INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
23, 69-90 (1986)
A SIMPLIFIED FINITE ELEMENT METHOD FOR LARGE
DEFORMATION, POST-BUCKLING ANALYSES OF LARGE
FRAME STRUCTURES, USING EXPLICITLY DERIVED
TANGENT STIFFNESS MATRICES
K. KONDOH AND S. N. ATLURI
Center for the Advancement of Computational Mechanics School of Civil Engineering, Georgia Institute of Technology,
Atlantu. Georgia 30332, U.S.A.
SUMMARY
In this paper, a simplified, economical and highly accurate method for large deformation, post-buckling,
analyses of frame-type structures is presented. Using the concepts of polar-decomposition of deformation, an
explicit expression for the tangent stiffness matrix of a member/element of the frame, that accounts for the
nonlinear bending-stretching coupling, is derived in closed form, at any point in the load-deformation path.
The ranges of validity of the present simplified approach are discussed. Several example problems to
demonstrate the feasibility of the present approach, over ranges of deformation that are well beyond those
likely to occur in practical large frame structures, are included.
1. INTRODUCTION
In the past few years there has been a renewed interest in the analysis of frame- and truss-type
structures, due to the current dreams of many to deploy very large structures in outer space. These
space-structures, generally of the frame type, are often envisioned to be as large as Manhattan
Island, and probably just as unwieldy! One of the primary topics of current interest is to analyse
parts of such structures, that may be subject to large local disturbances, with the ultimate goal of
controlling the dynamic deformations through active or passive mechanisms. This paper, however,
is limited, as a first step, to considerations of simple, yet highly accurate, methods of nonlinear
analyses of frames under quasi-static loads. While plane frames are treated in the present paper, the
extensions of the central methodology to space frames, and inelastic deformations, are subjects of
forthcoming publications.
Large deformation and post-buckling analyses of structures have been the subjects of extensive
research in the past decade or so (see, for instance, References 1-10). In all these studies, an
incremental approach, either of the ‘total Lagrangean type’ or of the ‘updated Lagrangean type’ is
employed. The incremental approach, in turn, is often based on the development, for each element,
of the so-called ‘tangent stiffness matrix’, which reflects all the nonlinear geometric and mechanical
effects. In most of the literature, the derivation of the ‘tangent stiffness matrix’ of an element (which
may be based, alternatively, on potential energy, complementary energy or general mixed-hybrid
formulations), in general, involves: (a) simple polynomial basis functions for displacements and/or
stress and moment resultants; and (b) Numerical integration of matrices (dependent on the
assumed basis functions and their spatial derivatives) over the domain of the element. The key
factors that determine the economic feasibility of the routine use of the above nonlinear analysis
0029-598 1/86/010069-22sO2.20
0 1986 by John Wiley & Sons, Ltd.
Received 15 August 1984
Revised 11 February 1985
70
K. KONDOH AND S. N. ATLURI
methods are: (i) the computational time involved in forming the tangent stiffness matrix of each
element (and, thus, of the entire structure) at each increment of external loading; (ii) the degree of
refinement of the finite element grid, when elements with simple polynomial basis functions are
used; and (iii) the techniques for solution of the system stiffness equations especially at or beyond
the critical (buckling) points in the load-path.
A majority of nonlinear analyses of typical engineering structures, and especially the frame-type
large space-structures, can be simplified if an explicit expression (i.e. without involving assumed
basis functions for displacement/stress and without involving element-wise numerical integrations) for the tangent stiffness matrix of an element (incorporating the effects of initial
displacements on the current stiffness)can be derived. Towards this end, the authors have recently
presented’ a method for explicitly deriving the tangent stiffness matrix of a truss-type structure,
wherein each of the members is assumed to carry only an axial load. This procedure for analysing
trusses has been demonstrated1’ to be not only inexpensive, but also highly accurate, in a wide
variety of problems involving very large deformations and highly nonlinear pre- and post-buckling
responses.
In this paper, we consider simplified procedures for largedeformation
of frame-type structures, wherein each member undergoes bending as well as stretching
deformations (which are coupled) and can carry axial loads as well as bending moments. Explicit
expressions for the coefficients of the tangent stiffness matrix of an element (applicable over a wide
range of deformations) are derived. By ‘explicit’ we mean that the procedure does not involve
assumptions of basis functions for the element nor of numerical integrations over each element.
