NUMERICAL INVESTIGATION BASED ON THE FINITE ELEMENT METHOD

OF THE POSTBIFURCATION BEHAVIOR OF THIN-WALLED

STRUCTURES

G. V. Isakhanov and A. B. Kritskii UDC 539.3

We worked an effective algorithm for investigating the postbifurcation solutions of systems of nonlinear equations

of equilibrium of thin-walled structures obtained on the basis of the finite element method. In the work we used

an eight-node isoparametric shell finite element (MPFE) of general form with three degrees of freedom in a node.

An example is the postbifurcation behavior of two classical structures: a cantilever strip loaded by a concentrated

force at the end, and a rectangular plate compressed uniformly by a distributed load on the end face. The obtained

results testify to the high accuracy and effectiveness of the method presented here.

The problem of investigating the branching of solutions of nonlinear equations is one of the most complex problems of

the theory of stability of structures. The methods of investigating the solutions of operator equations in the vicinity of branch

points are divided into two substantially different groups.

The first group originates from the work ofA. Poincar6 [1] and A. M. Lyapunov [2, 3]. In our case the bifurcation solutions

are constructed with the aid of the asymptotic representation of the solution according to the degrees of a small parameter which

can be the parameter of load as well as any generalized unknown of the model under consideration. The system of nonlinear

equations is divided into a number of systems of linear equations which, with a view to the Fredholm alternative, can be

successively solved.

Such an approach was used with success by Budianski and Hutchinson [4], Vorovich [5, 6], Ioss and Josef [7], Moser [8,

9], and Thompson [10, 11]. We also note that when there is information on one of the branches of the solution, the problem of

constructing bifurcation solutions can be greatly simplified [4, 7-9, 11-13].

Very effective algorithms for the construction of bifurcation solutions can be constructed when the solutions are repre-

sented in the form of a series of powers of the amplitude of the bifurcation characteristic shape [7].

The second group of methods is represented in publications by Vainberg and Trenogin [14, 15, etc.], Andronov et al. [16],

Krasnosel'skii et al. [17]. This alternative contains methods of going over from the initial operator equation to the investigation

of the equation of branching, a finite-dimensional equation or a system of several scalar equations accumulating all the peculiarities

of the solutions of the initial operator equation. The solutions of the initial equation are written by simple formulas according to

the solutions of the equation of branching. To divide the spaces of solutions and equations into a finite-dimensional and an infinite-dimensional part, many authors

[14-17] use the Lyapunov-- method. The equation of branching obtained as a result contains all the information on the branching

solutions. In that case the branching solutions can be constructed by the method of indeterminate coefficients in the form of a

series with fractional powers of the parameter. The exponents ar determined with the aid of a special geometric method, viz.,

Newton's diagram [14, 15, 18, 19]. To reduce the problem of constructing bifurcation solutions to the problem of finite dimensionality, some authors of

monographs [20, 21] use central manifold.

The two presented approaches of the second group are used with success in providing existence theorems. They can also

be use for constructing solutions [22-26] but in concrete applications they led to cumbersome calculations.

Kiev Institute of Civil Engineering, Kiev. Translated from Problemy Prochnosti, No. 7, pp. 79-89, July, 1993. Original

article submitted November 12, 1991.

0039-2316/93/2507-0533512.50 �9 1993 Plenum Publishing Corporation 533

After all, however, the division of the methods into groups is quite arbitrary, they are closely connected with each other,

but it should be emphasized that the equation of branching cannot always be written in explicit form, in actual applications methods

of the first group therefore play a more important role.

Let us consider the procedure of constructing bifurcation solutions.

Nonlinear deformation, stability, and also the supercritical behavior of thin-walled shell structures on the basis of the

finite element method (FEM) are described by a system of nonlinear algebraic equations which has the following structure:

AifUj + B # k V y ~ + Dq~tUiU~Ut - 2tPi = 0 , (1)

where U i is the vector of nodal displacements of the finite-element model (i, j, k, l = 1, 2 ..... N; N is the number of generalized

unknowns of the finite-element model); A~j, Bij k, DiN are the matrices of the coefficients of the linear, quadratic, and cubic terms

of the system of equations, respectively [26-28]; Pi is the vector of nodal loads; ~ is the load parameter.

