on the performance of mindlin plate elements in modelling ...rectangular and triangular plates...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 23, 1229% 1244 (1 986)

ON THE PERFORMANCE OF MINDLIN PLATE

MEDIUM INTERACTION: A COMPARATIVE STUDY ELEMENTS IN MODELLING PLATE-ELASTIC

R. K. N. D. RAJAPAKSE* AND A. P. s. SELVADURAI~

Department of Civil Engineering, Carleton University. Ottawa, Ontario, Canada K I S 5 B6

SUMMARY

This paper discusses the finite element analysis of the flexural interaction between an elastic plate and an elastic half-space. The applicability of certain Mindlin-type plate elements, based on reduced integration techniques, to this class of interaction problem is investigated. A series of numerical computations reveal that those elements which possess spurious zero-energy modes are deficient in modelling properly the interaction phenomenon. It is also found that the ‘heterosis’ plate element reported in the literature is capable of modelling the plate-elastic medium interaction very efficiently.

INTRODUCTION

The analysis of the interaction between structural elements, such as beams or plates, and elastic media is of importance to several branches of engineering. In the context of civil engineering, this class of problems provides useful analogues for the study of the interaction between structural foundations and the supporting soil Both analytical and numerical methods have been used quite extensively to study the interaction between structural elements and elastic media. In connection with the finite element study of these contact problems two approaches are possible. In the first approach both the plate and supporting elastic medium are represented by their finite element equivalents. This approach is particularly useful in instances where both the structural element and the elastic medium possess complicated geometry. In the second type of analysis the plate is represented by a finite element idealization and the response of the elastic medium, such as an elastic half-space or an elastic layer, is represented by an analytical equivalent. Both approaches have been applied quite extensively in the study of elastostatic contact problems.

The earliest application of finite element methods to the analysis of plates on elastic foundation is due to Cheung and Zienkie~icz.~ These authors considered the analysis of rectangular plates resting on an elastic half-space and subjected to uniformly distributed or concentrated loading. The finite element idealization of rectangular plates with free edges is achieved by choosing a rectangular plate element with three degrees of freedom at each nodal point. Smith5 has investigated the axisymmetric flexure of a circular plate resting on an elastic half-space by using an annular finite element. Similar results for the problems of plates resting on homogeneous and non- homogeneous media are given by Wood and Larnach6 and Wood.’ Finite element studies of

* Post-doctoral fellow ‘Professor and Chairman

0029-5981/86/101229-16$08.00 0 1986 by John Wiley & Sons, Ltd.

Received 17 April 1985 Revised 14 November 1985

1230 R. K. N. D. RAJAPAKSE A N D A. P. S. SELVADURAI

rectangular and triangular plates resting on an isotropic elastic half-space are presented by Svec and McNeice' and Svec and Gladwell.' A triangular plate element with 33 degrees of freedom is used to idealize the flexural behaviour of the plate. The displacement function is represented by a seventh degree polynomial. This particular triangular plate element preserved interelement continuity of slope and displacement.

Fraser and Wardle" have applied finite element techniques to examine the flexural behaviour of uniformly loaded plates resting on an elastic layer. The influence of friction and adhesion at the interface of the loaded raft foundations resting on homogeneous, non-homogeneous and transversely isotropic elastic media has been examined by Hooper,' 'J using finite element techniques. Non-linear effects due to loss of contact between the plate and the elastic medium are examined by Cheung and Nag13 and Fremond.14 References to further applications of finite element techniques to elastostatic contact problems are given by Sel~adurai,~ Desai and Christian,' Zienkiewicz,' Gudehusi6 and Valliappan et aE.17

In recent years, several efficient plate bending elements have been reported in the literature.'8-21 Among these, plate bending elements which are based on Mindlin's plate theory and reduced integration techniques are found to be extremely effective and efficient in application. Hughes et ak2' proposed a simple and efficient four-node quadrilateral element which uses bilinear shape functions. Using a reduced quadrature on the shear contribution, Hughes et ~ 1 . ~ ~ showed that this element performs excellently over the thin plate range. They also proposed techniques for handling both extremely thin and thick plates. Pugh et aLZ3 investigated the performance of five different quadrilateral elements by conducting a series of numerical simulations. Based on their investigation they concluded that a nine-node Lagrangian element with an appropriate reduced quadrature rule is nearly optimal as a generalized plate element. Hughes and CohenZ4 presented a plate bending element which they termed the 'heterosis' element. This element employs Lagrange shape functions and rotations and serendipity shape functions for the displacement.

