# Quadrilateral element in mixed FEM for analysis of thin shells of revolution

## Abstract

The purpose of study is to develop an algorithm for the analysis of thin shells of revolution based on the hybrid formulation of finite element method in two dimensions using a quadrilateral fragment of the middle surface as a discretization element. Nodal axial forces and moments, as well as components of the nodal displacement vector were selected as unknown variables. The number of unknowns in each node of the four-node discretization element reaches nine: six force variables and three kinematic variables. To obtain the flexibility matrix and the nodal forces vector, a modified Reissner functional was used, in which the total specific work of stresses is represented by the specific work of membrane forces and bending moments of the middle surface on its membrane and bending strains, and the specific additional work is determined by the specific work of membrane forces and bending moments of the middle surface. Bilinear shape functions of local coordinates were used as approximating expressions for both force and displacement unknowns. The dimensions of the flexibility matrix of the four-node discretization element were found to be 36×36. The solution of benchmark problem of analyzing a truncated ellipsoid of revolution loaded with internal pressure showed sufficient accuracy in calculating the strength parameters of the studied shell.

## Full Text

Introduction Finite element analysis of thin shells based on the displacement method (when the unknown nodal variables are displacements and their partial derivatives) has been developed quite well and is widely used today in various software suites. In [1], FEM is presented as an alternative to the finite difference method with justification of its advantages. It is widely used in the calculations of beams and frame structures [2], as well as multilayer plates and shells [3; 4], and in the calculation of three-dimensional structures and thick slabs [5; 6]. FEM was widely used in the formulation of the displacement method in the calculation of thin shells under elastic and elastoplastic deformation [7-11]. This method is also used in the analysis of the nonlinear deformation of plates and shells [12-15]. FEM is used in the formulation of the displacement method and in cases of large deformations during loading of plates and shells [16-18], as well as in shell stability calculations [19; 20]. In engineering stability problems, a mixed formulation based on the predictor-corrector scheme was proposed [21; 22]. To reach the appropriate level of accuracy for computing the unknown values, it is necessary to use the approximating expressions of class С(2), since the Cauchy relations for thin shells [23; 24] contain second order partial derivatives of the normal displacement vector. When using the mixed FEM (when the unknown nodal variables are displacements, axial forces and moments), approximating the unknowns with functions of classes С(0) and С(1) is sufficient. A major advantage of using the mixed FEM is the ability to compute stresses and deformations at an element node in terms of the evaluated unknowns of the element at this specific node. In contrast, to determine nodal stresses, FEM based on the displacement method requires calculating the unknowns at the adjacent nodes, which increases the computational error. In this regard, the most relevant problem now is the development of algorithms for linear and non-linear analysis of thin shells with the mixed FEM using curvilinear coordinates. Methods The Reissner functional is considered in the following definition [25; 26]: (1) where are the stresses and deformations at a point in the shell, which is located at vertical distance from the corresponding point of the middle surface; matrix represents the transformation matrix from vector to vector , which is composed based on the Hooke’s law for thin shells [23; 24]; is the row vector of displacement components of the middle surface point; is the external load vector. Stresses in functional (1) are expressed in terms of the forces of the middle surface [23; 24]: (2) where is the moment of inertia of the cross-section; is the height of the cross-section. Deformations of an arbitrary layer of the shell are determined in terms of membrane and bending strains by relations [23; 24]: (3) Physical and geometric expressions (2) and (3) may be represented in matrix form: (4) where Membrane and bending strains of the middle surface are defined by expressions [27]: (5) where are the basis vectors of a middle surface point; is the displacement vector of the middle surface point; is the difference vector of normal lines of the middle surface point in the deformed and undeformed states. Relationships (5) may be expressed in matrix form: (6) where is the differentiation and algebraic expressions matrix. Moments and forces at a point on the middle surface, which are contained in (2), may be expressed in terms of the values of these force unknowns at the nodes of the quadrilateral element using approximating bilinear functions with the following matrix product: (7) where are the bilinear functions of local coordinates of the quadrilateral finite element [27]. Deformations at a middle surface point (6) may expressed using bilinear functions as the following matrix product: (8) where represents tangential or normal displacement vector component. Considering (4), (6), (7) and (8), functional (1) may be represented as (9) By minimizing functional (9) with respect to , the following relation can be obtained: (10) where In order to minimize functional (7) with respect to unknown nodal displacements , equation (9) needs to be represented in the following form: (11) Minimizing (11) with respect to yields the following matrix expression: (12) By rearranging (10) and considering (12), it is possible to obtain the flexibility matrix and the nodal forces vector for the quadrilateral element in the following form: (13) Thus, the dimension of the flexibility matrix of the quadrilateral element is 3636, and the nodal unknowns vector contains 24 force and 12 kinematic factors, which are axial forces and moments and displacement components of a nodal point of the middle surface. Construction of the general flexibility matrix and nodal forces vector of the entire shell is conducted using the index matrix, which reflects the boundary conditions of the shell [28]. Results and discussion Calculation example. In order to verify the developed algorithm, a truncated ellipsoid of revolution, which is illustrated in Figure, was analyzed. Изображение выглядит как антенна Автоматически созданное описание Truncated ellipsoid of revolution The following initial data was adopted: ellipsoid shape parameters m; m; shell thickness m; modulus of elasticity MPa; Poisson’s ratio ; internal pressure MPa. Only 1/8 of the shell was analyzed due to ellipsoid having planes of symmetry. The results of the analysis are presented in Table, in which the numerical values of normal stresses of the middle surface at the support (m) and end (m) sections of the ellipsoid with different finite element grid are given. The results in Table imply that refining the grid leads to stable convergence of the computational process. However, convergence stability is a necessary, but not sufficient condition for the efficacy of the algorithm in regards to the real physical distribution of stress in the shell. To evaluate the objectiveness of the results, let us compute meridional stress at the support and end sections. The meridional stress at the support section can be obtained from the following equilibrium equation: (14) where are the radii of revolution of the ellipsoid at the support and the end sections respectively, besides m; m. Values of normal stresses in the middle surface of the ellipsoid Section Stress, MPa Node grid Analytical solution according to the Laplace equation 4141 6161 8181 101 101 121 121 Support, x = 0.0 m σ11 95.93 95.89 95.88 95.87 95.87 95.86 σ22 179.03 179.04 179.05 179.05 179.05 179.06 End, x = 1.2 m σ11 0.916 0.449 0.270 0.182 0.133 0.00 σ22 167.75 168.45 168.78 168.96 169.06 167.82 By substituting the initial data into (14), it is possible to obtain the following value of meridional stress at the support section: MPa. The meridional stress at the end section must be zero, since the right end of the shell is not loaded: MPa. Circumferential stress of the middle surface of the ellipsoid at the support and end sections may be expressed using the Laplace equation: (15) Radii of curvature and in (15) are defined by (16) where is the radius of revolution of the ellipsoid, is the second order derivative of the radius of revolution; . Thus, it is possible to obtain the analytical value of circumferential stress at the support and end sections of the ellipsoid from (15): (17) Substituting the corresponding initial data into (17) yields the values of the desired stresses: MPa; MPa. Conclusion By comparing the analytical values of meridional stress and circumferential stress computed with equations (14)-(17) and the values obtained via the developed algorithm, it can be concluded that the adequate level of accuracy of the finite element analysis has been reached, as the minimum computational error does not exceed 1%. The developed algorithm may be recommended for application in engineering practice for the analysis of thin shells.
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### Yuriy V. Klochkov

