Comparative analysis of the stress state of an equal slope shell by analytical and numerical methods

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Introduction The present paper is one more of a series of research articles on the study of the geometry and stress-strain state of torses of equal slope with a directrix ellipse by various methods of analysis under different loads and support conditions. To date, the authors have reviewed, analyzed, and drawn conclusions on the tensional state of the torse under the action of a linear uniformly distributed load directed along the generatrix at the upper edge of the shell [1], uniformly distributed load on the middle surface along straight generatrixes [2], and the shell self-weight [3]. The works [1-3] study the problem with simple (movable) supports of the ellipse at the base. The article [4] considers a rigid (fixed) support under the action of self-weight of the torse. A new structure in the shape of a torse of an equal slope is proposed in [5], and new results in geometric studies are shown in [6; 7]. The development of modern technologies and innovative structural design and construction methods is impossible without scientifically based methods of analysis, and research of mathematical and experimental models [8-10]. Along with numerical methods, there are also analytical methods for structures analysis, which engineers use, due to their complexity, only for a narrow class of thin-walled structures and elements [11]. The finite element method (FEM) is a numerical method for calculating the stress-strain state (SSS) of various types of structures. Due to the variety of finite element types and the possibility of modifying their sizes and shapes, this method has undeniable advantages for the analysis of structures of complex shapes, with holes or with stress concentration zones. The paper [12] proposes a method of shell design using triangular finite elements to increase the accuracy of the solutions. The work [13] reports an algorithm developed for strength analysis of large span thin-walled structures in the geometric nonlinear formulation. However, the FEM in comparison with the variational-difference method (VDM) does not consider the external and internal geometry for the determination of the stress-strain state of thin-shell spatial structures of complex shapes with rapidly changing geometrical characteristics [14]. The variational-difference method [15-17], also known as finite-difference energy method (FDEM) [15; 18-20], also belongs to the numerical calculation methods [21]. The VDM allows to consider the geometric parameters of the middle surface of shells for a more accurate determination of the SSS of the thin-shell structures. The history of VDM development begins with Courant's proposal in 1943 [15; 22; 23]. Houbolt in 1958 [18; 24], Griffin and Varga in 1963 [24; 25], Bushnell in 1973, and Brush and Almroth in 1975 [26] continued the development of this method. In the early 2000s Professor V.N. Ivanov and his PhD students developed SHELLVRM, a computer software based on the VDM for determining the SSS of certain types of plates and shells with middle surfaces described by analytical equations [14; 21; 27]. In 2015, Krivoshapko and Ivanov published the encyclopedia [28], where described over 600 analytical surfaces. Among an extensive variety of analytical surfaces, the torse shells of equal slope have a distinctive characteristic of unfolding onto a plane without folds [27]. This class of surfaces is used in many areas of industry [29; 30]. Method Torse shell of equal slope with an ellipse at the base A straight line moving in the normal plane of a flat directrix curve with a constant angle of inclination to the normal plane of the directrix forms a ruled surface of equal slope. The torse surface of equal slope with an ellipse at the base (Figure 1) is formed when the ellipse is set as a flat directrix curve. The basic properties of the surfaces of an equal slope are described in [11; 27]. These surfaces are surfaces of zero Gaussian curvature (K = 0) and also belong to the Monge surfaces [27]. The directrix ellipse is defined by parametric equations [11]: (1) The parameters a and b are the dimensions of the semi-axes of the directrix ellipse, and the parameter is within . Figure 1. Torse shell of equal slope with an ellipse at the base The parametric equations of the torse of equal slope with an ellipse at the base are [11] (2) The coefficients of the basic quadratic forms of this surface and its main curvatures are [11] ; (3) where ; . In this research the momentless theory (MLT) of shell analysis, the variational-difference method and the finite element method are applied to study a thin torse of equal slope with a directrix ellipse under the action of a uniformly distributed load q = 1 kN/m2 directed along the normal to the middle surface of the torse (internal pressure) (Figure 2). Consider the torse with the following geometric parameters a = 3 m, b = 2 m, α = 60° and u = 2 m. Boundary condition at the level u = 0 m is simple (movable) support and free edge is at the level u = 2 m. Figure 2. Torse under the action of internal distributed surface load To determine the parameters of the stress state of the torse (Figure 2) the momentless theory of shell analysis, the SHELLVRM program based on the VDM and the SCAD Office software based on the FEM are used. Differential equations of equilibrium of a momentless torse shell To determine the normal and tangential forces under the action of a uniformly distributed load acting in the direction normal to the middle surface of the torse (Figure 2), we obtain differential equations of equilibrium of the momentless theory in orthogonal curvilinear curvature lines [11]: (4) For this type of applied load on the studied torse of equal slope (Figure 2), we have X = Y = 0 and Z = q. The differential equations of equilibrium (4) are simplified as follows: (5) The forces S, Nu are equal to zero, i.e. S = 0 and Nu = 0 at the level u = 2 m. From the third differential equation of the system (5) we obtain an expression for the normal force

About the authors

Olga O. Aleshina

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: xiaofeng@yandex.ru
ORCID iD: 0000-0001-8832-6790

PhD, Assistant, Department of Civil Engineering, Academy of Engineering

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

Vyacheslav N. Ivanov

Peoples’ Friendship University of Russia (RUDN University)

Email: i.v.ivn@mail.ru
ORCID iD: 0000-0003-4023-156X

Doctor of Technical Sciences, Professor-Tutor, Department of Civil Engineering, Academy of Engineering

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

David Cajamarca-Zuniga

Catholic University of Cuenca

Email: cajamarca.zuniga@gmail.com
ORCID iD: 0000-0001-8796-4635

Docent of the Department of Civil Engineering

Ave Las Americas & Humboldt, Cuenca, 010101, Republic of Ecuador

References