Use of the integration by part to derive beam/plate equilibrium equations from elasticity with applications to beams

Abstract

This article proposes a rigorous mathematical justification of the derivation of the governing equations of any beam and plate theory by the three three-dimensional (3D) elasticity equations. This is done by the application of the integration by part theorem via weight residual-like methods. The F_is polynomials (subscripts i and s denote the i-direction and the generic s-term of the displacement expansion u_is, respectively) related to a given beam/plate theory are used as weight functions for each of the displacement unknown. To show how the method works a number of simple problems are discussed. These are related to the equilibrium equations and boundary conditions of bending beams according to shear deformation theory. Three methods for obtaining these equations are compared in some detail. The first two are the well-known methods used in mechanics of deformable bodies: (1) use of the Principle of Virtual Displacements; (2) use of the six cardinal ‘Newton’ equations of statics. The third method coincides with the one proposed and rigorously justified in this paper which is (3) the direct integration of the equations of 3D elasticity. Beam theories based on Lagrange polynomials are used as well. The latter makes it possible to immediately develop theories for layered beams and obtain so-called ‘layer-wise’ models, which are only applied here to the case of sandwich structures. Three appendixes discuss: A – The problem in which the derivative of one of the unknowns appears among the unknowns, as is the case with the Euler–Bernoulli theory; B – The extension to dynamics; C-Simple derivation of constitutive equations and related closed-form solution via Navier method.

Keywords:

• Stress resultants
• beams
• plates
• compact notations
• integration by part

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This procedure in case of weak form solutions would coincide to the application of the weight residual method, see for instance the books [5, 8].

2 The change in the sign is due to the fact the external loading is at the right-hand-side in the PVD equations and, it is positive. A simple change in the sign of all terms make the two set of equations the same.

3 It would be enough to substitute the first two equations with their sum and difference, respectively.