a l c u l a t e d  a n d  p l o t t e d  i n  F i g u r e  4 t o  
show t h e  h i g h  d e g r e e  of nonl inear i ty  of  the  problem.  
I n  a n a l y z i n g  t h e  r e s u l t s  (see Table I) i t  w a s  d e c i d e d  t h a t  a comparison  of 
the v a l u e s  c a l c u l a t e d  a g a i n s t  t h e  exact v a l u e s  as w e l l  as a comparison of the 
number of i t e r a t i o n s  r e q u i r e d  f o r  e a c h  method  would  be  of  use. The r a i s e d  
numbers b e s i d e  t h e  c a l c u l a t e d  d i s p l a c e m e n t s  i n  T a b l e  I r e p r e s e n t  t h e  number  of 
i t e r a t i o n s  r e q u i r e d  a b o v e  t h e  o r i g i n a l  estimate t o  r e d u c e  t h e  q u o t i e n t  of t h e  
ca l cu la t ed  d i sp lacemen t  and  the  e s t ima ted  d i sp lacemen t  to  the  to l e rance  ind i -  
ca t ed  a t  t h e  column heading. 
It shou ld  be  no ted  tha t  a t  t h e  b u c k l i n g  l o a d  t h e  t o l e r a n c e  r e q u i r e d  t o  
o b t a i n  two s i g n i f i c a n t  d i g i t s  a c c u r a c y ,  1 . 0 0 1 ,  l e d  t o  a 5 v s .  26 advan tage  in  
i t e r a t i o n s  f o r  t h e  new method.  However, t h e  new method w a s  edged  by t h e  o l d  
in  the  pos t -buckled  reg ion  by  a c o n s i s t e n t  4 v s .  7 margin .  The  reason  for  th i s  
w a s  a p p a r e n t l y  t h a t  t h e  l i n e a r  e x t r a p o l a t i o n  d i d  n o t  f o l l o w  t h e  c h a n g i n g  s t i f f -  
nes s  of t h e  s t r u c t u r e  v e r y  w e l l .  I f  so ,  a b e t t e r  a p p r o x i m a t i o n  would  be 
obta ined  wi th  a p a r a b o l i c  e x t r a p o l a t i o n .  
On t h e  w h o l e  t h i s  e x t r a p o l a t i o n  showed promise i n  t h i s  c a s e  b u t  n o t  of a 
c l e a r l y  d e c i s i v e  n a t u r e .  T h e r e f o r e ,  t h e  m o t i v a t i o n  f o r  a more  complex  example 
was e s t a b l i s h e d .  
Ring Buckling Problem 
This problem w a s  t o  d e t e r m i n e  t h e  d e f l e c t i o n  o f  a r ing  unde r  a uniformly 
l o a d e d  e x t e r n a l  p r e s s u r e  of v a r y i n g  v a l u e s .  The r i n g  w a s  modeled  through 90 
degrees  as shown i n  F i g u r e  5. The 90 degree  a rch  w a s  b roken  in to  two subs t ruc -  
t u r e .  The degrees  of  reedom  per  node were 
1. z 
2 .  R 
3 .  dZ/ds 
4 .  dR/ds 
w i t h  t h e  r o t a t i o n s  p o s i t i v e  as  shown by 0 i n  F i g u r e  5. The r i n g  was modeled 
w i t h  a modulus  of e l a s t i c i t y  of 2 . 1  x l o 6  kg/cm2 ( 3 0  x 106 psi)  and a r a d i u s  
of 51 cm ( 2 0  i n . ) .  F i n a l l y ,  a k i c k e r  f o r c e  w a s  a p p l i e d  a t  node 1 of subs t ruc -  
t u r e  1 i n   t h e   n e g a t i v e  R d i r e c t i o n   w i t h  a magnitude  of 1 .5  x kg. (3 .4  x 
10-6 l b )  . Obvious ly ,  th i s  w a s  s i m p l y  t o  f o r c e  t h e  r i n g  i n t o  a buckled mode 
w i t h o u t  a l t e r i n g  t h e  d e f l e c t i o n s  d u e  t o  t h e  p r e s s u r e  l o a d i n g .  
