limitations of thin plate theory
The performance of both 'thick' and 'thin' elements is frequently compared for the examples of clamped and simply supported square plates, though, of course, more stringent tests can and should be devised. Figure 5.15(a)-5.15(d) illustrates the behaviour of various elements we have discussed in the case of a span-to-thickness ratio ( L / t ) of 100, which generally is considered within the scope of thin plate theory. The results are indeed directly comparable to those of Fig. 4.16 of Chapter 4, and it is evident that here the thick plate elements perform as well as the best of the thin plate forms.
It is of interest to note that in Fig. 5.15 we have included some elements that do not fully pass the patch test and hence are not robust. Many such elements are still used as their failure occurs only occasionally - although new developments should always strive to use an element which is robust.
All 'robust' elements of the thick plate kind can be easily mapped isoparametrically and their performance remains excellent and convergent. Figure 5.16 shows isopara- metric mapping used on a curved-sided mesh in the solution of a circular plate for two elements previously discussed. Obviously, such a lack of sensitivity to distortion will be of considerable advantage when shells are considered, as we shall show in Chapter 8.
Of course, when the span-to-thickness ratio decreases and thus shear deformation importance increases, the thick plate elements are capable of yielding results not obtainable with thin plate theory. In Table 5.4 we show some results for a simply supported, uniformly loaded plate for two L / t ratios and in Table 5.5 we show results for the clamped uniformly loaded plate for the same ratios. In this example we show
Fig. 5.15 Convergence study for relatively thin plate ( L / t = 100): (a) centre displacement (simply supported, uniform load, square plate); (b) moment a t Gauss point nearest centre (simply supported, uniform load, square plate). Tables 5.1 and 5.2 give keys to elements used.
Performance of various 'thick' plate elements - limitations of thin plate theory 205
Fig. 5.1 5 Convergence study for relatively thin plate (L/t = 100): (c) centre displacement (clamped, uniform load, square plate); (d) moment at Gauss point nearest centre (clamped, uniform load, square plate). Tables 5.1 and 5.2 give keys to elements used.
Fig. 5.16 Mapped curvilinear elements in solution of a circular clamped plate under uniform load: (a) meshes used; (b) percentage error in centre displacement and moment.
also the effect of the hard and soft simple support conditions. In the hard support we assume just as in thin plates that the rotation along the support (0,) is zero. In the soft support case we take, more rationally, a zero twisting moment along the support (see Chapter 4, Sec. 4.2.2).
Table 5.4 Centre displacement of a simply supported plate under uniform load for two L / t ratios; E = 10.92, v = 0.3, L = IO, q = 1
Mesh, M L / t = IO; 11' x lo-' L i t = 1000; II' x IO-'
hard support soft support hard support soft support
2 4.2626 4.6085 4.0389 4.2397
4 4.2720 4.5629 4.0607 4.1297
8 4.2727 4.5883 4.0637 4.0928
16 4.2728 4.6077 4.0643 4.0773
32 4.2728 4.6144 4.0644 4.0700
Series 4.2728 4.0624
Table 5.5 Centre displacement of a clamped plate under uniform load for two L / t ratios; E = 10.92, v = 0.3, L = 10, 4 = 1
Mesh, M L / / = IO; 11' x 10-1 L / t = 1000; II' x IO-'
2 1.4211 1.1469
4 1.4858 1.2362
8 1.4997 I .2583
16 1.5034 1.2637
32 1.5043 1.2646
Series 1.499'6 1.2653
Performance of various 'thick' plate elements - limitations of thin plate theory 207 It is immediately evident that:
1. the thick plate ( L / t = 10) shows deflections converging to very different values depending on the support conditions, both being considerably larger than those given by thin plate theory;
2. for the thin plate ( L / t = 1000) the deflections converge uniformly to the thin (Kirchhoff) plate results for /lard support conditions, but for soft support condi- tions give answers about 0.2 per cent higher in center deflection.
It is perhaps an insignificant difference that occurs in this example between the support conditions but this can be more pronounced in different plate configurations.
In Fig. 5.17 we show the results of a study of a simply supported rhombic plate with L / t = 100 and 1000. For this problem an accurate Kirchhoff plate theory solution is available,6' but as will be noticed the thick plate results converge uniformly to a displacement nearly 4 per cent in excess of the thin plate solution for all the L / t = 100 cases.
