Non-stationary stress-strain state of the Timoshenko plate


DOI: 10.34759/trd-2022-125-05

Аuthors

Levitskiy D. Y.1*, Fedotenkov G. V.2**

1. PJSC Yakovlev , 68, Leningradskiy prospect, Moscow, 125315, Russia
2. ,

*e-mail: amtrak95@mail.ru
**e-mail: greghome@mail.ru

Abstract

Rapidly developing technical progress poses new, more complex and interesting tasks for engineers. This did not bypass the area of problems of the mechanics of a deformable solid body, and specifically the theory of plates. Plates and shells are extremely widely used in the construction of a wide variety of engineering structures.

At present, nonstationary problems in the theory of plates remain poorly studied.

In this work, vibrations of the Timoshenko plate under the action of non-stationary pressure are studied. investigated. The plate is assumed to be infinitely extended. To describe the movement of the plate, the well-known equations of the S.P. Timoshenko.

The solution method is based on the principle of superposition, according to which the normal displacements of the plate are a convolution of a given pressure with an influence function in spatial coordinates and time. The influence function for a plate is its formal displacements under the influence of a special type of pressure, namely, a unit concentrated force applied instantaneously in time. Mathematically, such a distribution is given by the product of the Dirac delta functions.

A spatial problem is considered in a Cartesian rectangular coordinate system. In this case, expansions in double trigonometric Fourier series and the integral Laplace transform in time are used to construct the influence function. The original coefficients of the expansion series are determined analytically using the second expansion theorem for the Laplace transform. Using the principle of superposition and the constructed original of the influence function, the solution of the problem of non-stationary oscillations of a rectangular Timoshenko plate, as well as displacement at a point under the influence of a distributed load, is obtained.

The paper investigates the response of the hinged Timoshenko plate to the impact of various non-stationary loads. For the solution, a numerical algorithm was developed and implemented on a computer. Examples of calculation of the deformed state of the plate are given.

Keywords:

Timoshenko plate, superposition method, influence functions, Fourier series, integral transformations, non-stationary load

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