A Triangular Shell Element Based on Higher-order Strains for the Analysis of Static and Free Vibration

This research paper proposes a new triangular cylindrical finite element for static and free vibration analysis of cylindrical structures. The formulation of the proposed element is based on deep shell theory and uses assumed strain functions instead of displacement functions. The assumed strain functions satisfy the compatibility equations. This finite element possesses only the five necessary degrees of freedom for each of the three corner nodes. The element's displacement field, which contains higher-order terms, satisfies the requirement of rigid-body displacement. The element's performance is evaluated using various numerical static and free vibration tests for cylindrical shell problems, including an analysis of the effect of shell openings on natural frequencies. The results of the developed element are evaluated in comparison with published analytical and numerical solutions. The new cylindrical element's formulation is straightforward. Compared to the degenerate nine-node shell element and other elements, the results of the present element have shown excellent accuracy and efficiency in predicting static and free vibration of curved structures. This element only requires the use of very coarse meshes to converge. In addition, the triangular shape of this element is more advantageous than the quadrilateral shape when the geometric domain of the structure is deformed or complicated.


Introduction
The numerical analysis of shell structures is often used to solve problems in engineering and industry. The finite element method is one of the popular methods used by researchers to simulate the behavior of curved structures [1]. Three types of finite elements are employed: first, the curved shell elements derived from general shell theory, such as Zienkiewicz [2], and Liang & Izzuddin [3]; Second, degenerated shell elements that were obtained from the threedimensional solid theory, as Abed-Meraim & Combescure [4] and Trinh et al. [5]; third, an approximate representation of the geometry by flat shell elements [6][7][8]. However, the necessity of using curved shell elements offers numerous advantages, as demonstrated by Jones & Strome [9]: Deriving structural stiffness equations does not involve any additional geometric approximations or coordinate transformations. In addition, using curved shell elements produces efficient elements and avoids problems such as slow convergence for strongly curved shells.
Therefore, the formulation of curved shell elements has received more attention, such as the rectangular element developed by Connor & Brebbia [10] and Cantin & Clough [11]. This cylindrical shell element had better responses 2097 for coarse meshes when the cylindrical shell was tested. The higher-order elements [12][13][14] are developed using the displacement formulation with additional degrees of freedom. There are also other works based on three-dimensional elements, such as the 20-node solid element for shell analysis [15] and the 3D finite element (SFR8) based on the space fiber rotation concept (SFR) developed by Ayad et al. [16]. However, the overall structural matrix has a substantially wider bandwidth when higher-order finite elements with more degrees of freedom are used. Furthermore, there is no link between the additional internal degrees of freedom and the associated generalized physical forces. This work has therefore given priority to the development of higher-order curved elements with only the necessary degrees of freedom.
The employment of finite elements based on assumed strain functions has provided several advantages [17], including the simplicity of satisfying the convergence criteria (constant deformations and rigid body motion). In addition, these independent assumed strains satisfy the compatibility equations, as well as the ability to have the displacement field reinforced by high-order terms without adding intermediate nodes or non-essential degrees of freedom. The strain approach was applied to develop finite elements in which imposed strains were proposed, and the corresponding displacement functions were obtained by simple integration of the strain-displacement relations. A brief review of the strain approach found in the literature for different elements is presented as follows. This approach was applied for isotropic plate bending analysis [17][18][19][20][21][22], functionally graded plates [23,24], composite plate materials [25], general plane elasticity problems [26][27][28][29], and three-dimensional analysis [30][31][32][33][34].
The contribution of the strain approach for curved shell elements has been shown by the formulation of the first cylindrical shell element based on deep shell theory [35]. This element has only five necessary external nodal degrees of freedom per node and is rectangular in plan. From the validation tests, this element shows superior convergence with coarse mesh compared to all other rectangular elements. Based on the shallow shell theory, several rectangular, cylindrical shell elements were formulated by Djoudi & Bahai [36][37][38]. The first was used for linear and geometric non-linear analysis. The second element was used to study how cut-outs affected the vibration behaviors of cylindrical panels, and the last element was used to calculate the natural frequencies of cylindrical panels. To improve the performance of strain-based finite shell elements, Bourezane [39] proposed a rectangular, cylindrical shell element with six degrees of freedom per node by introducing an additional rotational degree of freedom. The effectiveness of these elements was demonstrated, and an acceptable degree of accuracy was reached without using many elements. Therefore, all the cylindrical shell elements [35][36][37][38][39] based on the strain approach presented above are rectangular. The reasons mentioned above prompted the authors to use this approach to develop a new triangular cylindrical shell element.
In this research, a three-node triangular cylindrical shell element has been developed to analyze curved structures using the strain approach and deep shell theory. Only five degrees of freedom are used per node for the developed element called SBTDS (Strain Based Triangular Deep shell). This element is based on assumed strains satisfying the compatibility equations and the rigid body modes for displacements. Numerical integration has been used for calculating the element stiffness and mass matrices. Various examples of static and free vibration of curved structures were used to evaluate the results of the element (SBTDS), and the results then compared to previously published solutions.