Fig. 5.17 Skew 30' simply supported plate (soft support); maximum deflection at A, the centre for various degrees of freedom N . The triangular element of reference 15 IS used.
This problem is illustrative of the substantial difference that can on occasion arise in situations that fall well within the limits assumed by conventional thin plate theory ( L / t = loo), and for this reason the problem has been thoroughly investigated by BabuSka and Scapolla,62 who solve it as a fully three-dimensional elasticity problem using support conditions of the ‘soft’ type which appear to be the closest to physical reality. Their three-dimensional result is very close to the thick plate solution, and confirms its validity and, indeed, superiority over the thin plate forms. However, we note that for very thin plates, even with soft support, convergence to the thin plate results occurs.
5.1 0 Forms without rotation parameters
It is possible to formulate the thick plate theory without direct use of rotation parameters. Such an approach has advantages for problems with large rotations where use of rotation parameters leads to introduction of trigonometric functions (e.g. see Chapter 11). Here we again consider the case of a cylindrical bending of a plate (or a straight beam) where each element is defined by coordinates at the two ends. Starting from a four-noded rectangular element in which the origin of a local Cartesian coordinate system passes through the mid-surface (centroid) of the element we may write interpolations as (Fig. 5.18)
x Ni(J, V ) -%i + F(CT 7 ) -fj + N/c(E, 77) gk + N/(J, V ) -f/
Y = Ni(<: v)yi +.Nj(C, v ) j j + N!i(J, v ) j k + N/(t, q ) j /
(5.74) in which Ni, etc. are the usual 4-noded bilinear shape functions. Noting the rectangu- lar form of the element, these interpolations may be rewritten in terms of alternative parameters [Fig. 5.18(b)] as
(5.75) where shape functions are
NI(J) =;(I - a 7 N2(J) = f ( 1 + I ) (5.76) and new nodal parameters are related to the original ones through
(5.77) Since the element is rectangular = f 2 = 7 [Fig. 5.18(b)]; however, the above inter- polations can be generalized easily to elements which are tapered. We can now use isoparametric concepts to write the displacement field for the element as
(5.78)
Forms without rotation parameters 209
{I:}= Yx I'
L -
v i 0
1 v i 0 6 ,
0 0
t [{!i] [' 0 0 "I(i] (5.79)
Ni..x ~y Nix 0
f O v 0 0
0 0 6 7
0 z 1 Nj
P 4
0 = Ni Ni,s z;- Ni.x
- t -
Fig. 5.19 Simply supported 30" skew plate with uniform load (problem of Fig. 5.17); adaptive analysis to achieve 5 per cent accuracy; L / t = 100, v = 0.3, six-node element T6S3B3;15 0 = effectivity index, q = per- centage error in energy norm of estimator.
of Volume 1 and is left as an exercise for the reader. We do note that here it is not necessary to use a constitutive equation which has been reduced to give zero stress in the through-thickness ( y ) direction. By including additional enhanced terms in the thickness direction one may use the three-dimensional constitutive equations directly. Such developments have been pursued for plate and shell applications.63p66 We note that while the above form can be used for flat surfaces and easily extended for smoothly curved surfaces it has difficulties when 'kinks' or multiple branches are encountered as then there is no unique 'thickness' direction. Thus, considerable addi- tional work remains to be done to make this a generally viable approach.
5.1 1 Inelastic material behaviour
We have discussed in some detail the problem of inelastic behaviour in Sec. 4.19 of Chapter 4. The procedures of dealing with the same situation when using the
Concluding remarks - adaptive refinement 21 1 Reissner-Mindlin theory are nearly identical and here we will simply refer the reader
to the literature on the subject6796* and to the previous chapter.
5.1 2 Concluding remarks - adaptive refinement
The simplicity of deriving and using elements in which independent interpolation of rotations and displacements is postulated and shear deformations are included assures popularity of the approach. The final degrees of freedom used are exactly the same type as those used in the direct approach to thin plate forms in Chapter 4, and at no additional complexity shear deformability is included in the analysis.
If care is used to ensure robustness, elements of the type discussed in this chapter are generally applicable and indeed could be used with similar restrictions to other finite element approximations requiring C , continuity in the limit.
The ease of element distortion will make elements of the type discussed here the first choice for curved element solutions and they can easily be adapted to non-linear material behaviour. Extension to geometric non-linearity is also possible; however, in this case the effects of in-plane forces must be included and this renders the problem identical to shell theory. We shall discuss this more in Chapter 11.
In Chapters 14 and 15 of Volume 1 we discussed the need for an adaptive approach in which error estimation is used in conjunction with mesh generation to obtain answers of specified accuracy. Such adaptive procedures are easily used in plate bend- ing problems with an almost identical form of error e ~ t i m a t i o n . ~ ~
In Figs 5.19 and 5.20 we show a sequence of automatically generated meshes for the problem of the skew plate. It is of particular interest to note:
Fig. 5.20 Energy norm rate of convergence for the 30" skew plate of Fig. 5.17 for uniform and adaptive refinement; adaptive analysis to achieve 5 per cent accuracy.
1. the initial refinement in the vicinity of corner singularity;
2. the final refinement near the simply support boundary conditions where the effects Indeed, such boundary layers can occur near all boundaries of shear deformable plates and it is usually found that the shear error represents a very large fraction of the total error when approximations are made.
of transverse shear will lead to a ‘boundary layer’.
References
1. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Curved thick shell and membrane elements with particular reference to axi-symmetric problems. In L. Berke, R.M. Bader, W.J.
Mykytow, J.S. Przemienicki and M.H. Shirk (eds), Proc. 2nd Con$ Matrix Methods in Structural Mechanics, Volume AFFDL-TR-68-150, pp. 539-72, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH, October 1968.
2. S. Ahmad, B.M. Irons and O.C. Zienkiewicz. Analysis of thick and thin shell structures by curved finite elements. Int. J . Num. Meth. Eng., 2, 419-51, 1970.
3. S. Ahmad. Curved Finite Elements in the Analysis of Solids, Shells and Plate Structures, PhD thesis, University of Wales, Swansea, 1969.
4. O.C. Zienkiewicz, J. Too and R.L. Taylor. Reduced integration technique in general analysis of plates and shells. Int. J. Num. Meth. Eng., 3, 275-90, 1971.
5. S.F. Pawsey and R.W. Clough. Improved numerical integration of thick slab finite elements. Int. J. Num. Meth. Eng., 3, 575-86, 1971.
6. O.C. Zienkiewicz and E. Hinton. Reduced integration, function smoothing and non-con- formity in finite element analysis. J. Franklin Inst., 302, 443-61, 1976.
7. E.D.L. Pugh, E. Hinton and O.C. Zienkiewicz. A study of quadrilateral plate bending elements with reduced integration. Int. J . Num. Meth. Eng., 12, 1059-79, 1978.
8. T.J.R. Hughes, R.L. Taylor and W. Kanoknukulchai. A simple and efficient finite element for plate bending. Int. J . Num. Meth. Eng., 11, 1529-43, 1977.
9. T.J.R. Hughes and M. Cohen. The ‘heterosis’ finite element for plate bending. Computers and Structures, 9, 445-50, 1978.
10. T.J.R. Hughes, M. Cohen and M. Harou. Reduced and selective integration techniques in the finite element analysis of plates. Nuclear Engineering and Design, 46, 203-22, 1978.
11. E. Hinton and N. BitaniC. A comparison of Lagrangian and serendipity Mindlin plate elements for free vibration analysis. Computers and Structures, 10, 483-93, 1979.
12. R.D. Cook. Concepts and Applications of Finite Element Analysis, John Wiley, Chichester, Sussex, 1982.
13. R.L. Taylor, O.C. Zienkiewicz, J.C. Simo and A.H.C. Chan. The patch test - a condition for assessing FEM convergence. Znt. J . Num. Meth. Eng., 22, 39-62, 1986.
14. O.C. Zienkiewicz, S. Qu, R.L. Taylor and S. Nakazawa. The patch test for mixed formula- tions. Int. J . Num. Meth. Eng., 23, 1873-83, 1986.
15. O.C. Zienkiewicz and D. Lefebvre. A robust triangular plate bending element of the Reissner-Mindlin type. Int. J . Num. Metlz. Eng., 26, 1169-84, 1988.
16. O.C. Zienkiewicz, J.P. Vilotte, S. Toyoshima and S. Nakazawa. Iterative method for con- strained and mixed approximation: an inexpensive improvement of FEM performance.
Conzp. Meth. Appl. Mech. Eng., 51, 3-29, 1985.
17. D.S. Malkus and T.J.R. Hughes. Mixed finite element methods in reduced and selective integration techniques: a unification of concepts. Comp. Metlz. Appl. Meclz. Eng., 15, 63-81, 1978.
