# Abstract

The aim of the work is to perform a comparative analysis of the results of analyzing arbitrarily loaded shells of revolution using finite element method in various formulations, namely, in the formulation of the displacement method and in the mixed formulation. Methods. To obtain the stiffness matrix of a finite element a functional based on the equality of the actual work of external and internal forces was applied. To obtain the deformation matrix in the mixed formulation the functional obtained from the previous one by replacing the actual work of internal forces in it with the difference of the total and additional work was used. Results. In the formulation of the displacement method for an eight-node hexahedral solid finite element, displacements and their first derivatives are taken as the nodal unknowns. Approximation of the displacements of the inner point of the finite element was carried out through the nodal unknowns on the basis of the Hermite polynomials of the third degree. For a finite element in the mixed formulation, displacements and stresses were taken as nodal unknowns. Approximation of the target finite element values through their nodal values in the mixed formulation was carried out on the basis of trilinear functions. It is shown on a test example that a finite element in the mixed formulation improves the accuracy of the strength parameters of the shell of revolution stress-strain state.

# Full Text

1. Introduction The theory of deformation of solids has been developed in sufficient detail to date [1-2]. However, analytical obtaining of specific results is possible only in some cases, far from the practice of engineering calculations. Therefore, the development of approximate and numerical methods for calculating structural elements of engineering structures is an actual task. Among the modern methods of studying the stress-strain state of building structures, the numerical finite element method (FEM) based on the displacement method has become widespread recently [3-15]. It can be stated that the main disadvantages of this FEM formulation are the lack of continuity of the displacement derivatives on the edges and side surfaces of finite elements. The development of finite elements in the mixed formulation [16-25] allows to reduce the degree of approximating functions for expressing the desired quantities through nodal unknowns, makes it possible to fulfill the conditions for the continuity of stresses and displacements not only at the nodal points, but also on the edges and lateral surfaces of the discretization elements. The subject of the study is the stress-strain state of the shell of rotation under arbitrary loading. The purpose of the study is a comparative analysis of finite element algorithms for determining the strength parameters of the shell of rotation. To perform a comparative analysis of variants, finite element algorithms of the hexahedral finite element are developed in two formulations: in the formulation of the displacement method and in the mixed formulation. When obtaining the hexahedron stiffness matrix in the formulation of the displacement method, displacements and their first derivatives are used as nodal unknowns. When forming the matrix of the stress-strain state of a hexagon in a mixed formulation, displacements and stresses are taken as nodal unknowns. 2. Research methods To obtain the stiffness matrix of a hexagonal finite element, the displacement method formulation uses a functional based on the equality of the actual work of external loads on displacements and the actual work of internal stresses on deformations over the volume of the finite element. To approximate the desired values of the internal point of a finite element through nodal unknowns, third-degree Hermite polynomials were used. To obtain the stress-strain state matrix in a mixed formulation, we used a functional obtained by replacing the actual work of internal forces of the displacement method functional with the difference between the total work of internal forces and their additional work. Trilinear relations are used to approximate displacements and stresses through nodal unknowns. 2.1. Shell of revolution geometric parameters Position of an arbitrary point in the shell of revolution middle surface is defined by radius vector (1) where - unit vectors of the Cartesian coordinate system; - radius of revolution for the considered point of the middle surface; - angle measured counterclockwise from the vertical diameter. Basis vectors of an arbitrary point on the middle surface are determined by expressions (2) where Position of an arbitrary point of the shell located at a distance from the middle surface is defined by radius vector (3) Base vectors of the point are defined by differentiation (3): (4) Using (2) based on (4) the following matrix relations can be formed: (5) where Differentiating (4) taking into account (5), derivatives of the basis vectors of point can be defined by the components in the same basis: (6) where 2.2. Displacements and deformations Displacement vector of point from load action is represented by components in the basis of point : (7) where Derivatives of displacement vector (7) with respect to curvilinear coordinates are determined by expressions (8) Based on (6) relations (8) can be represented in the form (9) where - functions of components of the displacement vector and its derivatives, defined by the expressions … (10) Deformations are determined by the relationships of continuum mechanics: (11) Taking into account (4) and (9), we can form the matrix relationship (12) where - matrix of algebraic and differential operators. 