Investigation of the accuracy and convergence of the results of thin shells analysis using the PRINS program

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Introduction Shell structures are widely used in various fields of technology - construction engineering, machine-building, aircraft construction, shipbuilding and so on. Fundamental questions of the shell analysis theory have been studied in detail in the works of domestic and foreign authors [1-5]. In these works, equations were obtained that completely describe the stress-strain state of thin shells of arbitrary shape under arbitrary loading. However, these equations do not have a common analytical solution. Various authors have obtained particular solutions for shells of a relatively simple form. The search for such solutions is underway at the present time [6-9]. The most famous of them, tested by long-term practice of their use, are given in reference and educational literature [10; 11]. The finite element method, which appeared in 1956, became a universal tool for calculating shells of arbitrary shape [12]. The versatility of the method is provided by the fact that the shell surface is represented as a set of elements of a simple geometric shape, triangles and/or quadrangles, which can be both flat and curved. Attempts were made to construct a curvilinear finite element on the basis of the shell theory [13], but it was impossible to make such element universal. Therefore, at present, for the calculation of shells, either plane finite elements are used, built on the basis of the plate bending theory [14; 15], or curvilinear ones, built on the basis of the general theory of elasticity [16]. The finite element method has been implemented in various computer programs. Those, who have been thoroughly verified, use the confidence of calculators and designers. The subject of research in this article is created by Professor V.P. Agapov computer program PRINS, the development of which is carried out by Professor V.P. Agapov together with his followers. The theory and practical implementation of shell finite elements used in this program is briefly described, and numerous examples of calculation of shells of various shapes are given. Method In the PRINS program, plane triangle and quadrilateral finite elements, implemented in single-layer and multilayer versions, are used for the calculation of thin shells. Since PRINS is intended for calculations of both linear and nonlinear deformable structures, the fundamental position in the development of finite elements was to obtain the simplest mathematical formulations. This circumstance is explained by the need to use rather dense finite element (FE) grids in the calculations, on the one hand, and the need for multiple recalculation of the stiffness characteristics of elements in the process of nonlinear problems solving, on the other. Therefore, the simplest triangle (Figure 1, a) with linear approximating functions for membrane displacements and a function in the form of an incomplete cubic polynomial for deflections (1) was taken as the basis for the shell elements constructing. (1) A finite element with such displacement functions has well known to specialists disadvantages [14], the main of which are non-invariance with respect to the local coordinate system and the lack of compatibility of rotations of the normal with adjacent elements at the boundaries that do not coincide with the local axis. However, an advanced triangular finite element can be built on its basis. Two such elements are used in the PRINS program - multilayered EL34 and single-layered EL36. When developing multilayer element, the technique proposed by Professor Agapov [17] was used, and in the development of single-layer element the method described in the work of Clough and Tocher [14], was realised. The main idea in both cases is to use three so-called subtriangles with approximating displacement functions, taken in the form (1), to obtain the characteristics of a given triangular FE. Methodology of professor Agapov is illustrated in Figure 2, and the Clough and Tocher method is shown in Figure 3. In both cases, the characteristics of subtriangles are initially formed in their local axes, then converted to axes common for a given triangle, summed and averaged. a b c Figure 1. To create laminated shell FE: a - triangular FE in local coordinates xm-ym; b, c - cross section a b Figure 2. To calculate the bending stiffness matrix: a - triangular FE with nodes i, j, k; b - subtriangles in local axes Figure 3. R.W. Clough & J.L. Tocher triangular finite element formation As shown in [14; 17], the triangular elements thus improved have the property of invariance with respect to the coordinate axes and complete compatibility of displacements and rotations with adjacent elements at all boundaries. The characteristics of a quadrangular FE are obtained by summing and averaging the characteristics of four triangles according to the scheme shown in Figure 4. Figure 4. Forming a quadrangular finite element A detailed description of the methods for obtaining all the characteristics of flat finite elements necessary for the calculation of shells can be found in [14; 15; 17]. The purpose of this work is a comparative analysis of shell FE implemented in PRINS program, as well as analysis of the accuracy and convergence of the results