The structural behavior of an elastic beam-column placed with a gap between two nonlinearity elastic layers each resting on a rigid foundation was examined. The beam-column was laterally supported at both ends and subjected to a uniform transverse load and axial compression. Its slenderness was such that the axial compressive force exceeds the amount that would be necessary to buckle it as a simple supported column. The elastic layers were represented by an elastic foundation with a strongly nonlinear specific reaction taken as a rapidly increasing function of the layer compression. The analytical model developed simulated the entire pattern of the deflection and stress state including layer and end support reactions, under gradually increasing axial force

Johns, T. G.

Kuznetsov, E. N.

English

NASA Technical Reports Server

POST-BUCKLING  BEHAVIOR  OF A BEAM-COLUMN 
ON A NONLINEAR  ELASTIC  FOUNDATION  WITH A GAP' 
Edward  N.  Kuznetsov 
Battelle  Columbus  Laboratories 
Thomas G. Johns 
Battelle  Houston  Operations 
SUMMARY 
The  subject  of  this  paper  is  the  structural  behavior  of an elastic  beam- 
column  placed  with  a  gap  between  two  nonlinearly  elastic  layers  each  resting 
on a  rigid  foundation.  The  beam-column  is  laterally  supported  at  both  ends 
and  subjected  to  a  uniform  transverse  load  and  axial  compression.  Its 
slenderness  is  such  that  the  axial  compressive  force  exceeds  the  amount  that 
would  be  necessary  to  buckle  it  as  a  simply  supported  column.  The  elastic 
layers  are  represented  by an elastic  foundation  with  a  strongly  nonlinear 
specific  reaction  taken  as  a  rapidly  increasing  function  of  the  layer  com- 
pression.  The  analytical  model  developed  simulates  the  entire  pattern  of  the 
deflection  and  stress  state  including  layer  and  end  support  reactions,  under 
gradually  increasing  axial  force. 
INTRODUCTION 
There  are  many  cases  when  a  primary  buckling  mode  occurring  at  the  onset 
of buckling  cannot  develop  freely  (References  1-2)  because  of  changing  con- 
straint  or  support  conditions.  Such  is,  in  particular,  the  case  of  a  column 
with  lateral  supports  arranged  with  gaps,  etc.  In  this  case,  the  post-buckling 
deflection  is  constrained  laterally  and  the  axial  force  can  be  increased  by  far 
in  excess  of  its  first  critical  value.  As  a  result,  the  structural  behavior 
is  characterized  by  a  sequence  of  alternating  gradual  changes in the  deformed 
configuration  and  rather  abrupt  jumps fromone equilibrium  configuration  to 
another.  A  similar  behavior  pattern  was  observed  (Reference 3 )  for  a  com- 
pressed  plate. 
The  subject  of  this  paper  is an elastic  beam-column  placed  with  a  gap 
between  two  nonlinearly  elastic  layers  each  resting on a  rigid  foundation 
(Figure 1). The  beam-column  is  laterally  supported  at  both  ends  and  subjected 
This  study was  sponsored  by  the  American  Gas  Association  Pipeline  Research 
Committee . 
1 65 
https://ntrs.nasa.gov/search.jsp?R=19800024259 2020-03-21T15:56:54+00:00Z
to  a  uniform  transverse  load  and  axial  compression. The  elastic  layers  have  a 
strongly  nonlinear  specific  reaction  taken  as  a  rapidly  increasing  function  of 
the  layer  compression. The problem  consists in the  analytical  evaluation  of  the 
stress  state  and  deflection  under  gradually  increasing  axial  compression. 
The  deflection  of  the  beam-column in consideration is small  enough  to 