2.3. Relationships between deformations and stresses Hooke’s law is represented in curvilinear coordinate system by expressions (13) where is the first invariant of strain tensor; is the first invariant of stress tensor; are covariant and contravariant components of metric tensor; are covariant and contravariant components of strain tensor; are covariant and contravariant components of stress tensor; are Lame parameters; is modulus of elasticity; is Poisson’s ratio. On the basis of relations (13) the following matrix expression is formed: (14) where 2.4. Shell of revolution finite element in displacements method formulation The finite element is taken in the form of a hexahedron with nodes For performing numerical integration, hexahedron is mapped onto a cube with local coordinates changing within the limits . Displacements and their derivatives in local coordinates are taken as nodal unknowns. Approximation of displacement for the inner points of the finite element was performed on the basis of Hermite polynomials of the third degree by the matrix expression below: (15) where Vectors of the nodal unknowns in local and global coordinate systems are related by the following matrix ratio: (16) where Matrix is formed on the basis of differential relations (17) where indicates components of the displacement vector . Using (15), deformations (12) are defined in matrix form (18) In order to construct the stiffness matrix for the finite element, a functional based on the equality of work done by internal and external forces is used: (19) where is element volume; - specified load application surface; - components of external load vector. Taking into account (14), (15), (16) and (18), functional (19) is given by expression (20) After performing minimization of functional (20) the following is obtained: (21) where is finite element stiffness matrix; - nodal forces vector. 2.5. Shell of revolution finite element in mixed formulation Displacements and stresses of the hexahedral finite element are taken as nodal unknowns: (22) For approximating target quantities through nodal values, bilinear functions are adopted: (23) where refers to values . Based on (22) and (23) the following matrix relations are formed: (24) Deformations (12) based on (24) can be written in matrix form: (25) For obtaining the finite element deformation matrix a functional, obtained from (19) by replacing the actual work of internal forces by the difference between total and additional work of internal forces, is used: (26) Based on (24) and (25), functional (26) for the finite element is written as (27) After variation of functional (27) by nodal unknowns and the following systems of equations are obtained: (28) where Systems (28) are represented in traditional FEM formulation (29) where is finite element deformation matrix; - nodal unknowns row; - finite element nodal loads row 3. Results and their analysis The stress state of cylindrical shell fixed at the ends and loaded by internal pressure of intensity was determined. The following initial data were specified: radius of middle surface = 1.0 m; generatrix length = 0.5 m; wall thickness = 0.02 m; = 5 MPa; modulus of elasticity MPa; Poisson’s ratio = 0.3. The results of the analysis are presented in Tables 1 and 2. The values of stresses in the direction of cylinder axis are given at the points: 1 - located in the fixed end; 2 - located at a distance from the fixed end; 3 - in the midspan. Analytical model is represented by a single strip of hexahedral elements along the cylinder axis. The first column in Table 1 shows the number of rows of finite elements along the thickness of the cylinder, the second column shows the number of nodal points in the axial direction and in the direction of shell thickness. The remaining columns show numerical results of the stresses in the direction of the cylinder axis in inner and outer fibers, respectively, at points 1, 2, 3. Analysis of numerical results in Table 1 shows convergence of the computing process when using FEM in formulation of the displacement method. Differences in the results of calculations for point 1 are explained by the difference in boundary conditions for finite elements in the specified formulations, namely, in the displacement method, boundary conditions are assigned for the derivatives of displacements, and in the mixed formulation, boundary conditions for displacements are assigned. At point 2, which is at a distance from the fixed support, the results stabilized by the mixed method already with one finite element in thickness, and in the displacement method, it monotonically tends to the same numerical values with an increase in the number of finite elements in thickness. Table 1 Numerical