As a rule the solutions of system (1) are investigated on the basis of the method of parameter continuation. Here it is

indispensable to check whether the trajectory of equilibrium of the thin-walled structure contains limit points and bifurcation

points at which the determinant of the linearized matrix Aij ~ of system (1) ~ From the point of view of the investigation

the most complex case of loss of stability corresponds to the branching of equilibrium solutions.

We note that the structure of the system of equations of equilibrium of a shell (1) obtained on the basis of the FEM is

determined by the correlation between the deformations and displacements of the nonlinear theory of elasticity (its form also

remains unchanged for structures with different geometric shape), the boundary conditions and spatial configuration of the load;

this makes it possible to derive universal relations and algorithms for the investigation of bifurcation solutions of a broad class

of problems.

Let the point (U ~ )~~ = (U, ~ U2 ~ ..... U o, )~o) be a bifurcation point. For investigating the branching solutions it is

expedient to rewrite the equations of equilibrium (1) by expressing their terms in terms of the increase of the load parameter g

and of the increase of the displacement vector u i so that

2 .-.= )t o +/.Z , U i = U~i + u i . (2)

Substituting (2) into Eq. (1) we have

0 A~u? + Bi/kuyu k + O~klUjUkU l -- PPi = 0 , (3)

where the coefficients of the matrices Aij ~ Bijk ~ and D ~ ijk/ are determined by the coefficients of the matrices Aij , Bij k and Dijkl (1)

and the vector of the generalized coordinates of the bifurcation point Ui ~ The matrix A~j ~ is the matrix of the system of equations

linearlized in the vicinity of the bifurcation point, and it is degenerated.

We assume that the vector function fi ~ = fi i(g) is the known solution of the system (3) in some interval g. Then the function

fi~ satisfied Eq. (3) in the vicinity of the bifurcation point (0, 0), and the following equality applies:

A~ uj + ~ k "i uk + ~ L .j-k~t - ~' P~ = 0 . (4)

Since in the vicinity of the bifurcation point there exist several solutions, one of which is known, we represent the vector

of the Solution of system (3) in the form of the sum of the known solution fi i and the new required function fi i:

where

u~ = ui Oz) + ~ i , (5)

u,. (~) = ~,. g + wi/~2, (6)

7/i ~ . 1,~. i 1 d 2 ui(/U) l (7) = d/z o' =~" d~2 o"

tn the theory of bifurcation it is accepted practice to call the performed operation (2)-(7) reduction (or adduction) to the

local form. We note that the replacement of the variables (5)-(7) is nonlinear. This transformation introduces a new nonlinear regularity of change of scale along the axes of the generalized coordinates.

As a result of the substitution of (5) into Eq. (3) with (4) and (6) taken into account, we obtain

~4i] "U.i + Bi]k "U~k + bijkl "jUkUl = 0 , (8)

534

- D ~ where the coefficients of the matrices ~t ~j, B ~jk a n d / ) ~jkt are determined by the coefficients of the matrices A,:/~ Bijk ~, and ~j~a

(3), the vectors Vi and Wi (7), and they are functions of the parameter g [29], The homogeneous system of equations (8) is the initial system for constructing the bifurcation solutions. Let us consider

the case when in accuracy one eigenvalue of the matrix A~ ~ onto the eigenvector corresponding to the zero eigenvalue.

e = ('~i" SO, (9)

which determines the most effective alternative of the algorithm,

We will seek the solution of system (8) in the form of power series of the small parameter e:

~ (s) = hie + ii~ ~z + Vi ~ + . . . .

:'(~) = 4 ~ + /~ ez + :/.~3 + . . . . (lO)

Substituting (10) into (8) and equating the sum of terms of equal order to zero, we have

A~:&: = O; (11a)

+ DLi~V:ukul~ - , . + D}~2k:~:kul. + D~izyhkP'l] + ,~2[B~/k~V:h k +

( l ib )

+ + + + + + + t = o .

We investigate possible distorted states of equilibrium in the vicinity of the bifurcation point.