justify  the  use of.the conventional  linearized  expression  for  the  curvature. 
However,  there  are  two  other  sources  of  nonlinearity  (nonlinearly  elastic 
layers  and  the  presence  of  a  gap)  which  were  fully  accounted  for.  Note,  that 
the  presence  of  a  transverse  load  makes  the  problem  nonhomogeneous so that  it 
is not  a  bifurcation  problem. 
ANALYTICAL  FORMlTLATION AND SOLUTION  METHOD 
Under  the  above  assumptions,  vertical  equilibrium  of  the  beam-column 
requires  that 
F(Y) EIY(X)I~ + PY(X)II + Q(Y(~)) - G = 0 (1) 
Here Y is  the  elastic  deflection of the  beam-columny E1 is its  flexural 
rigidity,  is  the  axial  compression  force, Q(Y) is  the  foundation  reaction  per 
unit  length  as  a  function of Y, G is  the  distributed  transverse  load  (assumed 
uniform  and  constant), x' is the  column  longitudinal  axis  and  prime  denotes 
differentiation  with  respect  to  x. 
The  elastic  foundation  reaction  is  taken in the  following  form: 
where k and  n  are  given  constants  and  c is the  gap  size.  Thus,  the  foundation 
reaction  is  proportional  to  a  power of the  beam-column  penetration  into  the 
elastic  layer. 
The  self-correcting  finite  increment  method  (Reference 4 )  will  now  be 
applied  for  solving  Equation (1). To  this  end  the  compression  force  P  is 
given an infinitesimal  increment p which  results in some  infinitesimal  varia- 
tion y(x) of  deflection  Y (x) : 
F(Y+y) F E1 (Y+y) + (P+p)  (Yt-y) 'I + Q (Y+y) - G = 0 .  IV (3)  
Specializing Q(Y) in  accordance  with  Equation (2) yields 
F (Y) + EIy  IV + Py" + pY1' +{ 0 
kn (Y-c)  n-ly 
- G  = 0 .  
1 66 
The  operation  performed  is  known  as  Frechet  differentiation. It resulted 
in  a  linearized  equation i unknown  variation y(x) and  is  rigorous  for  in- 
finitesimal  increments  only.  The  equation  is  extrapolated  to  small  but  finite 
increments  which  permits itsuse in  a  step-by-step  solution  procedure. A 
solution  obtained  at  each  step is  an  approximate  one.  Therefore,  employing 
it  as a starting  point  for  the  next  step  introduces  some  error in addition  to 
that  resulting  from  the  next  step  itself.  This  is  partially  offset  by  a 
correction  which  consists in retaining  the  first  term in Equation ( 4 ) .  For  an 
exact Y(x), this  term,  according  to  Equation (1) would  be an identical  zero. 
Since in reality  the  solution  obtained  after  the  m-th  step, 
Y"(X) = Ym-l(x) + ym(x) , 
is  approximate,  it  does  not  turn F(Y) into  zero.  Retaining  this  term in 
Equation ( 4 )  compensates  for  the  error  of  a  current  step  solution  thus 
preventing  both  systematic  and  occasional  errors  from  passing  to  the  next  step 
and  accumulation. 
The  solution  of  linearized  Equation ( 4 )  is  sought in the  form  of  a  linear 
combination of several  approximating  functions  satisfying  the  boundary  con- 
ditions  of  the  problem: 
where R is  the  beam  half-length  (Figure 2). As is  readily  seen,  only 
symmetric  configurations  of  the  beam  are  taken  into  consideration.  This  was 
done  because  the  particular  case  of  interest  is  characterized  by  a  relatively 
big  transverse  load,  which  precludes  the  antisymmetric  configurations  from 
occurrence  at  the  early  stages  of  post-buckling  deformation. (The uniform 
transverse  load  would  perform  zero  mechanical  work  over  antisymmetric  dis- 
placements). 
The  Galerkin  method is now  applied. It requires  substituting  the  above 
