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Plane stress and plane strain
Two-dimensional elastic problems were the first successful examples of the application
of the finite element method.''2 Indeed, we have already used this situation to
illustrate the basis of the finite element formulation in Chapter 2 where the general
relationships were derived. These basic relationships are given in Eqs (2.1)-(2.5)
and (2.23) and (2.24), which for quick reference are summarized in Appendix C.
In this chapter the particular relationships for the plane stress and plane strain
problem will be derived in more detail, and illustrated by suitable practical examples,
a procedure that will be followed throughout the remainder of the book.
Only the simplest, triangular, element will be discussed in detail but the basic
approach is general. More elaborate elements to be discussed in Chapters 8 and 9
could be introduced to the same problem in an identical manner.
The reader not familiar with the applicable basic definitions of elasticity is referred
to elementary texts on the subject, in particular to the text by Timoshenko and
G ~ o d i e r , ~ whose notation will be widely used here.
1. M.J. Turner, R.W. Clough, H.C. Martin, and L.J. Topp. Stiffness and deflection analysis
2. R.W. Clough. The finite element in plane stress analysis. Proc. 2nd A X E Con5 on Electro-
3. S. Timoshenko and J.N. Goodier. Theory of Elasticity. 2nd ed., McGraw-Hill, 1951.
4. S.G. Lekhnitskii. Theory of Elasticity of an Anisotropic Elastic Body (Translation from
Russian by P. Fern). Holden Day, San Francisco, 1963.
5. R.F.S. Hearmon. An Introduction to Applied Anisotropic Elasticity. Oxford University
6. O.C. Zienkiewicz and J.Z. Zhu. The superconvergent patch recovery (SPR) and adaptive
finite element refinement. Comp. Methods Appl. Mech. Eng., 101, 207-24, 1992.
7. B. Boroomand and O.C. Zienkiewicz. Recovery by equilibrium patches (REP). Internat.
J. Num. Meth. Eng., 40, 137-54, 1997.
8. O.C. Zienkiewicz, Y.K. Cheung, and K.G. Stagg. Stresses in anisotropic media with
particular reference to problems of rock mechanics. J. Strain Analysis, 1, 172-82, 1966.
9. G.N. Savin. Stress Concentration Around Holes (Translation from Russian). Pergamon
10. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, B. Schrefler, and T. Shiomi. Computational
Geomechanics. John Wiley and Sons, Chichester, 1999.
11. O.C. Zienkiewicz and Y.K. Cheung. Buttress dams on complex rock foundations. Water
Power, 16, 193, 1964.
12. O.C. Zienkiewicz and Y.K. Cheung. Stresses in buttress dams. Water Power, 17, 69, 1965.
13. K. Terzhagi. Theoretical Soil Mechanics. Wiley, 1943.
14. O.C. Zienkiewicz, C. Humpheson, and R.W. Lewis. A unified approach to soil mechanics
problems, including plasticity and visco-plasticity. Int. Symp. on Numerical Methods in Soil
and Rock Mechanics. Karlsruhe, 1975. See also Chapter 4 of Finite Elements in Geomechanics
(ed. G. Gudehus), pp. 151-78, Wiley, 1977.