The concept of a ‘polar-decomposition’ of the deformation is employed to decompose the
arbitrary deformation of the element into rigid rotations and pure stretches. The present work is
based on the assumptions: (i) arbitrarily large rigid translations and rotations of each
member/element of the frame are accounted for; (ii) the local relative (non-rigid) rotations of a
differential segment of a member/element of the frame are moderately small, and that their squares
enter to the expression for axial stretch, in a manner analogous to that in the well-known Von
Karman theory for plates, and (iii) the nonlinear coupling between the bending and stretching
motions of the member/element is inherently accounted for. The ranges of validity of these
approximations are critically examined, Several numerical examples dealing with the nonlinear
pre- and post-buckling responses of frames are presented. In these examples, the arc-length
method’ *-14 is employed to solve the system stiffness equations. It is demonstrated that in most of
the cases a single element, with the presently derived explicit stiffness matrix, is adequate to model
each member of the frame; and that the methodology is not only inexpensive, but is also highly
accurate even for ranges of deformations that are well beyond those likely to be encountered in
practice.
The remainder of the paper is organized as follows: Section 2 gives a detailed account of the
present procedure for an explicit evaluation of the tangent stiffness matrix of a frame-member, at
any point in the nonlinear load-deformation path; the ranges of validity of the present simplified
approach are delineated in Section 3; several (five)numerical examples are presented and discussed
in Section 4; and concluding comments are made in Section 5.
2. EXPLICIT EVALUATION OF TANGENT STIFFNESS FOR A FRAME MEMBER
In this work, each member of a frame, over ranges of its deformation that are likely to occur in
practice, shall be sought to be modelled, for the most part, by a single finite element. The frame-type
structures discussed herein are assumed to remain elastic. Only a conservative system of
concentrated loads are assumed to act at the nodes of the frame.
LARGE FRAME STRUCTURES
71
(After Deformation)
-N
\
T
(Before Deformation)
,
z
I
2
I
I
Figure 1. Nomenclature for kinematics of deformation of a frame-member
2.1. Kinematics of deformation of a member: decomposition to stretch and rotation
Consider a typical frame member, modelled here as a beam element, that spans between nodes 1
and 2 as shown in Figure 1. The element is considered to have a uniform cross-section and to be of
length I before deformation. The co-ordinates z and x are the local co-ordinates for the undeformed
element, as shown in Figure 1. The functions w(z) and u(z) denote the displacements, at the
centroidal axis of the element, along the co-ordinate directions z and x,respectively. Also, as shown
in Figure 1, (i) gis the angle between the straight line joining the nodes of the deformed element and
the z axis (and thus, is the rigid rotation of the member/element), and (ii)6* is the angle between the
tangent to the deformed centroidal axis and the line joining the two nodes of the deformed element.
From the polar decomposition t h e ~ r e m , 'the
~ relations between the point-wise stretch E, the
point-wise rotation 8, and the displacements w(z) and u(z) in the element, are
(1 +e)sin6=where
du
dz
e=g+e*
Integrating (la, b) along the length of the element, one obtains
j l ( 1 - t &)cos8dz = 1
+ KJ
[ l ( l +&)sin@dz=G
where
72
K. KONDOH AND S. N. ATLURI
and
6 = u / , = f- uI,=o = u2 - u ,
From Figure 1, it is easy to see that
j l ( 1 + E ) cos B*dz = 1 + 6
and
j l ( 1 + &)sin8*dz = 0
where 6 is the total ‘stretch’ of the element (see Figure 1). Substituting for 8 from (2) in (3a, b) and
using (5a, b), one obtains
( 1 + 6) cos B = 1 + Gj
and
( I + 6) sin 17 = 6
From (6a, b) one obtains
6=[(1
+
+ (6)2]”2 - 1;
and Q= tan-’
(A)
___
The non-rigid (‘relative’) rotations at nodes 1 and 2, denoted hereafter as ST and @, respectively,
are defined, using (2) and (8) as
6:
= e*I,=, =el,=, - 8= 8, -tan-’
(94
and
2.2. Relations between the stretch and relative rotation, and axial force and
bending moment, at the nodes of an individual frame member or finite element
In preparation for the task of deriving an explicit expression for the tangent stiffness matrix that
is valid for a wide range of deformations of a frame member, we first derive certain explicit relations
between the kinematic variables of stretch and relative rotations, and the mechanical variables of
l=---
-6
L
73
LARGE FRAME STRUCTURES
axial force and bending moments of an individual frame member, or of a finite element if more than
one finite element is contemplated for modelling an individual member. These ‘generalized’forcedisplacement relations for an individual member/(element) are also intended to be valid over a
range of deformations that may be considered as ‘large’.
