Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method's effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: Equation(1)(1) −σij,i=Pj,i,j=x,y,z(1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; Equation(2)(2) −Nxx,x=qx,−Nyy,y=qy.(2) the stress resultants are used to replace stress components in the latter equations; Equation(3)(3) ux(x,y,z)=ux0(x,y)+z ux1(x,y)+ezux2(x,y)uy(x,y,z)=uy0(x,y)+z uy1(x,y)+z3uy2(x,y)uz(x,y,z)=uz0(x,y)+z uz1(x,y)+z cos (2zh) uz2(x,y).(3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; Equation(4)(4) δu:−Tx,x−Ty,y−Tz,z=P(4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.