Calculation of thin-walled axisymmetrically loaded structures of the AIC taking into account FEM-based physical nonlinearity

The article describes an algorithm for calculating an axisymmetrically loaded shell structure with a branching meridian, taking into account elastic-plastic deformations when loading based on the deformation theory of plasticity without assuming that the material is incompressible during plastic deformations. The correct relations which determine the static conjugation conditions of several revolution shells in the joint assembly are used. A comparative analysis of finite element solutions is presented for various options plasticity matrix development at the loading stage.


Introduction
The theory of thin shells is widely used [1][2][3][4][5][6]. Due to difficulties in obtaining analytical solutions based on the theory of shells for the designed thin-walled elements, numerical methods have become widespread [7][8][9][10]. The action of external loads in the presence of shell meridian branching can cause plastic deformations. Taking into account the nonlinear work of the material, one can evaluate the work of the structure which can develop more economical solutions [11,12]. When determining the relationships between stress and strain increments based on the deformation theory of plasticity, we used the functional dependence between the average stress and average strain obtained from the tension experiments.

Shell geometry
The position of arbitrary point M 0 of the middle surface of the axisymmetrically loaded shell of revolution is determined by the radius vector o =xi+rk, (1) where х is the axis coordinate; r the revolution radius, being function of х;i and kunit vectors of the Cartesian coordinate system.
By differentiating (1) with respect to the meridional coordinate s, we can determine the unit vector tangent to the meridian of the shell at arbitrary point M 0 of the middle surface 1 0 = , 0 = ( + , ) , . (2) The unit vector of the normal to the shell surface at arbitrary point M 0 is determined by the vector product 0 = 1 0 × = − , + .
As a result, deformations and curvatures of the arbitrary point of the shell of revolution included in (14) for j loading steps are expressed by components of the displacement vector and their derivatives

The final element at the loading stage
Under the axisymmetric loading, the use of a onedimensional finite element is most effective. A fragment of the shell, bounded by two parallel planes perpendicular to the axis of rotation is displayed on the linear element with coordinates of the nodes = −1 and = 1, where − the local coordinate used for the numerical integration (Fig. 1).
The meridional (s) and axial (x) coordinates of the shell were considered as functions of the local coordinate Nodal unknown variables of the finite element for j loading steps are represented by components of the displacement vector and their derivatives in the local coordinate system where {ʋ } Т = {ʋ 1 ʋ 1 ʋ, 1 where q a meridional ʋ 1 or normal ʋ component of the displacement vector. The displacements of the internal FE point at the (j + 1) loading step are expressed by the relations

Relations between strains and stresses at the loading step
To take into account the physical nonlinearity of the material used, the relations of the deformation theory of plasticity are used [14] ε 0 = 3 2 where ε 0 = For axisymmetric deformation, relations (29) can be written as Coefficient K in (30) was determined in [14] using the assumption that, as a result of plastic deformations, the volume does not change.
This coefficient was determined from a simple tensile experiment based on the relations (1 + µ), As a result we have As can be seen, coefficient K_1 is a function of the parameters of the curve of the material deformation diagram.
Relations for strain increments through stress increments are obtained by differentiating (30)

Based on relation (33), a matrix expression is formed
where [ ]the matrix of the stress-strain state of the material.

The stiffness matrix at the loading stage
When deriving the FE stiffness matrix at the (j + 1) loading stage of loading, we used the functional F written on the basis of the equality of the work of external and internal forces at a possible displacement

Results
As an example, the problem was solved to determine the stress-strain state of a complex-jointed shell structure shown in Figure 2.

Fig. 2. Revolution shell with a branching meridian
The following data were used: internal pressure q=0.7 MPa; Poisson's ratioµ = 0.32; elastic modulus Е=7·10 5 MPa; radius of a cylindrical shell Rc=0.9 m; cylinder length Lc=0.8 m; shell thickness t=0.01m; cone lengths Llк=1.1 m. and Luc=0.2 m; angle α=45 0 , angle β=30 0 . The left edge of the structure has a hinged bearing (Fig. 2). A sufficient convergence of the computational process took place when each branch of the structure was divided into 12 FEs.
In the software implementation of the step procedure, for simplicity of calculations, a strain diagram with linear hardening was used When articulating the three shells of revolution at the branching point of the meridian, the correct kinematic and static conjugation conditions were used: the invariance of the displacement vectors, as well as the equality of the angles of rotation of the normals to the median surfaces before and after deformation [15].
The calculation was performed in two versions. In the first version (Table 1), when deriving the plasticity matrix at the loading step, we used coefficient K, which determines the relationship between the average linear stress σ 0 and the average linear strain ε 0 , without taking into account compressibility at elastic plastic deformations. In the second version (Table 2), coefficient K1 determined in (32) was applied. The number of loading steps in the first and second versions varied from 30 to 100. The results of finite element solutions are presented in Tables 1 and 2. They show the numerical values of the meridional stresses at the points of the calculated structure: in the reference section (x = 0.0) and in the branch node of the meridian (x = 0.8 m). Values of the parameters are given for internal ( ) and external ( ) fibers, and for the points of the middle surface of the revolution shell . The branch node represents stresses of the elements of adjacent shells: 1 , 1 , 1for the lower cone and 2 , 2 , 2for the upper one.

Discussion
The results were verified according to the convergence of the computing process and the comparison of the values of the controlled parameters of the stress-strain state with the values calculated using well-known formulas obtained from the equilibrium equation.
The analysis showed that in the reference section of the structure, the convergence of the computational process is satisfactory for both calculation options.
At the branching node of the meridian, that is, in the zone of stress concentration, the results of finite-element solutions differ from each other in the first and second versions of the calculation. In the first version, a significant increase in the values of the meridional stresses in the branching node is observed, which is not reliable with the value of the specified load. With an increase in the number of loading steps in the first embodiment, the controlled parameters of the stressstrain state increase in their absolute values so that the convergence of the computational process is not observed. For the lower cone with a number of steps equal to 100, an increase in the meridional stress indices by 50% is observed in comparison with the values when the number of steps n = 70.
In the second version, there are no sharp increase in stresses in the branching node. In addition, in the second version, according to the calculation scheme (Fig. 2), the lower cone works in tension, and the upper one works in compression. For the first version of the calculation, this correspondence is not observed. With an increase in the number of loading steps in the second version, the values of the meridional stresses at the branching node of the meridian change insignificantly, which indicates the satisfactory convergence of the computational process.
In the reference section, the value of the meridional stress at the points of the middle surface can be calculated based on the equilibrium condition ( = 23.11 (41) As can be seen from the Tables, both options allow us to obtain the exact solution of the reference section. The calculation error does not exceed 1%.

Conclusion
It can be concluded that the algorithm for the formation of a plasticity matrix at the loading stage, when calculating complex shell structures, allows us to obtain finite element solutions acceptable for the design engineering practice. They can be recommended for using in design systems when analyzing the stress-strain state of agricultural facilities. The relationship between the average linear stress σ 0 and the average linear strain ε 0 during the plasticity matrix formation can be implemented in the form (32).