y(x) into  Equation ( 4 ) ,  multiplying  it  by  one  of  the  approximating  functions 
and  integrating  the  product  over  the  beam  length.  This  results in a  system  of 
N linear  algebraic  equations  in  unknown  parameters  y  with  coefficients i 
R 
and  free  terms 
a = (DiEI-P)Di t + sS(x) sin 2R dx 2 (2i-1)rx ii 0 
R 
aij = J S(x)sin (2i-1)nx sin (2j-1)rx dx 
0 2R  2R 
R 
a = - D ~ Y ~ ~  + JF(~) sin (2i-1) rx 2R io 0 
( 8 )  
1 67 
where 
n 
S(x) = kn[Y  (x)-cln-' 
N' 
Obviously,  making  the 
D.Y.sin (2i-l)~x 
1 1  2R + Q [Y(x)l-G 
(10) 
axial  force  increments  smaller  improves  the 
accuracy  of  the  solution,  but  increases  the  number  of  solution  steps. A 
reasonable  compromise  was  achieved  by  arranging  intermediate  iterations in 
which  Equation ( 4 )  was solved  without  incrementing  the  axial  force  (i.e., 
setting p = 0). In these  iterations,  the  pattern  of  the  beam  interaction  with 
the  elastic  layers  is  refined  for  the  fixed  magnitude  of  the  axial  force.  Only 
after  some  assigned  level  of  accuracy  is  reached,  the  axial  force is given its 
next  increment. 
The  specificity  of  the  problem in consideration  is  that  more  than  one 
equilibrium  configuration  may  correspond  to  a  given  axial  force.  To  determine 
whether  other  equilibrium  configurations  exist in the  vicinity  of  the  original 
one,  the  following  approach  is  employed.  Upon  achieving  the  convergence  of 
the  internal  iterations  for  a  fixed  value  of  the  axial  force,  the  system  is 
perturbed  by  giving  the  deflection  some  random  distortion  and  internal  itera- 
tions  are  performed  once  again.  This  may  result in overcoming  the  energy 
barriers  separating  the  possible  equilibrium  configurations  and  increases  the 
likelihood  of  solution  convergence  to  the  most  stable  configuration. 
The  perturbation  is  physically  meaningful:  it  reflects  imperfections  in 
material  properties,  system  geometry,  load  application  and  many  other  factors 
not  accounted  for  explicitly.  The  magnitude  of  the  distortion  presumably 
correlates  with  the  mentioned  imperfections. 
A computer  program  implementing  the  above  features  was  written  and 
applied  to  the  analysis  of  a  precompressed  cryogenic  pipeline. 
NUMERICAL  RESULTS AND DISCUSSION 
The  concept  of  preshortening  a  cryogenic  pipeline  by  compression  is 
intended  to  reduce or eliminate  the  need for  thermal  expansion/contraction 
devices.  The  concept  involves  the  compression  of  one  pipe  (inner,  conveying 
pipe)  within  another  (casing  pipe). The  pipes  are  separated  by  thermal 
insulation  and an air  gap  (clearance)  exists  between  the  insulation  and  the 
inner  or  the  outer  pipe.  The  magnitude  of  the  compressive  force  is  limited 
by  the  amount  that can be  tolerated in the  inner  pipe  wichout its local 
inelastic  buckling  as  a  cylindrical  shell  (Reference 5) .  Under  this  force, 
1 68 
tigated  to  determine  the 
effect  of  design  variables  such  as  the  gap  size,  pipe  length,  insulation 
elastic  properties,  etc.,  upon  the  maximum  stresses,  lateral  reactions,  and 
amount  of  absorbed  compressed  length. 
The  following,  rather  typical  data  were  used in one of the  numerical 
examples : 
0 pipe  length - 100 m  (328  ft) 
0 pipe  outer  diameter - 46.7 cm (18 inches) 
0 wall  thickness - 0.9525 cm  (3/8  inch) 
0 clearance - 2.54 cm (1 inch) 
0 permissible  stress - 320 MF'a (46  ksi). 
