Numerical-analytical method in solving the problem of creep of a shallow shell

Objective. The paper presents general equations of the moment theory of thin shallow shells with a relatively small rise above the plane of their projection taking into account creep deformation. The problem of the stress-strain state of the shell with boundary conditions is considered. At the edges, the shell is connected to diaphragms that are absolutely rigid in their plane and flexible from it. Resolving equations are obtained for calculating shallow isotropic and orthotropic shells taking into account creep deformations. The problem is reduced to a system of two fourth-order differential equations with respect to deflection and stress function. Method. The solution is given by the numerical-analytical method in the MATLAB software package. The nonlinear Maxwell-Gurevich equation is used as the equation of state between creep deformations and stresses. To determine creep deformations, a linear approximation of the first derivative with respect to time (Runge-Kutta method) was used. To verify the solution to the problem, a shell made of secondary PVC was calculated using the grid method. The method has been tested by comparing the solution with the calculations of other well-known researchers. Result. A program has been developed for calculation in the MATLAB package with the ability to vary the initial data and output a graph of the dependence of displacements and stresses on time. It has been established that stresses and internal forces in an orthotropic shell of the same shape as for an isotropic one are subject to stress redistribution: normal stresses increase, and tangential stresses decrease. Longitudinal and shear forces remain almost constant; stress changes occur mainly due to the redistribution of bending and torque moments. Conclusion. The proposed approach can be applied to the analysis of the stress-strain state and bearing capacity of a reinforced concrete shell as well. There are no restrictions on boundary conditions and types of loading, and the beam material can be not only polymers and composites for construction purposes, but also concrete.

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