TOachieve the above purpose, we consider a beam-column, as shown in Figure 2. The equation
of equilibrium in the transverse direction of the beam may be ~ r i t t e n ’ ~as
,’~
d28*
1
(MI - M,)cosB* - N sin B*
El--dS2 1+6
=0
where
(E = Young’s modulus, I = cross-sectional moment of inertia)
M,
M , =MI,,,
=
In the above, S is the curvilinear co-ordinate along the centroidal axis of the deformed member, and
B* has been defined earlier. For the type of problems contemplated, we assume that the
deformation of the frame as a whole is such that the non-rigid rotation B* in each individual
member/(or its elements) of the frame may be considered as being small. Under this assumption,
(10) may be approximated as
d28* 1
El- - ( M i - M 2 ) - NB* = O
dS2 1
The boundary conditions are
The total axial ‘stretch’ of the beam may be
6=
(jb[ +
1
1
=(Ncos
as
B*
+ Q sin B*)
I >
cosO*dS - 1
(14)
where
and Q is shown in Figure 2. For the type of deformations contemplated here, i.e. 8” being small,
(14) may be approximated as
Thus, the nonlinear term (e*)2is retained in the axial stretch relation as, for instance, in the Von
Karman plate theory. E~ations,(12),(13a, b) and (16)form the basis of the present derivation of the
relations between the generalized displacements and forces in the element.
We first define non-dimensional axial force, and bending moment, denoted as n and m
respectively, through the relations
74
K. KONDOH AND S. N. ATLURI
Now, when n GO, the solution of equations (121, (€3a,b) is
i'
@ * = -.....-
(-n)
I
sin[ ( (-n)'i2
1
+ { --(-n)+
n ) l i 2 ~ ]-
cosecf - n)1/2cos[ ( -
[
cot( cos ( f - n)1/2
;] j
n)II2-
m,
n)'/2~]]m2
(-n)"2
Equation (18) leads to the following relations between tfie relative rotations (6: and 82)at the ends
of the member, and the correspond~ngbending moments (m,and m2):
and
Also, using (18) in (16), we obtain
On tfie other hand, when iz > 0, one may obtain, instead of equations (18), (19a, b) and (ZO), the
following equations for @*,O:, 6; and 6, respectively:
{
c~th[(n)''~]}
cosech[(n)"']
m , + -(n)'is
(n)'i2
1
{ln
~osech[(n)~/~]) 1 ~oth[(n)'/~]
m , + -mz
q= --+
n
{
(n)'i2
and
(n)lP
]
The sets of equations (19a,b) and (20)and (22a,b) and (23)may be written in a more convenient
form by decomposing the kinematic and mechanical variabies of the beam into 'symmetric' and
'antisymmetric' parts, as
and
6: = e(0T
+ 0;);
and 6: = 3(6: - 6;)
ma = (m,- m2); and
+ m2)
wl, = (m,
75
LARGE FRAME STRUCTURES
where subscripts a and s refer to ‘antisymmetric’ and ‘symmetric’ parts, respectively (it is worth
while recalling the sign conventions for 0” and m as shown in Figure 2). In terms of these variables,
equations (19a, b) as well as equations (22a, b) may be written as
QZ=h,m,
and O:=h,ms
(26%b)
wherein
(i) for n < 0:
ha=---- 1
1
1 cot[
( - n) 2( - n)ll2
~
]
and h,
;
=
2( - n)1/2
tan[
i-v:i]
(27a, b)
and (ii) for n > 0
h =--+1
1 coth[
n 2(n)1’2
~ ] ;
and h,
=
1
2(n)
~
l2
(i]
tanh[ n)’”
(28a, b)
Also, in terms of the new variables, equations (20) and (23) may be rewritten, in a unified form, as
6 1 dh,
1 = 2 -)m:
dn
-
-(
+
where, for n d 0,
dha1
-__-____
dn ( - n)2
--__
dh,
hi
[7
(-
1 cosec2
8( -n)
dh, - 1
dn n2
1 cosech2[
8n
1 sech2
dn -8n
(294
+ __
EA
1 dh
(0*)2 N
! s
2 ( d n ) h: +EA
dh, (0:)’
- _1 ____
-+=2(dn)
N
+($)m:
[~]
n)U2
1
~] -
-
(29b)
(294
4( __
4,iCOLh[
1
- __
1 tanh
dn3/2
[(nz’2]
~
],
(290
Equations (26a, b) and (29)are the sought-after relations between the generalized displacements
and forces at the nodes of an individual frame member, for the range of deformations considered. In
connection with equations (26a, b) and (29),it is worth while to recall that: (i) N is in the direction of
the straight line connecting the nodes of the frame-member after its deformation (see Figure 2), and
(ii)the parameters 6,BT and 0; are calculated from equations (7),(9a) and (9b)which are valid in the
presence of arbitrarily large rigid motions (translations and rotations) of the individual member.