An examination of the literature indicates that although these plate bending elements have been extensively investigated, their applicability to the study of contact problems involving plates and elastic media requires further attention. The present study, which investigates the applicability of Mindlin type plate elements to the study of plate-elastic medium interaction, is motivated by the efficient performance displayed by such elements in the study of purely plate bending problems. Results of extensive numerical experiments indicated that the plate bending elements proposed by Hughes et aLZ2 and Pugh et aLZ3 fail to model contact interaction between the plate and the half- space region. The plate bending element proposed by Hughes and C ~ h e n , ' ~ on the other hand, performs more accurately and efficiently in the modelling of the contact problem. The causes of improvement and loss in accuracy of the respective plate bending elements are interpreted from the element characteristics. The effects of complete quadrature in integration are also examined for plate elements reported in References 22 and 23. The paper presents numerical results for the problem of contact between a centrally loaded plate and an elastic half-space. The methodology presented in the paper could be further extended to cover the analysis of contact between a plate and an elastic layer.

FINITE ELEMENT ANALYSIS

The structural element is modelled by a Mindlin plate element which has three degrees of freedom at each nodal location. These are the displacement win the z-direction, the rotation 6, about the y- axis and the rotation 6, about the x-axis (Figure 1). It is assumed that the displacements of the plate and the displacements of the half-space region are compatible over the entire region of the plate.

MINDLIN PLATE ELEMENTS 1231

Figure 1. Definition of plate degrees of freedom

mar discretized by plate elements; E, , up, h, /-

grid formed on f r e e s u r f a c e of half s p a c e by nodes connected

X

Figure 2. Finite clement idealization of plate-elastic medium system: *-nodes at which continuity of w exists


This compatibility between the displacement of the plate and the half-space region at the four nodal locations (Figure 2) of each element ensures a restricted form of bilateral contact, The restricted form arises from the fact that continuity of 0, and 8, is not enforced at the nodal locations. It may be noted here that the analytical expressions for loading of a half-space by a (convex) region (S) of uniform or distributed surface normal traction (these are the contact stresses) result in singular behaviour of slopes at the boundary of S. Hence continuity of slopes at nodaf locations cannot be enforced.

Plate elements In the present study we employ the Mindlin plate elements reported in References 22-24. The

characteristics of these plate elements and their formulations are fully documented in these articles. For completeness, the major characteristics of the bilinear, Lagrange and heterosis elements are summarized in Figure 3. The explicit forms for serendipity and Lagrange shape functions expressed in terms of element local co-ordinates are given by Zienkiewicz.' The formulations of the element stiffness matrix and equivalent nodal force vectors follow standard procedures given in Reference 15. Appropriate quadrature rules which are selected for integration are also summarized in Figure 3.

Elastic medium In the present study we restrict our attention to situations where the elastic plate is in contact

with an isotropic elastic half-space. Cheung and Zienkiewicz' derived the stiffness matrix of the half-space region by using Boussinesq's fundamental solutionZ5 for the loading of a half-space

tagrange" Weterosis" I * " Bi lineor"

w Shape functions serendipity Logrange serendipity

serendipity Lagrange Lagrange ox and 9, Shape functions

3x3 bending 2x2 Shear

3x3 bending 2 x2 Shear

2 x 2 bending I integration scheme (Gauss) I x I Shear

2 I 0 no. of spurious zero- energy modes

constraint index 1 4 3

Figure 3. Mindlin plate elements considered in the present study: *---us, ff,, 6, degrees of freedom; 0-4, and (2, degrees of freedm


region by a concentrated normal force. Their method consists of computing the displacement ,fij at node i due to a unit normal load at nodej. Since Boussinesq's fundamental solution results in singular behaviour for fi i , Cheung and Zienkiewicz4 used the solution corresponding to uniform normal loading over a rectangular area to compute fii. This value can also be obtained by replacing the rectangular region of uniform loading by an equivalent circular region. The authors have also developed a simplified scheme by which fij and fii can be computed for the case of an elastic layer which is underlain by a rigid stratum. The layer exhibits bonded contact at this interface. Details of these developments are given in Reference 26.

We note here that irrespective of the thickness of the supporting elastic medium, fij can be evaluated quite accurately either using analytical or numerical schemes. The stiffness matrix for the soil medium denoted by K, is obtained by a simple inversion of the matrix fij, i.e.

