INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 4, 583-586 (1972)

ON THE RIGID DISPLACEMENT CONDITION

P. CLBMENT AND J. DESCLOUX

Department of Mathenratics, Ecole Polytechnique FJdJrale, Lausanne, Switzerland

SUMMARY This paper shows the importance of the rigid displacement condition when the finite element method is used for structures having little rigidity. The theoretical analysis is confirmed by numerical results obtained with a curved beam.

INTRODUCTION Consider an elastic structure with no boundary condition for which the mathematical model includes the expression of the energy of deformation W(6) in terms of the displacement 6. We say that the mathematical model satisfies the rigid displacement condition if the energy of deformation is nul l for any rigid displacement. Most models commonly used satisfy this require- ment; however Dupuis* has remarked that the model for shells established by Green and Zerna in Reference 1 does not satisfy the rigid displacement condition. Consider now a finite element model of this structure (using the same expression W(6) for the energy of deformation). We say that this model satisfies the rigid displacement condition if any rigid displacement of the structure can be represented exactly. If a sequence of finite element models is defined, it is then meaningful to speak (for a particular case of loads and boundary conditions) of the convergence of approximate solutions towards the ‘exact’ solution. In Reference 2 it is claimed that a necessary condition of convergence is that the models of the sequence satisfy the rigid displacement condition. This statement is, in general, wrong; however, it is true for some particular structure and finite elements, as Goelf has proved in Reference 3. In the following we shall say that a structure with defined boundary conditions has ‘little rigidity’ if there exists a ‘large displacement’ with corresponding ‘little’ energy of deformation.

The purpose of this paper is to show that for a structure of little rigidity it is advantageous to use finite element models having the rigid displacement property. Our argument will be partly heuristic, partly mathematical. In order to avoid abstractness, we shall consider only a very particular case of curved beam, although the same kind of analysis is valid for other structures- shells for example.

A CURVED BEAM We consider a curved beam in the xy plane. Let s be the curvilinear abscissa, u(s) = (u,(s), uy(s)) be the displacement at the extremity of r(s), #(s) be the angle between the x axis and the tangent of the beam. If L denotes the length of the beam, the energy of deformation is defined by

~~~

* G . Duptiis, Brown University, Division of Engineering, Providence, Rhode Island 02912, USA. t J.-J. Goel, Department of Mathematics, Faculte des Sciences, Universite de Fribourg, Switzerland.

Received 12 January I970 Revised 12 March I971

0 1972 by John Wiley & Sons, Ltd.

583

584 P. CLBMENT AND J . DESCLOUX

X

Figure 1 . Description of the beam

where E is the modulus of elasticity, J the moment of inertia, F the section area and

d ti d iiY w = -sin+-'+cos+-

ds ds

one can verify that this mathematical model satisfies the rigid displacement condition. Let for any vectorial functions f(s) (0 < s < L),

P(f) can be reasonably defined as a memure of the displacement u.

conditions From now on we assume that the beam is fixed at x = 0 and x = L, i.e. satisfies the boundary

U,(O) = U,(O) = u,(L) = u,(L) = 0

Let Q be the set of vectorial functions, f(s) = (f,(s), ,fJs)) such that f , and fY are continuous, have a first piecewise continuous derivative and finally satisfy the boundary conditions

f,(O) =f,(O) =f&) =f,(L) = 0

One can look among all displacements of measure 1 which displacement induces the less energy of deformation; let

and + E Q, P(+) = I , be a displacement for which the minimum is reached. Following our 'definition' of the first section, the curved beam will have 'little rigidity' if h is small; this is effectively the case in our example where X = 0.066 t/m2. ( A and \L are the smallest eigenvalue and corresponding eigenfunction of the differential equations of the displacement; if p is the

RIGID DISPLACEMENT CONDITION 585

density and if we choose p = 7.7 kg/dm3, T,,,, = 2rJ(pF/A) is the greatest period of the free oscillation. Here T,,, = 1.81 sec)

TWO FINITE ELEMENT MODELS

Briefly, we can say that a finite element model defines a finite dimensional subspace of Q.

Model I The interval 0 6 s 6 L is divided in n equal sub-intervals. On each sub-interval u,(s) and uy(s)

are cubic polynomials in s; the parameters are the values of u, and uy and of their first derivatives with respect to s at the nodes. One can verify that this model does not satisfy the rigid displace- ment condition.

Model 2 Instead of s, one uses x as an independent variable. The interval 0 6 s < L is divided in n equal

sub-intervals. On each sub-interval u,(x) and u,(x) are cubic polynomials in x ; the parameters are the values of u, and u!, and their first derivatives with respect to x at the nodes. One can verify that this model does satisfy the rigid displacement condition. (This is a particular case of a general method due to Dupuis which allows to find elements for curved beams and shells satisfying the rigid displacement req~irernent.)~

\L2 are defined by

Let ill and Q, be the subset of !2 relative to the models 1 and 2 respectively; A,, A,,

Now we can express our fundamental heuristic argument. Consider the displacement +: any small portion of the beam is displaced ‘almost’ rigidly since its energy of deformation is very small; consider for which the model does not satisfy the rigid displacement property; any small portion of the beam is not displaced rigidly; ‘consequently’ one can expect that W(ql , ) = Al+ A whence for \L2 the same argument would lead to the conclusion W(+J = A,- A; this conjecture is verified in our case for 8 6 n 6 24 (Table I).

Table I. Ratio of eigenvalues

n

4 8

12 16 20 24 32 40 48 64 80

2.68 1.90 1.68 1.55 1 *42 1.31 1.15 1.07 1.04 1.02 1.01

1.64 1.12 1.05 1.02 1.01 1.01 1 .oo 1 .oo 1 .MI 1 .oo 1 ~ 0 0

AN INEQUALITY FOR THE APPROXIMATION BY FINITE ELEMENTS

Letf(s) denote the density of force per unit length acting on the beam; let W(S)EQ be the resulting ‘exact’ displacement, wi(s) E !Ji be the resulting ‘approximate’ displacement obtained by model i = 1,2.

586 P. CLEMENT AND J. DESCLOUX

From the variational principles it is well known that W(w,) < W(w); consequently, we can write

and from definition of hi we finally get

For i = 1 (first model) and f(s) such that W(w)-hP(w) then P(wl)<P(w) since hi& A, i.e. the approximation w1 will be entirely wrong.

This circumstance is satisfied in our example when the beam is loaded by a horizontal force F = I t acting at the top D (Figure 1); indeed one has W(w)/P(w) = 0.068 t/m2; the exact horizontal displacement is ul) = 0.443 m; denoting by u ~ , ~ (i = 1,2), the horizontal displacement obtained at D by model i, we give in Table I1 the ratio uD,Juu for i = 1,2 and several values of n ; for n<32 the finite element model I gives poor results; however, for n248 the results become acceptable and in fact for n --f co, one has convergence.

Table 11. Ratio of the displacements

I? I b , I I U D U f l , 2 / I l D _ _ - -- -_ - -

4 0.248 0.655 8 0.409 0.863

12 0.460 0.935 16 0.565 0.965 20 0.639 0.980 24 0.714 0.996 32 0.842 0.998 40 0.920 1 .ooo 48 0.954 1 .ooo 54 0.976 1.000 80 0.980 1.000

REFERENCES

I . A. E. Green and W. Zerna, Theoretical Elasticity. Oxford University Press, 1963. 2. 0. C. Zien kiewicz, The Finite Element Methodin Structuraland Continuous Mechanics, McGraw-Hill, London,

3. J.-J. G o d , ‘Construction of basic functions for numerical utilisation of Ritz‘s method’, Nuni. Math. 12,

4. G . Dupuis and J.-J. Goel, ‘A curved finite element for thin elastic shells‘, lnt . J. Solids Struct. 6 , 1413-1428

5. J . Khanna, ‘Criterion for selecting stiffness matrices’, A l A A Jnl, 3, 1976 (1965). 6. J . Khanna and R. F. Hooley, ‘Criterion for selecting stiffness matrices’, A I A A Jnl, 4, 2105-21 11 (1966). 7. W. E. Haisler and J. A. Stricklin, ’Rigid-body displacements of curved elements in the analysis of shells

1967.

435-447 (1968).

( I 970).

by the matrix-displacement method’, A l A A Jnl, 5, 1525-1 527 (1967).