Thus, while the local stretch (pure strain) and relative rotation (non-rigid) of a differential-element
of an individual frame-member may be small, the individual member as a whole (and as a part of
the overall frame) may undergo arbitrarily large rigid motion. Hence, the ‘generalized’ forcedisplacement relations embodied in equations (26af,(26b) and (29) remain valid in the presence of
arbitrarily large rigid motion of the individual member of the frame. Also, it is important to note
that the present relations for each element account, as in the Von Karman plate theory, for the
nonlinear coupling between the bending and stretching deformations, as seen from equations (26a,
b) and (29).
76
K. KONDOH AND S. N. ATLUKI
The limits of validity of the relations (26a, b) and (29) are discussed and illustrated in Section 3 of
the present paper. However, certain special features of these relations are briefly discussed here.
First, note that when N = 0, the solution of (12)for the boundary conditions shown in Figure 2 may
be written as
0*=-$(m,-m
and
s
:s:
6 = -From (30) it is seen that
s2
)--ml-+~(2ml+m2)
l2
1
(0*)2dS
0: =&ma;
0:
(or, ha = & and
= $m,
ks = f )
and
Also, when the axial force N is of magnitude ( - 4n2EZ)/12(i.e. the Euler buckling load of tfixedfixed’ column), h,-+O.However, this situation is unlikely to arise in a practical frame structure.
2.3. Tangent stiffness matrix of a frame memberlelement
Recall that each member of the frame is sought, for the most part, to be treated herein as a beam
column as discussed in the previous subsection. In the case of extreme, and what may from a
practical viewpoint be considered as ‘pathological’, deformations, each member of the frame may
be modelled, say, by two or three elements utmost.
Note from equations (26a, b), the ‘flexibility’coefficients k, and h, are highly nonlinear functions
of the axial force (as given in (27a, b) for n 6 0, and in (28a, b) for n > 0). However, unless ha and k,
are equal to zero, one may invert (26a, b) to write the ‘force-displacement’ relations
Recalling the definition of non-dimensional moments as in (1 7b), one may express the strain energy
due to bending as
(34)
On the other hand, as explained earlier, in the limit as N + [ ( -4n2EZ)/I2], h,+O. Thus, the
inversion of (26b) to obtain (33b) is not meaningful. As in the classical problem of incompressibility,” in such a case one may use a ‘mixed’ form for the bending energy of the symmetric mode,
through the well-known Legendre contact transformation,’ as
and treat both m, and 0: as variables. However, the case of N +( - 4dE1)/12, i.e. the Euler buckling
LARGE FRAME STRUCTURES
77
load of a ‘fixed-fixed’ beam is unlikely to occur in a practical framed-structure. Thus, without loss
of generality, we restrict ourselves to considering the strain energy in the form (34).
Now we consider the strain-energy due to axial stretch of the member. Note from (29) that 6 is
related in a highly nonlinear fashion to the axial force N as well as the moments rn, and m,.The
inversion of this relation in explicit form, in order to express the axial force N as a function of 6,
appears impossible. With a view towards carrying out this inversion of the 6 vs. (N, rn) relation
incrementally, later on, we express the strain energy due to stretching, denoted here as x,, in a
‘mixed’ form, using the well-known concept of a Legendre contact transformation,’* as
Note, however, that in view of the dependence of h, and h, on n as in (26,28), there is coupling
between ‘bending’ and stretching variables. The internal energy in the member due to combined
bending and stretching, is
The condition of vanishing of the first variation of x (denoted here as n*)in (37)due to a variation in
N (denoted as N*), i.e., the condition
leads, clearly, to the relation between 6 and the generalized forces as given (29b). The reason for
using the ‘mixed form’ for the stretching energy in (36)is now clear from the result in (38). By using a
similar mixed form for the increment of stretching energy, the incremental axial stretch-vs.incremental generalized force relation can be derived in a manner analogous to that used in
obtaining (38) from (37). This incremental relation, which is, by definition, piecewise linear, may
easily be inverted, as demonstrated in the following. Also, it is shown in the following that equation
(37) forms the basis for generating an explicit form for the ‘tangent-stiffness’ of the member.
The increment of the internal energy of the member, denoted as An, involving terms up to second
order in the ‘incremental’ variables AS,*, AS,*, A N and A6 can be seen, from (37), to be
(AN)’+AN6+NA6+ANA6
(39)
In the above, it should be recalled that h, and h, are functions of N .
Now, using equations (4a, b), (7),(S), (9a, b) and (24a, b), the incremental quantities AS,*, AS: and
A6 may be expressed in terms of (wl, w2, u l , u2,S,, 0,) and/or their increments. Henceforth, we use
the notation for vector ( d ” ) that
Ld”J
= Lw1, ~ 2~ %
1 ~, 2 2 0 1 ~, Z J m e m b e r
(40)
78
K. KONDOH AND S . N. ATLURI
as shown in Figure 1. In terms of the increment {Ad"), equation (39) may be written as
+
A x =$LAd"fLAddJ{Adm) ANLAad]{hdm]
+ +A,,(AN)2 + LBdJ(Adm) + & A N
where
I
I
I
I
i
I
i
I
i
I
L
i
t
j
r-----------I
I
I
Sym.