Theoretical Considerations
Consider the curved triangular element shown in Figure 1. The center (O) of the hypotenuse of the element is the origin of the curvilinear coordinates x, y, and z (y=R ). The present element is formulated using deep shell theory, and the strain displacement equations in a system curvilinear coordinates are given [35].
As the displacements, U, V, and W are used to represent the six deformations given in Equation 1, these displacements have to verify the compatibility equations written as follows: The displacement modes of rigid bodies are determined by equating Equation 1 to zero, and then after integration, the following displacement fields, U, V, and W, are calculated: This element has five degrees of freedom ( , , , = , = − ) at each of the three nodes. Moreover, hence the displacement functions should contain only 15 constants where six constants (α1, α2,……α6) having used for the rigid body modes, and the remaining constants (α7, α8,…… α15) are distributed among the six suggested strains in the following manner: The terms for the assumed strains in brackets (Equation 4) must be added to verify the compatibility equations (Equation 2). Then, the strain functions expressed in Equation 4 are replaced in Equation 1, and the displacement functions that are obtained after integration are added to the corresponding expressions in Equation 3 to obtain the complete displacement functions:  sin 22 Equations 4 and 5 describing the element's displacement and strain functions are written in matrix form, respectively.
The element nodal displacements vector {qe} is connected to the vector of constants by the transformation matrix [C], which is given in the Appendix I, as follows: Equation 10 can be used to derive the constant parameters vector {α} as follows: Equation 11 is substituted for Equations 6 and 7 to produce the following result: With; By using the conventional expression, the stiffness and mass matrices ([K e ], [M e ]) may be derived, respectively: where {q} and {F} are respectively structural nodal displacements and structural nodal forces vectors whereas  is the angular frequency.

Numerical Validation
To evaluate the accuracy and efficiency of the formulated element (SBTDS), several numerical examples of static and free vibration analysis are examined.

Square Pinched Cylinder with Free Edges
The pinched cylinder illustrated in Figure 2 is the first problem to be solved. The literature frequently uses this test case as one example to evaluate finite elements' convergence. Only one-eighth of the cylinder is modelized with a variety of meshes for reasons of symmetry ( Figure 3); the geometrical, mechanical characteristics, boundary, and symmetry conditions are represented in Figure 2, where two cases can be distinguished for the cylinder thickness and applied loads. Tables 1 and 2 show the normal displacement results WC at point C, which illustrate the high precision obtained by the present element. The results of the developed element are similar to the analytical solution [40]. However, a divergence of results is observed for the Djoudi element, which is based on the shallow shell theory (Figure 4).

Curved Cantilever Beam
The second example is the curved cantilever beam clamped at one end and loaded at the other free end ( Figure 5). The geometrical parameters and the values of Poisson's ratio, Young modulus, and load are shown in Figure 5. The results obtained for the deflection at the z-direction (Table 3) are compared with the theoretical solution given by Macneal & Harder [41] and with other finite elements [35,36,42]. Figure 6 shows the convergence of deflection at the z-direction for the curved beam. The proposed element (SBTDS) gives excellent results even for a small number of elements. The reference solution is reached by this element for a 1×4 mesh ( Figure 6). Figure 6 shows that the SBTDS element produces more accurate results than those given by Djoudi element [36].

