On Solution of the Problem of Bending of Orthotropic Plates on the Basis of Bimoment Theory

This paper is devoted to the development of new theory of orthotropic thick plates with account of internal forces, moments and bimoments. An equation of motion of plates is described by two systems with nine equations each. Boundary conditions depended on displacements, forces, moments and bimoments are given. An exact solution of the bending of thick plate under the effect of sine load is built. Numerical results for maximal values of displacements and stresses of the plate are obtained.


Introduction
Specified theories of plates are widely used in analysis of structure elements.Review and general technique for constructing a specified theory can be found in [1] [2].In spatial case of bending and vibrations along the thickness of the plate, the displacements vary according to nonlinear law, and classical theories of plates and shells become unacceptable.In general case the field of displacement of thick plates does not obey to any simplifying hypotheses.It is necessary to take into account all the components of the tensor of stress and strain: ( ) , , , 1, 3 ; by them we will introduce tensile and crosscutting forces, bending and torsion moments and the concept of bimoments [3], generated due to nonlinear law of distribution of displacements in cross-sections of the plate.
This article briefly describes a method of constructing a theory of plates with bimoments.Determinant correlations of forces, moments, bimoments and the equations of motion in relation to these types of force factors are given.

Statement of the Problem
Consider orthotropic thick plate of constant thickness 2 H h = and dimensions , a b in plan.Introduce the signs: 1 2 3 , , E E E -elasticity modulus and 12 13 23 , , G G G -shear modulus; 12 13 23 , , ν ν ν -Poisson's ratio of plate material.
Introduce Cartesian system of coordinates 1 2 , x x and z.Axis oz is directed vertically downwards.Let dis- tributed surface normal and tangential loads are applied to the lower and the upper face surfaces of the plate z h = and z h = − .Normal loads in oz axis we will designate as ( ) ( ) , q q + − , tangential loads in direction 1 2 , q q k + − = .Vlasov B. F. [4] has built an exact analytical solution of this problem in trigonometric series.
Components of the vector of displacement are determined by the functions of three spatial coordinates and time

(
) ( ) ( ) , , , , , , , , , , , u x x z t u x x z t u x x z t .Components of the tensor of strain are determined by Cauchy correlation.The plate is considered as a three-dimensional body, its material obeying Hooke's generalized law: , where As an equation of motion of the plate we will use three-dimensional equations of motion of the theory of elasticity: ( ) here ρ -is a density of plate material.
Boundary conditions on the lower and the upper surfaces z h = and z h = − have the form:

Method of Solution
Methods of constructing a bimoment theory of plates is based on displacements expansions in infinite series, Hooke's generalized law (1), three-dimensional equations of the theory of elasticity (2) and boundary conditions on face surfaces (3).Components of the vector of displacements are expanded in Macloren's series in the form [3]- [7]: B A -are unknown functions of two spatial coordinates and time:  Offered bimoment theory of plates [5]- [8] is described by two unrelated problems, each of which is formulated on the basis of nine two-dimensional equations with corresponding boundary conditions.It should be noted that proposed bimoment theory of plates presents a two-dimensional theory of elastic orthotropic layer, which is deformed in general three-dimensional form.
The first problem is described by two equations relative to longitudinal and tangential forces, by four additionally constructed equations in relation to bimoments and three equations obtained from the boundary conditions (3) on the basis of expansion (4).The forces, moments and bimoments of the plate are determined by nine unknown kinematic functions from relationships [5]- [8]: We will get the equations of equilibrium relative to longitudinal and tangential forces by integrating two first equations of the theory of elasticity in coordinate z (2): where 11 12 22 , , N N N -are longitudinal and tangential forces determined from relationships -terms of equations with external load.
On the basis of force expression (7) two Equations ( 6) include three unknown functions 1 2 , ,r ψ ψ . To derive additional equations we will introduce bimoments, generated in tension and cross compression of the plate.Longitudinal and tangential bimoments are determined by expressions: 11 22 12 , , T T T : Introduce intensities of transversal bimoments 13 23 , p p and 13 23 , τ τ on tangential stresses 13 23 , σ σ : ( ) ( ) Introduce intensities of normal bimoments 33 p and 33 τ on normal stress 33 σ in the form of relations: Equations in relation to longitudinal and transversal bimoments, acting in plate plane are obtained in the form: By using series (4) and Formulas ( 5) are obtained expressions for series' (4) coefficients ( ) Equations ( 6), (11), ( 12) and (13) make a combined system of differential equations of motion, which consists of nine equations relative to nine unknown functions: 1 2 1 1 1 2 , , , , , , , , u u r W ψ ψ β β γ .The second problem is described by two equations of moments, one equation of crosscutting forces, three equations of bimoments and three equations, obtained from boundary conditions (3) on the bases of expansion (4).Here forces, moments and bimoments are determined relative to nine unknown kinematic functions in the form [5]- [8]: The first three of these equations are the ones relative to bending, torsion moments and an equation relative to crosscutting forces, the rest three equations are derived in relation to bimoments.
Multiplying the first and the second equations of the theory of elasticity by coordinate z and integrating it by z, we will obtain an equation of equilibrium in moments and forces: , . 2 2 Integrating the third equation of the theory of elasticity by coordinate z (2), we will obtain an equation of equilibrium in forces: Bending and torsion moments are determined in the form: ( ) Expressions for crosscutting forces have the form: In Equations ( 15) terms with external load are determined by the following formula: To derive other equations we will introduce the following bimoments, generated at bending and shear of the plate.Bimoments 11 22 12 , , P P P are determined by the following formula: ( ) Intensity of transversal tangential and normal bimoments 13 23 , p p   and 33 p  are determined by expressions ( ) Equations relative to bimoments at bending and transversal shear are derived in the form: From expressions for series (4) and Formulas (14) relations for series' coefficients ( ) ( ) ( ) functions and boundary conditions (3) lets represent like next equations: The system of differential equations of motion ( 15), ( 20), ( 21) and (22) makes combined system of nine equations relative to nine unknown functions 1 2 1 2 1 1   , , , , , , , ,  u Formula to determine the displacements and stresses in the layers of the plate z h = − and z h = are: Thus, two unrelated problems of bimoment theory of thick plates are formulated in the paper.An accuracy of bimoment theory is defined in dependence on the number of held terms of the series (4).In construction of equations of equilibrium eight terms are held, while for expressions (13) and ( 22) six terms of each series are held (4).The first equation in ( 13) and the second equation in ( 22) are built up to the fourth order relative to small parameter of the plate 10 H a .The second Equation in (13) and the first Equation in ( 22) are built up to the sixth order relative to the parameter.