Equation (11a) has a trivial solution: a i = 0, which according to (5) and (10) corresponds to deformation of the structure along the ascending branch of the trajectory of equilibrium (6). Another bifurcation solution of Eq. (1 ta) of interest to us is the

eigenvector Si of the matrix A~j ~ which has a zero eigenvalue. Therefore for the first term of the expansion (10) of the bifurcation solution we write

ui = S i . (12)

To calculate the largest terms of the series (t0) we have to solve the sequence of the linear equations (11 b), (11 e). However,

the matrix A~j ~ has zero eigenvalue, and an accordance with the Fredholm alternative, the systems (1 lb) and (1 tc) have a solution

only if the vector of the free terms of the equations under consideration is orthogonal to the bifurcation proper form Si. Bearing

this fact in mind, we obtain from (1 l b) an expression for a, and from (11 c) for //:

/, = - + k@k) 1 ; ( t3a)

The relations (13a) and (13b), derived on the basis of the condition of the Fredholm alternative, make it possible with a

view to (9) to solve successively Eqs. (I lb) and (I lc) for the vectors i1 i and ~ . The vectors of the first and second derivatives

(Vi and t-u i) of the ascending branch of the solution fi ~(l,t) for the toad parameter g, which are used in the calculation of the

vectors of the free terms of systems (13a) and (13b), are found from the equations

A~V: - Pi = 0 (ViS~') = O; (14a)

535

/ 1 2

/ \ ,

' ~ t 3 :I0-7 m - t5-~-0,5 e 0.5 ;,0 ~,S 2,0 0,5 1,0 f, Su~ a,%|

d r

0

,r

8 iz to m24~ ~FE

Fig. 1. Results of the stress analysis of a cantilever: a) calculation diagram; b,

c) dependences of the vertical (UA 1') and horizontal (uA 2') displacement, respec-

tively, of the free end of the beam on the load parameter %; d) graphs of

convergence.

A}~j + ~ k ~ Y k = 0 (~iS;) = 0. (14b)

The solutions of the degenerated systems of linear equations (14a), (1 lb), (14b), and (11c) are constructed with the aid

o fa speciaIly devised algorithm which was described in [30].

The bifurcation solution obtained in the form (5), (6), and (10) satisfies the initial nonlinear operator (1) within some

range of the small parameter. The supercritical branches of the solution can be further investigated on the basis of the stranded

procedure of the method of parameter continuation.

In the present work we use an octagonal isoparametric shell finite element of general shape with three degrees of freedom

in the node. The relations for the coefficients of the system of equations (1) were derived on the basis of the moment procedure

of finite of equations (1) were derived on the basis of the moment procedure of finite elements (MPFE) which, with equivalent

computer time and capacity, is more accurate than the other finite-element procedures [26-28].

As an example let us consider the postbifurcation behavior of two classical structures: a cantilever strip loaded by a

concentrated force at the end (Fig. la) and a rectangular plate compressed by a uniformly distributed load on the end face (Fig. 2a).

It is known that in beams subjected to bending in one plane the height of the cross section is usually larger than the width,

and consequently the moments of inertia relative to the principal axes differ substantially from each other. If the difference between

the moments of inertia of the bent beam in the plane of greatest rigidity is large, a situation may arise where the plane form of

bending becomes unstable. The beam begins to bend in another plane, too, and becomes twisted which, as a rule, causes an abrupt

increase of deformations and leads to failure of the structure.

536

a

, , t----'----

P

/ / 3 -

2 - /

0"- - - 0,~- 0~ 12 1,0 2D 8/t b

Fig. 2. Dependence of the error 6 = NN/PP on the amplitude of

the bifurcation solution e. (The dashed line in Fig. 2b was plotted

without regard to "u' .)

On the basis of the analytical approach the authors of [31-34] examined a number of simple structures of this type with

some kinds of boundary conditions and loads; they obtained expressions for determining the critical load. For a cantilever loaded

by a concentrated force at the end (Fig. la) the critical load is determined by the following formula:

~ c r p _ 4,01E,/v (15) 12 " qGY t / E d ; ,

where EJy is the bending rigidity relative to the vertical axis; GJ t is the torsional rigidity; 1 is the length of the beam.

The literature does not contain any information on investigations of the supercritical behavior of thin-walled beams.

