Accepted Manuscript Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory R. Ansari, E. Hasrati, A.H. Shakouri, M. Bazdid-Vahdati, H. Rouhi PII: DOI: Reference: S0020-7462(18)30078-7 https://doi.org/10.1016/j.ijnonlinmec.2018.08.007 NLM 3065 To appear in: International Journal of Non-Linear Mechanics Received date : 6 February 2018 Revised date : 11 August 2018 Accepted date : 16 August 2018 Please cite this article as: R. Ansari, E. Hasrati, A.H. Shakouri, M. Bazdid-Vahdati, H. Rouhi, Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory, International Journal of Non-Linear Mechanics (2018), https://doi.org/10.1016/j.ijnonlinmec.2018.08.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory a a b R. Ansari*, a E. Hasrati, a A. H. Shakouri, a M. Bazdid-Vahdati, b H. Rouhi Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran Abstract The present work is concerned with the application of the variational differential quadrature (VDQ) method [Appl. Math. Model., vol. 49, pp. 705–738], in the area of computational mechanics, to the nonlinear large deformation analysis of shell-type structures. To this end, based on the six-parameter shell model, the functional of energy in quadratic form is derived based on Hamilton’s principle which is then directly discretized by the VDQ technique. The formulation of article is presented in a general form so that it can be readily used for different structures such as beams, annular plates, cylindrical shells and hemispherical shells under various loading conditions. In order to reveal the accuracy of developed solution strategy, it is tested in several popular benchmark problems for the geometric nonlinear analysis of shells. The results show that the present numerical method is capable of yielding highly accurate solution in the nonlinear large deformation analysis of shells. It is also easy to implement due to its compact and explicit matrix formulation. Keywords: Large deformation analysis; Six-parameter shell model; Variational differential quadrature method 1. Introduction Shells are fundamental engineering structures with various applications in different fields [13]. Hence, studying their mechanical behaviors has been the subject of many research works up to now. In particular, the large deformation analysis of shells is a challenging topic in the related literature. One can follow this field of research from the work of Sabir and Lock in *Corresponding author. Tel. /fax: +98 13 33690276. E-mail address: r_ansari@guilan.ac.ir (R. Ansari). 1 1972 [4] to the recently published papers [5-15]. Using different shell theories and proposing different solution methods are the main focus of these works. In the following, some of the important papers are reviewed. Simo and co-workers [16-19] developed the six-parameter shell model which is known as the geometrically exact shell model. They proposed the continuum foundation of the geometrically exact shell theory obtained by reduction of the three-dimensional theory using the one inextensible director kinematic assumption [16]. Atluri [20] focused on alternate stress and conjugate strain measures and mixed variational formulations involving rigid rotations for the analysis of finitely deformed shells. Pietraszkiewicz [21] derived the equilibrium equations together with four geometric and static boundary conditions for an entirely Lagrangian nonlinear theory of shells. Schmidt [22] studied the geometrically nonlinear first approximation Kirchhoff-Love type theories for thin elastic shells undergoing small strains accompanied by moderate, large or unrestricted rotations. Braun et al. [23] and Büchter et al. [24] developed the seven-parameter shell theory. In [23], arbitrary complete three-dimensional constitutive equations without manipulation in nonlinear shell analysis were used, and the seven-parameter formulation was extended to study the response of laminated structures with arbitrarily large displacements and rotations. The research of Bischoff and Ramm [25] revealed that using the six-parameter shell formulation leads to the unexpected Poisson locking problem. A large number of studies are also associated with different finite element formulations and shell elements [4, 8, 9, 10-12, 26-31]. For example, Sansour and Kollmann [9] compared various kinds of four-node and nine-node elements of the finite element method (FEM) including hybrid stress, hybrid strain and enhanced strain elements for investigating the large deformation of shells within the framework of the seven-parameter shell theory. Arciniega and Reddy [28, 29] studied the large deformation responses of shells made of isotropic, laminated composites and functionally graded materials (FGMs). They utilized high-order elements (e.g. Q25, Q49 and Q81) to avoid the locking issue as well as using mixed interpolation methods like enhanced assumed strain elements. Amabili [32] developed a geometrically nonlinear theory for shells of generic shape considering third-order thickness and shear deformation and rotary inertia including eight parameters. Alijani and Amabili [33] investigated the geometrically nonlinear static and vibrations of rectangular plates taking full nonlinear