obtained with their help. Results and discussion To verify the above-described finite elements, we present a number of numerical calculations performed in PRINS program. Single-layer shallow shell. We consider a shallow shell, the middle surface of which is an elliptical paraboloid (Figure 5, a) with the following initial data: a = b = 10 m, h = 10 cm, f1 = f2 = 0.5 m, E = 3×104 MPa, q = 1 kPa. The shell rests on transverse diaphragms that are rigid in its plane and flexible out of plane. Figure 5. To the calculation of a shallow shell The middle surface of the shell considered is described by equation: . (2) This surface is formed by moving a line along a line. The authors estimated the accuracy and analyzed the convergence of the calculation results obtained with using of triangular and quadrangular elements of a single-layer shell (type EL36). For these purposes, a total of twelve finite element schemes of the considered shell were built (Figure 6) with different mesh densities: 10×10, 14×14, 20×20, 30×30, 36×36, 40×40. An analytical solution to this problem is given in the manual [11]. a b c Изображение выглядит как зонт, аксессуар, постельное белье Автоматически созданное описание Изображение выглядит как аксессуар, зонт, трусы Автоматически созданное описание Изображение выглядит как аксессуар, зонт Автоматически созданное описание d e f Изображение выглядит как зонт, аксессуар, постельное белье Автоматически созданное описание Изображение выглядит как зонт, аксессуар Автоматически созданное описание Изображение выглядит как зонт, аксессуар, постельное белье Автоматически созданное описание Figure 6. Finite element models of a shallow shell: a, b, c - for calculating with rectangular FE; d, e, f - for calculating with triangular FE Comparison of the results obtained in the PRINS program with the analytical calculation data was carried for the vertical displacement and for the total stresses at the upper (σu) and lower (σl) surfaces at the central point of the shell. Stresses were calculated by the formula (3) where N and M are the lineal membrane force and the lineal bending moment in the shell section respectively. The orientation of the shell surfaces is determined by the direction of the local zm axis of finite elements, the positive direction of which at the central point coincides with the direction of the global z axis in Figure 5. The results of the numerical calculation of the shallow shell are presented in Table 1. Table 1 Calculation results of the shell FE mesh Vertical displacement at the center point of the shell w, m Stresses at the upper and lower surfaces at the center point of the shell FEM solution Analytical solution FEM solution Analytical solution Δw, % σxx,u = σyy,u, kPa Δσ, % σxx,u = σyy,u, kPa σxx,l = σyy,l, kPa σxx,l = σyy,l, kPa 14×14 (3) -0.000235 -0.000221 6.33 -146.24 0.58 -145.39/-113.6 -122.01 7.9 30×30 (3) -0.00022 0.45 -145.8 0.28 -118.2 3.72 14×14 (4) -0.00023 4.07 -146.12 0.5 -121.88 7.29 30×30 (4) -0.000222 0.45 -145.82 0.29 -116.18 2.27 40×40 (4) -0.0002215 0.22 -145.6 0.15 -114.34 0.65 The displacement and stress fields in the shell are shown in Figure 7. a b Изображение выглядит как зонт, аксессуар, трусы Автоматически созданное описание c d Изображение выглядит как зонт, аксессуар, трусы Автоматически созданное описание Изображение выглядит как зонт, аксессуар, трусы Автоматически созданное описание Figure 7. Calculating results of a shallow shell: a - deformed shell scheme, total displacement fields, m; b - normal stress fields σxx, kPa; c - normal stress fields σyy, kPa ; d - shear stress fields τxy, kPa As you can see from the Table 1, PRINS program provides an equally stable solution using both triangular and quadrangular FE. With a relatively coarse FE mesh, the calculation error is less than 6%. The displacement convergence graph is shown in Figure 8. Figure 8. Convergence graph of calculation results for displacements Figure 9. To the calculation of a spherical shell Spherical shell under uniform pressure. The spherical shell clamped at the edges and loaded with the uniform load (Figure 9) is considered. The initial data are as follows: R = 2.28 m, α = 35°, h = 7.6 cm, E = 3×104 MPa, An analytical solution of this problem by the Steuermann - Geckeler method is given in [11]. The shell verification calculation was performed by a shell finite element (type EL36) at different FE mesh densities: 8×32, 12×48, 16×64, 32×128 (Figure 10). The results of the spherical shell analysis, obtained with the aid of PRINS program, are shown in Figure 11. The graphs of the convergence of the calculation results for meridional bending moments and circumferential normal stresses are shown in Figures 12 and 13 respectively. The data presented in these figures show the high accuracy of the finite elements used for calculations. Figure 10. Finite element models of a spherical shell a b c d Изображение выглядит как текст Автоматически созданное описание Figure 11. Calculating results of a spherical shell: a - deformed shell scheme, total displacement fields ×106, m; b - normal stress fields σxx, kPa; c - normal stress fields σyy, kPa; d - bending moments fields Mx, kNm/m Figure 12. Convergence graph of the calculation results by the meridional bending moments Mx Figure 13. Convergence graph of calculation results for circumferential stresses σy Tank consisting of a