K s = f i j ' (1) The matrix equation governing the elastostatic interaction between the thick plate and the isotropic elastic half-space can be reduced to the form

(2)

where (F} is the generalized force vector, K, is the stiffness matrix of the plate, (a} is the vector of nodal degrees of freedom and C, and C , are constants which depend on the elastic constants of the half-space region and the elastic constants and the geometry of the plate region.

(Fl = CIK, + C,KsG4

COMPUTER PROGRAM

Based on the concepts outlined in the preceding sections, the authors have developed a computer program to analyse the elastostatic contact between a plate of arbitrary geometry and an isotropic elastic half space or an elastic layer. The contact is assumed to be bilateral and smooth. The program also has the ability to combine beam and plate elements to reflect regions with high stiffness, which may be encountered in foundation systems. The main structure of the program is similar to the program STAP described by Bathe.27 A variety of other features, such as Winkler type foundations, rotational restraints at the boundary of the plate and the effects of distributed and local loadings, can be accommodated. A detailed description of the program, including a listing of the source program, a users manual, examples of data files, sample problems etc., will appear in a forthcoming publicationz6 on the structural analysis of raft foundations.

DISCUSSION OF RESULTS

Prior to the presentation and discussion of the results it is appropriate to define a relative stiffness parameter ( K G ) for the plate-elastic medium system. A number of such parameters can be chosen, and a detailed account of these expressions is given by Sel~adurai.~ In the present discussion we adopt the relative stiffness parameter K G defined by Gorbunov-Posadov and Serebrjanyi,28 i.e.

12741 - v,") E K, = (3)

where 21, 2b and h, are, respectively, the length, breadth and thickness of the rectangular plate. The limiting case of a rigid raft is obtained whenz8

KG < 8 f(l/b)'" (4)


Following References 2, 3 and 28 it can be shown that the uniform displacement of a rigid square plate (dimensions 21 x 2E) resting in smooth contact with an elastic half-space is given by

0*913(1- $)Pg W g =

2 1 4

where Po is the magnitude of the concentrated load acting at the centre. The three different plate bending elements considered in the present study are based on the

theory of Mindlin plates, where the degrees of freedom w, 0, and 8, are considered as independent quantities. In the application of these plate elements to model thin plates, the constraints (0, - dw/ax) = 0 and (0, - dw/dy) = 0 have to be employed to obtain realistic solutions. It can be shown that the formulation based on Mindlin’s plate theory leads to a system of equations of the form

c r 2 .JL

(F)=Kb+ 12(1+ ’ Vp)h,2 KS (4 where vp is Poisson’s ratio for the plate material, h, is the plate thickness, K, and K, are the stiffnesses due to bending and shear deformation, respectively, and L is a reference length (e.g. length or width) of the plate.

For a thin plate, L2/h; -+ a. To obtain a proper solution for equation (6) under such conditions K, should be singular, otherwise shear deformations dominate and the system would be overconstrained. The technique of reduced i n t e g r a t i ~ n ~ ~ - ~ ~ , ~ ~ - ~ ensures the singularity of K,, and a proper solution to (6) is obtained over the ‘thin plate’ range. However, we should ensure that the total stiffness matrix [K, + (5L2/12(1 + v,)h;)Ks] is non-singular.

the reduced integration Mindlin plate element has one disadvantage. Owing to the low quadrature employed in evaluating K,, the total stiffness [K, + (5L2/12(1 + v,)h,2)K,] tends to exhibit zero-energy modes in addition to the normal rigid body modes. In other words the total stiffness matrix is rank deficient. As presented in Figure 3, the ‘bilinear’ and ‘Lagrange’ elements have two and one additional zero-energy modes, respectively. However, the ‘heterosis’ element is shown to have the correct rank and hence the correct number of rigid body modes. It may also be observed that the global stiffness matrix resulting from a displacement-based finite element analysis is singular prior to the introduction of boundary constraints. Once the requisite boundary conditions are applied, the rows and columns corresponding to the restrained degrees of freedom are eliminated, the resulting matrix is non-singular and a realistic solution is obtained. In situations where suficient boundary constraints are not invoked, the singularity of the stiffness matrix is not removed and the resulting displacements follow mechanisms.