terms corresponding to Green–Lagrange strain–displacement relations into account based on the second-order thickness deformation theory. Gutiérrez Rivera et al. [34] proposed a twelve-parameter shell finite element for large 2 deformation analysis of composite shell structures via third-order thickness stretch kinematics. Also, Amabili [35] developed an eight-parameter theory with higher-order shear deformation considering geometric non-linearities in the in-plane and transverse displacements. Recently, Faghih Shojaei and Ansari [36] presented a numerical method in the area of computational mechanics called as variational differential quadrature (VDQ). The main idea of this technique is based upon the accurate and direct discretization of the energy functional in the structural mechanics. In VDQ, by introducing an efficient matrix formulation and via an accurate integral operator, the discretized governing equations are directly obtained from the weak form of equations with no need for the analytical derivation of the strong form. The VDQ method provides an alternative way to discretize the energy functional, which avoids the local interpolation and the assembly process existing in similar methods like FEM. The VDQ method is general so that it can be used for any engineering structure that is modeled through energy methods in the context of weak form formulation. Also, it is easy to implement due to its compact and explicit matrix formulation. Accordingly, the VDQ method, due to its simplicity and accuracy, has been utilized in several research works for solving various problems of beams, plates and shells [37-44]. In the present article, the VDQ method is adopted to analyze the large deformation behaviors of shell-type structures in the context of the six-parameter shell theory [16-19]. It is worth mentioning that the seven-parameter shell model is used in the finite element analyses to overcome the locking problem. But, the six-parameter shell model is used herein as the VDQ method is locking-free. Also, it should be mentioned that this work focuses on thin shells. The article is organized as follows. In Section 2, the nonlinear strain-displacement relations and constitutive relations for the large deformation analysis of six-parameter shells are derived. In Section 3, the obtained relations in Section 2 are matricized. In Section 4, the quadratic representation of energy functional is derived based on Hamilton’s principle. In Section 5, the VDQ method is introduced for the discretization of quadratic representation of energy functional in a direct way. Finally in Section 6, several well-known benchmark problems given by Sze et al. [10] are considered to show the accuracy and efficiency of the VDQ method for geometric nonlinear analysis of shells. 3 2. Six-parameter shell model In this section, the basic relations of the six-parameter shell model are presented. The reader is referred to [45, 46] for more details on the six-parameter shell theory. Since the coordinates of a point in the convective coordinate system remain constant, the deformation gradient tensor can be written as ⊗ ⊗ where ⊗ (1) is the normal vector of mid-surface, 1,2,3 and 1,2. In addition, and denote spatial coordinate system base vectors and material coordinate system base vectors, respectively. According to Eq. (1), the right Cauchy-Green deformation tensor ( ) and Greenst. Venant strain tensor ( ) are obtained as ⊗ 1 2 (2) 1 2 in which (3) ⊗ and are the metric tensors in spatial coordinate system and material coordinate system, respectively. Considering the mid-surface displacement vectors in the spatial coordinate system ( ) and material coordinate system ( ) as follows , , 0, (4) ,0 ,0 , 0,0 (5) and the base vectors of the material coordinate system and spatial coordinate system on the mid-surface as , , │ │ , , , │ , │ , │ (6) (7) one can obtain the displacement vectors of material coordinate system as , (8) where denotes curvilinear surficial coordinates, is time, and the coordinate in the thickness direction is denoted as . Based on the six-parameter shell theory, the displacement vector in the spatial coordinate system can be defined as , (9) in which is an unknown vector which is called “director”. Now, the material coordinate system base vectors can be rewritten as , , , (10) , 4 , where is the shifter tensor which makes a transformation between the base vectors and ; ⊗ (11) Using the following relations , (12) 2 (13) and according to Eq. (12) and defining │ , ⊗ (14) ⊗ ⊗ (15) the deformation gradient tensor ( ) can be recast as (16) Introducing the displacement vector ( ) and difference vector ( ) as follows , (17) one can obtain , , ⊗ ⊗ (18) ⊗ (19) in order to achieve strains, by substituting Eq. (16) into (2), the right Cauchy-Green deformation tensor ( ) can be attained as (20) Accordingly, the strain tensor (E) is obtained as follows 1 2 1 2 1 2 By ignoring the terms including (21) , Eq. (21) can be rearranged as (22) Based on Eq. (13) one has 1 2 1 2 (23) (24) After decomposing Eqs. (23) and (24) one can write ⊗ ⊗ ⊗ ⊗ 5 (25) ⊗ ⊗ ⊗ ⊗ (26) where 1 2 , ⋅ , 1 2 1 2 ⋅ , ⋅ , (27) ⋅ , ⋅ ⋅ , 1 2 ⋅ , ⋅ 1 2 ⋅ ⋅ , , ⋅ ⋅ , , , ⋅ , , ⋅ ⋅ , , ⋅ , , ⋅ , (28) 0 Considering a Cartesian coordinate system with base vectors , the components of the displacement vector and difference vector can be presented as , (29) and the base vectors in curvilinear coordinate can be given as , , ⋅ , (30) Consequently, one can represent the components of nonlinear strains of six-parameter shells in the Cartesian coordinate as follows (31) where 1 2 , 1 2 , , , , , (32) 1 2 6 1 2 , , , 1 2 , , , , , , , , , (33) , 0 where , 1,2 and 1,2,3. In addition, the displacement vector is given by (34) in which , directions, and , are the displacement projections on the mid-surface along , , denote the rotations about , , , , axis, respectively. which based on Eq. (31) can be recast as Also, the constitutive relation is (35) where (36) 2 2 2 (37) (38) , , 0, , 0 2 (39) It is worth mentioning that considering some extensions and modifications, the presented shell theory can be applied in different fields [47]. Surface stress phenomena [48], thinwalled structures made of micropolar materials [49], thin-walled structures made of viscoelastic materials [50-53], shells with initial stresses [54] and shells/plates with phase transition [55, 56] are some examples. 3. Matrix representation The relations defined in the previous section will be represented in a novel matrix form in the current section in order to be used in Hamilton's principle. The elasticity matrix can be rewritten as the following form based on Eq. (38) 7 (40) . in which by calculating using Eqs. (38) and (39), the elasticity matrix can be recast for different geometries (see the Appendix). Moreover, based on Eq. (37), the strain vector is given by 2 2 2 (41) Based on Eqs. (31) and (32), the strain-displacement relations can be defined as , (42) where 1 2 , in which and 1 2 , (43) stand for linear and nonlinear strains, respectively. The vector of kinematic parameters is expressed as , , (44) and the displacement vector can be recast as , in which (45) stands for an 3 ⋯ , 1,2, … ,18 ⋯ , 1,2, … ,45 0 0 0 (46) 0 0 (47) 0 where ∂ denotes ( 3 identity matrix. By defining the following relations and stands for the Kronecker delta, one can obtain the linear strains ) as follows (48) , , , , , , , 8 (49) ⊗ ⊗ (50) in which ⊗ denotes the Kronecker tensor product, stands for an 6 is defined in Eq. (30). Also, the nonlinear strains ( 6 identity matrix and ) can be obtained as follows ∘ (51) in which ∘ denotes the Hadamard product and 〈 ∘ in which 〈∎〉 1 2 〉 〈 〉 ∎ . Accordingly, 〉 〈 〈 1 〈 2 ∘ can be written as 〉 〈 〉 (52) can be expressed as follows 〉 (53) where 〈 in which 〉 〈 , and 〉 (54) are defined as follows (55) 1 2 1 2 , (56) 1 2 ; ; ; ; ; ; ; , (57) ; ; ; ; ; In addition, using Eq. (47) ; , ; , 1,2 1,2, … ,6 can be introduced as ; (58) ; , ; 9 , ; ; ; , ; ; , ; , ; ; 4. Hamilton's Principle According to Hamilton’s principle for a deformable body, one can write 0 where , and (59) are the kinetic energy, strain energy and the work of external forces, respectively. The kinetic energy vanishes for the bending problem. The strain energy is formulated as (60) in which the strain energy density 1 ⋅ 2 is proposed in a quadratic matrix representation as 1 2 (61) Based upon Eqs. (35) and (42), Eq. (61) can be recast as follows 1 2 1 2 1 2 (62) Inserting Eq. (43) into (62) leads to 1 2 1 2 1 2 Therefore, the variation of strain energy becomes 1 2 1 2 10 (63) (64) It should be noted that (65) √ where and stand for the determinant of the shifter tensor and base vectors, respectively, which are defined for each geometry, specifically. The integration on the volume reduced to surface is through introducing following matrices , , (66) 1,2 So, Eq. (63) can be rewritten as 1 2 1 2 (67) The variation of work of external loads is given by ⋅ in which (68) shows the force vector. Substituting Eq. (45) into (68) gives (69) where . 5. VDQ discretization In the VDQ method [36], the quadratic form of energy functional is directly discretized using matrix differential and integral operators. The matrix operators are presented in the Appendix. Calculating the components of the vector at the grid points which are selected from the Chebyshev-Gauss-Lobatto grid [57] yields the discretized displacements vector . Following the technique of VDQ discretization, the discretized form of Eqs. (67) and (65) on the space domain can be derived as 1 2 1 2 ⊗ where coordinates and (70) (71) ⊗ , , 1,2 . Let 〈 〉, where be the integral operator along the in-plane is defined in the Appendix. One can write the discretized form of Eq. (54) as follows 11 〈 〉 where 〈 and 〉 (72) are the discretized forms of and , respectively, and are presented by Eqs. (85-87). By defining following block matrices 1 2 1 3 (73) (74) 2 (75) 3 2 (76) ⊗ in which (77) and denote the discretized linear stiffness matrix and the discretized external forces vector, respectively. In addition, and are the discretized nonlinear parts of the stiffness matrix. Accordingly, Eqs. (70) and (71) are rewritten as (78) (79) By inserting Eqs. (78) and (79) into Eq. (59) (Hamilton’s principle), one arrives at 0 (80) Solving Eq. (80) using the Newton-Raphson scheme results in finding which is the vector of unknown degrees of freedom and