Author for correspondence.
Email: klotchkov@bk.ru
ORCID iD: 0000-0002-1027-1811
SPIN-code: 9436-3693

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

26 Universitetskii Prospekt, Volgograd, 400002, Russian Federation

### Valeria A. Pshenichkina

Email: vap_hm@list.ru
ORCID iD: 0000-0001-9148-2815
SPIN-code: 3399-0668

Doctor of Technical Sciences, Professor, Head of the Department of Building Structures, Foundations and Reliability of Structures, Faculty of Construction and Housing and Communal Services

28 Leninskii Prospekt, Volgograd, 400005, Russian Federation

### Anatoliy P. Nikolaev

Email: anpetr40@yandex.ru
ORCID iD: 0000-0002-7098-5998
SPIN-code: 2653-5484

Doctor of Technical Sciences, Professor of the Department of Mechanics, Faculty of Engineering and Technology

26 Universitetskii Prospekt, Volgograd, 400002, Russian Federation

### Olga V. Vakhnina

Email: ovahnina@bk.ru
ORCID iD: 0000-0001-9234-7287
SPIN-code: 3593-0159

Candidate of Technical Sciences, Associate Professor of the Department of Higher Mathematics, Electric Power and Energy Faculty

26 Universitetskii Prospekt, Volgograd, 400002, Russian Federation

### Mikhail Yu. Klochkov

Email: m.klo4koff@yandex.ru
ORCID iD: 0000-0001-6751-4629
SPIN-code: 2767-3955

postgraduate student, Department of Building Structures, Foundations and Reliability of Structures, Faculty of Construction and Housing and Communal Services

28 Leninskii Prospekt, Volgograd, 400005, Russian Federation

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Copyright (c) 2023 Klochkov Y.V., Pshenichkina V.A., Nikolaev A.P., Vakhnina O.V., Klochkov M.Y.