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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.

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 [

The authors in [

This paper gives a brief description of the technique of constructing a theory of plates with consideration of bimoments generated due to displacements distribution of cross-section points by a non-linear law. Here the equations of bimoments are built with the equation of three-dimensional dynamic theory of elasticity, described on face surfaces of the plate. The bimoments are introduced in stress dimensions and are characterized by the intensity of generated bimoments. We would use the designations and determinant correlations of forces, moments, bimoments and equations of motion relative to these force factors.

Unlike bimoment theory in [

Determinant relationships of forces, moments, bimoments and equations of motion relative to these force factors are given.

Consider an orthotropic thick plate of constant thickness

When building an equation of motion the plate is considered as a three-dimensional body and all components of stress and strain tensors:

The components of strain tensor

For orthotropic plate, the Hooke’ law, in a general case, is written as:

where

As an equation of motion of a plate we would use three-dimensional equations of dynamic theory of elasticity:

where

Boundary conditions on lower and upper face surfaces of the plate

Here

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

The displacements in stresses in upper

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

Introduce the external loads for the first problem

The expressions of longitudinal and tangential forces are written as [

, (9.а)

The intensities of the bimoments

The intensity of the bimoment

The equations of motion relative to longitudinal and tangential forces and bimoments from tangential and normal stresses have the form [

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

The second problem of bimoment theory consists of the equations for bending moments, torsional moments, shear forces relative to six kinematic functions

Introduce the generalized external loads for the second problem

Bending, torsional moments and shear forces, which are rigorously built, have the form [

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 [

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

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

Here the intensities of the bimoments

The second system of equations obtained from the equations of the theory of elasticity (3) is written in the form:

Here

The intensities of the bimoments

The intensities of the bimoments

Here

The expressions of the intensities of the bimoments

The expressions of the intensities of the bimoments

Write down the formulae to determine the displacements on the face surfaces of the plate

Formulae for stresses on the face surfaces of the plate

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).

Note, that the expressions of intensities of the bimoments (10), (27), (28), (29), (30), (31) and (32) are built for the first time and are new in the theory of plates.

Consider the boundary conditions of a discussed problem for the thick plates.

1) On the border of the plate the displacements are zero. On the edges of the plate

2) On the border

3) On the border

Boundary conditions on the border

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).

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:

where

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).

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

For square plates with dimensions

of external effect

Calculations show that when the value of the frequency of external effect

0.0000 | −0.0158 | 0.8560 | 0.9402 | 0.0113 | 0.4591 |

0.3000 | −0.0190 | 0.9882 | 1.0769 | −0.0108 | 0.5277 |

0.4000 | −0.0226 | 1.1302 | 1.2240 | −0.0356 | 0.6010 |

0.5000 | −0.0294 | 1.4039 | 1.5074 | −0.0852 | 0.7419 |

0.6000 | −0.0463 | 2.0663 | 2.1937 | −0.2098 | 1.0816 |

0.7000 | −0.1365 | 5.5665 | 5.8213 | −0.8909 | 2.8704 |

0.0000 | −0.0751 | 2.6428 | 2.7386 | −0.1047 | 0.7774 |

0.1000 | −0.0802 | 2.7835 | 2.8825 | −0.1263 | 0.8155 |

0.2000 | −0.1007 | 3.3478 | 3.4601 | −0.2158 | 0.9676 |

0.3000 | −0.1783 | 5.4420 | 5.6053 | −0.5650 | 1.5275 |

0.0000 | −0.3067 | 8.3941 | 8.5060 | −0.3487 | 1.2598 |

0.1000 | −0.3993 | 10.4739 | 10.6163 | −0.6033 | 1.5244 |

0.1300 | −0.5105 | 12.9496 | 13.1296 | −0.9163 | 1.8364 |

0.1600 | −0.8082 | 19.5290 | 19.8115 | −1.7691 | 2.6596 |

0.1700 | −1.0499 | 24.8541 | 25.2206 | −2.4678 | 3.3234 |

0.1800 | −1.5560 | 35.9866 | 36.5297 | −3.9369 | 4.7087 |

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

Kinematic equations serve to determine the generalized displacements

The equations (26) are determined by the solution of the system of linear algebraic equations relative to coefficients of the series (5)

Based on these studies, we would note that using the method of expansion in a series as part of three-dimen- sional 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.

Мakhamatali K. Usarov,Davronbek М. Usarov,Gayratjon T. Ayubov, (2016) Bending and Vibrations of a Thick Plate with Consideration of Bimoments. Journal of Applied Mathematics and Physics,04,1643-1651. doi: 10.4236/jamp.2016.48174