Plates and beams of asymmetric structure in thickness

The theory of plates and beams of asymmetric structure in thickness is constructed. It is shown that, in general, the complete problem for this kind of a structure is not divided into a plane problem and a bending problem. Here stress-strain state of plates and beams of asymmetric structure in thickness is analyzed by the mathematical method without using any simplifying assumptions. Simple theories were obtained for practical applications by an asymptotic method. As examples, the dynamic problems for a two-layer beam and a plate with a gradient of properties in thickness were calculated.


Introduction
Plates and beams are widely used in constructions. To ensure that they work reliably, they must be calculated correctly. Often they have an asymmetrical structure in thickness. Such structures do not have a middle plane, which is a plane of symmetry in terms of its geometric and physical parameters. Examples of such constructions are asymmetric layered structures, structures with a gradient of properties in thickness, and asymmetric constructions with variable thickness. It is known that the complete problem for a symmetrical beam or plate is divided into two problems. They are a plane problem and a bending problem. A different situation occurs for asymmetric plates and beams. In this case the complete problem, generally speaking, is not broken up into a plane problem and a bending problem. So the elasticity relations for forces and moments simultaneously contain both tangential and flexural deformations. In addition, the inertia of rotation must be taken into account in the equations of motion. Many papers are devoted to arbitrary laminated thin-walled structures and structures with gradient of properties in thickness [1], [2]. In most of them, either numerical methods or models based on some hypotheses are used.

Formulation of the problem and initial equations
First we obtain two-dimensional (2D) equations for the plate. The cross-section of the plate related to the Cartesian coordinate system is shown in figure 1. As the initial equations, we take the three-dimensional (3D) elasticity equations. They are written as follows Equations of motion

Strain-displacement formulas
Hooke's law, recorded in a form convenient for the future, resolved relative to the main stresses ) ( , , ) ( , ) (   22  11  33  33  3  3  33  2   1  1  2  1 Here nm  are the components of the stress tensor, nm e are the components of the strain tensor, E is the modulus of elasticity,  is the Poisson ratio.
As a result of the asymptotic analysis of 3D equations, which for displacements and deformations coincides with that performed in [3], with an accuracy up to values of the order  ) ( (s is the variability of stress  strain state with respect to the coordinates 1 x and 2 x ) we obtain the following expansions for displacements and deformations with coordinate Here k is the number of the layer for which h is the thickness of the k-th layer.
Integrating 3D equations of motion with respect to the coordinate 3 x with allowance for formulas (3), we obtain the equations of motion of the plate , Formulas (7) and (8) refer to plates with a gradient of properties along the thickness and arbitrary laminated plates, respectively Using the equations of consistency of deformations strain -displacement formulas (2) and formulas 11 12 11 12 11 12 we write the equations of motion in the following form From the equation (9) we find 0 z . The plane 0 3  x in this coordinate system will be called the neutral plane. In the theory of plates with a gradient of properties, it plays the same role as the middle plane for isotropic plates. The formula (9) for two-layered structure was first obtained in [4]. We rewrite the equations of motion and the elasticity relations of the plate, putting

Asymptotic analysis of 2D equations
For asymptotic analysis, we turn to dimensionless variables and dimensionless unknown quantities. As is customary in asymptotic methods, we will perform an asymptotic scale extension with respect to the variable i x . We denote the variability of the stress-strain state with respect to the coordinates by s Suppose that, as in the case of homogeneous plates, the complete problem is conditionally divided into two problems  the problems of quasi transverse vibrations and the quasi tangential vibrations.

Quasi transverse vibrations
We assume that the vibrations are caused by the uniform normal load Z . Suppose that the deflection is much greater than the tangential displacements For the unknown quantities we take the following asymptotic representation:   In the formulas (13) all quantities with asterisks are dimensionless and of one asymptotic order. We substitute the asymptotic representation of the unknown quantities into equations (10) and perform scale extension (12). As a result we obtain equations with respect to dimensionless unknown quantities, in which the order of each term of the equation is specified. Discarding small terms with the assumed accuracy (1), we obtain the following system of equations: The upper indices ) (b and ) (t indicate the quantities of the quasi transverse and the quasi tangential vibrations problems, respectively. Equations (14) are equations of quasi transverse vibrations, but the problem can not be considered purely bending, since the tangential forces .At the second stage we solve the quasi tangential problem with taking into account the quantities found in solving the quasi transverse problem.

Quasi tangential vibrations
We assume that the vibrations are caused by the uniform tangential load i X . We consider that the tangential displacements are much greater than the deflections w v i  . For the unknown quantities we take the following asymptotic representation:  x Moments (17) and inertia of rotation are calculated by arithmetic operations after solving the problem for quasi tangential vibrations. We will solve the complete problem in two stages. At the first stage we solve the problem (16), (17), discarding in the equations of motion the terms in square brackets. At the second stage, in the problem for quasi transverse vibrations, we take into account the forces (15) and the inertia of rotation The theory of beams is a special case of the plate theory. The similar results for smart structures were received in [5]. We write the systems of equations for quasi tangential vibrations The position of the neutral axis is found from the equations In statics the complete problem for a beam is exactly divided into a plane problem and a bending problem.
In the dynamics, the partition is conditional: quasi transverse (quasi tangential) vibrations generate quasi tangential (quasi transverse) vibrations. The relationship between both kinds of vibrations is realized through the inertia of rotation. Especially dangerous are vibrations with frequencies close to their natural frequencies. In this case, as you approach any natural vibration frequency, all forces, moments, deformations, and displacements increase indefinitely.

Problem 1
Consider a two-layer beam. The beam with rigidly clamped edges makes vibrations under the action of a uniform tangential load X .The length of the beam is equal to l, the thickness is h, the layer with the number 1 is three times thinner than the layer with the number 2 (

Problem 2
We consider axisymmetric forced harmonic vibrations of a circular plate with rigidly clamped edge R r  under the action of a uniform normal load Z. Let the properties of the material of the plate vary in thickness according to a linear law  we find the first four natural dimensionless frequencies: 3.20, 6.36, 9.44 and 12.58. Using received formulas, we calculate the bending moment and the tangential force. Imagine the results of the account in the form of graphs. The change in the moment * r G and the force *

Conclusions
We have established the following: For the thin-walled structures under consideration with asymmetric properties of the material over the thickness, the complete problem in general is not divided into a plane problem and a bending problem. This connectedness of quasi transverse and quasi tangential vibrations leads to an unlimited growth of all unknown quantities (forces, moments, stresses, deformations, displacements) when the frequency of vibrations approaches any natural vibrations frequency. A simple method for calculating the stressstrain state of plates and beams of an asymmetric structure is developed. This method can be used to determine complex material properties.