the displacements values at grid points. It should 0, is used as the be noted that, the solution of the linear part of Eq. (80), which is initial guess for the Newton-Raphson method. Only un-symmetric matrix-part of above that is defined using Eq. (75). The symmetric counterpart of relation is 〉 〈 Substituting with leads to is derived as 〉 〈 (81) which is a symmetric matrix. The Jacobian of stiffness parts of Eq. (80) is obtained as (82) Furthermore, the linear and nonlinear discretized strain matrices are given by ; ; , ; ⊗ , ⊗ ; , ; ⊗ (83) ; in which ⨂ , ⨂ , ⨂ 12 (84) where and are the differentiation matrix operators in and Appendix). Besides, where and and indicate identity matrices with the size are the number of grid points in and directions (see the and , directions, respectively. Also, one has ; ; ; ; ; ; ; , (85) ; ; ; ; ; ; , ; ; , ; ; , ; ; , ; ; , ; , ; (86) ; where ⊗ , ⊗ , ⊗ (87) It is worth mentioning that the present numerical approach can be extended to study the behavior of shells based on the micropolar theory as the studies presented in [58, 59]. 6. Benchmark tests In this section, the accuracy of developed solution methodology is revealed through eight popular benchmark problems on large deformation behaviors of cantilevers, plates, cylindrical shells and spherical shell. It should be noted that the cantilever is modeled as a long rectangular plate. Comparisons are made with the results reported in [10]. The results 13 are represented in the form of load–defection curves. Base vectors, strains and elasticity matrix for various geometries are given in the Appendix. 6.1. Cantilever subjected to end shear force and uniformly distributed load As the first benchmark problem, a cantilever under the end shear force distributed load and uniform is considered as shown in Figs. 1a and 1b. It should be noted that the cases of uniformly distributed loads on beams and plates are very difficult to be treated numerically as shown in [60, 61]. The material and geometrical properties are assumed as 1.2 where 10 Pa , , , , and respectively. Also, 0, 10 m , 1 m , 0.1 m are Young’s modulus, Poisson’s ratio, length, width and thickness, 4 N⁄m and 4 N⁄m . In Fig. 1c, the end shear force and uniformly distributed load are plotted versus the vertical and horizontal tip defections. As can be seen, there is an excellent agreement between the present results and those given by Sze et al. [10]. Moreover, the deformed configurations of cantilever subjected to the maximum end shear force and uniformly distributed load are indicated in Figs. 1d and 1e, respectively. Moreover, Table 1 provides a comparison between the convergence rate of present approach and that of [10]. In this table, the number of load increments (NINC) and the total number of iteration (NITER) to reach the converged solution are presented. It is observed that the NITER of the present approach is smaller than that of [10]. 6.2. Cantilever subjected to end moment In this case study, a cantilever subjected to end moment is considered as shown in Fig. 2a. It is assumed that 1.2 10 Pa , 12 m , 0, 1 m , 0.1 m , 50 ⁄3 N⁄ The end moment-vertical/horizontal tip defections curves and the deformed configuration of cantilever under the maximum moment are shown in Figs. 2b and 2c, respectively. Fig. 2b clearly indicates the reliability of the VDQ method. 14 Similar to Table 1, Table 2 is given to compare the convergence characteristics of present approach with those of [10] in the case of cantilever subjected to end moment. Again, it is seen that the VDQ method needs lesser computational effort. 6.3. Slit annular plate subjected to lifting line force According to Fig. 3a, a slit annular plate is selected in this benchmark test. The line force is applied at one end of the slit, whereas its other end is fully clamped. The material and geometrical parameters are chosen as 21 10 Pa , 6 m , 0, 10 m , 0.3 m 0.8 N force⁄length . Fig. 3b shows the variation Besides, it is considered that of load parameter with the vertical defections at the tips of the slit, A and B. The deformed configuration of the slit annular plate subjected to the maximum force is also illustrated in Fig. 3c. 6.4. Hemispherical shell subjected to alternating radial forces As the fourth benchmark problem, the large deformation of a hemispherical shell subjected to alternating radial forces is analyzed. In Fig. 4a, a hemispherical shell with an 18◦ circular cutout at its pole is shown. As indicated, alternating radial point forces s at 90◦ intervals are applied to the hemispherical shell. The parameters are assumed as 6.825 10 m , 10 Pa , 0.04 m , 0, 400 N It should be noted that owning to symmetry, one-quarter of the shell is modeled. According to Fig. 4b which shows the load-deflection curves of the shell, the present results agree well with those of [10]. 15 6.5. Pullout of an open-ended cylindrical shell Fig. 5a shows an open-ended cylindrical shell pulled by a pair of radial forces s. It is assumed that 10.5 10 Pa , 0.3125, 4.953 m , 10.35 m , 0.094 m , 40000 N In Figs. 5b and 5c, the load-deflection curves and deformed configuration are shown. Due to symmetry, one-eighth of the shell is modeled. 