Bending and Vibrations of a Thick Plate with Consideration of Bimoments

The paper is dedicated to the development of the theory of orthotropic thick plates with consideration of internal forces, moments and bimoments. The equations of motion of a plate are described by two systems of six equations. New equations of motion of the plate and the boundary conditions relative to displacements, forces, moments, and bimoments are given. As an example, the problems of free and forced oscillations of a thick plate are considered under the effect of sinusoidal periodic load. The problem is solved by Finite Difference Method. Eigenfrequencies of the plate are determined, numeric maximum values of displacements, forces and moments of the plate are obtained depending on the frequency of external force. It is shown that when the value of the frequency of external effect approaches the eigenfrequency, there occurs an increase in displacement, force and moment values; that testifies a gradual transition of the motion of plate points into the resonant mode.


Introduction
Theory of plates and shells has a special place in design of structural elements.Specified theories of plates are built by many authors.All existing specified theories of plates are developed on the basis of a number of simplifying hypotheses.An overview of the main statements and common methods of constructing an improved theory of plates and shells can be found in the works of S. A. Ambartsumyan [1], K. Z. Galimov [2], Sh.K. Galimov [3], Kh.M. Mushtari [4] and others.Static problem of the bending of a thick isotropic plate in threedimensional theory of elasticity is considered by B.F. Vlasov in [5], which gives an exact analytical solution in
When building an equation of motion the plate is considered as a three-dimensional body and all components of stress and strain tensors: are taken into consideration.The components of displacement vector are the functions of three spatial coordinates and time

(
) ( ) ( ) , , , , , , , , , , , u x x z t u x x z t u x x z t .The components of strain tensor ij ε are determined from Cauchy relation as: For orthotropic plate, the Hooke' law, in a general case, is written as: where 11 12 33 , , , E E E  are the elastic constants, determined through Poisson ratio and the moduli of elasticity in the form [14] [15].
As an equation of motion of a plate we would use three-dimensional equations of dynamic theory of elasticity: where ρ is a density of plate material.Boundary conditions on lower and upper face surfaces of the plate z h = and z h = − are: Here ( ) ( ) ( ) , , q q q + + + are distributed external loads, applied to upper and lower face surfaces of the plate z h = and z h = − along the direction of 1 2 , , ox ox oz coordinates axes.

Method of Solution
The methods of building the bimoment theory of plates are based on Cauchy relation (1), generalized Hooke's law (2), three-dimensional equations of the theory of elasticity (3), boundary conditions on face surfaces (4).A proposed bimoment theory of plates is also described by two non-connected problems, each of which is formulated on the basis of six two-dimensional equations of motion with corresponding boundary conditions.The components of displacement vector are expanded into Maclaurin infinite series in the form: Here ( ) , B A are unknown functions of two spatial coordinates and time ( ) , , , , In a general case, these functions are determined according to the formulae: The displacements in stresses in upper z h = − and lower points z h = in plate fibers we would designate as ( ) ( ) ( ) The first problem of bimoment theory describes tension-compression and transverse reduction of the plate, and the second one-the bending and transverse shear of the plate.Determinant relationships and corresponding equations of motion of the plate in the first and second problems are briefly described below.
The first problem is described by the forces and bimoments with six generalized functions 1 2 1 2 , , , , , u u r W ψ ψ , which are determined by relationships: Introduce the external loads for the first problem 2 2 q q q q q q q q q The expressions of longitudinal and tangential forces are written as [12]- [15]: The intensities of the bimoments 13 23 , p p from tangential stresses 13 23 , The intensity of the bimoment 33 p from normal stress 33 σ is written in the form: The equations of motion relative to longitudinal and tangential forces and bimoments from tangential and normal stresses have the form [12]- [15]: Note, that the expressions of force factors ( 9), (10), and hence, the equations of motion of the system (11), ( 12) is rigorously built.This system consists of three equations relative to six unknown functions 1 2 1 2 , , , , , u u r W ψ ψ  .As could be seen, three equations are missed.If in expressions (9.а) the terms 13 13