Solution of Tests Problem
As an example consider the problem of static bending of the plate, loaded by normal load: ( ) along the upper face surface z h = − , where 0 q -is a load parameter.Let the ends of the plate rest on the ends, then 1 const x = and 2 const x = , and we have the conditions : The   , , Solution of the Equations ( 6), (11), ( 12) and (13), satisfying boundary conditions (24), is written in the form: where 15), (20), ( 21) and ( 22), satisfying boundary conditions (24), has the form: Substituting solution (27) into Equations ( 6), ( 11), (12) (13), we will obtain the system of linear algebraic equations relative to nine unknown constants 1 2 9 , , , C C C  .In similar way, substituting solution (28) into Equations (15), (20), ( 21), ( 22), we will obtain one more system of linear algebraic equations relative to nine unknown constants.
Analysis for orthotropic square plate with elastic characteristics is conducted [1]:  1 and Table 2 give dimensionless numerical results of calculations of displacements and stresses in upper and lower layers of the plate.Maximal values of displacements and stresses of the plate are reached in face surfaces of the pate and are determined by solutions of the first and the second problems.
Table 3 and Table 4 give dimensionless numerical results of displacements and stresses in the middle surface.
It should be noted that coefficients of the series with zero indices are displacements ( ) , , B B A , characterize rotary angle of normal vector and strain of transversal com- pression of middle surface of the plate.
Analysis has shown that the values of normal displacement 3 u vary considerably along the thickness of the plate.In [4] [8] it is shown that maximal values of stresses and displacements, obtained according to bimoment theory for isotropic plates with high accuracy agree with calculations of an exact solution [4].

Conclusion
So on the basis of expansion method, a theory of plates is improved by consideration of bimoments.In the case of spatial deformation of the plate along its thickness, there nonlinear laws of displacements distribution occur, without any simplifying hypotheses.Consequently, existing specified theories of plates and shells, built with a number of simplifying hypotheses could not be used in development of methods of calculation of stresses and displacements of thick plates and shells under the effect of various types of external influences.Calculations of thick plates from anisotropic materials with low strength characteristics could not be made on the basis of classical or specified existing theory.In such cases it is advisable to conduct calculations based on rigorous methodologies developed on the basis of the theory of plates and shells, which takes into account all the components of stress and strain tensor ( ) , , , 1, 3 functions and boundary conditions (3) let represent as next equations:

Table 1 .
Values of displacements and stresses in the upper layer of the plate.

Table 2 .
Values of displacements and stresses in the lower layer of the plate.

Table 3 .
Values of displacements and rotary angle of a normal in middle surface of the plate.

Table 4 .
Values of stresses in middle surface of the plate.