values of parameters for stress-strain state of cylindrical shell when using elements in displacement method formulation Number of rows of elements through cylinder's thickness Mesh discretization Cross section Point 1 Point 2 Point 3 Stress, MPa 1 381.14 -308.97 298.26 -221.91 -57.04 141.07 396.74 -298.97 303.02 -211.51 -56.83 140.99 430.50 -269.26 302.17 -212.05 -56.61 140.88 467.52 -236.44 301.87 -211.73 -56.46 140.81 505.40 -202.51 301.66 -211.46 -56.33 140.74 2 444.24 -329.74 331.64 -253.09 -73.52 157.34 464.29 -323.12 342.49 -260.16 -73.83 157.77 489.77 -326.93 342.70 -261.20 -73.78 157.80 516.03 -328.81 342.71 -260.89 -73.69 157.76 540.37 -326.81 342.64 -260.69 -73.62 157.72 3 472.09 -347.74 334.54 -262.45 -75.68 159.53 499.49 -349.06 345.85 -267.92 -76.03 160.00 527.69 -352.34 347.01 -267.10 -76.03 160.07 552.45 -352.12 346.93 -266.94 -75.95 160.03 578.18 -351.27 346.82 -266.77 -75.88 160.00 4 491.13 -354.73 338.14 -265.33 -76.91 160.78 526.89 -355.26 348.14 -270.49 -77.43 161.42 561.89 -355.87 350.38 -270.86 -77.50 161.56 586.72 -355.73 350.32 -270.78 -77.45 161.54 611.58 -355.93 350.22 -270.64 -77.39 161.51 7 523.33 -362.29 348.61 -268.24 -78.25 162.14 584.38 -363.68 350.40 -274.48 -79.00 163.04 642.43 -365.46 354.14 -275.20 -79.19 163.29 677.46 -365.73 354.56 -275.23 -79.19 163.32 704.12 -365.73 354.53 -275.16 -79.16 163.31 Table 2 Numerical values of parameters for stress-strain state of cylindrical shell when using elements in mixed formulation Number of rows of elements through cylinder's thickness Mesh discretization Cross section Point 1 Point 2 Point 3 Stress, MPa 1 472.47 -381.34 373.02 -288.59 -72.05 159.57 479.93 -390.65 365.74 -277.22 -79.57 165.13 480.76 -392.41 366.43 -278.84 -81.44 166.05 480.60 -392.55 366.27 -278.98 -81.76 166.07 480.43 -392.54 366.11 -278.98 -81.87 166.03 2 532.16 -335.43 364.93 -292.25 -74.68 160.75 543.99 -334.06 370.30 -278.72 -81.31 165.52 544.53 -332.87 369.31 -279.13 -82.49 166.10 543.17 -333.25 368.38 -279.17 -82.55 166.05 542.10 -333.71 367.80 -279.22 -82.51 166.00 3 516.60 -365.22 365.88 -293.70 -75.69 160.87 541.69 -384.38 363.23 -275.69 -81.81 165.84 556.19 -394.50 364.47 -278.87 -82.82 166.42 559.89 -397.72 364.49 -279.98 -82.84 166.34 561.38 -399.15 364.43 -280.50 -82.80 166.26 4 519.91 -350.43 363.74 -296.82 -76.28 161.07 553.90 -367.90 361.51 -274.04 -82.07 165.86 583.84 -381.24 363.74 -277.62 -82.99 166.41 592.53 -385.93 364.13 -278.75 -82.99 166.33 595.59 -388.04 364.09 -279.27 -82.92 166.24 7 517.45 -354.04 364.49 -298.62 -76.85 161.26 558.47 -382.77 358.64 -271.14 -82.31 165.97 616.05 -413.21 361.39 -275.66 -83.07 166.43 645.52 -425.87 363.43 -277.76 -83.06 166.35 659.74 -431.69 364.19 -278.77 -83.01 166.28 Analysis of numerical results in Table 2 shows more rapid convergence of computational process when using finite element method in the mixed formulation. It is explained by the fact that in the mixed finite element the stresses are consistent not only at the nodes of finite elements, but also on their faces. In the finite elements of the displacement method, there is no deformation compatibility along the faces. 4. Conclusion The accuracy of determining the strength parameters of the shell of revolution and the convergence of computational process are higher when using finite elements in the mixed formulation. This is due to the fact that when obtaining the deformation matrix of this finite element, the degree of approximating functions for approximating the desired values of the inner point of the finite element through the nodal unknowns in the mixed formulation is lower than in the displacement method formulation. The compatibility condition of the target quantities in the displacement method formulation is satisfied only at nodal points. The aforementioned compatibility conditions are absent on the edges and faces of hexahedral finite elements. When using finite elements in the mixed formulation, the compatibility conditions for displacements and stresses are satisfied not only at nodal points, but also on the edges and faces of the hexahedral element.

### Natalia A. Gureeva

Financial University under the Government of the Russian Federation

Author for correspondence.
Email: aup-volgau@yandex.ru
49 Leningradskii Ave, Moscow, 125993, Russian Federation

Doctor of Physics and Mathematics, Associate Professor of the Department of Mathematics

### Yuriy V. Klochkov

Email: aup-volgau@yandex.ru
26 Universitetskii Ave, Volgograd, 400002, Russian Federation

Doctor of Technical Sciences, Professor, Head of the Department of Higher Mathematics of the Electric Power and Energy Faculty

### Anatoly P. Nikolaev

Email: aup-volgau@yandex.ru
26 Universitetskii Ave, Volgograd, 400002, Russian Federation

Doctor of Technical Sciences, Professor of the Applied Geodesy, Environmental Engineering and Water Use Department of the Ecology and Melioration Faculty

Email: aup-volgau@yandex.ru
26 Universitetskii Ave, Volgograd, 400002, Russian Federation

Candidate of Technical Sciences, Associate Professor of the Applied Geodesy, Environmental Engineering and Water Use Department of the Ecology and Melioration Faculty

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