In all previous studies the performances of Mindlin plate elements were evaluated on the basis of solutions derived for conventional plate bending problems. Here we use the term conventional to denote the class of plate bending problems where elementary boundary conditions (e.g. simply supported, clamped) were available to produce a well-constrained system such that the resulting solutions were admissible. In the finite element modelling of the contact between a plate and an elastic half-space, no boundary constraints are available within the plate region. Instead the addition of the stiffness matrix of the half-space region makes the final global stiffness matrix non- singular. It may be noted that the stiffness matrix of the half-space is always non-singular due to the fact that the displacements of the half-space region do not contain any rigid body displacement modes. Thus in the present study we discuss the performance of Mindlin plate elements in a situation which has not been reported previously.

Even though it is considered as very efficient and accurate in the modelling of thin

MINDLXN PLATE ELEMENTS 1235

3.0

2 .o

I .o

- Heterosls [24], 4x4 mesh K, = 471.23 (lo4)

x Rorzaque [so] KG = 47.12 (los)

X 0.5 1.0 - 0 28,

0 i 0 0.5

X - 212

I .o

Figure 4. Displacement along the centre line for a centrally loaded square plate resting on an elastic half-space (v, = V, = 0.15)

For reasons which are explained subsequently, we first consider the performance of the ‘heterosis’ element. Figures 4(a) and 4(b) show the normalized vertical displacements along the centre line of a centrally loaded square plate resting on an isotropic elastic half-space. The displacements are normalized with respect to the uniform settlement of a centrally loaded rigid square plate given by equation (5). Figure 5 shows the displacements along the edge Y= 0. The results presented in Figure 4 correspond to three different values of the relative rigidity parameter kl,, representing flexibfe, as well as nearly rigid, plates for two different values of the ratio hp/21.

In all the cases considered (Figures 4 and 51, displacement profiles represent smooth curves and approach the correct limit of uniform settlement as the value of K , decreases. It is interesting to note that the condition for a rigid plate as given by equation (4) is very nearly satisfied. In the absence of any analytical solution for the case of an elastic plate, we employ the plate element presented by Razzaque3’ to model the same problem. As can be seen from Figures 4 and 5, we obtain excellent agreement with those results obtained by using the ‘heterosis’ element.

It should be noted here that the plate element given in Reference 32 is based on the classical thin plate theory and always results in stiffness matrices with correct rank. The ‘heterosis’ element, even


3 .O - Heterosis I241

O i I

X 0.5 28 - 0 1 .0

Figure 5. Deflection along the edge y = 0 for a centrally loaded square plate resting on an dastic half-space (vp = Y, = 0.15)

0.95 t-- I

.a

0.80 }

0.70 I

0 0.1 0.2 0.3 0.4 0.5 I

EF Figure 6. Convergence of displacement for a centratIy loaded rigid square plate resting on an elastic half-space

though based on the Mindlin plate theory, has the correct rank and hence the correct number o rigid body modes. Thus it has the capability to model interaction problems very accurately and tf achieve the correct rigid body mode as K , decreases.

In order to assess the efficiency and accuracy of the 'heterosis' element with respect to modeHin of the contact between the plate and the supporting elastic medium, we compare its performanc with certain results reported by Fraser and Ward1e.l' These authors examine the problem interaction between a centrally loaded square plate and an elastic half-space. Figure 6 shows tl. convergence study for the problem. It is evident that the present scheme has faster convergence ar greater accuracy than the scheme used by Fraser and Wardfe." It should be also mentioned th the heterosis element is much easier to formulate and c o ~ p u t a t ~ o ~ a ~ ~ y convenient than the schen used in Reference 10.


1.8

Figure 7 shows the contact stress distribution along the centre line of the centrally loaded rigid square plate. The approximate results developed by Gorbunov-Posadov and SerebrjanyiZ8 are also presented in Figure 7 for the purpose of comparison. Except for locations near the edges of the plate, where the analytical solution yields singular stresses, the present finite element study gives results which compare favourably with the approximate analytical scheme. It is possible to obtain accurate results showing the singular nature near the edges of the plate by employing a very refined mesh near the edges of the plate. This procedure, however, would make the analysis quite expensive, such refinements are usually neglected in most practical applications.