LARGE FRAME STRUCTURES
79
By setting to zero the variation of An in (41) with respect to A N , one obtains the relation
[And](Ad"}
+ B, + A,,AN
=0
(48)
Thus, (48)is the incremental counterpart of the 6 vs. the generalized force relation obtained in (38).
Unlike the nonlinear relation in (38), however, the piecewise linear relation (48) can be inverted to
express AN in terms of generalized displacements, as
1
1
A N = --LARdJ(Adm}--Bn
Ann
Ann
Substituting (49) into (41), one obtains the internal energy expression
(49)
A X = ~ [ A d " ] [ K " ] ( A d " }+ [ A d m ] ( R m )
where
[K"] = tangent stiffness matrix of member/element
(R") = internal generalized force vector for member/element
1
- ___ Bn { A n d )
(52)
Ann
Recall that the tangent stiffness matrix and the internal force vector are written in the member coordinate system as shown in Figure 1. Thus, it is necessary to transform (d"} from a member coordinate system to a system (global) co-ordinate system in the usual fashion.
It should be emphasized once again that equations (42)-(44) give an explicit expression for the
tangent stiffness matrix [K"] of (51); and, likewise, equations (43)-(46) give an explicit expression
for the internal generalized force vector (R").
The assembled tangent stiffness equations for the frame are solved using the now well-known
'arc-length' method.12-14
=fBd}
3. RANGES OF VALIDITY OF THE PRESENT SIMPLIFIED APPROACH
Recall that the principal assumptions underlying the present development of an explicit expression
for the tangent-stiffness matrix of an element are as follows: (i) arbitrarily large rigid translations
and rigid rotations of the element are accounted for; (ii) however, the local (non-rigid) rotation $* is
restricted to be small such that sin $* N $* and cos $* = 1; (iii) the local axial stretch E is restricted to
be small; (iv) the nonlinear coupling between the bending deformation (characterized by $*) and
stretching deformation (characterized by total axial stretch 6) is accounted for, as seen from (a) the
dependence of ha and h, in (26a, b) and (27a, b) on the axial force n and likewise, (b) the dependence
of axial stretch 6 on the moments (rotations) as seen from equation (29).
We now critically examine the ranges of validity of the above assumptions underlying the
present developments. The most severe test is to see if a single element, based on the present
formulation as given in Section 2, can model the entire history of post-buckling behaviour of a
column subjected to axial compressive load. Clearly, the ability of a single element to characterize
the entire post-buckling behaviour can be seen to be limited by assumptions (ii) and (iii) above.
These two assumptions clearly show that a single element can predict only the initial post-buckling
response, i.e. as long as $* << 1 and 6/1<< 1. This can be seen by considering, for instance, a 'pinnedpinned' (but axially movable) column subject to axial load N alone. The Euler buckling load for
= 0.8333(in')
i
\
\
I
I
\
\
\
\
e
\
+
: s
II
II
Figure 3(a). Problem definition for an
eccentrically loaded column
e/l = 0.001
E = 107(psi)
I = 100.0 (in)
I
A = 1.0 (inz)
005
010
2 Elements
0 I5
-
0
Present
020
0.25
Theoretical ("Elastica")
8 Elements
-+- 4 Elements
- +-
--+--I Element
030
6, /I ,S,/I
Figure 3(b) Dependence of axial and transverse displacements on axial
load for a buckled column under axial eccentric load convergence of
solution
0 01
000
,
, 2 0 N ~
8 Elements1
Elements
02
04
01
02
__ Theoretical ("Elastica")
0
a 2
06
03
t
04
08
$,/I
'I
Figure 3(c) Results similar to those in Figure 3(b),except for a wider
range of deformations
00
0 0
I. 0
I.5
2. 0
2.5
3.0
3.5
N/N,
LARGE FRAME STRUCTURES
81
this beam, in non-dimensional form, is
n = --.n
2
(53)
We consider the initial post-buckling response. Let Ap( ) denote the increment of the quantity ( )
immediately after the Euler buckling load is reached. The boundary conditions of the problem are
Apma= Apms = 0
(54)
From equations(26a, b), (27a, b) and (29b),it may be seen that in the immediate vicinity of the Euler
buckling load,
Ape,*= 0
(55)
and
even while Apn = 0. Thus, when a single element is used to model post-buckling response of a beam,
while the nonlinear coupling between bending and stretching is accounted for, the slope of the postbuckling response curve [(6/1)vs. n] is not correctly accounted for. This is the inherent limitation of
assumptions (ii) and (iii) above and is further illustrated in the numerical example below.
