Mathematical modeling and computation of composite cylindrical shells under axisymmetric loading

The paper suggests an approach of mathematical modeling of composite cylindrical shells under axisymmetric load. The stress-strain state of each layer is characterized by a system of forces, moments, strains, and displacements adopted in the classical theory using Kirchhoff–Love’s hypotheses (L. I. Balabuha – I. V. Novozhilov variant). The study of the stress-strain state is accomplished in accordance with the statements and hypotheses of the complex structures deformation theory taking into account shear stresses in joints. There have been built mathematical models for composite structures in the form of a circular cylindrical shell, closed in a transverse direction and bounded by two plane sections perpendicular to the axis of the cylinder. Computation of a two-layered cylindrical hinged shell under axisymmetric load has been performed. 1Introduction Composite structures formed by layers of materials with different physical and mechanical properties are used as protective constructions in oil and gas industry complexes. The connection of individual layers is made up by continuous or discrete type finite rigidity braces. A shift of layers under loadispossible; therefore, the stress-strain state is studied in terms of multilayered structures with delamination. The above research area is related to the theory of composite structures [1,2,3] for composite rods, plates and shallow shells. The theory of complex structures deformation in terms of shearing stresses in joints is applied to composite cylindrical shells. According to the theory [1] composite shells are structures consisting of several layers with finite rigidity braces pliable in the longitudinal direction and absolutely rigid in the transverse direction. 2Mathematical model A composite cylindrical structure consisting of separate i+1 layers and joints is under consideration. It is assumed that deflection w is the same for all the package layers, and each individual shell is regarded as i-th layer thicknessh. Corresponding author: irina_donkova@mail.ru, i.a.donkova@utmn.ru DOI: 10.1051/ , 04012 (2017) 71060 MATEC Web of Conferences matecconf/201 106


1Introduction
Composite structures formed by layers of materials with different physical and mechanical properties are used as protective constructions in oil and gas industry complexes. The connection of individual layers is made up by continuous or discrete type finite rigidity braces. A shift of layers under loadispossible; therefore, the stress-strain state is studied in terms of multilayered structures with delamination. The above research area is related to the theory of composite structures [1,2,3] for composite rods, plates and shallow shells. The theory of complex structures deformation in terms of shearing stresses in joints is applied to composite cylindrical shells. According to the theory [1] composite shells are structures consisting of several layers with finite rigidity braces pliable in the longitudinal direction and absolutely rigid in the transverse direction.

2Mathematical model
A composite cylindrical structure consisting of separate i+1 layers and joints is under consideration. It is assumed that deflection w is the same for all the package layers, and each individual shell is regarded as i-th layer thicknessh (i) .
The stress-strain state of each structure layer is determined by a system of forces, moments, deformations and displacements, used in the classical theory applying Kirchhoff-Love's hypotheses (L.I. Balabuha -I.V. Novozhilov variant). The hypotheses are not applied to the whole package; thickness metric change is not regarded [6].
To describe the structure behavior the curvilinear orthogonal right-handed coordinate system is used: x, φ, z (i) . The middle surface of the i-thshell layer is related to coordinate surfacez (i) =0. Coordinate x, φlines coincide with the middle surface main curvatures lines, axisz (i) is straight. With x , the distance along the generatrix is M The shell thickness is measured from the middle surface in the from the middle surface to the joint are entered; the distance between the median surfaces is . In the selected coordinate system, the displacement of a point on the middle surface of each i-thlayer has components The deformation parameters of the composite cylindrical shell i-thlayers median surfaces in displacements will be represented as: i are tangential deformation parameters; The values of deformations in the equidistant surfaces, located at distance z i) ( from the i-thlayer middle surface will be represented as [3]: Generalized Hooke's law is used as physical correlations for a structure material considering orthotropy.
After integrating by thickness the stresses at an arbitrary point of the i-thlayerwe move to linear forces, acting at the level of the middle surface of the same layer: Where rigidity characteristics To record shear stresses along the generatrix in the i-thjoint the following formulas are used [1]: are the differences of longitudinal displacements of adjacent layers in the i-thjoint area in the direction of xandφcoordinates respectively. Differential equilibrium equations in displacements for the i-thlayer, given shear stresses, will be represented in accordance with the methodology [1,3,4,5]. The order of differential equations system determines the number of boundary conditions.
There have been regarded composite structures in the form of a circular cylindrical shell, closed in a transverse direction and bounded by two plane sections perpendicular to the axis of the cylinder. Boundary conditions have been specified on each of the two transverse layers along coordinate х . The periodicity conditions of the required function along coordinate φmust be met.
In case of axisymmetric deformation of a composite cylindrical shell the system of differential equations is simplified [4]: .
The equilibrium equations of a two-layer composite cylindrical shell under axisymmetric deformation will be represented as: Where C is the distance between the layers middle surfaces. The boundary conditions for a composite structure with hinging ends (with L x х ; 0 ) are:

3Asymptotic representation of the solution
The system of differential equations is solved by decomposition of required functions and the load vector into trigonometric series which, with given boundary conditions, has the following form:  . Before facility commissioning the shell is tested for tightness and strength with pressure of MPa 575 . 0 . The shell parameters are as follows: length L is 68 m,internal diameter is 45 m, thickness of the concrete layer h (1) is 0.7 m, thickness of the inner steel lining h (2) is 12 m. The boundary conditions correspond to hinging ends. The mechanical properties of isotropic layers are as follows:

Conclusions
The solution of the problem of bending of a closed two-layer cylindrical shell with steel layers under axisymmetric load showed that with rigidity exceeding mm Н 3 10 5 the structure can be regarded as single, without taking into account slippage between layers. The variation of load distribution for a closed composite cylindrical shell showed that the interlayer brace rigidity under evenly distributed load significantly affects the stressstrain state at the near-anchorage area, and under local load at the central area. To assess the accuracy of the computation results exact solutions for a single-layer structure under axisymmetric loading (rigidity factors Kof, Ko 0) are used. For example, if the interlayer brace rigidity of a two-layer shell is equal to zero, the bending rigidity is defined as the sum of rigidities of each individual layer. Provided that they coincide, i.e.the two layers are of the same thickness and material, we obtain a differential equation for cylindrical shell axisymmetric deformation [5]. The developed mathematical model in the form of differential equations allows studying the stress-strain state of composite cylindrical shells with different design parameters and load types.