2D Elastic Bending Equations for Thick Plates

Abstract—Currently, bending deformations of lithospheric plates and bending vibrations of structures in earthquakes are studied based on the Kirchhoff–Love theory for thin plates with a thickness-to-length ratio h/L < 1/10 formulated by G. Kirchhoff in 1850. However, even for long oceanic plates, the effective h/L ratio is about 1/8. Therefore, the paper considers the possibility of using bending theories for thick plates. In engineering, for calculating bends of thick plates, along with numerical solutions of general elasticity equations, the Timoshenko’s (1922) and Reissner’s (1945) equations found by variational method have been used for the last 80 years. However, in papers, textbooks and reference books on the theory of elasticity, these equations are given with notes indicating their approximate nature and systematic error due to neglect of transverse deformation at bending. In this paper, we derive a system of two-dimensional (2D) second-approximation bending equations for thick plates by direct transformation of the initial general elasticity equations using a method of successive approximations. It is noteworthy that the obtained second-approximation equations refining the Timoshenko and Reissner equations do not become more complicated, since only the numerical coefficient in the differential equation for the plate deflection function changes and additive terms are introduced in the algebraic expressions for stresses and displacements. Significantly simplifying the solution compared to the general partial differential equations of elasticity, the derived ordinary differential bending equation neglects only small terms of higher than third order of smallness (h/L)3. The comparison of the solutions of the new equations with the test analytical solution of the exact general equations of elasticity has shown their complete coincidence to the fourth order of smallness. For thick plates at h/L = 1/3, compared to the exact solutions of the general elasticity equations, the solutions of the Kirchhoff equation give a systematic error for the deflection function up to 20%, the Timoshenko–Reissner’s equations up to 5%, while the new refined equations have an uncertainty of solutions below 1%. The paper presents the example of using the obtained equations for a refined calculation of the bending of oceanic plates, in which the solution is obtained in an analytical form.