We consider the problem of an ‘e1astica’-a simply-supported (but axially movable) beam, of
length 1, that is subjected to an axial compressive load, with a load eccentricity of (l/lOOO) from the
undeformed axis of the beam, as shown schematically in Figure 3(a).The beam has a square crosssection of area 1.0 (in.2),1 = 100.0 (in.), E = 10’ (psi) and I = 08333 (in.”). For testing the range of
deformations, over which the present explicit expression for tangent stiffness matrix of an element
are valid, the beam is idealized, successively, by 1,2,4 and 8 elements over its length, respectively.
As shown in Figure 3(a), 6, is the total axial ‘stretch’ of the beam, while 6, is the transverse
displacement at midspan, in the post-buckling range. Figure 3(b) shows the dependence of (6, and
6,) on the axial load N , for the range ofdeformations [ (6,/1) and (6,/1)] 2: 0.30. On the other hand,
Figure 3(c) shows the dependence of (6, and 6,) on the axial load N , for a much wider range of
deformations, viz. (6,/1) N 1.0 and (6,/1) 2: 0.5.
From Figures 3(b) and 3(c),it is seen that while a ‘singleelement’ representation of the entire beam
does account for the nonlinear coupling between bending and stretching (as seen from the large
values of 6, and 6, at N 2: N E ) ,the slope of the post-buckling response curve for N vs. (6, or 6,) is
not accurately represented. On the other hand, Figures 3(b) and 3(c) clearly indicate that even a
two-element representation of the beam yields results for post-buckling response, that are in close
agreement with the classical ‘elastica’solution, even for very large deformations of the order, (6,/1)
2: 0.4 and (6,/l) N 0.5. It is also seen from Figures 3(b) and 3(c) that a four-element representation
produces solutions that are in exact agreement with the ‘elastica’solutions for the entire range of
deformations considered. The reason for this excellent behaviour of the ‘two-’ or ‘multi-’element
models of the elastica is due to the fact that the present element development can account for
artbitrarily large rigid motions, even if for element-wise small-strain motions, as discussed under
assumptions (i) to (iv) at the beginning of this section.
It is worth pointing out that while arbitrarily large deformations (such as the straight beam
folding into a circle) have been considered in the present example of a single beam, when a practical
frame-structure is considered, it is unlikely that each of its members will undergo such gross
deformations. Thus, inasmuch as the present element development accounts for nonlinear
bending-stretching coupling, it may be sufficient to model each member of the frame by only one
or two of the present elements, whose stiffness matrices are given explicitly in Section 2. The
following five numerical examples illustrate this assertion.
82
K. KONDOH AND S. N. ATLURI
I
I
12 943(1n)
E = I 0 3 x 107(1b/~n2)
Figure qa). Schematic of Williams' toggle
-
b)
70.0
Present
(I Element per Member)
Wood Ef Zienkiewicz's
( 5 Elements per Member)
\
600
+
Williams'
Experimental
/
i
50 0
400
300
20 0
100
00
/
00
1000
01
2000
3000
02
03
4000
5000
04
05
6000
06
7000 R(lb)
07
8(m)
Figure qb). Variations of load-point displacement and support reaction with load, for Williams' toggle in the postbuckling range
LARGE FRAME STRUCTURES
83
4. ASSORTED NUMERICAL EXAMPLES
The first example is that of the so-called Williams' toggle frame, which was first treated by
Williams' and later analysed by Wood and Zienkiewicz5 and Karamanlidis, Honecker and
K n ~ t h eA. ~schematic of the structure is shown in Figure 4(a). The structure has a semi-span of
12.943(in.),a raise of 0.386 (in.),and is composed of two identical members, each with a rectangular
cross-section of width 0.753 (in.), depth of 0243 (in.), and E = 1-03 x lo7 (psi). Each member of the
frame is modelled by a single element of the type derived in Section 2. Figure 4(b) shows the
presently computed relation between the external load P and the conjugate displacement 6, and
also that between P and the horizontal reaction ( R )at the fixed end. Also, shown in Figure 4(b) are
the comparison experimental results of Williams' as well as the numerical solutions obtained by
Wood and Z i e n k i e ~ i c zExcellent
.~
agreement between all the three sets of results may be noted.
However, the efficiency of the present method is clearly borne out by the facts that: (a) the present
solution uses one element to model each member, while Reference 5 uses five elements to model
each member; and (b) no numerical integrations are used to derive the tangent stiffness of the
element during each step of deformation, since an explicit expression for such is given in Section 2.