Let us consider a thin-walled cantilever (strip) with retangular cross section loaded at the free end by a concentrated force

(Fig. la) . The length of the beam i = 1.0 m, the height of the cross section h = 0.1 m, thickness t = 0.005 m, modulus of elastici ty

E = 2.1011 Pa, Poisson ratio v = 0. The concentrated force P is applied at the center of gravity of the cross section of the beam.

The investigation of the convergence of the solutions with condensation of the network showed that 2 • 24 finite elements suffice

for an approximation of the cantilever beam.

The graphs of the dependence of the vertical and horizontal displacements of the free end of the beam (point A) on the

load parameter are presented in Fig. lb, c. When the load is increased from zero to the critical value (the segment 0-1 of the curve)

a state of plane stress and strain develops in the beam; the correlation between the load parameter and the vertical displacement

of the free end of the beam is almost linear, there are no displacements along the z 1'-axis.

An analysis of the linearized system of equations of the FEM revealed the bifurcation point on the tracjectory of

equilibrium of the beam (Fig. lb, c) at whose transition the state of plane stress and strain of the cantilever beam becomes unstable.

It should be noted that the critical value of the load parameter obtained on the basis of the FEM (~L:? = 1243.4 N) is about 8%

higher than the corresponding value obtained by formula (15) (%orp = 1166.3 N) because an examination of the cantilever from

the posit ions of the three-dimensional approach, which was used in the present work, can describe its state of stress and strain

more fully. When factors such as shear, warping of the section, elongation, etc. are taken into account, the degeneration of the

nonlinear operator occurs at a larger load.

The numerical investigation carried out on the basis of the newly devised method shows that in the problem under

consideration there is a one-sided supercritical bifurcation [7], i.e., the branching solutions in the vicinity of the bifurcation point

are stable. Consequently, at a load exceeding the critical value the structure under consideration passes to the bifurcation branch

of the trajectory of equilibrium. We note that the branching solutions in this problem are symmetrical about the %-axis.

537

TABLE 1. Processor Time on an ES-1061 Computer Required for Different Stages

of Solution of the Problem

Stage of solution of the problem Processor time on Es-106l, min

Plotting of subcritical segment of tracjectory of equilibrfum 0,1649 Determination of type of singular point 0,0887 Amended position of bifurcation point 8,4867 Investigation of initial super- critical behavior 0,6223 Plotting of supercritical segments of trajectory 12,7917 Total 22.1543

A study of the magnitude of the vicinity, where the bifurcation solution of (5)-(6)-(10) is correct, showed that with an

amplitude e < 0.007 m. which represents 1.4 times the thickness of the beam, the error (the ratio of the sums of squares of the

discrepancies to the sum of the squares of the components of the load vector) does not exceed 1.10 -4 (Fig. 2b).

It should be emphasized that the supercritical behavior of the cantilever is characterized by an abrupt increase of

displacements out of its plane, and also by development of a spatial state of stress and strain. According to the results of the

investigation of the supercritical behavior of the cantilever, in deformation on the bifurcat ion branch a slight increase of the load

brings about a great increase of its displacements. In plotting the supercritical segment of the trajectory of equilibrium (Fig. la,

b) on the basis of the method of parameter continuation we chose as conductor unknown the horizontal displacement of the end

of the beam (which has maximal increase on the bifurcation branch).

Our investigation showed that the supercritical behavior of cantilevers is of a very complex nature. On the postbifurcation

segment of the trajectory of equilibrium at a load about 1.5% larger than the critical one we discovered the limit point 2 (Fig. lb,

c) upon whose passage the solution becomes unstable. Therefore, after the bifurcation point 1 has been passed, the beam can

withstand a very slight increase of the toad only; at the same time the bending and torsional deformations greatly increase.

The ability of the beam under consideration to withstand a load somewhat exceeding the bifurcation load is probably due

to the "inclusion" of shear stresses induced by torsional deformations. Upon passage through the l imit point the derivative of the

load-deflection curve (Fig. Ib, c) changes its sign and the trajectory of equilibrium begins to descend, which means that at a load

exceeding the l imit value the transition of the beam to another state of equilibrium proceeds jumpwise.

The required processor times on an ES-1061 computer for different stages of the solution are shown in Table 1.