2
, 2 E W E W are omitted, then we would obtain two equations of motion of classic theory of plates in the form (11), since the equation of motion (12) becomes isolated and fail.
The second problem of bimoment theory consists of the equations for bending moments, torsional moments, shear forces relative to six kinematic functions 1 2 1 2 , , , , , , determined by formulae: ( ) ( ) Introduce the generalized external loads for the second problem 2 2 q q q q q q q q q Bending, torsional moments and shear forces, which are rigorously built, have the form [12]- [15]: The system of equations of motion of the second problem consists of two equations relative to bending, torsional moments and one equation relative to shear force and it is written in the form [12]- [15]: Note, that the expressions of forces and moments (16), hence, the equations of motion of the system (17), (18) are rigorously built.Similar to the first problem, here three equations are missed.The system of equations of motion ( 17), (18) consists of three equations relative to six unknown functions 1 2 1 2 , , , , , . If in expressions of forces and moments, into the equations of motion ( 17) and (18) conventionally introduce k G k G , (where z k is a shear coefficient), then an equation of motion of plates could be obtained according to Timoshenko's theory.
To complete the systems ( 11), ( 12) and ( 17) and ( 18) it is necessary to build two more systems, with three equations in each.Write down three equations of motion of the theory of elasticity (3) on face surfaces of the plate z h = − and z h = + .Adding and subtracting the equations of the theory of elasticity (3) on face surfaces of the plate z h = − and z h = , and taking into account the Hooke's law (2), surface conditions (4) and desig- nations ( 6), ( 7) and ( 13), (14), two independent systems with three equations in each could be obtained.The first of these systems describes the first problem and has the form: Here the intensities of the bimoments 11 22 12 , , σ σ σ -under transverse reduction and tension-compression of the plate, generated due to 11 22 12 , , σ σ σ are: The second system of equations obtained from the equations of the theory of elasticity (3) is written in the form: Here 11 22 12 , , σ σ σ    are the intensities of the bimoments under transverse bending and shear for the second problem generated due to the stresses 11 22 12 , , σ σ σ : The intensities of the bimoments * * * 13 23 33 , , σ σ σ    , generated due to the stresses 33 13 23 , , σ σ σ , under transverse shear and bending are written in the form: The intensities of the bimoments , , , , , , The expressions of the intensities of the bimoments * * * 13 23 33 , , σ σ σ are determined by the solution of the sys- tem of linear algebraic equations relative to coefficients of Maclaurin series ( ) ( ) ( )  , which are obtained by the substitution of the series (5) into the conditions on face surfaces at z h = − and z h = (4) and designations ( 6), (7).
The expressions of the intensities of the bimoments * * * 13 23 33 , , σ σ σ    are determined by the solution of the sys- tem of linear algebraic equations relative to coefficients of Maclaurin series ( ) ( ) ( ) =  , which are obtained by the substitution of the series (5) into the conditions on the face surfaces at z h = − and z h = (4) and designations ( 13), (14).
Write down the formulae to determine the displacements on the face surfaces of the plate z h = − and z h = + : Formulae for stresses on the face surfaces of the plate z h = − and z h = have the form: Maximum values of displacements and stresses of the plate are reached on the face surfaces of the plate and are determined by the solutions of the first and second problems by the formulae (33) and (34).
Consider the boundary conditions of a discussed problem for the thick plates.
On the border of the plate the displacements are zero.On the edges of the plate 1 const x = and 2 const x = the conditions should be as follows: 2) On the border 1 const x = the plate is supported.The following conditions should be satisfied: 0, 0, 0, 0, Boundary conditions on the border 2 const x = are similarly written.When studying the problem of transverse bending and shear it is enough to consider only the second problem with the equations of motion (17), ( 18), ( 23), (24) and boundary conditions (35)-(40).

Solution of Tests Problem
As an example, consider the forced harmonic vibrations of a cantilever rectangular plate fixed on both ends under the effect of harmonic periodic external load: 0, 0, 0, 0, sin sin sin x x q q q q q q q t a b where 0 0 0 , , q ω β is an amplitude, frequency and the mode of vibration of an external load, respectively.Note, that if 0 0 ω = , we obtain the problem of static bending of the plate.
Substituting (41) into ( 8) and (15) determine the load terms of the equation of motion.For a plate fixed on both ends the boundary conditions are written in the form (35) and (36).

Numeric Results
First determine eigenfrequencies of the plate.After dividing the variables by spatial coordinates and time, the problem is solved by Finite Difference Method.The step in spatial coordinates is the value of eigenfrequency is 1 p = .Table 1 shows the results obtained for the displacements, moments and forces in fixed square plates the values of the displacements, forces and moments dramatically increase; this testifies of gradual transition of the motion of plate points into resonant mode.As seen, an abrupt increase in the values of displacements, forces and moments could be observed.
Table 2 and Table 3 show numeric values of displacements, moments and forces, calculated for the fixed square plates with dimensions Calculations show that when the value of the frequency of external effect 0 ω approaches eigenfrequency, an increase in the values of displacements, forces and moments is observed; this testifies of gradual transition of the motion of plate points into resonant mode.
Kinematic equations serve to determine the generalized displacements 1 2 , u u   .The equations (26) are determined by the solution of the system of linear algebraic equations relative to coefficients of the series (5) ( ) ( ) ( ) =  , which are obtained by the substitution of the series (5) into the conditions on the face surfaces at z h = − and z h = (4) and designations ( 13), (14).

Conclusion
Based on these studies, we would note that using the method of expansion in a series as part of three-dimensional dynamic theory of elasticity, a two-dimensional bimoment theory of orthotropic thick plates was developed and the equations of motion of the plate relative tot forces, moments and bimoments were built.It is shown that the problem in the general case is reduced to the definition of twelve unknown functions of two spatial coordinates and time.New expressions to determine the forces, moments and bimoments of the plates were built, as well as the methods for solving the problems of free and forced vibrations of plates based on Finite Difference Method.

3 )
On the border 1 const x = the plate is free of supports.The following conditions should be satisfied

.
When the value of the frequency of external effect 0 ω approaches the eigenfrequency 1 0.7469 p =

Table 3 .
DisplacementsCalculations show that the equations of motion of a plate (23) may be substituted by kinematic conditions relative to tangential stresses: