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3.
Int 4x2x2
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(a)   
Model I - distorted

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4.
4.
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(b)   
Model III - distorted   
Figure 12: Large deflection analysis of a cantilever using distorted elements

Table 8 Results for large deflection analysis of a cantilever using distorted elements 

<table><tr><td rowspan="2"></td><td colspan="3">Model I (distorted)</td><td colspan="3">Model III (distorted)</td></tr><tr><td>step 2</td><td>step 5</td><td>step 8</td><td>step 2</td><td>step 5</td><td>step 8</td></tr><tr><td> $\phi^{FEM}/\phi^{analyt}$ </td><td>0.13</td><td>0.13</td><td>0.13</td><td>0.95</td><td>0.84</td><td>0.76</td></tr><tr><td> $u^{FEM}/u^{analyt.}$ </td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.89</td><td>0.68</td><td>0.56</td></tr><tr><td> $w^{FEM}/w^{analyt}$ </td><td>0.10</td><td>0.11</td><td>0.12</td><td>0.95</td><td>0.86</td><td>0.81</td></tr><tr><td> $\phi^{analyt}$ </td><td>18°</td><td>45°</td><td>72°</td><td>18°</td><td>45°</td><td>72°</td></tr></table>

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P
2a
h
2a
R1
R2
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(a) Spherical shell

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| Central deflection, Wc | Central load, (P/1000) |
| ---------------------- | ---------------------- |
| 0                      | 0                      |
| 50                     | 30                     |
| 100                    | 45                     |
| 150                    | 50                     |
| 200                    | 40                     |
| 250                    | 35                     |
| 300                    | 55                     |
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(b) Non-linear load displacement curve.   
Figure 13 Geometric non-linear response of a spherical shell. O, Horrigmoe; —, Leicester; ●, nine 4-node elements; □, one 16-node element Int 4×4×2

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ε
ε
102.
54.
0.54
0.5
4
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(a) Stiffened plate

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| Vertical displac. of center | τ/τ_CR |
| --------------------------- | ------ |
| 0.004                       | 0.95   |
| 0.008                       | 1.00   |
| 0.012                       | 1.00   |
| 0.016                       | 1.00   |
| 0.020                       | 1.00   |
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(b) Large deflection response   
Figure 14 Non-linear response of a stiffened plate. $E=2.1\times10^{6}$ ; v=0.3

6 Bathe, K. J. and Ho, L. W. A simple and effective element for analysis of general shell structures, J. Comput. Struct., 13, 673–682 (1980)   
7 Bathe, K. J. and Hô, L. W. Some results in the analysis of thin shell structures, Nonlinear Finite Element Analysis in Structural Mechanics, (Ed. W. Wunderlich et al.), Springer-Verlag, Berlin (1981)   
8 Batoz, J. L., Bathe, K. J. and Ho, L. W. A study of three-node triangular plate bending elements, Int. J, Num. Meth. Eng., 15, 1771–1812 (1980)   
9 Batoz, J. L. and Ben Tahar, M. Evaluation of a new quadrilateral plate bending element, Int. J. Num. Meth. Eng., 18, 1655–1677 (1982)   
10 Bercovier, M., Hasbani, Y., Gilon, Y., and Bathe, K., J., On a finite element procedure for nonlinear incompressible elasticity, Hybrid and Mixed Finite Element Methods, (Ed, S. M. Atluri et al.), John Wiley, New York (1983)   
11 Flügge, W. Stresses in Shells, 2nd edn, Springer-Verlag, Berlin (1973)   
12 Forsberg, K. and Hartung, R. An evaluation of finite difference and finite element techniques for analysis of general shells, Symp. High Speed Computing of Elastic Structures, IUTAM, Liège (1970)   
13 Fung, Y. C. Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey (1965)   
14 Gallagher, R. H. Problems and progress in thin shell finite element analysis, Finite Elements in Thin Shells and Curved Members, (Ed. D. G. Ashwell and R. H. Gallagher), John Wiley, New York (1976)   
15 Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd edn, Oxford University Press (1968)   
16 Horrigmoe, G. Finite element instability analysis of free-form shells, Report 77-2, Division of Structural Mechanics, The Norwegian Institute of Technology, University of Trondheim, Norway (1977)   
17 Hughes, T. J. R. and Liu, W. K. Nonlinear finite element analysis of shells: Part I, Three-dimensional shells, J. Comput. Meth. Appl. Mech. Eng., 26, 331–362 (1981)

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hinged
immovable edge
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(a) 4-node shell model

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t
R
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(b) Axisymmetric model

18 Irons, B. M. and Razzaque, A. Experience with the patch test for convergence of finite elements. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, (Ed. A. K. Aziz), Academic Press, New York (1972)   
19 Kråkeland, B. Nonlinear analysis of shells using degenerate isoparametric elements, Finite Elements in Nonlinear Mechanics, Vol. 1, (Ed. P. G. Bergan et al.), Tapir Publishers (Norwegian Institute of Technology, Trondheim, Norway) (1978)   
20 Leicester, R. H. Finite deformations of shallow shells, Proc. Am. Soc. Civil Eng., 94, (EM6), 1409–1423 (1968)   
21 Lindberg, G. M., Olson, M. D. and Cowper, G. R. New developments in the finite element analysis of shells, Q. Bull. Div. Mech. Eng. and the National Aeronautical Establishment, National Research Council of Canada, Vol. 4 (1969)   
22 MacNeal, R. H. A simple quadrilateral shell element, J. Comput. Struct. 8, 175–183 (1978)   
23 Noor, A. K. and Peters, J. M. Mixed models and reduced/selec-

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| Vertical displac. of center | P     |
| --------------------------- | ----- |
| 0                           | 0     |
| 1                           | 1500  |
| 2                           | 2500  |
| 3                           | 1000  |
| 4                           | 1200  |
| 5                           | 1300  |
| 6                           | 1400  |
| 7                           | 1500  |
| 8                           | 1600  |
| 9                           | 1700  |
| 10                          | 1700  |
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(c) Elastoplastic load-displacement curve   
Figure 15 Response of elastic-perfectly plastic circular plate subjected to a concentrated load, P, at its centre. TLF abbreviates use of total Lagrangian formulation and MNO abbreviates use of materially non-linear-only formulation. R=100, t=1; $E=2.1\times10^{6}$ ; $E_{T}=0.0$ ; $\nu=0.3$ ; $\sigma_{\nu}=1000$ . Circular plate response; —, axisymmetric model;   
●, 4-node shell model

tive integration displacement models for nonlinear analysis of curved beams, Int. J. Num. Meth. Eng., 17, 615–631 (1981)   
24 Ramm, E. and Sattele, J. M. Elasto-plastic large deformation shell analysis using degenerated elements, Nonlinear Finite Element Analysis of Plates and Shells, (Ed. T. J. R. Hughes), AMD-Vol. 48, Am. Soc. Mech. Eng., New York (1981)   
25 Report AE 83-5, ADINA System Verification Manual, ADINA Engineering, Västerås, Sweden and Watertown, Mass. (1983)   
26 Timoshenko, S. P. and Gere, J. M. Theory of Elastic Stability, 2nd edn, McGraw-Hill, New York (1961)   
27 Washizu, K. Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford and New York (1968)   
28 Wempner, G., Talaslidis, D. and Hwang, C.-M. A simple and efficient approximation of shells via finite quadrilateral elements, J. Appl. Mech., 49, 115–120 (1982)   
29 Zienkiewicz, O. C. The Finite Element Method, McGraw-Hill, New York (1977)