Let us now investigate the supercri t ical behavior of a rectangular plate compressed by a uniformly distributed force on

the end face. It is known that for plates under compression the critical load can be determined very accurately on the basis of the

theory of linear stability [35-37]. For a rectangular plate compressed by a uniformly distributed end-face force in two directions

the critical pressure is determined by the formula [31, 35, 38]

2Crq = (1 + [b/a] 2) x2D b2 , (16)

where a is the length, b the width of the plate d = Et3/12(1 - v 2) is the flexural r igidity; t is the thickness of the plate; E is the

modulus of elasticity; v is the Poisson ratio.

The publications by Vorovich [6, 39, 40], Srubshchik and Trenogin [25], Srubshchik and Yudovich [41] contain a

qualitative analysis of the behavior of compressed plates with different shape after loss of stability. These authors proved the

existence of bifurcation solutions, determined the number of branched in different cases, analyzed qualitatively the special features

of the behavior of solutions and their stability. This kind of investigation permits qualitative evaluations of the supercritical

behavior of thin-walled structures but for the stress analysis of complex shaped shells numerical methods are indispensable.

Let us consider a plate that is rectangular in plan and spin-supported, with sides 1 and 2 m long, 0.002 m thick, compressed

in two directions by a uniformly distributed load on the end face. The material of the plate obeys Hooke's law, and the corresponding

elastic constants adopted are E = 2-10 ~l Pa, v = 0.

Investigations [31, 36, 38] showed that a rectangular plate compressed in two directions loses stability by transition to

bifurcation form of equilibrium which has the property of symmetry about the x 2' and xY-axis. This makes it possible to examine

a quarter of the plate with the corresponding conditions of symmetry (Fig. 3a). According to the data of the investigation of the

convergence of the approximate solutions a network of 8 • 8 finite elements suffices for the approximation of a quarter of the plate.

538

11

b ~'I0 ~ c ).~.10 3

o,t~- o, i I I I t I ,! n l T i I

- 3 - 2 - ! 0 I f ~ t . f O "3, u ~,--~ m ! 9 5t,t . 'IO~m

0 ~ 0 b 2 MPa ZSt ~b~ MPa S z a'

Z 2 ' d

~b~ MPa

-___..O, ff / l

ZZ'

Fig. 3. Nonlinear behavior of a rectangular plate: a) finite-element representation

of a quater of the structure; b, c) graphs of load vs. deflection for points O, A, and

B; d) curves of the bending stresses at the instant of origin of plastic deformations

at the center of the plate.

The curve of load vs. deflection of the plate (point A) is shown in Fig. 3b. When the load does not exceed the critical

value, the plate is in a momentless state of plane stress and strain, all the finite elements of the plate are equally stressed, and

their state of stress increases almost linearly with increasing load.

An analysis of the linearized system of equations of the FEM revealed on the trajectory of equilibrium the bifurcation

point 1 at a load %~rq = 1.645.103 (Fig. 3b, c). This shows that at a load exceeding the critical load (~,q > ~Yq) the plane form of

equilibrium of the plate becomes unstable.

The study of the magnitude of the vicinity in which the bifurcation solution (5)-(6)-(10) is correct showed that with an

amplitude e < 0.009 m, which is about half the thickness of the plate, the error does not exceed 1.10 4.

Numerical investigation of the bifurcation solutions based on the method worked out in the present work confirmed the

known fact that in the trajectory of equilibrium of a compressed rectangular plate there is also one-sided symmetric supercritical

bifurcation, i.e., the branching solutions are stable. Consequently, when the load exceeds the critical value, the plate passes to

the bifurcation branch of the trajectory of equilibrium.

Loss of stability of the plate is accompanied by an abrupt increase of the flexural form of equilibrium of the plate, and

in view of that the plotting of the supercritical segment of the trajectory of equilibrium requires much more computer time than

the other stages of solution of the problem.

539

We note that in modeling of the nonlinear behavior of thin-walled structures in the present work, the origin of plastic

deformation is checked by the method of parameter continuation which in the devised software is carried out on the basis of the

Huber-Mises criterion. In the presented problems (Figs. 1, 3) the origin of plastic deformations was discovered after loss of

stability of the structure, by deformation along the bifurcation branch.

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541