Figure  3  shows  the  evolution  of  the  elastic  deflection  of  the  pipe  as  the 
compression  force  grows.  Diagram  3a  is  the  sagged  configuration  of  the  pipe 
resting on the  elastic  layer  almost  uniformly  compressed. As the  axial  force 
is  increased,  the  pipe  bends  and  develops  progressively  increasing  waviness 
(3b  and  c)  till  the  end  segment  ''snaps  through"  (3d).  At  this  moment  the  pipe 
assumes  another  equilibrium  configuration  which  continues  its  evolution in 
further  loading (3e). 
The  results  of  numerical  experiments  confirmed  the  role  of  systematic 
perturbations  applied  during  the  analysis in order  to  obtain  equilibrium 
states  with  lower  total  energy.  In  all  cases  the  self-correcting  finite 
increment  method  provided  a  rapid  convergence  of  the  computation  process. 
From  the  viewpoint of the  precompression  concept  it  was  important  to 
establish  the  role  of  the  gap  between  the  pipe  and  insulation.  Conceivably,  a 
wider  gap  could  even  be  an  advantage  since  it  would  provide  more  room  with 
which  to  absorb  the  ''excess"  pipe  length. A parametric  study  showed,  however, 
an adverse  effect  of  the  gap on the  relative  compression  of  the  pipe:  the 
wider  the  gap,  the  greater  the  portion  of  the  material  strength  spent  on 
bending  stress.  Interestingly,  the  maximum  stress  (composed  of  the  axial  and 
bending  stresses)  does  not  grow  monotonically  with  the  compression  force. 
The  performed  study  also  revealed  the  role of the  pipe  length. A s  shown 
in  Figure 4 ,  the  amount  of  compression  that  can  be  absorbed  without  exceeding 
the  permissible  stress  increases  for  shorter  lengths of pipe.  The  limiting 
case  is  the  pipe  length  at  whichoverallbuckling  does  not  occur  at  all. 
1 69 
REFERENCES 
1. Thompson, J.M.T.  and Hunt, G. W.,  "A General  Theory  of  Elastic  Stability," 
Wiley,  London,  1973. 
2. Budiansky, B.,  "Theory of Buckling  and  Post-Buckling  Behavior  of  Elastic 
Structures,"  Advances  Appl.  Mech.,  Vol. 14, Academic  Press,  1974. 
3. Nakamura, T.  and Uctani, K . ,  Int. J. Mech.  Sci.,  Vol. 21, No. 5, 1979. 
4. Mescall, J., "Numerical  Solutions of Nonlinear  Equations  for  Shells of 
Revolution,"  AIAA  Journal,  Vol. 4, No. 11, 1966. 
5. Volmir, A. S., "Stability of Elastic  Systems,"  Moscow,  1963.  (WPAFB 
Translation AD 628508, 1966). 
1 70 
P-)- 
Figure 1.- Beam-column between two layers of 
elastic  foundation  with gap. 
Figure 2.- Buckled column segment with partial  
contact with foundation. 
171 
0 
876 
(193) 
I751 
(386)  
3067 
( 6 7 6 )  
3355 
(740) 
Figure 
. .  
""""_ 
t 
E 
' E  
I! 
\ 16 
W 
E 
0 """""""""""" - 
n 
""" 
a. 
b. 
C. 
d. 
320 (45.6) 
W 
V 
E 
0 e. - W 
0 
0 5 IO 15 20 25 30 35 40 45 50 
Pipe Length, meters 
- - Undeformed elastic layer surface 
X 88 Location and magnitude of the maximum stress, 
MPa (ksi) 
3.- Development of elastic  deflected  shape  with  increasing  axial load. 
172 
I .75 1 I I I 
u = 320 MPa (46 ksi) 
c = 2.54 cm ( I  inch) 
m 
0 
2 
ti 
1.50 - c - .- 
c 
0 
v) 
u) 
.- 
E 1.25 - 
E 
- 
0 
0 
I .oo I 1 I I 
0 
~- 
25 (82)  5 0  (164) 75  (246) 100 (328) 
Pipe Length, m(f t )  
Figure  4 . -  Amount of compression that  can be absorbed without  exceeding 
t h e  a l l o w a b l e  stress v e r s u s  p i p e  l e n g t h .  
173 


Post-buckling behavior of a beam-column on a nonlinear elastic foundation with a gap

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