The deformation problem of a circular elastic flexible plate under variable load

It is well known that for the solutions of equations with variable coefficients the methods of nonlinear mathematics are applied. These methods allow to solve analytically significant amount of problems given in the classic monographs of the leading experts in the mechanics of deformable bodies, geomechanics and geotechnics, theoretical mechanics, thermoelasticity, heat conduction theory and other areas. In many cases these remarkable monographs contain rather general nonlinear or with variable coefficients differential equation processes [1-7]. Solutions in such publications are being built for a large extent simplified equations or differential equations set. The present work gives analytical solutions and graphics assigned problems method of partial discretization of nonlinear differential equation. In the work [8] obtained an analytic solution of the problem bending of a thin elastic component axisymmetric inhomogeneous plate with a hole, each component of which has a variable stiffness and working in a nonuniform temperature field. In the work [9] investigated the stress-strain state the cylindrical shell of variable and constant in thickness due to distributed the load. In both works solving problems were obtained by method partial discretization nonlinear the differential equations, since this method allows you to get solve the problem without any simplifications. In this paper presents the analytical solutions and graphs problem about deformation round elastic, flexible plate loaded by a variable load by method the partial discretization of nonlinear differential equations. Round and ring plates as elements of machinebuilding objects meet quite often. These constructions can be attributed turbine disks, the bottom of the combustion chambers of jet engines, gas turbines, boilers, tanks, reservoirs, etc.


Introduction
It is well known that for the solutions of equations with variable coefficients the methods of nonlinear mathematics are applied.These methods allow to solve analytically significant amount of problems given in the classic monographs of the leading experts in the mechanics of deformable bodies, geomechanics and geotechnics, theoretical mechanics, thermoelasticity, heat conduction theory and other areas.In many cases these remarkable monographs contain rather general nonlinear or with variable coefficients differential equation processes [1][2][3][4][5][6][7].Solutions in such publications are being built for a large extent simplified equations or differential equations set.The present work gives analytical solutions and graphics assigned problems method of partial discretization of nonlinear differential equation.
In the work [8] obtained an analytic solution of the problem bending of a thin elastic component axisymmetric inhomogeneous plate with a hole, each component of which has a variable stiffness and working in a nonuniform temperature field.In the work [9] investigated the stress-strain state the cylindrical shell of variable and constant in thickness due to distributed the load.In both works solving problems were obtained by method partial discretization nonlinear the differential equations, since this method allows you to get solve the problem without any simplifications.
In this paper presents the analytical solutions and graphs problem about deformation round elastic, flexible plate loaded by a variable load by method the partial discretization of nonlinear differential equations.
Round and ring plates as elements of machinebuilding objects meet quite often.These constructions can be attributed turbine disks, the bottom of the combustion chambers of jet engines, gas turbines, boilers, tanks, reservoirs, etc.

Formulation of the problem
Let us suppose that an alternating load acts on the circular plate.The plate is warmed irregularly along the radius and thickness (Fig. 1).

Fig. 1 Round plate with pinched contour
In the present work is obtained an analytic solution to the axisymmetric bending of a thin plate with variable mechanical characteristics, differential equations, which in general have the form [10]: where

The general solution of the problem
For the illustration obtaining a particular solution of the deformation problem of circular elastic flexible plate we assume that the modulus of elasticity, coefficient of thermal expansion and temperature vary according to the laws: where   , T r z is temperature, varia- bles are that depend on the the radius and the plate thickness.1) and ( 2) can be written as: .20 1 Discretizing of the second term of Eq. ( 6) and the third term of the Eq. ( 7) we will obtain: The solutions of Eqs. ( 8) and ( 9) can be written as: (11) Two differential equations of the third and fourth orders should have seven boundary conditions.But here we can be limited with six conditions as the function  does not interest us: it is enough to define its first derivative with respect to r.
The problem is solved for rigidly fix the points Fig. 2 Schemes of supports: a -rigidly fix the points the supporting the contour; b -hinged support along the contour of the plate the supporting the contour (Fig. 2, a) and hinged support along the contour of the plate (Fig. 2, b).

Solution of the problem for rigidly fixed points of the supporting contour plate
In rigid fixing adopt the following boundary conditions [11] Using the boundary conditions (13) one can find the integration coefficients after that the solution can be written as: where

Solution of the problem for hinged support along the contour the plate
For pin joint support we adopt the following boundary conditions [11]: Using the boundary condition (17) one can find the integration coefficients after that the solution can be written as: where From the solutions ( 14), ( 15) and ( 18), (19) using the mathematical induction method we find the formulas for  

The results of calculations
Using mathematical software for engineering calculations MathCAD the graphics profiles of bending of the circular plate along the radius with rigid and pin joint support along the contour are plotted (Figs. 3 and 4).Fig. 3 The profile bending of the radius a circular plate for rigidly fixing Fig. 4 The profile bending radius of the circular plate for case of simply supported The same problem was solved for the case when a uniformly distributed normal load acts on a plate.Curve changes are given in Fig. 5 for comparison at normal and variable loading. deflection at the r q q r q z 1 0   ) (

Conclusion
Expressions of the bending moments and effort in a median surface of a plate for two cases of fixing of a plate on a contour are received.
Deflections of a round plate (Figs.3-5) shows deformation under the influence of constants and variable loadings for various boundary conditions.Thus, the method partial discretization of nonlinear differential the equation gives the chance to solve the differential equations for any law of change of mechanical characteristics and to receive the corresponding picture of deformations.
Proceeding from this can be formulated the novelty of research which is to obtain a solutions to the system of equations for the first time analytical method of partial discretization of nonlinear equations.Furthermore, these solutions are completely satisfied to studied system of the differential equations and boundary conditions are shown in high precision of the obtained data.
(Work is was performed as part of the scientific program for basic research RK Ministry of Education "Solution new mathematical methods of nonlinear differential and integro-differential equations of fundamental and applied problems of mechanics of solid and deformable solids" (Contract for research № 851 from 02.03.2012, the grant funding for scientific research).

V.B. Rystygulova THE DEFORMATION PROBLEM OF A CIRCULAR ELASTIC FLEXIBLE PLATE UNDER VARIABLE LOAD S u m m a r y
The general view of the solution of a bend of a thin axisymmetric plate with the variable mechanical characteristics are received.The task is solved by a method of partial discretization of the nonlinear differential equations.Some versions of the solution of problems of deformations of a plate of a constant and variable thickness for various fixing of a plate on a contour are considered.
are coefficients, which determine phy-siccal characteristics of the plate.Using the changes of laws parameter (4) the differential Eqs. ( :

Fig. 5
Fig. 5 Deflection curves of the change:   deflection at the const q z  ,  1