Clamped Cylindrical Shell with Rigid Diaphragm
A different test case is a cylindrical shell clamped with a rigid diaphragm under two opposing concentrated loads. The performance of the shell elements in simulating complex membrane state problems dominated by bending is evaluated in this test. The geometrical, mechanical characteristics, loadings, boundary, and symmetry conditions are presented in Figure 7. One-eighth of the shell is considered for idealization. The results obtained for both the normal displacements at point C and tangential displacements at point are compared with the theoretical solution [43] and with other finite elements CHA element [44]), ASH element [35], and Djoudi element [36]. The results of the SBTDS element (Table 4) are similar to those of the other elements. Graphical representations of these results are shown in Figures 8 and 9. The new triangular element (SBTDS) and the other rectangular elements give almost the same results for deflection.

Convergence of Mesh Discretization
In this test, we study the present element's convergence rate due to the domain discretization of a clamped cylindrical panel. The geometrical and mechanical characteristics of the panel are as follows: l = 1 m, r = 2 m, t = 0.005 m, =0.5 rad, Young's modulus E = 208 × 10 9 N/m 2 , Density = 7833 kg/m 2 , and Poisson's ratio is v = 0.29.
The results of the first and second natural frequencies are reported in Tables 5 and 6 against of the total number of elements and compared with the results of Djoudi element [37] and the theoretical solution [45]. In this test, we notice the high accuracy obtained by the present element, and its convergence to the theoretical solution is more rapid than that of Djoudi element [37].

Clamped Cylindrical Panel
Another test case considered is a clamped cylindrical panel, and the geometry and material characteristics are illustrated in Figure 10. The results of the clamped panel frequencies obtained using a mesh of 10 ×10 are presented in Table 7 with analytical solution [46], numerical solution [47], and other finite elements, LAG9; nine-node shell element [48], ASL9; assumed strain shell element [49] and nine nodes degenerated shell element [50]. The frequencies obtained with the proposed element are better than those obtained with LAG9, ASL9, and the degenerate nine-node shell element, which is a very expensive element.

Effect of Central Openings on the Natural Frequencies of Cylindrical Panels
This example of cylindrical panels with a central opening ( Figure 11) clamped along all four edges, treated by the Djoudi [35], is analyzed to study the effect of the openings on natural frequencies. The geometry and material properties of the panel are illustrated in Figure 11. Figure 12 compares the natural frequencies of the current element SBTDS to those of the Djoudi element [35]. It should be noted that the width of the hole has the same impact on the natural frequency for both elements and that the current element's numerical results agree with the Djoudi element's result.

Conclusions
A three-node triangular cylindrical shell element is proposed using assumed strains and the deep shell theory. This element has only external degrees of freedom, three translations, and two rotations at each corner node. The displacement field of the developed element is calculated by integrating the assumed strain functions that satisfy the compatibility equations. The numerical integration is used for the evaluation of the element stiffness and mass matrices. The performance and accuracy of the developed element have been verified with various numerical examples in static and free vibration of cylindrical structures. The following advantages can be concluded from the numerical results of the current element:  In comparison to elements containing internal nodes, such as the nine-node element, this element is simpler, with simply corner nodes and the five necessary exterior degrees of freedom;  It has a rapid convergence rate to the exact solutions for static and free vibration analyses;  The triangular shape of this element is more advantageous than the quadrilateral form because it facilitates meshing when the geometric domain of the structure is complicated;  High accuracy and good performance have been obtained using the present element with only coarse meshes.
The results of this triangular cylindrical shell element show monotonic convergence and are in excellent accord with the analytical solutions and the results of various elements available in the literature. In perspective, this element can be applied to functionally graded shells, composite shell materials, and non-linear problems of shell structures.

Data Availability Statement
The data presented in this study are available in the article.

Appendix I
The 15×15transformation matrix [C] for the present element SBTDS is given as: where (xi, i, yi), (i=1, 2, 3) are the coordinates of the three-element node.