Figure 8 shows a comparison of M,, along the line y = 0.931 (i.e. the line joining Gauss points near to the centre line) with equivalent results given by Gorbunov-Posadov and Serebrjanyi” for the variation of flexural moment along the centre line (y = E ) . These results confirm the efficiency of the ‘heterosis’ element in predicting accurately secondary quantities such as contact stresses and stress resultants.

a

I .4

1.0

- 4qk2

0.6

0.2

6x6 finite element mesh (n.36)

o Gorbunov Pasadov and -Present study

Serebr jonyi [28]

X 0.5 I .o - 0 2 1

Figure 7. Contact pressure along the centre line of a centrally loaded rigid square plate

1 6x6 finite element mesh I I .o

0.8

0.6

0.4

0.2

0.5 X

0 - 2L

I .o

Figure 8. Comparison of M,, for a rigid square plate

1238 R. K. N. D. RAJAPAKSE AND A. P. S. SELVADURAI

(a)

3.0

2.0

I .o

0

(b) 3.0

2 .O w(w,&f

WO

t .O

X 0.5 2 1 - 1.0

B i - h e a r (red. integration 6 x 6 mesh for plate

K,= 3.01 (to51 hp , / 26 40

0 X 0.5 I .o 2L - 0

Figure 9. Deflection along the centre line for a centrally loaded plate resting on an elastic half-space { Y ~ = vl = 0.15)

3.0 Bi-linear [22] (reduced integration)

2*o hp 1 - * - w(x,o) 21 40

WO

I .o

X 0.5 2a - 0 1.0

Figure 10. Deflection along the edge y = 0 for a centrally loaded plate resting on an elastic half-space (Y, = vs = 0.15)


Figures 9(a) and 9(b) show the normalized vertical displacement along the centre line of the plate obtained by modelling the plate by using the ‘bilinear’ plate element. Note that in these Figures we construct displacement profiles using straight lines to be consistent with the shape functions. The results are presented for the same K , and hp/2E ratios as in the case of the ‘heterosis’ element. Figure 10 shows the displacement along the edge y = 0 for a plate with h,/21 = 1/40. For the case of a relatively flexible plate (KG = 471-23, 301.59) the displacement along the centre line agrees closely with that presented in Figure 4. However, the deflection profiles presented in Figure 10 for these K , values contradict equivalent results presented in Figure 5. As can be seen from Figures 9 and 10, with the decrease of value of K , the displacement profiles tend to deviate from a smooth curve and follow a rather questionable pattern. It is evident from Figure 10 that even for K,= 301-59, the displacement along the edge y=O tends to deviate from a smooth curve and exhibits a profile consistent with a typical mechanism. Moreover at low values of K , ( K , = 4.71, 3.01), instead of taking the uniform displacement limit appropriate for a rigid plate, the displacement follows a zig-zag pattern and shows the presence of a mechanism.

It should be noted here that the stiffness matrix derived from the ‘bilinear’ element is rank deficient and has two additional spurious zero-energy modes. In the absence of sufficient restraints these modes act up to generate spurious results. From the results presented in Figures 9 and 10 for a rigid plate (the lowest K , value), the displacement profiles tend to follow that of an hour-glass mode which is one of the additional spurious zero-energy modes associated with the ‘bilinear’ element. From the above discussions, it is clear that the bilinear element proposed by Hughes et fails to model reliably the elastostatic flexural interaction between a thin plate and elastic half-space over a wide range of relative rigidities.

From the above discussion, it is clear that the inadequacy of ‘bilinear’ element stems from its rank deficiency which activates one of the spurious zero energy modes. In conventional plate bending problems this unacceptable situation is avoided by the presence of precise boundary restraints which suppress the activation of spurious zero-energy modes. If we employ 2 x 2 integration on the shear component, an eigen-analysis shows that the ‘bilinear’ element has the correct rank and there are no spurious zero-energy modes. Noting this fact we reconsider the same problem modelled by a ‘bilinear’ element with a 2 x 2 shear integration rule.

The results are presented in Figure 11 and it is clear that the system behaves well for the range of K , values considered and approaches the correct rigid body limit as K , decreases. However a more careful analysis of results shows that the limit of a rigid indentor or plate is achieved well before that given by equation (4). In Figure t 1 the values given in parenthesis next to the value of K , represent the value of the ratio E,/E,. It is clear that the system behaviour is governed more by the ratio E J E , rather than the global value of K,. For instance, if the system behaviour is governed by K , then the displacement corresponding to the highest values of K , (for which the plate is more flexible) in Figure 11 should be larger than those given in Figure 4. Instead, the displacements are smaller, indicating a stiffer system, because the shear stiffness is no longer singular, owing to the use of the correct integration rule.