6.6. Pinched semi-cylindrical shell In this case study, the large deformation problem of a pinched semi-cylindrical shell is solved. Fig. 6a shows the semi-cylindrical shell subjected to an end pinching force at the middle of the free-hanging circumferential periphery. Note that half of the shell is modeled owning to symmetry. It is considered that 2068.5 Pa , 101.6 m , 0.3, 304.8 m , 3 m , 2000 N The results of analysis are shown in Figs. 6b and 6c. 6.7. Pinched cylindrical shell mounted over rigid diaphragms In this benchmark test, a pinched cylindrical shell mounted on rigid end diaphragms, over which the in-plane displacements and are restrained, is considered (see Fig. 7a). It is assumed that 30 10 Pa , 100 m , 0.3, 200 m , 1 m , 12000 N Because of symmetry, one-eighth of the shell is modeled. The results of analysis are shown in Figs. 7b and 7c. 16 6.8. Hinged cylindrical isotropic roof Fig. 8a indicates a hinged semi-cylindrical roof subjected to a central pinching force. Onequarter of the roof is modeled due to symmetry. The considered parameters are as follows 3102.75 Pa , 2540 m , 0.3, 254 m , 12.7 m , 0.1 rad , 3000 N The variation of load parameter versus deflection and deformed configuration are portrayed in Figs. 8b and 8c. 6.9. Comparison to theories with thickness deformation In order to compare the present six-parameter shell theory to theories with thickness deformation, the present results are compared to those of Alijani and Amabili [33] for the case of a simply-supported aluminum square plate under uniformly distributed load in Fig. 9. They studied the geometrically nonlinear static and vibrational behaviors of rectangular plates by considering all of the nonlinear terms corresponding to the Green–Lagrange strain– displacement relations based on the second-order thickness deformation theory. They also carried out another analysis using the ANSYS three-dimensional modeling. The geometrical parameters, material properties and magnitude of uniform load for the comparison study are considered as follows: 70 GPa , 0.1 m , 1.05 0.3, 0.001 m , 10 Pa The essential boundary conditions at the edges for an immovable simply-supported plate can be presented as: at 0, 0 at 0, 0 17 The schematic of the plate and deformed configuration for the maximum uniformly distributed load are shown in Figs. 9a and 9b, respectively. Also, comparisons are made between the present results and those given in [33] for the non-dimensional maximum transvers deflection at the center of the plate, ⁄2 , ⁄2 ⁄ , the maximum rotation, 0, ⁄2 , and the dimensionless thickness deformation, Δ ⁄ , in Figs. 9c-9e, respectively. Fig. 9c shows the non-dimensional maximum transvers deflection at the center of the plate, ⁄2 , ⁄2 ⁄ , versus the non-dimensional uniformly distributed load, , for different theories. It can be seen that the results of six-parameter shell theory agree well with those reported in [33]. Fig. 9d indicates that the maximum rotation, 0 and at ⁄2, versus the non-dimensional uniformly distributed load based on different theories. One can see that, the present maximum rotation curve is in reasonable agreement with those of 2TS-NL and ANSYS 3D. It is worth noting that based on the definitions given in [33], 2TS denotes the second-order thickness stretching theory and NL stands for the retaining nonlinearities and thickness deformation which can be defined as Δ . According to Fig. 9e, the non-dimensional thickness deformation at the center of the plate ( /2) for 1428.6 is converged for different theories, while it is divergent close to the edges. Also, it should be noted that the present six-parameter shell theory can only take the first-order thickness deformation parameter into account. 7. Conclusion In this paper, in the context of the six-parameter shell theory, an efficient solution strategy was proposed to analyze the nonlinear large deformation of shells. The idea of paper was based on the VDQ method, a recently developed technique to simplify numerical analysis of structures. First, the nonlinear strain-displacement relations and constitutive relations for the large deformation analysis of the six-parameter shell model were derived and represented in suitable matrix forms. Then, according to Hamilton’s principle and by means of the VDQ discretization technique, the quadratic form of energy functional was directly discretized. In order to show the accuracy of developed formulation, eight well-known benchmark problems were considered, and comparisons were made between the present results and those of FEM reported in the literature. Also, comparisons were made between the results of six-parameter shell theory and those of theories with thickness deformation. It was revealed that the VDQ method is capable of solving various problems on the large deformation analysis of shell-type 18 structures with high accuracy. Easy implementation, general formulation, absence of locking problem, accuracy and low computational cost are the main advantages of approach proposed in this paper. 