Equations of Elastic 2D-Bending of Thick Plates

At present, bending deformations of lithospheric plates and bending vibrations of structures during earthquakes are studied on the basis of the theory of bending of thin plates with a thickness-to-length ratio h/L 1/10 formulated by Kirchhoff in 1850. However, even for long oceanic plates, the effective ratio h/L is about 1/8. Therefore, this paper considers the possibility of using theories of bending of thick plates. In technology, the equations of Timoshenko (1922) and Reissner (1945) found by the variational method have been used for calculating the bending of thick plates (along with numerical solutions of the general elasticity equations) over the past 80 years to the present day. However, in articles, textbooks and reference books on elasticity theory, these equations are given with indications of their inaccuracy and systematic error due to the neglect of transverse deformation during bending. In this paper, a system of equations for 2D bending of thick plates of the second approximation is derived by directly transforming the original general equations of elasticity using the method of successive approximations. It is noteworthy that, by refining the Timoshenko and Reissner equations, the obtained equations of the second approximation do not become more complicated, since only the numerical coefficient in the differential equation for the bending function is changed and additive terms are added to the algebraic expressions for stresses and displacements. Significantly simplifying the solution compared to the general elasticity equations in partial derivatives, the obtained differential ordinary equation of bending neglects only small terms above the third order of smallness (h/L)³. Comparison of the solutions of the obtained equations with the test analytical solution of the exact general elasticity equations showed their complete coincidence with an accuracy of up to the fourth order of smallness. For thick plates at h/L = 1/3, compared to exact solutions of general elasticity equations, the solutions of the Kirchhoff equation give a systematic error for the bending function of up to 20%, the solutions of the Timoshenko–Reissner equation - up to 5%, and the obtained refined equations have an inaccuracy of solutions of less than 1%. An example of using the obtained equations for a refined calculation of the bending of oceanic plates is given, in which the solution is obtained in analytical form.