As t h e r e  w a s  no e x a c t  s o l u t i o n  o t h e r  t h a n  t h e  known c o l l a p s e  l o a d ,  t h e  
tolerance chosen,  1 .001,  w a s  t h a t  which  gave two s i g n i f i c a n t  d i g i t s  a c c u r a c y  
for   the   bar -spr ing   problem.  The r e su l t s  obta ined  are shown i n   T a b l e  11. The 
p o i n t  a t  which. t h e  s t r u c t u r e  would  "collapse" w a s  4.22  kg/cm2  (60 p s i ) .  A s  can 
b e  s e e n ,  t h e  r e s u l t s  were qui te   remarkable  as t h e  s t r u c t u r e  became s o f t e r .  A t  
4.18 kg/cm2 (59.5 p s i )  t h e  number  of i t e r a t i o n s  r e a c h e d  by t h e  o l d  method were 
n o t  e n o u g h  y e t  t o  s a t i s f y  t h e  t o l e r a n c e  r e q u i r e m e n t  o f  1.001.  The  authors 
s u s p e c t  t h a t  a n o t h e r  50 t o  100 i t e r a t i o n s  would have been required.  
132 
CONCLUDING 
t h i s  p a p e r  
REMARKS 
w a s  t h e  c o s t  of s o l u t i o n  o f  l a r g e  
n o n l i n e a r  s t r u c t u r a l  e q u a t i o n s .  T h i s  d i f f i c u l t y  h a s  b e e n  and i s  be ing  
r e sea rched ;  however ,  t he  d i r ec t ion  of most  present  research  i s  a p p a r e n t l y  con- 
c e r n e d  w i t h  t h e  s e c o n d  p a r t i a l  of t h e  s t r a i n  e n e r g y  e x p r e s s i o n  ( s t i f f n e s s  
matr ix) .   This   paper   implies   and  subsequent   research by t h e  a u t h o r s  s u p p o r t s  
t h e  s u p p o s i t i o n  t h a t  t h e  f i r s t  p a r t i a l  of t h e  s t r a i n  e n e r g y  e x p r e s s i o n  ( r e s i s t -  
i n g  f o r c e )  i s  n o t  b e i n g  f u l l y  u t i l i z e d  i n  t h e  d e t e r m i n a t i o n  of t h e  new esti- 
mated  displacement  needed f o r  impl ic i t  methods .  It may w e l l  b e  d e t e r m i n e d  t h a t  
updat ing  and/or  re formula t ion  of t h e  s t i f f n e s s  m a t r i x  is  o c c u r r i n g  f a r  t o o  
o f t e n  i n  p r e s e n t  s o l u t i o n  t e c h n i q u e s .  
133 
REFERENCES 
1. Mondkar D. P. and  G. H. Powell,  "Evaluation of Solution  Schemes  for  Non- 
linear  Structures."  Computers & Structures,  Vol. 9,  1978, pp.  223-236. 
2. Matthies, H. and G. Strang,  "The  Solution of Nonlinear  Finite  Element  Equa- 
tions."  International  Journal  for  Numerical  Methods  in  Engineering, 
V O ~ .  14, pp. 1613-1626,  Nov.,  1979. 
3 .  Strang, G. and H. Matthies,  "Numerical  Computations  in  Nonlinear  Mechanics." 
NSF  Report,  MCS  78-12363. 
4 .  Dennis,  J. E.  and  J.  J.  More:  "Quasi-Newton Methods,  Motivation  and  Theory." 
SIAM Review,  Vol. 19, pp. 46-89,  1977. 
5. Crisfield,  M.  A., "A Faster  Modified  Newton-Raphson  Iteration."  Computer 
Methods  in  Applied  Mechanics  and  Engineering, Vol. 20, pp. 267-278,  1979. 
6 .  Felippa, C.  A.,  "Procedures  for  Computer  Analysis of Large  Nonlinear  Struc- 
tural  Systems."  Large  Engineering  Systems,  ed.  by A. Wexler,  Pergamon 
Press,  London, pp. 60-101, 1976. 
7. Jones, R. F., "Incremental  Analysis of Nonlinear  Structural  Mechanics  Prob- 
lems  with  Applications  to  General  Shell  Structures."  Naval  Ship  Research 
and  Development  Center  Report 4142, 1973. 
