Calculation of infinite plate on an undeformed foundation under the action of local loading by the analytical method

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Hurtovyi O. H., Tynchuk S. O., Uhryn L. S. № 2 (59) 65-71 Image Image

A refined model of symmetrical stress-strain state (SSS) plate about its mid-surface has been used for calculating the SSS of infinite plate on an undeformed foundation. The double-thickness plate bilaterally symmetrically loaded about its mid-surface has been considered. The plate is formed by symmetric supplementing it with regard to the contact surface and the foundation.

The refined model has been obtained according to the method suggested by Professors O.F.Ryabov and V.G.Piskunov. According to the method, kinematic ratios of classical plate theory are substituted into the Hooke’s law, the obtained stresses are integrated into the equations of equilibrium of the three-dimensional theory of elasticity, which allows to obtain refined expressions for the stresses of transverse shear and transverse compression.The Hooke and Cauchy law is again applied here, resulting in the refinement of the kinematic relations in the first and subsequent iterative approximations.

Such a diagram models the frictionless slip of the plate over surface of its contact with the foundation. The rigid contact of the plate with the foundation is modelled by introducing an additional thin practically undeformed layer. The refined continual model has been used to take into account the deformations of transverse shear and transverse compression in high iterative approximation.The model has one transverse shear function and two transverse compression functions. The system of four calculated differential equations has a general twelfth order of differentiation which does not depend on the number of layers in the plate.

An analytical solution for an infinite plate under the action of concentrated force in the middle has been obtained. The deflections in the cross-section of the plate under the concentrated force, both for a homogeneous isotropic plate, and for a plate with a stratification defect have been calculated which is considered as a multilayer plate with a thin layer of slip. The features of deformation of plates on a rigid foundation under the action of a concentrated load have been analyzed.

Keywords: infinite plate, a refined model, transverse shear, transverse compression, analytical method.

doi: 10.32403/1998-6912-2019-2-59-65-71

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