An analysis of annular plate in curvilinear non-orthogonal coordinates with the help of equations of a shell theory1

Article history Received: August 7, 2020 Revised: October 15, 2020 Accepted: November 16, 2020 Abstract The complete system of equations of a linear theory of thin shells in curvilinear non-orthogonal coordinates proposed in the paper was taken as the basis of the investigation. Earlier, this system was used for static analysis of a long developable helicoid. In the article, this system is applied for the determination of stress-strain state of annular and circular plates under action of the external axisymmetric uniform load acting both in the plane of the plate and out-of-their plane. Presented results for annular plate given in the non-orthogonal coordinates expand a number of problems that can be solved analytically. They can be used as the first terms of series of expansion of displacements of degrees of the small parameter if a small parameter method is applied for examining a long tangential developable helicoid.


Introduction
It is known, that the simplest equations of theory of thin shells are turned out for middle shell surfaces given in principle curvatures. But sometimes, it is very difficult to set a surface in principle curvatures and one must use governing equations of a theory of thin shells in curvilinear non-orthogonal coordinates. The complete system of equations in curvilinear non-orthogonal coordinates was proposed by A.L. Goldenveizer [1]. This system contains internal "pseudoforces", "pseudomoments", and Christoffel's symbols. The system of equations, presented by Ya.M. Grigorenko and A.M. Timonin [2], is written in tensor form.
The system, proposed by the author, contains internal forces and moments usual for engineers and is free from Christoffel's symbols [3]. Hereinafter, the equations presented in a paper [3] will be used.

An aim of investigation
Having the systems of the governing equations of a theory of thin shells, set by different scientists, it is desirable to use them for solution of problems of bending of plates and for solution of plane problems of their analysis, or to apply the governing equations of a theory of thin shells for solution of test examples of analysis of plane elements. This approach is illustrated by an example of reducing of general equations of a theory of thin shells in curvilinear arbitrary coordinates to the equations for analysis of tangential developable helicoid, and after for analysis of annular plates under action of axisymmetric uniform load of two types.

Governing equations of a theory of thin shells in curvilinear non-orthogonal coordinates
A system of the governing equations of a theory of thin shells in curvilinear non-orthogonal coordinates, proposed by the author, has the following form [3]: -six equilibrium equations: where X, Y, Z -external uniform surface load in the direction of mobile orthogonal axes x, y, z, and the х-axis coincides with the tangent to the u-coordinate line; -six geometrical equations suggested by A.L. Goldenveizer and presented in a monography [1], the first three of which after submission of Christoffel's symbols in them can be expressed as [3] Remaining three equations for the determination of change of curvatures κ u and κ v and torsion κ uv have rather complex form, for example, for shallow shells Eight physical equations connecting internal forces N u , N v , S u , S v , Q u , Q v and moments M u , M v , M uv , M vu and components of tangential and bending deformations ε , ε , ε , κ , κ , κ u v between themselves can be written as [3] ctg Eh M (4) where χ ≠ π/2 is the angle between the coordinate lines u and v; ν = Poisson's ratio; h = thickness of shell.
A vector of displacements can be written as The forces (a) and moments (b), per unit length, needed for equilibrium The forces and moments, per unit length, needed for equilibrium are shown in Figure and are positive as shown.