19 Appendices: A. Differential and integral operators The differential and integral operators used in the VDQ method are presented here. First, the operators are given in general form using the differential quadarature technique and then they are defined for each geometry, specifically. A.1. Differential operator Consider the following column vector ⋯ where is the value of (A-1) at each grid point , and denotes the number of grid points. The values of -th derivative of at each point is defined as (A-2) ⋯ By means of the generalized differential quadarature (GDQ) method, it is rewritten as (A-3) in which the differentiation matrix operator is evaluated at grid point values through the following relation [62] 0 , , , 1 2, … , 1,2, … , 1 (A-4) 1 , where , 1,2, … , and is identity matrix, and is (A-5) , The grid points distribution is selected according to the Chebysev-Gauss-Lobatto grid because of its high convergent in the GDQ method [57] which can be expressed as 20 1 1 2 1 1 cos , 1,1 which can be mapped onto This distribution gives points on 2 2 (A-6) 1,2, ⋯ , , as (A-7) , A.2. Integral operators Using the Taylor series and the GDQ technique, an accurate row vector integral operator can be constructed as [63] ξ ξ ξ , ξ ξ ξ ξ (A-8) ξ with ξ 2 ξ 1 ! ,…, ξ 2,3, … , ξ ξ 1 ! 2 ξ ,…, ξ 2 ξ 1 ! , 1 (A-9) where denotes the GDQ differential operator or the weighting coefficients matrix of the th-order derivative, and signals a column vector with values of the arbitrary function at , components containing the nodal 1,2, … , . The integral operators for different geometries are given in the following. Integral operator in the direction ( ) and direction ( ) for cylindrical shell can be presented as follows (A-10) (A-11) in which denotes the radius of shell. Also, integral operator in the direction ( ) and direction ( ) for annular plate are defined as follows ∘ (A-12) (A-13) where is the grid points in direction ⋯ In addition, for the hemispherical shell, integral operator in the direction ( ) can be given as 21 direction ( ) and ∘ , cos (A-14) (A-15) in which stands for the radius of shell and is the vector of grid points in direction ⋯ and integral operator in the and directions can be introduced as (A-16) (A-17) A.3. Generalization to N-Dimensional space , The nodal value of function ⋯ , , ,…, , , ,…, where , , ,…, ,…, ,…, in the N-Dimensional space can be given by ⋯ , , ,…, , , ,…, , , ,…, ⋯ , 1, … , 1, … , ⋮ 1, … , , , ,…, (A-20) is the number of grid points in each dimension and is the number of dimensions. By means of matrix differential operator of Eq. (A-3), the partial derivative of function in - dimensional space is approximated by the following relation ⋯ , ,…, ⊗ … Additionally, the integral operator of ⋯ , ,…, ⊗ ⋯⊗ ⊗ (A-21) -dimensional space is formulated as … ⊗ ⊗ ⋯⊗ ⊗ … (A-22) Thus, for a two-dimensional problem, the integral operator can be presented as ⊗ (A-23) 22 B. Base vectors, strains and elasticity matrices for various geometries The relations of base vectors, strains and elasticity matrices are given for each geometry here. B.1. Cantilever The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system ( ) for a cantilever can be defined as follows , , , , , , , (B-1) , It should be noted that the Cartesian ( , , ) and curvilinear coordinates ( ) coincide in this geometry. Also, the determinant of the shifter tensor is given as 1 (B-2) Based on Eq. (30) and using the given base vectors in this geometry, the coefficients can be calculated as 1, 0, (B-3) , , , , , 0 , (B-4) and accordingly, the components of nonlinear strains of six-parameter cantilever in the Cartesian coordinate can be represented as follows 1 2 1 2 , , , , , , , , , , 2 , 2 , , , , 2 , , , , , , , , , , , (B-5) 1 2 , , , , , , , , , , , , , , 2 , 2 , , , , , , , , (B-6) , , 23 , , , , , , , 2 , , , , 0 Furthermore, the components of metric tensor ( 1, ) for a cantilever can be presented as 1, 1 (B-7) using Eq. (38), on can attain the elasticity matrix as Consequently, by calculating follows 2 0 0 2 0 0 0 0 0 0 0 0 0 0 . (B-8) 2 B.2. Annular plate The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system ( ) for an annular plate can be presented as , , , , , cos where , sin , sin cos , (B-9) denotes polar coordinate system and the determinant of the shifter tensor ( ) is defined as follows 1 (B-10) Similarly, the coefficients for annular plate are given as cos sin sin cos 0 , 0 0 , Using 0 , , 1 , (B-11) 0 , and based on Eqs. (32) and (33), the strain components for an annular plate can be presented as cos sin , sin , , cos 1 2 , , 1 2 , , , (B-12) , , 24 , , 2 sin cos , , 2 , 2 , cos , sin , , , , , , , cos sin sin , , cos , , , , 1 2 cos sin , sin 2 , , cos , sin , cos , , , , , , , , , , 2 , , , , 2 , , , , , , cos , , , , , , , sin , , , , , , , (B-13) , 0 Also, the components of metric tensor ( ) and the elasticity matrix ( ) in this geometry can be presented as follows 1 1, 2 2 , 1 0 0 0 0 0 0 0 0 0 0 0 (B-14) (B-15) 0 . 