Abstract
In the previous chapter, we exploited the idea that the strain energy U stored in an elastic structure must be equal to the work done by the external loads, if these are applied sufficiently slowly for inertia effects to be negligible.
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Notes
- 1.
The equation appears to be dimensionally inconsistent, but we must remember that the loading comprises a traction of unit magnitude. The dimensions could be made explicit by writing \(\sigma _{xx}=\sigma _{yy}=\sigma _{zz}=S\), where S is a constant with dimensions of stress which will be found to cancel in the final expression for \(\varDelta V_1\).
- 2.
J. D. Renton (1991), Generalized beam theory applied to shear stiffness, International Journal of Solids and Structures, Vol. 27, pp. 1955–1967, argues that the correct shear stiffness for a beam of any cross section should be that which equates the strain energy associated with the shear stresses \(\sigma _{xz}, \sigma _{yz}\) and the work done by the shear force against the shear deflection. For the rectangular beam of §9.1, this is equivalent to replacing the multiplier on \(b^2/a^2\) in (9.13)\(_{2,3}\) by \(2.4(1\!+\!\nu )\). This estimate differs from (9.13)\(_3\) by at most 3%.
- 3.
R. T. Shield (1967), Load-displacement relations for elastic bodies, Zeitschrift für angewandte Mathematik und Physik, Vol. 18, pp. 682–693.
- 4.
An approximate solution to this problem for a punch of fairly general plan-form is given by V. I. Fabrikant (1986), Flat punch of arbitrary shape on an elastic half-space, International Journal of Engineering Science, Vol. 24, pp. 1731–1740. Fabrikant’s solution and the above reciprocal theorem argument are used as the bases of an approximate solution of the general smooth punch problem by J. R. Barber and D. A. Billings (1990), An approximate solution for the contact area and elastic compliance of a smooth punch of arbitrary shape, International Journal of Mechanical Sciences, Vol. 32, pp. 991–997.
- 5.
- 6.
See for example, Problem 38.10.
- 7.
J.N.Goodier, Proceedings of the 3rd U.S. National Congress on Applied Mechanics, (1958), pp. 343–345.
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Barber, J.R. (2022). The Reciprocal Theorem. In: Elasticity. Solid Mechanics and Its Applications, vol 172. Springer, Cham. https://doi.org/10.1007/978-3-031-15214-6_38
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