References 2 13 18. B. Fraeijs de Veubeke. Displacement and equilibrium models in finite element method. In
O.C. Zienkiewicz and G.S. Holister (eds), Stress Analysis, pp. 145-97. John Wiley, Chichester, Sussex, 1965.
19. K.-J. Bathe. Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996.
20. W.X. Zhong. FEM patch test and its convergence. Technical Report 97-3001, Research Institute Engineering Mechanics, Dalian University of Technology, 1997, [in Chinese].
21. W.X. Zhong. Convergence of FEM and the conditions of patch test. Technical Report 97- 3002, Research Institute Engineering Mechanics, Dalian University of Technology, 1997, [in Chinese].
22. I. BabuSka and R. Narasimhan. The BabuSka-Brezzi condition and the patch test: an example. Comp. Meth. Appl. Mech. Eng., 140, 183-99, 1997.
23. O.C. Zienkiewicz and R.L. Taylor. The finite element patch test revisited: a computer test for convergence, validation and error estimates. Comp. Meth. Appl. Mech. Eng., 24. T.J.R. Hughes and W.K. Liu. Implicit-explicit finite elements in transient analyses: part I
and part 11. J. Appl. Mech., 45, 371-8, 1978.
25. K.-J. Bathe and L.W. Ho. Some results in the analysis of thin shell structures. In W. Sun- derlich et a/. (eds), Nonlinear Finite Element Analysis in Structural Mechanics, pp. 122-56, Springer-Verlag, Berlin, 198 1.
26. O.C. Zienkiewicz, Z. Xu, L.F. Zeng, A. Samuelsson and N.-E. Wiberg. Linked interpola- tion for Reissner-Mindlin plate elements: part I - a simple quadrilateral element. Int. J . Nunz. Meth. Eng., 36, 3043-56, 1993.
27. Z. Xu, O.C. Zienkiewicz and L.F. Zeng. Linked interpolation for Reissner-Mindlin plate elements: part 111 - an alternative quadrilateral. Int. J . Num. Meth. Eng., 37, 1437-43, 1994.
28. F. Auricchio and R.L. Taylor. A shear deformable plate element with an exact thin limit.
Conip. Meth. Appl. Mech. Eng., 118, 393-412, 1994.
29. Z. Xu. A simple and efficient triangular finite element for plate bending. Acta Mechanica Sinica, 2, 185-92, 1986.
30. D.N. Arnold and R.S. Falk. A uniformly accurate finite element method for Mindlin- Reissner plate. Technical Report IMA Preprint Series No. 307, Institute for Mathematics and its Application, University of Maryland, College Park, MD, 1987.
31. T.J.R. Hughes and T. Tezduyar. Finite elements based upon Mindlin plate theory with particular reference to the four node bilinear isoparametric element. J. Appl. Mech., 46, 32. E.N. Dvorkin and K.-J. Bathe. A continuum mechanics based four node shell element for
general non-linear analysis. Engineering Computations, 1, 77-88, 1984.
33. K.-J. Bathe and A.B. Chaudhary. A solution method for planar and axisymmetric contact problems. Int. J . Num. Meth. Eng., 21, 65-88, 1985.
34. H.C. Huang and E. Hinton. A nine node Lagrangian Mindlin element with enhanced shear interpolation. Engineering Conzputations, 1, 369-80, 1984.
35. E. Hinton and H.C. Huang. A family of quadrilateral Mindlin plate elements with substi- tute shear strain fields. Computers and Structures, 23, 409-31, 1986.
36. O.C. Zienkiewicz, R.L. Taylor, P. Papadopoulos and E. Oiiate. Plate bending elements with discrete constraints; new triangular elements. Conzputers and Structures, 35, 505- 22, 1990.
37. K.-J. Bathe and F. Brezzi. On the convergence of a four node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation. In J.R. Whiteman (ed.), The Mathematics qf’Finite Elements and Applications, Volume V, pp. 49 1-503, Academic Press, London, 1985.
38. H.K. Stolarski and M.Y.M. Chiang. On a definition of the assumed shear strains in forniu- lation of the C0 plate elernents. European Jo~trrzal of’ Mechanics, A/Solids, 8. 53-72, 1989.
149, 523-44, 1997.
587-96, 1981.
