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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.

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 [

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.

Consider orthotropic thick plate of constant thickness

Introduce Cartesian system of coordinates

metric series.

Components of the vector of displacement are determined by the functions of three spatial coordinates and time

where

As an equation of motion of the plate we will use three-dimensional equations of motion of the theory of elasticity:

here

Boundary conditions on the lower and the upper surfaces

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 [

here

Offered bimoment theory of plates [

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 [

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

On the basis of force expression (7) two Equations (6) include three unknown functions

Introduce intensities of transversal bimoments

Introduce intensities of normal bimoments

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:

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 [

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:

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

Intensity of transversal tangential and normal bimoments

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

The system of differential equations of motion (15), (20), (21) and (22) makes combined system of nine equations relative to nine unknown functions

Formula to determine the displacements and stresses in the layers of the plate

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

order relative to the parameter.

As an example consider the problem of static bending of the plate, loaded by normal load:

The values

here

Solution of the Equations (6), (11), (12) and (13), satisfying boundary conditions (24), is written in the form:

where

Solution of the Equations (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

Analysis for orthotropic square plate with elastic characteristics is conducted [

and kinematic functions

Analysis has shown that the values of normal displacement

1/3 1/4 1/5 1/6 1/10 | 0.5282 1.1970 2.3131 3.9885 18.5117 | 0.7210 1.5193 2.7671 4.5727 19.5965 | 2.6566 5.8987 11.7869 21.5969 135.5503 | −2.9471 −4.7690 −7.1799 −10.1601 −27.6397 | −1.5908 −2.2905 −3.1620 −4.2010 −10.2253 | 0.7326 1.1947 1.7875 2.5103 6.7043 |

1/3 1/4 1/5 1/6 1/10 | −0.5119 −1.2028 −2.3388 −4.0325 −18.6193 | −0.7871 −1.6414 −2.9410 −4.7958 −20.0046 | 2.2516 5.4911 11.3784 21.1880 135.1412 | 2.5562 4.4799 6.9456 9.9581 27.4884 | 1.3123 2.0546 2.9474 4.0074 10.0410 | −0.7618 −1.2509 −1.8578 −2.5886 −6.7951 |

1/3 1/4 1/5 1/6 1/10 | −0.0296 −0.0355 −0.0409 −0.0463 −0.0693 | −0.0773 −0.0970 −0.1168 −0.1370 −0.2198 | −0.2054 −0.2062 −0.2061 −0.2058 −0.2051 | −0.2079 −0.7391 −1.7176 −3.2571 −17.2440 | −0.5186 −1.2758 −2.4808 −4.2421 −19.0810 | 2.4322 5.4460 11.7303 21.6553 136.2752 |

1/3 1/4 1/5 1/6 1/10 | 0.7033 1.0019 1.2986 1.5921 2.7469 | 0.6492 1.0019 1.0357 1.2274 1.9995 | −0.4904 −0.4965 −0.4985 −0.4992 −0.4999 | −0.0627 −0.0582 −0.0555 −0.0537 −0.0509 | 0.0089 −0.0093 −0.0210 −0.0285 −0.0410 | −0.0438 −0.0552 −0.0612 −0.0647 −0.0702 |

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