1 34 
TABLE 1 
BAR-SPRING PROBLEM 
Load 
( lb )  Method ( 4  1.1 1.01  1.001  1.0001 1.0 00  
kg Exact 
1.4 Old 0.59781 0.5915; 0. 59643 0.59785 4 0. 59786 €I 0.59787 7 
(3.0 New 0.5915  0.5964  0.5978  0.5978  0.5978 
2.7 Old 2.5400 2.01772  2.45454 l3 2.53205 2. 53926 39 2.54006 
(6.0) New 2.2329  2.5018  2.5373  2.5400  2.5400 
4.1 Old 4.4821 4 .  49331 4.48061 4.48211 
(9.0) New 4.4788  4.4816  4.4821 4 .4821 4.4821 
4 
26 51 
1 4 .  48211  4.48213. 1 
5.4 Old 5.0800  5. 07923 5.08005  . 7  5. 08008 5.08009 5 
(12.0) New 5.0828  5.0803  5.0800  5.0800  5.0800 
~~ 
6.8 Old 5.4907 5. 49431 5.49075  . 7 5. 49079 5.490710 4 6 
(15.0) New 5.4948  5.4910 5.4907  5.4907  5.4907 
8.2 Old 5.8143 5. 81761 5.81465  . 37 5. 81439 5.814311 
4 7 
(18.0) New 5.8176  5.8148  5.8146  5.8143 5.8143 
*Buckling Load 
Note: When the  number of i t e r a t i o n s  is  less than (2) ,  there  i s  NO di f fe rence  between t h e  new 
and old methods. 
TABLE I1 
RING PROBLEM (1,001) 
Load  Substructure 1 (cm) Substructure 2 (cm) 
kg 1 cm2  Node 1 Node 1 
(psi) Method  Iterations D.O.F. 2 D.O.F. 2 
.5  Old 
( 7 )  New 
2 
4 
- .78547 E-03 -.41397  E-03 
-.78555  E-03 -.41397  E-03 
1 .5  Old 5 -.25537 E-02 - . l o 4 6 8  E-02 
( 2 1 )  New 4 - .25545 E-02 - . l o 4 5 9  E-02 
2 . 5  Old 9 - .49439 E-02 - . l o 6 0 7  E-02 
( 3 5 )  New 4 - .49472  E-02  - . lo576 E-02 
3 . 5  Old 23   - . lo230  E-01  .18 1 7  2 E-02 
( 4 9 )  New 3 - . lo237  E-01  , 1 8 2 4 3  E-G2 
3 . 9  Old 5 4  
( 5 6 )  New 3 
~~ ~~ 
- . 22451  E-01 .12827  E-01 
-.22597  E-01 .12972  E-01 
4 . 2  Old 149* -. 88354  E-01,k . 77978  E-Ol* 
(59 .5 )  New 4 -.95669  E-01 .85268  E-01 
*Maximum  number of iterations  allowed. 
Convergence  not  yet  satisfied. 
DISPLACEMENT 
F- 
- 
FORCE 
Figure  1.- D i r e c t  i t e r a t i o n  method. 
DISPLACEMENT 
Figure  2.- D i r e c t  i t e r a t i o n  method with 
e x t r a p o l a t i o n  m o d i f i c a t i o n .  
137 
7- 
2.54 cm 
(1 in.) 
P 
$ A€ = 7.03 X IO5 kglcm2 ( IO7 psi) 
Q 
Q 
$ 
2 
\ Ks = 1.07 kg/cm + (6 Iblin.) 
? 
2 
? 
\ 
\ 
\ 
\ -  
Figure  3.  - Bar-spring problem. 
DISPLACEMENT (in.) 
0 0.5  1 .o 1.5  2.0  2.5 
10.0 I I I I 
- 20 
7.5 - 
- 15 
5.0- 
(kg) -10 
n 
w 
-5 
I I 
3.0 4.5 6.0 
I -0 
I 
LOAD 
(Ib) 
DISPLACEMENT (cm) 
Figure  4.- E x a c t  s o l u t i o n  t o  bar-spr ing problem. 
1 38 
2 
Figure 5.- Ring problem. 
1 39 


Study of solution procedures for nonlinear structural equations

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