Issue 
BIO Web Conf.
Volume 17, 2020
International ScientificPractical Conference “Agriculture and Food Security: Technology, Innovation, Markets, Human Resources” (FIES 2019)



Article Number  00199  
Number of page(s)  5  
DOI  https://doi.org/10.1051/bioconf/20201700199  
Published online  28 February 2020 
Calculation of thinwalled axisymmetrically loaded structures of the AIC taking into account FEMbased physical nonlinearity
^{1}
Volgograd State Agrarian University, 400002 Volgograd, Russia
^{2}
Financial University under the Government of the Russian Federation, 125167 Moscow, Russia
^{*} Corresponding author: arsen82@yandex.ru
The article describes an algorithm for calculating an axisymmetrically loaded shell structure with a branching meridian, taking into account elasticplastic 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.
© The Authors, published by EDP Sciences, 2020
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The theory of thin shells is widely used [1–6]. Due to difficulties in obtaining analytical solutions based on the theory of shells for the designed thinwalled elements, numerical methods have become widespread [7–10]. 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.
2 Materials and methods
2.1 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 (1)
where x is the axis coordinate; r the revolution radius, being function of x;i and k  unit 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(2)
The unit vector of the normal to the shell surface at arbitrary point M^{0} is determined by the vector product(3)
Equalities (2) and (3) are presented in the matrix(4)
where .
Derivatives of local basis vectors are determined by expressions(5)
In the matrix form of expression (5) they are as follows(6)
where .
From (4) we can obtain the matrix dependence(7)
Given (7) expression (6) is written as follows(8)
In the process of shell deformation, the point of its middle surface is considered in three positions: initial (point M^{0}), after loading steps j (point M, displacement vector v), and after the (j + 1)th loading step (M*, displacement vector Δv). At a distance ζ from the middle surface, the following pointsM^{0ζ},M^{ζ}ИM*^{ζ} correspond to it. When taking into account the hypothesis of direct normals, the position of these points is determined by the following radius vectors(9)
Displacement vectors vand ∆v, included in (9), for the FE under study are presented in the basis of point M^{0} (10)
By differentiating (10) along the curvilinear coordinate s with allowance for (5), we have(11)
where are functions of vectors v, ∆v and coefficients of matrix [n], included in (8).
Differentiation (9) determines the basis vectors of an arbitrary point of the shell in a deformed state(12)
The unit vectors of the normals in the deformed state are expressed by the vector product(13)
Deformations and their increments in an arbitrary layer of the shell spaced at a distance ζ from the middle surface can be determined on the basis of equations of continuum mechanics [13](14)
The covariant components of metric tensors included in (14) in the initial and deformed states are determined by the corresponding scalar products(15)
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(16)
where
Similarly, for step (j + 1) in a geometrically linear formulan(17)
(16) and (17) can be written as
Deformations and curvatures of the point of the middle surface and their increments at the loading stage included in (18) are expressed by matrix ratios(19)
where matrix of differential and algebraic operators.
2.2 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 η = 1 and η = 1, where η – 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 η (20)
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(21)
where .
At step (j+1), nodal variable parameters are determined as follows(22)
where
Derivatives in the local coordinate system are determined through derivatives in the global coordinate system as follows(23)
where q a meridional υ^{1} or normal υ component of the displacement vector.
The displacements of the internal FE point at the (j + 1) loading step are expressed by the relations(24)
The matrix {φ}^{T} included in (24), (25) is determined through the чepeзHermite polynomials of the third degree(26)
Thus, expressions (24) and (25) can be written in matrix form(27)
By differentiating (24) and (25), taking into account (23), the derivatives of the components of the displacement vector in the global coordinate system are determined as follows:(28)
where
Fig. 1. Discretization element at the loading stage 
2.3 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](29)
where – average linear strain; – average linear stress; ε_{i}  strain intensity; σ_{i}  stress intensity; δ_{ij} – Kronecker symbol.
For axisymmetric deformation, relations (29) can be written as(30)
where μ Poisson’s ratio.
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(31)
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)(33)
Based on relation (33), a matrix expression is formed(34)
where [D]  the matrix of the stressstrain state of the material.
2.4 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(35)
where – the matrixrow of strain increments at the loading stage;
 matrixrows of stresses and their increments in an arbitrary layer of the shell of revolution;
– the matrixrow of increments of the components of the displacement vector at the (j + 1)th loading stage;
– surface load and its increments at the (j + 1) th loading stage.
The strain increment matrix can be represented as(36)
The increments of stresses in an arbitrary layer of the shell of revolution are as follows(37)
where [C_{pl}] is determined from з (34).
The increments of the components of the displacement vector at the (j + 1) th loading step are determined according to (27).
When substituting expressions (27), (36) and (37) into functional (35) and performing its minimization with respect to nodal unknowns, the expression(38)
where – the stiffness matrix of the final element of the shell of revolution at the loading stage; – the vector of external loads at the loading stage;
– NewtonRaphson correction.
3 Results
As an example, the problem was solved to determine the stressstrain state of a complexjointed shell structure shown in Figure 2.
The following data were used: internal pressure q=0.7 MPa; Poisson’s ratioμ = 0.32; elastic modulus E=7·10^{5} MPa; radius of a cylindrical shell R_{c}=0.9 m; cylinder length L_{c}=0.8 m; shell thickness t=0.01m; cone lengths L_{IK}=1.1 m. and L_{uc}=0.2 m; angle α=45°, angle β=30°. 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 (39)
where σ_{0} = 200 MPa; ε_{T} = 0.0023496 m; p = 18087.03 MPa.
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 K_{1} 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:  for the lower cone and  for the upper one.
Fig. 2. Revolution shell with a branching meridian 
Calculation option 1.
Calculation option 2.
4 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 stressstrain state with the values calculated using wellknown 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 finiteelement 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 (Fig. 2)(40)
Substituting the initial data in (39), we have(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%.
5 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 stressstrain 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).
Acknowledgments
The study was financially supported by the Russian Fundamental Research Fund and Volgograd Region Administration No. 1941340002 r_a.
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All Tables
All Figures
Fig. 1. Discretization element at the loading stage 

In the text 
Fig. 2. Revolution shell with a branching meridian 

In the text 
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