Static thermal stability of a shallow geometrically irregular shell made of orthotropic temperature-sensitive material

A flat orthotropic geometrically irregular shell of constant torsion, whose thermomechanical parameters are linearly dependent on temperature, is considered. When the temperature reaches a certain value, the change in the shape of the equilibrium occurs abruptly, which causes a change in the initial geometry of the shell. These temperatures are called critical.
For practice, the relationships connecting the critical temperatures with the geometrical and thermomechanical parameters of the geometrically irregular shell are of considerable interest. The solution of the problems of static thermal stability of geometrically irregular shells usually begins with an analysis of their initial momentless state. Tangential forces caused by shell heating are defined as solutions of a system of singular differential equations of momentless thermoelasticity. These efforts are contained in the Brian or Reissner forms in the equations of static thermal stability and the further solution of the problem essentially depends on their structure.
In this paper, the solution of singular momentless thermoelasticity is found by elementary functions. Using the method of displacement functions, the equations of moment thermoelasticity, written in the components of the displacement field, are reduced to a single singular differential equation in partial derivatives of the eighth order depending on the temperature, which is assumed to be constant. The solution is written as a double trigonometric series. The coefficients of the series, based on the Galerkin procedure, are determined as solutions to a linear homogeneous algebraic system of equations. From the equality to zero of the determinant of this system, an algebraic equation of the fifth degree is obtained for the relative critical temperature. The smallest positive real root of which is the desired temperature. A quantitative analysis of the influence of the geometrical and thermomechanical parameters of the geometrically irregular shell on the value of the critical temperature is carried out.