The next two examples concern frames, with two and three members, respectively, which bring
out rather fascinating features of responses of frames. For these examples, experimental results
were reported by Britvec and Chilver,16while theoretical solutions for the buckling load and postbuckling responses were also reported in References 16 and 17. In these two examples, each of the
members has a rectangular cross-section of width 1.0 (in.),depth 0.0625 (in.) and 1 = 20.0(in.). Also,
the buckling load of each member, when considered individually as a pinned-pinned column, is 8.1
(1b). The schematics of the two examples are given in Figures 5(a) and 6(a), respectively. In both
these examples, each member of the frames is modelled by a single element.
Each of the structures shown in Figures 5(a) and 6(a), respectively, has two distinctly different
post-buckling load-displacement curves, corresponding to the two types of buckling modes,
designated, respectively, as (E) and (B) in the insets of Figures 5(b) and 6(b). Each of the modes (a)
and (p)may be excited by considering a corresponding type of load eccentricity, designated also as
cases (a) and (P), respectiveiy, in Figures 5(a) and 6(a).
I
I = 20 (in)
Cross-secttonal Area of Members = 0 0625 (in2)
Euler Buckling L o a d of Each Member (Treoted
as a Pinned- Pinned Beam) = 8 I (lb)
Case I
e/l = -0001 (Mode ( a ) )
C a s e 2 e/l = 0001 (Mode
(p))
Figure 5(a). Schematic of Bntvec and Chilver's two-bar frame
84
I8 0
160
I4 0
12.0
10 0
80
00
10
20
30
S(m)
Figure 5(b). Two modes of post-buckling deformation for problem of Figure 5(a)
Figure 5(b) shows the presently computed post-buckling P vs. 6 relation for the two-member
frame of Figure 5(a), along with the experimental and analytical results reported in References 16
and 17. The present results agree excellently with those in References 16 and 17, except for the mode
(p)deformation, in which case the present results are close to the analytical results of References 16
and 17, while experiment appears to predict a much stiffer response than either the present results
or the analytical results of References 16 and 17. Similar observations apply to the results given in
Figure 6(b) for the post-buckling response of the three-member frame of Figure 6(a).
The fourth example is that of a right-angled frame, shown schematically in Figure 7(a). This
structure was first studied experimentally and analytically by Roorda2' and Koiter,2 and later
analysed by Argyris and DunneZ2to demonstrate the imperfection sensitivities of structures.
Recently, this problem was also analysed in Reference 9. The dimensions and material properties of
the members are identical to those used in Reference 9 and are indicated in Figure 7(a). Based on
the experience with the example of a beam considered in Section 3 (see Figure 3a), each of the
members in Figure 7(a) is modelled by two elements of the present type, derived in Section 2. Five
different cases of load eccentricity, with e as marked in Figure 7(a) being given the values (ell)=
00001; 0.01; 0.05; - 0001; - 0.01, respectively, are considered. Figure 7(b) shows the presently
85
LARGE FRAME STRUCTURES
2 0 (in)
Cross-sectionol Area of Members = 0 0625 (rn2)
Euler Buckiing Load of Each Member (Treated
as a Pinned-Pinned 8eamJ = 8 I (Ib)
Case I e l l = -0001 (Mode
( 0 ) )
Case 2 e/i
(p))
:
0001 iMode
Figure @a). Schematic of Britvec and Chilver’s three-bar frame
I
30 0
25 0
-
Present
(IEiement per Member)
-
Britvec’s Analytical
m
0
Experimental
2QO
15 0
100
00
L
00
10
20
30
6hn)
Figure 6(b). Two modes of deformation of the three-bar frame
, E= 10000 0 (PSI)
I =I0000(in)
Figure 7@). Eccentrically loaded right-angled
frame
F$ =13.885E1/i2( e . 0 )
, I = 600000 (In4)
A - 6 0 0 0 (rn2)
IP
-10
1
0
10
20
30
0.00
00
02
04
06
08
10
12
14
P/pE
0.05
0.10
0.15
-
Q.20
e/l = 0.01
Wi
e/l = 0 01
Figure 7(b). Variation of corner rotation of a right-angled frame for various Figure 7jc). Variation of vertical displacement (of the corner) with
values of load eccentricity, (e/l), in the post-buckling range
load, for various values ofload eccentricity @/I), in the post-buckling
range
-20
en = 0.00)
87
LARGE FRAME STRUCTURES
2P
I
-
B =I0 (in)
, H = 0 25 (in)
I :85(ln)
, E -16 (PSI)
Figure 8(a). Schematic of a four-member square frame
Jpi'
Present (4 Elements)
4. c
3C
2.c
I.0
0.c
00
02
04
06
08
Figure 8(b). Variation of displacements 6 , and 6, (see Figure 8a) with load for a square frame
88
K. K O N D O H A N D S. N. ATLURI
Jv'
El
+Present (4 Elements)
4c
30
20
1.0
M/PI
0.0
0.0
0.2
0.4
0.6
0.8
Figure 8(c). Variation of moments M I and M , (at points 1 , 2 in Figure 8a) with load, for a square frame
4 1154
2 4183
I6474
09391
I
Figure 8(d). Deformation profiles at various load levels [or a square Frame
LARGE FRAME STRUCTURES
89
computed results (for each of the five e values) for the P vs. 0 (the rotation at the corner, as defined in
Figure 7a) relations, along with the available numerical results of Karamanlidis, Honecker and
