It is well known that in approximate analysis of elasticity problems lower and upper bounds to the true strain-energy content can be obtained by the alternative assumptions of compatible displacement fields or of equilibrating stress fields. While formulations based on the first are relatively easy to achieve those based on the second present many difficulties. In this paper it is shown how by virtue of the analogy between plane-elasticity and slab-deflection problems compatible-displacement formulations in either one can be used to generate equilibrating formulations in the other. This should result in a direct application of existing programmes to a wider range of problems.
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References
1.
Fraeijs de VeubekeB.‘Displacement and equilibrium models in the finite element method’, Stress analysis (Ed. ZienkiewiczO. C.HolisterG. S.) 1965, Chapter 9 (John Wiley and Sons, Inc., New York).
2.
CloughR. W.‘The finite element method in structural mechanics’, ibid., Chapter 7.
3.
ZienkiewiczO. C.CheungY. K.The finite element method in structural and continuum mechanics1967 (McGraw-Hill Book Co., Inc., New York and London).
4.
MichellJ. H.‘On the direct determination of stress in an elastic solid with application to the theory of plates’, Proc. Lond. math. Soc.189931, 100.
5.
MindlinR. D.‘The analogy between multiply connected slices and slabs’, Q. appl. Math.19464, 279; 1947 5, 238.
6.
ZienkiewiczO. C.CruzC.‘The use of the slab analogy in the determination of thermal stresses’, Int. J. mech. Sci.19624, 285.
7.
SouthwellR. V.‘On the analogues relating flexure and extension of flat plates’, Q. Jl Mech. appl. Math.19503, 257.
8.
FungY. C.‘Bending of thin elastic plates of variable thickness’, J. aeronaut. Sci.195320, 455.
9.
WeingartenG.cited by LoveA. E. H. In Mathematical theory of elasticity1944, §156 (Dover Publications, Inc., New York).
10.
VolterraV.‘Sur l'équilibre des corps élastiques multiplement connexes’, Annls Éc. norm.190724, 401.
11.
SanderG.Fraeijs de VeubekeB.‘Upper and lower bounds to structural deformations by dual analysis in finite elements’, Air Force Technical Report AFFDL-TR-199, 1966.
12.
Fraeijs de VeubekeB.‘Upper and lower bounds in matrix structural analysis’, AGARDograph196472, 165.
13.
Fraeijs de VeubekeB.‘Bending and stretching of plates. Special models for upper and lower bounds’, Conf. Matrix Meth. struct. Mech.1965 (AFFDL, Dayton, Ohio).
14.
ZienkiewiczO. C.GerstnerR. W.‘A stress function approach to interface and mixed boundary condition problems’, Int. J. mech. Sci.19602, 93.
15.
MindlinR. D.SalvadoriM. G.Handbook of experimental stress analysis (Ed. HetenyiM.) 1950, Chapter 16 (John Wiley and Sons, Inc., New York).
