Calculation of the rotation shells on axisymmetric load taking the creep into account

. The paper presents a derivation of governing equations and the method of calculation of shells of revolution taking the creep into account. Moment theory is considered. The problem is reduced to a system of two second order differential equations. We also present an example of the calculation of reinforced concrete shell on the effect of its own weight. Viscoelastic model of hereditary concrete aging is used.


Introduction
Axisymmetric problem is one of the important problems of structural mechanics and theory of elasticity. Examples of axially symmetric load acting on the shell of revolution are the dead weight, uniform snow load, the fluid pressure in the tank, and so on.
In this case, resolving equations of the theory of thin shells are greatly simplified. In some cases calculation can be carried out on the membrane theory. However, membrane theory can not account for the edge effect, which occurs in support zone [1]. Calculation of the shells by bending theory is linked with the solution of systems of differential equations with variable coefficients [2][3]. With regard to the calculation based on the creep, currently there are only some partial solutions [4][5][6][7][8].

Derivation of resolving equations
We consider the rotation shell, which is in terms of axisymmetric stress state. Element produced by cutting the two neighboring meridional planes and two normal sections is shown in Fig. 1.
For this element equilibrium equations can be written as [3]:

Fig. 1. Equilibrium of rotation shell element
Integrating the second equation in (1) in the range of 0 M to φ we will get: -the rotation angle of the normal.
Physical equations can be written as: We express from (6) stresses through the strains: Bending moments are defined as follows:  (1), we obtain the first governing equation for the functions α and V: For deriving the equation of compatibility of strains, we differentiate the second equation in (4) by φ: Substituting the expression for the longitudinal forces (12) in the expression for the deformation of the middle surface (14), and then the middle surface deformation compatibility equation of deformation (11), we obtain the second governing equation:

Results and discussion
Diagram of changes of the maximum value of the deflection w is shown in Fig. 3. The greatest value of the deflection at t → ∞, and t = τ 0 differ in 1.95 times. Redistribution of internal forces during creep was not observed. The difference between the highest values of the bending moments Mφ at t = τ 0 and t → ∞ is only 1.29%.

Summary
Obtained equations allow calculation of the shell of rotation of arbitrary shape at any load and an arbitrary law of the connection between stresses and creep strains. The calculation of a concrete shell in the form of hyperboloid showed that concrete creep does not have a significant impact on the magnitude of internal forces.