Knotheg (i.e. for (ell)= 0.01 and 0.05) and the analytical results of Koiter” for the case of zero
eccentricity (e = 0) of the load. From Figure 7(b), it is seen that when the imperfection (e) is very
small (i.e. e 2: rt: 0.001), the present solutions agree excellently with those of Koiter,’ (e = 0), in the
range of small deformations ($ N 10 degrees). However, the present numerical results indicate that
the structure stiffens gradually, as the post-buckling deformation progresses. This apparent effect
of very large deformations is also confirmed by Roorda’s2* experimental results. Thus, the present
results appear to be accurate over a wide range of deformations. Moreover, for the values of (ell)=
N 0.001 and (ell)= 005, the present results are in excellent agreement with those of Karamanlidis
et al.’ However, it should be remarked that the present solutions are based on using four elements
to model this two-member frame, while Reference 9 uses 18 elements to model the same frame. To
provide a further insight into the post-buckling response, and imperfection sensitivity, of this
simple frame, the presently computed variations of the displacement 6 (see Figure 7a) with load P,
for each of the five values of load eccentricity, e, are shown in Figure 7(c).
The final example concerns a four-member frame subjected to point loads, as sketched in Figure
8(a). The geometric and material data of the members, which are identical, is given in Figure 8(a).
Because of symmetry, one-quarter of the frame (the rectangle 1-2-3) alone is modelled by using
four elements (two each in segments 1-2 and 2-3, respectively, as in Figure 8a). The presently
computed variations of displacements 6, and 6, (as defined in Figure 8a) with the applied load P
are shown in Figure 8(b),along with comparison results of Lee et aLZ3and the theoretical results.24
The variations of the presently computed moments M , and M, (at points 1 and 2, respectively)
with the applied load are shown in Figure 8(c),along with the theoretical results.24 Figures 8(b)and
8(c) illustrate the excellent accuracy of the present simplified method. Lastly, the profiles of
deformation of the frame at various levels of applied load P are sketched in Figure 8(d).
5. CONCLUSIONS
In this paper, a simple and effective way of forming the tangent stiffness matrix has been presented
for finding the large-deformation and post-buckling response of large frame-structures of the type
that are contemplated for use in outer space.
The salient features of the present methodology are:
1. An explicit expression is given for the ‘tangent stiffness’matrix of an individual element (which
may then be assembled in the usual fashion to form the ‘tangent stiffness matrix’ of the frame
structure). The formulation that is employed accounts for (a) arbitrarily large rigid rotations and
translations of the individual element, and (b) the nonlinear coupling between the bending and
axial stretching motions of the element.
2. The presently proposed simplified methodology has excellent accuracy in that only one element
may be sufficient,in most cases (ofpractical interest in the behaviour ofstructural frames), to model
each member of the frame structure. Inasmuch as the relative (non-rigid) rotation of a differential
segment of the present element is restricted to be small, a single element alone is not enough to
model the post-buckling response of an entire beam column undergoing excessively large
deformations as in an elastica. However, when considered as a part of a practical frame structure,
the situation of each member of the frame undergoing abnormally large deformations, as in
elastica, represents a pathological case.
3. Because of features 1 and 2, the present method is a computationally inexpensive method to
90
K. KONDOH AND S. N. ATLURI
analyse frame-structures and, therefore, is of considerable potential applicability in analysing large
space-structures of the type that are currently being considered for deployment in outer space.
While the studies herein have been restricted to large elastic deformations of planar frames, the
natural extensions of the present methodology for space-frames undergoing large elastic as well as
inelastic deformations are subjects of forthcoming companion papers.
ACKNOWLEDGEMENTS
The results presented herein were obtained during the course of investigations supported by the
Wright-Patterson Air Force Base, USAF, under Contract F33615-83-K-3205 to Georgia
Institute of Technology. The authors acknowledge this support as well as the encouragement
received from Dr. N. S . Khot. It is a pleasure to record here our thanks to an associate, Ms. J. Webb,
for her assistance in the preparation of this typescript.
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EM6, 217-255 (1963).
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deflections’, Q. J . Mech. Appl. Math., 17(4), 451-469 (1964).
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