Equations of Elastic Bending of Lithospheric Plates
摘要
At present, the bending of the lithosphere is being studied on the basis of Kirchhoff’s thin plate bending equations (1845). However, they give an error of ~15% in the bending of thicker blocks where the thickness to length ratio is h/L = 1/4. More accurate equations of bending of thick plates by Timoshenko (1922) and Reissner (1945) are used in engineering. But they are still not accurate, as they do not take into account transverse compression in bending. The authors have obtained new, more accurate equations of thick plates bending of the second approximation, taking into account all deformations at bending, which makes them, unlike the Timoshenko–Reissner equations, mathematically completely correct. By refining the Timoshenko and Reissner equations, the new second approximation equations do not get more complicated, since only the numerical coefficient in the ordinary differential equation for the bending function is changed and explicit additive terms in the algebraic expressions for longitudinal stress and longitudinal and transverse displacements are added. Compared to the more complex general partial elasticity equations, the resulting bending equations neglect only small fourth-order terms (h/L)4, which, even for thick plates with a thickness of one-quarter of the length, account for about only one percent. A universal analytical solution of the new equations, which gives a more accurate stress and strain field, has been obtained as an application to oceanic plate bending.