39. H.K. Stolarski and M.Y.M. Chiang. Thin-plate elements with relaxed Kirchhoff con- straints. Znt. J. Nurn. Meth. Eng., 26, 913-33, 1988.
40. R.S. Rao and H.K. Stolarski. Finite element analysis of composite plates using a weak form of the Kirchhoff constraints. Finite Elements in Analysis and Design, 13, 191-208,
1993.
41. K.-J. Bathe, M.L. Bucalem and F. Brezzi. Displacement and stress convergence of four MITC plate bending elements. Engineering Computations, 7, 29 1-302, 1990.
42. E. Oiiate, R.L. Taylor and O.C. Zienkiewicz. Consistent formulation of shear constrained Reissner-Mindlin plate elements. In C . Kuhn and H. Mang (eds), Discretization Methods in Structural Mechanics, pp. 169-80. Springer-Verlag, Berlin, 1990.
43. E. Oiiate, O.C. Zienkiewicz, B. Sukrez and R.L. Taylor. A general methodology for deriving shear constrained Reissner-Mindlin plate elements. Znt. J. Num. Meth. Eng., 44. P. Papadopoulos and R.L. Taylor. A triangular element based on Reissner-Mindlin plate
theory. Znt. J . Num. Meth. Eng., 30, 1029-49, 1990.
45. G.S. Dhatt. Numerical analysis of thin shells by curved triangular elements based on discrete Kirchhoff hypotheses. In W.R. Rowan and R.M. Hackett (eds), Proc. Symp. on Applications of FEM in Civil Engineering, Vandervilt University, Nashville, TN, 1969.
46. J.L. Batoz, K.-J. Bathe and L.W. Ho. A study of three-node triangular plate bending elements. Znt. J . Num. Meth. Eng., 15, 1771-812, 1980.
47. J.L. Batoz. An explicit formulation for an efficient triangular plate bending element. Znt. J . Nurn. Meth. Eng., 18, 1077-89, 1982.
48. J.L. Batoz and P. Lardeur. A discrete shear triangular nine d.0.f. element for the analysis of thick to very thin plates. Znt. J. Num. Meth. Eng., 28, 533-60, 1989.
49. T. Tu. Performance of Reissner-Mindlin elements, PhD thesis, Department of Mathe- matics, Rutgers University, New Brunswick, NJ, 1998.
50. A. Tessler and T.J.R. Hughes. A three node Mindlin plate element with improved trans- verse shear. Comp. Meth. Appl. Mech. Eng., 50, 71-101, 1985.
51. A. Tessler. A Co anisoparametric three node shallow shell element. Comp. Meth. Appl.
Mech. Eng., 78, 89-103.
52. A. Tessler and S.B. Dong. On a hierarchy of conforming Timoshenko beam elements.
Computers and Structures, 14, 335-44, 1981.
53. M.A. Crisfield. Finite Elements and Solution Procedures for Structural Analysis, Volume I , Linear Analysis, Pineridge Press, Swansea, 1986.
54. L.F. Greimann and P.P. Lynn. Finite element analysis of plate bending with transverse shear deformation. Nuclear Engineering and Design, 14, 223-30, 1970.
55. P.P. Lynn and B.S. Dhillon. Triangular thick plate bending elements. In First Znt. Con$
Struct. Mech. in Reactor Tech., p. M 615, Berlin, 1971.
56. Z. Xu. A thick-thin triangular plate element. Znt. J . Nurn. Meth. Eng., 33, 963-73, 1992.
57. R.L. Taylor and F. Auricchio. Linked interpolation for Reissner-Mindlin plate elements:
58. F. Auricchio and R.L. Taylor. A triangular thick plate finite element with an exact thin 59. M.A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, Volume 1, John 60. E.L. Wilson. The static condensation algorithm. Znt. J . Nurn. Meth. Eng., 8, 199-203, 61. L.S.D. Morley. Skew Plates and Structures, International Series of Monographs in Aero- 62. I. BabuSka and T. Scapolla. Benchmark computation and performance evaluation for a
33, 345-67, 1992.
part I1 - a simple triangle. Znt. J. Nurn. Meth. Eng., 36, 3057-66, 1993.
limit. Finite Elements in Analysis and Design, 19, 57-68, 1995.
Wiley, Chichester, Sussex, 1991.
1974.
nautics and Astronautics, Macmillan, New York, 1963.
rhombic plate bending problem. Znt. J . Num. Meth. Eng., 28, 155-80, 1989.