Developable helicoid
Parametric equations of evolvent (developable) helicoid can be written as is the lead of a helix u = 0 (cuspidal edge); v is an angle measured from an axis Ох; a is a radius of a cylinder on which the helical cuspidal edge is lying.
In this case, coefficients of the fundamental forms of the surface (5) and its principal curvatures (k u ; k v ) can be written as and also The u-coordinate lines coincide with the straight generatrixes of the helicoid but the v-lines are co-axis helixes. The formulae (6) show that conjugated (M = 0) non-orthogonal (F ≠ 0) system of curvilinear coordinates is used.
For a developable helicoid (5) with coefficients of the fundamental forms (6), the equations (1)-(3) become simpler. Substituting the geometrical equations (2), (3) into the formulae of Hooke's law (4) but the results into the equilibrium equations (1) one can obtain a system of three differential equations in partial derivatives of the 8 th order. This was made in a monograph [3]. But this system of equations was not solved analytically.
A system of three differential equations in partial derivatives of the 8 th order, presented in a work [4], was reduced to a system of three ordinary differential equations of the 8 th order for a long developable helicoid. It was assumed that all components of stress-strain state of a long developable helicoid depend only on the u-coordinate, i. e. all derivatives with respect to parameter v are equal to zero. In this case, a problem of determination of components of stress-strain state yields to analytical solution. With the help of small parameter method, it was solved in works [4; 5]. With the help of asymptotic semianalytical method, it was solved in a work [6]. Analogous approach was used in a paper [7] for analysis of a long right helicoid. Two methods of  [8] where the equations of A.L. Goldenveizer [1] and the equations (1)-(4) were applied. Now five types of helicoidal shells are known [9]. All these shells can be analyzed with the help of presented equations (1)-(4) [10; 11].

Governing equations for thin annular plates in curvilinear non-orthogonal coordinates
For annular plates, we have b = 0, therefore, formulae (1)-(7) are simplified and become m = a; A = 1; F = a; B 2 = a 2 + u 2 ; Substituting the values (8) into formulae (1)- (7) gives the possibility easily to obtain corresponding governing equations for annular plate or its fragment subjected to arbitrary uniform surface load or to linear load along contour of the plate acting in the plane of the plate.
One can obtain a system of three differential equations in partial derivatives in displacements and reduce them to a system of two differential equations in displacements of the same order. But to solve analytically the obtained systems it is not possible yet. Of course, this problem can be solved with the help of FEM [12].
Assume that an annular plate is subjected to an axisymmetric load. Then, all parameters of stress-strain state of the plate will not depend on the v-coordinate, i. e.
where С 1 , С 2 , С 3 , С 4 are the unknown constants of the integration, that can be determined after satisfaction of boundary conditions. Substituting a value of displacement (14) in the expression for the determination of BQ u gives 8 2 , i. e.

.
At last, we can find

Applied load is in the plane of the plate
Examine loading annular or circular plate (a = 0) by axisymmetric load acting in the plane of the plate. In this case, we use the first two equations of equilibrium (9). After substitution the first fourth physical equations in them and after some transformations, equations (9) take the following form: Integration of two last expressions gives where D 1 , D 2 , D 3 , D 4 are the unknown constants of the integration. The last two equations give the possibility to find parameters of deflections (12). Substituting these parameters into the physical equations (11) one can find the internal forces and moments in an annular or circular plate subjected to an external uniform axisymmetric load acting in the plane of the plate.
Assume that X = Y = 0 then from the last two expressions we determine The last expression can be applied if uniform loads X, Y are absent, but displacements given in advance or external moments or forces given in advance are known on the plate edges.
In a paper [13], the analogous axisymmetric problem was solved by using finite difference method.

Conclusion
If one assumes that a radius of the inner edge of the annular plate a = 0 then B = u, χ = π/2 and the wellknown equations for circular plate in the polar coordinates would be obtained [14].
It shows that governing equations of a shell theory (1)-(4) assumed as a basis of presented investigation are correct. The results obtained for annular plate in non-orthogonal coordinates widen a class of problems which now can be solved analytically. Examining a long tangential developable helicoid with the help of a small parameter method [15], it is possible to use the solutions (14) and (15) as the first terms of series of expansion of displacements of degrees of the small parameter. Using equations (8)- (15) it is possibly to obtain analytical solutions for plates with different axisymmetric static load and support conditions presented in a paper [16].
The additional information on the application of annular and circular plates and on numerical and analytical analysis of thin and thick plates is given in manuscripts [17][18][19][20].