cylindrical part and a spherical dome. Let us consider a dome-shaped shell, turning into a cylindrical one, under the action of uniform pressure (Figure 14). The initial data are as follows: Rm = Rt = 20 m, h = 50 cm, E = 3×104 MPa, The tank is rigidly fixed in the base. Изображение выглядит как часы Автоматически созданное описание Figure 14. To the calculation of a tank The finite element schemes of the reservoir were constructed using triangular and quadrangular shell elements (type EL36) and had the following parameters: 12×24, 18×36, 24×48, 30×60 (Figure 15). Figure 15. Tank finite element models The stress state of the reservoir under consideration can be determined using the membrane theory of shells (the edge effect is not considered). Normal stresses in the cylindrical part are equal: (3) The stresses in a spherical dome are determined by the equation: (4) The results of the reservoir analysis with the aid of PRINS program are presented in Table. 2. Table 2 Tank calculating results FE mesh Cylindrical shell Spherical shell FEM solution Analytical solution FEM solution Analytical solution σm, kPa σt, kPa σm, kPa σt, kPa σm, kPa σt, kPa σm, kPa σt, kPa 12×24 9900 19 800 10 000 20 000 9950 9990 10 000 10 000 18×36 9960 19 900 9960 10 000 24×48 9980 20 000 9980 10 000 30×60 9990 20 000 9990 10 000 The displacement and stress fields in the tank are shown in Figure 16. a b c Изображение выглядит как клетка, легкий Автоматически созданное описание Figure 16. Tank calculating results: a - deformed tank scheme, total displacement fields, m; b - normal stress fields in the meridional direction σm, kPa; c - normal stress fields in the circumferential direction σt, kPa As you can see from the Table 2, the results obtained using the PRINS practically coincide with the analytical solution according to the membrane theory. Flat layered cylindrical panel. The calculation of a flat layered cylindrical panel rested on transverse diaphragms that are rigid in its plane and flexible out of plane (Figure 17, a) is presented. Panel dimensions: a1 = 1 m, a2 = 2 m, R1 = 3 m. The cross-section of the panel consists of five layers symmetrically located relative to the middle surface (Figure 17, b). The characteristics of the layers are as follows: h1 = 0.5 cm, h2 = 1.5 cm, h3 = 1.6 cm, E1 = 7×104 MPa, E2 = 2.6×104 MPa, E3 = 195 MPa, q = 35 kPa. The calculations were carried out using a triangular multilayer FE (EL34). Four different finite element meshes were used: 6×12, 9×18, 12×24, 15×30 (Figure 18). a b Figure 17. To the calculation of a laminated cylindrical panel: a - general view; b - cross-section of the panel Изображение выглядит как постельное белье Автоматически созданное описание Figure 18. Finite element models for calculating a laminated panel Table 3 Calculation results of laminated cylindrical panel FE mesh Vertical displacement at the center point of the shell w, m Forces in the middle of the panel FEM solution Analytical solution Error, Δw, % FEM solution Analytical solution My, kNm/m Ny, kN/m My, kNm/m Ny, kN/m 6×12 0.000459 0.000455 3.086 -15.8 3.05 -15.29 0.88 9×18 0.000458 3.08 -15.5 0.65 12×24 0.000456 3.075 -15.4 0.2 a b Изображение выглядит как квадрат Автоматически созданное описание c d Изображение выглядит как текст Автоматически созданное описание Изображение выглядит как текст Автоматически созданное описание Figure 19. Calculating results of a laminated cylindrical panel: a - deformed shell scheme, total displacement fields, m; b - normal force fields Nx, kN/m; c - normal force fields Ny, kN/m; d - shear force fields Nxy, kN/m An analytical calculation of the problem under consideration is given in the book [18]. The results of numerical calculations are presented in Table 3. The displacement and stress fields in a laminated panel are shown in Figure 19. The calculation error ranges from 3.5 to 0.2%, depending on the dimension of the FE mesh. Conclusion The principles of the shell finite elements constructing described in this article were implemented in PRINS program. On the basis of numerous verification tests, it has been established that the finite elements (type EL36 and type EL34) used for single-layer and multilayer shell analysis have a fast convergence and have a sufficiently high accuracy. For rectangular planar shallow shells with side length l, the optimal size of the finite element that provides the required solution accuracy with significant savings in computing resources is. For the calculation of cylindrical and spherical shells, the size of the finite element is recommended to be taken within . PRINS program can be effectively used by specialists from design and scientific organizations to solve a wide class of engineering problems.

About the authors

Vladimir P. Agapov

Moscow State University of Civil Engineering (National Research University)

Email: agapovpb@mail.ru
ORCID iD: 0000-0002-1749-5797

Doctor of Technical Sciences, Professor of the Department of Reinforced Concrete and Masonry Structures

26 Yaroslavskoye Shosse, Moscow, 129337, Russian Federation

Alexey S. Markovich

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: markovich-as@rudn.ru
ORCID iD: 0000-0003-3967-2114

Candidate of Technical Sciences, Associate Professor of the Department of Civil Engineering, Academy of Engineering

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References