Comparison of Figures 1 l(a) and 1 1(b) further confirms the fact that the governing parameter is the ratio E,/E,, instead of the K , relevant to a thin plate, which also includes the dependence on hp/l. Owing to the correct integration employed on the shear contribution, the element behaviour resembles more closely that of a continuum element rather than a thin plate element. Thus the application of a 2 x 2 integration rule on shear quadrature eliminates the danger of obtaining solutions corresponding to mechanisms. However, it fails to represent the behaviour of a thin plate resting on an elastic medium.

Next we consider the performance of the Lagrange element reported in Reference23, in modelling the plate- elastic medium interaction. The deflection profiles along the centre line are

1240 R. K. N. D. RAJAPAKSE AND A. P. S . SELVADURAI

Bi-lineor (full integrotion) 6 x 6 mesh for plote

0 0 X s

0.5 1.0

Bi-linear (full integration)

6 x 6 mesh for plote 3.0 -

W ( X , l ) - WO

0 . 0 0.5 I .o

X - 21

Figure 11. Deflection along the centre line for a centrally loaded plate resting on an elastic half-space (v, = v, = 0.15)

presented in Figure 12. The behaviour is rather questionable for all values of K , and h,/21 considered in the study. The deflections along the edge y = 0 presented in Figure 13 also follow similar unacceptable behaviour. It should be noted here that the Lagrange element with 2 x 2 shear integration rule is rank deficient and has one spurious zero-energy mode.

It is interesting to note here that the displacements obtained from the Lagrange element agree closely with those obtained by using the ‘heterosis’ element (Figure 4) at all element edge modes. As shown in Figure 13, such behaviour is also observed along the edge y = 0. Note that the plate is connected to the underlying elastic medium only at the edge nodes. The mid-side and centre nodes of plate elements have no restraints imposed on them by the supporting medium. This lack of restraints together with the fact that the element itself is rank deficient makes the mid-side and centre nodes behave as unsupported pin joints giving rise to a ponding effect between the edge nodes (see Figures 12 and 13). Thus the ‘Lagrange’ element reported in Reference 23 fails to model the interaction between a thin plate and an elastic medium. The change of integration rule for shear contribution from 2 x 2 to 3 x 3 results in an element with correct rank, and hence no spurious zero-energy modes are incorporated.


0 " 0.5 I .o

(b) 12

10

8

4x4 mesh for ptore

(red. integration)

L wcx Ll we 4

2

0

-2 0 0.5

X - za. I .O

Figure 12. Deklection along the centre fine for a centrally loaded plate resting on an elastic half-space (v, = v, = 0.15)

3.0

2 .o

WO

I .o

0

- 1.0 0 0.5

X - 21

I .O

Figure 13. Deflection along the edge y = 0 for a centrally loaded square plate [v, = P,, = 0.151

1242 R. K. N. D. RAJAPAKSE AND A. P. S . SELVADURAI

3.0

2.0 w(xQ) wo

I .O

Lagrange f full integration)

K g O . 4 7 (lo’) J

0 < X 0.5 2R - 0 1.0

(b l 3.0 Lagrange (full integration)

&t WO

0 X 0.5 I .o - 212

Figure 14. Deflection along the centre line for a centrally loaded square plate (v, = v, = 0.15)

Figures 14(a) and 14(b) show the displacement profifes along the centre fine for different K‘, and hpj2E values considered in this study. The displacement curve represents smooth variations and approaches the correct ‘rigid limit’ as K , decreases. Comparison of the magnitudes of the displacements with those in Figures 4 and 11 shows that the element is much softer than the ’heterosis’ element or the fully integrated ‘bilinear’ element. It is also noted that the ‘rigid limit’ is governed more by KG than the ratio EpjE,, showing the presence of flexural behaviour. Although, as in the case of the ‘bilinear’ element, the change of shear integration rule eliminates the effects due to rank deficiency it still fails to model the behaviour which is characteristic of a thin plate resting on an elastic medium.

