Numerical-analytical calculation of a cylindrical reservoir taking into account creep

. The article proposes a numerical-analytical solution to the problem of axisymmetric loading of the closed cylindrical shell, taking into account the creep of the material. The calculation is performed using the functions of A.N. Krylov in combination with the method of Euler and Runge-Kutta of the fourth order. Comparison with the solution using the finite difference method is presented.


Introduction
In paper [1] we earlier considered the problem of calculating the axis symmetrically loaded circular cylindrical shell (figure 1) taking into account the creep of the material. The problem was reduced to a fourth-order differential equation with respect to deflection w: where 3 2

12(1 )
Eh D    -cylindrical stiffness, E and ν -respectively, the modulus of elasticity and the Poisson's ratio of the material, h -shell thickness, (1) was solved numerically by the finite difference method. However, in this problem, there is a pronounced edge effect at the base of the shell, and therefore a sufficiently dense mesh is required for the correct calculation.
In this article, we propose the numerical-analytical method that allows one to more accurately take into account the edge effect in the support zone.

Methods
Let us introduce the dimensionless coordinate ξ determined by the formula: where α is a coefficient determined by the formula: The transition from derivatives with respect to x to derivatives with respect to ξ is carried out as follows: The solution to equation (5) can be represented as: where Y1, Y2, Y3, Y4 are the functions of academician A.N. Krylov: 1 cosh sin sinh cos ; 2 1 sinh sin ; 0 w -the particular solution of equation (5) without taking into account creep, * w -an additive to a particular solution related to creep.
The function 0 w can be expressed through the functions of A.N. Krylov by integral: For a triangular diagram of water pressure, the load function in dimensionless coordinates is written as: After substituting (9) into (8), the particular solution 0 w takes the form: The function * w is defined as follows: To integrate expression (11), a uniform mesh with respect to x is introduced. Integration is performed numerically using the trapezium or Simpson method. The angle of rotation is written as: The bending moment is determined as follows: 2  1 3  2 4  3 1  4 2  3  4  2  5 2 * * 2 ( 4 4 ) .
x x The boundary conditions for the shell rigidly clamped at the base are: Substituting the boundary conditions (15) into expressions (6), (12) -(14), we obtain: Thus, at each time step, we have two equations with two unknowns to determine the constants C3 and C4.
The creep deformations at each step are determined from the deformations and stresses at the previous step using the fourth-order Runge-Kutta method or Euler method. The procedure for determining creep deformations is described in more detail in [2][3][4][5][6][7][8][9].

Results and Discussion
The polymer shell from recycled PVC was calculated with the same initial data as in [1] : h = 1 cm, l = 3 m, R = 2 m, γ = 10 kN/m 3 , E = 1480 MPa, ν = 0.3. The nonlinear Maxwell-Gurevich equation was used as the creep law, which in cylindrical coordinates, taking into account axial symmetry, is written in the form: where * 0  is the initial relaxation viscosity, * m is the velocity modulus.
Rheological parameters of PVC at the temperature of 20 °С: high elasticity modulus E∞ = 5990 MPa, velocity modulus m* = 12.6 MPa, initial relaxation viscosity 0   = 9.06 • 10 5 MPa • min -were determined previously in paper [10]. Figure 2 shows the graphs of the time variation of the maximum deflection value (at the point x = 0.32 m), obtained using the Euler method for a different number of time steps n. This figure also shows a curve calculated on the basis of the Runge-Kutta method when the time interval is divided into 5 segments. The Runge-Kutta method has a significantly faster convergence, in the considered problem, 16 times fewer steps were required to obtain an acceptable result, however, when using it, at each step, it is required to determine the stresses 4 times. Comparison of the results obtained using finite differences method (FDM) and numerical-analytical calculation is presented in Table 1.