2 B.3. Hemispherical Shell One can define the base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system ( ) for a hemispherical shell as 25 , , , , , sin , cos sin sin cos cos cos cos , , where cos , (B-16) , sin sin denote spherical coordinates and the determinant of the shifter tensor ( ) can be presented as (B-17) 1 Using the relations of the base vectors ( and ) and Eq. (30), the coefficients in this geometry are given as sin cos cos sin cos sin cos cos cos , sin cos , , cos sin , sin cos cos 0 sin sin sin sin cos cos (B-18) , cos , 0 and accordingly, the nonlinear strains of six-parameter spherical shell in the Cartesian coordinate can be attained as sin cos 1 2 cos 2 cos , , sin cos cos , cos cos , , sin 2 cos cos cos cos , cos cos , 1 2 , , cos , , sin , sin , , cos , , , , sin sin , sin sin , cos , sin , , sin , (B-19) , cos cos cos sin , cos , , sin sin , cos cos cos , sin , , sin 2 sin , , 26 cos , , sin cos , 1 2 , sin sin cos sin , sin cos sin sin cos sin sin , 2 sin 2 cos , cos sin sin sin , , , , , (B-20) , cos , , cos , sin , , , cos , , , cos , cos , , , sin , cos cos sin , , cos , , sin cos cos sin , , , cos , cos , , sin sin , , sin , , , sin , sin , , cos cos cos , , cos , cos sin , , , , cos , cos cos cos cos cos , , cos , cos 2 sin , cos , sin , , 0 Also, the components of metric tensor ( ) and the elasticity matrix ( ) are written as follows 1 1 cos , 2 cos 2 cos , 1 0 0 0 0 0 0 0 0 0 0 0 cos cos . (B-21) cos (B-22) 0 2 B.4. Cylindrical shell The base vectors in the Cartesian coordinate system ( ) and curvilinear coordinates system ( ) for a cylindrical shell can be given as 27 , , , , , , (B-23) , sin cos cos , sin in which ( , , ) denote the cylindrical coordinate system and the determinant of the shifter tensor ( ) can be defined as 1 (B-24) Using Eq. (30) one can write 1 0 0 0 sin , 0 cos cos , 0 , , 0 , sin 0 sin (B-25) , 0 , cos and the six-parameter nonlinear strains for a cylindrical shell can be written as follows , 1 2 , , , 2 , , 1 2 , , , , 2 , , 2 , , , , , , , , , , , , , , (B-26) , , , , 1 2 , , , , , , , , , , , , , , (B-27) , , , 2 , , , 2 , , , , , , , , , , , , 28 , , , , , , , 2 , , , , , 0 In addition, one can write the components of metric tensor ( 1 1, , 1 and based on Eqs. (38) and (40 ) the elasticity matrix ( 2 2 0 0 0 0 0 0 0 0 0 0 0 (B-28) ) can be presented as follows (B-29) 0 . ) as 2 29 References [1] Ventsel, E., and Krauthammer, T., 2001, “Thin plates and shells, Theory, analysis and applications,” Marcel Dekker, New York. [2] Pietraszkiewicz, W., and Gorski, J. (eds.), 2013, “Shell Structures: Theory and Applications,” Proc. 10th SSTA Conf. Poland. [3] Zingoni, A., 1997, “Shell structures in civil and mechanical engineering: Theory and Closed-form Analytical Solutions,” Thomas Telford, London. [4] Sabir, A. B., and Lock, A. C., 1972, “The applications of finite elements to large deflection geometrically nonlinear behaviour of cylindrical shells,” Brebbia, C. A. and Tottenham, H. (eds.) In Variational methods in engineering: proceedings on an international conference held at the University of Southampton, 25th September, 1972. Southampton University Press. [5] Chróścielewski, J., Makowski, J., and Stumpf, H., 1992, “Genuinely resultant shell finite elements accounting for geometric and material non linearity,” Int. J. Numer. Meth. Eng., 35, pp. 63-94. [6] Başar, Y., Ding, Y., and Schultz, R., 1993, “Refined shear-deformation models for composite laminates with finite rotations,” Int. J. Solids Struct., 30, pp. 2611-2638. [7] Betsch, P., Menzel, A., and Stein, E., 1998, “On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells,” Comput. Meth. Appl. Mech. Eng., 155, pp. 273-305. [8] Eberlein, R., and Wriggers, P., 1999, “Finite element concepts for finite elastoplastic strains and isotropic stress response in shells: theoretical and computational analysis,” Comput. Meth. Appl. Mech. Eng., 171, pp. 243-279. [9] Sansour, C., and Kollmann, F., 2000, “Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements,” Comput. Mech., 24, pp. 435-447. [10] Sze, K., Liu, X., and Lo, S., 2004, “Popular benchmark problems for geometric nonlinear analysis of shells,” Finite Elem. Anal. Des., 40, pp. 1551-1569. 30 [11] Massin, P., and Al Mikdad, M., 2002, “Nine node and seven node thick shell elements with large displacements and rotations,” Comput. Struct., 80, pp. 835-847. [12] Negahban, M., Goel, A., Marchon, P., and Azizinamini, A., 2009, “Geometrically exact nonlinear extended-reissner/mindlin shells: fundamentals, finite element formulation, elasticity,” Int. J. Comput. Meth. Eng. Sci. Mech., 10, pp. 430-449. [13] Burzyński, S., Chróścielewski, J., Daszkiewicz, K., and Witkowski, W., 2016, “Geometrically nonlinear FEM analysis of FGM shells based on neutral physical surface approach in 6-parameter shell theory,” Compos. Part B: Eng., 107, pp. 203-213. [14] Chróścielewski, J., Sabik, A., Sobczyk, B., and Witkowski, W., 2016, “Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory,” ThinWalled Struct., 105, pp. 207-219. [15] Nguyen-Thanh, N., Zhou, K., Zhuang Areias, X., Nguyen-Xuan, H., Bazilevs, Y., and Rabczuk, T., 2017, “Isogeometric analysis of large-deformation thin shells using RHTsplines for multiple-patch coupling,” Comput. Meth. Appl. Mech. Eng., 316, pp. 11571178. [16] Simo, J. C., and Fox, D. D., 1989, “On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization,” Comput. Meth. Appl. Mech. Eng., 72, pp. 267-304. [17] Simo, J. C., Fox, D. D., and Rifai, M. S., 1989, “On a stress resultant geometrically exact shell model. Part II: The