Stress resultants governing equations of any beam theory by direct manipulation of 3D equilibrium

Abstract

In this article, the author shows that the equilibrium equations of any one-dimensional theory can be obtained by direct manipulation of the indefinite 3D elasticity equations: It is found that there is no need to refer to Newton-type or Lagrange-type equations to write the equilibrium in terms of stress resultants. Such ‘direct manipulation’ is illustrated by considering various problems, from the simplest to the most complex. First, the bar case is analyzed and the well-known equilibrium normal stress resultant equation is obtained: second, the Timoshenko beam theory is tackled; finally, the case of arbitrary higher order beam theories based on the Carrera Unified Formulation is treated. The equations of a new, unprecedented higher-order beam theory are derived directly from the indefinite equilibrium equations as an example of the capabilities of the proposed method. The manipulation introduced consists of the following points: (i) the 3D elasticity is first written for each degree of freedom (dof) of the beam theories; (ii) the respective stress resultants are used to replace stress components in the latter equations; (iii) the derivatives on the beam axis z do not affect the nature of the indefinite equilibrium equations and these derivatives are applied directly to the respective stress resultants; (iv) the derivatives on the section coordinate x – y should 1 – change the sign of the respective stress resultants, 2 – be applied to the functions of x, y used to define the stress resultants. These points have been proved by referring to the principle of virtual work and splitting the differential operators of the strain-displacement relations into two parts: over the section derivatives along the beam axis derivatives. Some possible further generalizations are discussed in the appendix; in particular, the extension to is briefly discussed.