Mixed formulation for thick plates .1 The approximation

The problem of thick plates can, of course, be solved as a mixed one starting from Eqs (5.6)±(5.8) and approximating directly each of the variables h, S and w

0.0044

3x 3 Gaussian integration of all terms 4x 4 Gaussian integration of all terms

Exact thin plate

3x 3 Gaussian integration of all terms 4x 4 Gaussian integration of all terms

Exact thin plate solution 0.00406

Exact thin plate solution 0.00127 wcD/qL4 wcD/qL4

Fig. 5.5 Performance of cubic quadrilaterals: (a) serendipity (QS) and (b) lagrangian (QL) with varying span-to-thickness, L=t, values.

180 `Thick' Reissner±Mindlin plates

independently. Using Eqs (5.6)±(5.8), we construct a weak form as

We now write the independent approximations, using the standard Galerkin procedure, as

w ˆ N_w~w h ˆ N_~h and S ˆ N_s~S

w ˆ N_w~w h ˆ N_~h and S ˆ N_s~S …5:23†

though, of course, other interpolation forms can be used, as we shall note later.

After appropriate integrations by parts of Eq. (5.22), we obtain the discrete symmetric equation system (changing some signs to obtain symmetry)

0 0 E

The above represents a typical three-®eld mixed problem of the type discussed in Sec. 11.5.1 of Volume 1, which has to satisfy certain criteria for stability of approx-imation as the thin plate limit (which can now be solved exactly) is approached.

For this limit we have

 ˆ 1 and H ˆ 0 …5:26†

In this limiting case it can readily be shown that necessary criteria of solution stability for any element assembly and boundary conditions are that

n_‡ n_w5 n_s or _Pn_‡ n_w

Mixed formulation for thick plates 181

When the necessary count condition is not satis®ed then the equation system will be singular. Equations (5.27) and (5.28) must be satis®ed for the whole system but, in addition, they need to be satis®ed for element patches if local instabilities and oscilla-tions are to be avoided.^13ÿ15

The above criteria will, as we shall see later, help us to design suitable thick plate elements which show convergence to correct thin plate solutions.

5.3.2 Continuity requirements

The approximation of the form given in Eqs (5.24) and (5.25) implies certain continu-ities. It is immediately evident that C₀ continuity is needed for rotation shape functions N_(as products of ®rst derivatives are present in the approximation), but that either N_w or N_s can be discontinuous. In the form given in Eq. (5.25) a C₀ approximation for w is implied; however, after integration by parts a form for C₀ approximation of S results. Of course, physically only the component of S normal to boundaries should be continuous, as we noted also previously for moments in the mixed form discussed in Sec. 4.16.

In all the early approximations discussed in the previous section, C₀continuity was assumed for both h and w variables, this being very easy to impose. We note that such continuity cannot be described as excessive (as no physical conditions are violated), but we shall show later that very successful elements also can be generated with discontinuous w interpolation (which is indeed not motivated by physical considera-tions).

For S it is obviously more convenient to use a completely discontinuous interpola-tion as then the shear can be eliminated at the element level and the ®nal sti€ness matrices written simply in standard ~h, ~w terms for element boundary nodes. We shall show later that some formulations permit a limit case where ^ÿ1 is identically zero while others require it to be non-zero.

The continuous interpolation of the normal component of S is, as stated above, physically correct in the absence of line or point loads. However, with such interpola-tion, elimination of ~S is not possible and the retention of such additional system variables is usually too costly to be used in practice and has so far not been adopted.

However, we should note that an iterative solution process applicable to mixed forms described in Sec. 11.6 of Volume 1 can reduce substantially the cost of such additional variables.¹⁶

5.3.3 Equivalence of mixed forms with discontinuous S interpolation and reduced (selective) integration

The equivalence of penalized mixed forms with discontinuous interpolation of the constraint variable and of the corresponding irreducible forms with the same penalty variable was demonstrated in Sec. 12.5 of Volume 1 following work of Malkus and Hughes for incompressible problems.¹⁷ Indeed, an exactly analogous proof can be used for the present case, and we leave the details of this to the reader; however, below we summarize some equivalencies that result.

182 `Thick' Reissner±Mindlin plates

Thus, for instance, we consider a serendipity quadrilateral, shown in Fig. 5.6(a), in which integration of shear terms (involving ) is made at four Gauss points (i.e. 2  2 reduced quadrature) in an irreducible formulation [see Eqs (5.16)±(5.20)], we ®nd that the answers are identical to a mixed form in which the S variables are given by a bilinear interpolation from nodes placed at the same Gauss points.

This result can also be argued from the limitation principle ®rst given by Fraeijs de Veubeke.¹⁸ This states that if the mixed form in which the stress is independently interpolated is precisely capable of reproducing the stress variation which is given in a corresponding irreducible form then the analysis results will be identical. It is clear that the four Gauss points at which the shear stress is sampled can only de®ne a bilinear variation and thus the identity applies here.

The equivalence of reduced integration with the mixed discontinuous interpolation of S will be useful in our discussion to point out reasons why many elements mentioned in the previous section failed. However, in practice, it will be found equally convenient (and often more e€ective) to use the mixed interpolation explicitly and eliminate the S variables by element-level condensation rather than to use special integration rules. Moreover, in more general cases where the material properties lead to coupling between bending and shear response (e.g. elastic±plastic behaviour) use of selective reduced integration is not convenient. It must also be pointed out that the equivalence fails if  varies within an element or indeed if the isoparametric mapping implies di€erent interpolations. In such cases the mixed procedures are generally more accurate.

5.4 The patch test for plate bending elements