CONCLUSIONS

The finite element method offers an efhient scheme for the study of contact between elastic media. In particular, the class of elastostatic problems pertaining to contact between structural elements such as beams and plates and elastic media can be conveniently handled by finite element schemes. (i) The application of finite element techniques to this dass of problems can be enhanced by


approaches which determine the stiffness matrix of the supporting elastic medium, such as half- space or an elastic layer, by analytical means. (ii) The selection of an appropriate plate bending element to analyse this class of elastostatic contact problems is a crucial part of the modelling problem. (iii) A number of elementary and higher-order plate bending elements have been reported in the literature. The ‘bilinear’, ‘Lagrange’ and ‘heterosis’ elements fall into the latter category of higher order elements. In the context of the bilinear element, the term ‘higher order’ is meant to signify the fact that the element is based on Mindlin’s plate theory. Although the performance of these elements in replicating conventional plate bending behaviour is well documented, their applicability to the study of contact problems is not reported in the literature. (iv) This paper presents, for the first time, a study in which the behaviour of higher order elements in modelling elastostatic contact between thin plates and elastic media is examined. (v) It is shown that both the ‘bilinear’ and ‘Lagrange’ elements fail to give reliable results for the elastostatic contact problem for a thin plate, owing to the lack of adequate boundary restraints and the activation of spurious zero- energy modes. (vi) The ‘heterosis’ elements furnish reliable results for this class of elastostatic contact problems. Their use is advocated.

ACKNOWLEDGEMENTS

The authors are grateful for the valuable comments by the referees which led to significant improvements in the presentation of the results.

REFERENCES

1. C. S. Desai and J. T. Christian (eds), Numerical Methods in Geotechnical Engineering, McGraw Hill, New York, 1977. 2. G. M. L. Gladwell, Contact Problems in the Classical Theory ofelasticity, Sijthoff and Noordhoff, Amsterdam, 1980. 3. A. P. S. Selvadurai, Elastic Analysis of Soil-Foundation Interaction, Developments in Geotechnical Engineering,

4. Y. K. Cheung and 0. C. Zienkiewicz, ‘Plates and tanks on elastic foundations-an application of the finite element

5. I. M. Smith, ‘A finite element approach to elastic soil-structure interaction’, Canadian Geotech. J . , 7,95-105 (1970). 6. L. A. Wood and W. J. Larnach, ‘The Effects of soil-structure interaction on raft foundations’, Proceeding of the

7. L. A. Wood, ‘The economic analysis of raft foundation’, Int. j . numer. anal. methods geomech., 1, 397-405 (1977). 8. 0. J. Svec and G. M. McNeice, ‘Finite element analysis of finite sized plates bonded to an elastic halfspace’, Comp.

9. 0. J. Svec and G. M. L. Gladwell, A triangular plate bending element for contact problems’, Int. J . Solids Struct., 9,

10. R. A. Fraser and W. J. Wardle, ‘Numerical analysis of rafts on layered foundations’, GPotechnique, 26,613-630 (1976). 11. J. A. Hooper, ‘Analysis ofa circular raft in adhesive contact with a thick elastic layer’, Gdotechnique, 24,561-580(1974). 12. J. A. Hooper, ‘Foundation interaction analysis’, in C. R. Scott (ed.), Developments in Soil Mechanics-1, Applied Science

13. Y. K. Cheung and D. K. Nag, ‘Plates and beams on elastic foundations-linear and nonlinear behaviour’, Geotechnique,

14. M. Fremond, ‘Dual formulation for potential and complementary energies, Unilateral boundary conditions. Applications to the finite element method’, in J. R. Whiteman (ed.), The Mathematics of Finite Elements and Applications, Academic Press, London, 1972, pp, 175-188.

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method‘, Int. J . Solids Struct., 1, 451-461 (1965).

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Publishers, London, 1978, Chap. 5, pp. 149-211.

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15. 0. C. Zienkiewicz, The Finite Element Method, 3rd edn, McGraw-Hill, New York, 1977. 16. G. Gudehus (ed.), Finite Elements in Geomechanics, Wiley, New York, 1977. 17. S. Valliappan, S. J. Hain and 1. K. Lee (eds), Analysis ofsoil Behaviour and its Applications to Geotechnical Structures,

18. J. K. Batoz, K. J. Bathe and L. Ho, ‘A study of three-node triangle plate bending elements’, Int. j . numer. methods eng.,

19. R. H. Gallagher, ‘Geometrically nonlinear shell analysis’, Proc. Int. Con5 in Nonlinear Solid and Structural Mechanics,

20. R. H. Gallagher, ‘Shell elements’, Proc. World Congress on F.E.M. in Structural Mechanics, Vol. 1, Bournemouth,

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15, 1771-1812 (1980).

CoI-l/Col-26, Geilo, 1977.

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