linear theory; computational aspects,” Comput. Meth. Appl. Mech. Eng., 73, pp. 53-92. [18] Simo, J. C., Fox, D. D., and Rifai, M. S., 1990, “On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory,” Comput. Meth. Appl. Mech. Eng., 79, pp. 21-70. [19] Simo, J. C., Rifai, M. S., and Fox, D. D., 1990, “On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through-the-thickness stretching,” Comput. Meth. Appl. Mech. Eng., 81, pp. 91-126. [20] Atluri, S. N., 1984, “Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely 31 deformed solids, with application to plates and shells—I: Theory,” Comput. Struct., 18, pp. 93-116. [21] Pietraszkiewicz, W., 1984, “Lagrangian description and incremental formulation in the non-linear theory of thin shells,” Int. J. Non-linear Mech., 19, pp. 115-140. [22] Schmidt, R., 1984, “On Geometrically Non-Linear Theories for Thin Elastic Shells,” In: Axelrad E.L., Emmerling F.A. (eds) Flexible Shells. Springer, Berlin, Heidelberg [23] Braun, M., Bischoff, M., and Ramm, E., 1994, “Nonlinear shell formulations for complete three-dimensional constitutive laws including composites and laminates,” Comput. Mech., 15, pp. 1-18. [24] Büchter, N., Ramm, E., and Roehl, D., 1994, “Three‐dimensional extension of non‐ linear shell formulation based on the enhanced assumed strain concept,” Int. J. Numer. Meth. Eng., 37, pp. 2551-2568. [25] Bischoff, M., and Ramm, E., 1997, “Shear deformable shell elements for large strains and rotations,” Int. J. Numer. Meth. Eng., 40, pp. 4427-4449. [26] Hughes, T. J., and Carnoy, E., 1983, “Nonlinear finite element shell formulation accounting for large membrane strains,” Comput. Meth. Appl. Mech. Eng., 39, pp. 6982. [27] Bischoff, M., Bletzinger, K. U., Wall, W. A., and Ramm, E., 2004, “Models and Finite Elements for Thin‐Walled Structures,” Encyclopedia of computational mechanics, John Wiley & Sons. [28] Arciniega, R. A.,, and Reddy, J. N., 2007, “Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures,” Comput. Meth. Appl. Mech. Eng., 196, pp. 1048-1073. [29] Arciniega, R. A., and Reddy, J. N., 2007, “Large deformation analysis of functionally graded shells,” Int. J. Solids Struct., 44, pp. 2036-2052. [30] Pimenta, P. D. M., and Campello, E. D. M. B., 2009, “Shell curvature as an initial deformation: a geometrically exact finite element approach.” Int. J. Numer. Meth. Eng., 78, pp. 1094-1112. 32 [31] Gruttmann, F., 2011, “Nonlinear Finite Element Shell Formulation Accounting for Large Strain Material Models,” In Recent Developments and Innovative Applications in Computational Mechanics (pp. 87-95). Springer Berlin Heidelberg. [32] Amabili, M., 2015, “Non-linearities in rotation and thickness deformation in a new thirdorder thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells,” Int. J. Non-linear Mech., 69, pp. 109-128. [33] Alijani, F., and Amabili, M., 2014, “Non-linear static bending and forced vibrations of rectangular plates retaining non-linearities in rotations and thickness deformation,” Int. J. Non-linear Mech., 67, pp. 394-404. [34] Gutiérrez Rivera, M. E., Reddy, J. N., and Amabili, M., 2016, “A new twelve-parameter spectral/hp shell finite element for large deformation analysis of composite shells,” Compos. Struct., 151, pp. 183-196. [35] Amabili, M., 2014, “A non-linear higher-order thickness stretching and shear deformation theory for large-amplitude vibrations of laminated doubly curved shells,” Int. J. Non-linear Mech., 58, pp. 57-75. [36] Faghih Shojaei, M., and Ansari, R., 2017, “Variational differential quadrature: A technique to simplify numerical analysis of structures,” Appl. Math. Model., 49, pp. 705–738. [37] Ansari, R., Torabi, J., and Faghih Shojaei, M., 2017, “An efficient numerical method for analyzing the thermal effects on the vibration of embedded single-walled carbon nanotubes based on the nonlocal shell model,” Mech. Adv. Mater. Struct., DOI: http://dx.doi.org/10.1080/15376494.2017.1285457. [38] Ansari, R., Torabi, J., and Faghih Shojaei, M., 2016, “Free vibration analysis of embedded functionally graded carbon nanotube-reinforced composite conical/cylindrical shells and annular plates using a numerical approach,” J. Vib. Control, DOI: https://doi.org/10.1177/1077546316659172. [39] Ansari, R., Torabi, J., Faghih Shojaei, M., and Hasrati, E., 2016, “Buckling analysis of axially-loaded functionally graded carbon nanotube-reinforced composite conical panels using a novel numerical variational method,” Compos. Struct., 157, pp. 398-411. 33 [40] Ansari, R., Torabi, J., and Faghih Shojaei, M., 2017, “Buckling and vibration analysis of embedded functionally graded carbon nanotube-reinforced composite annular sector plates under thermal loading,” Compos. Part B, 109, pp. 197-213. [41] Ansari, R., Shahabodini, A., and Faghih Shojaei, M., 2016, “Vibrational analysis of carbon nanotube-reinforced composite quadrilateral plates subjected to thermal environments using a weak formulation of elasticity,” Compos. Struct., 139, pp. 167-187. [42] Ansari, R., Torabi, J., and Faghih Shojaei, M., 2016, “Vibrational analysis of functionally graded carbon nanotube-reinforced composite spherical shells resting on elastic foundation using the variational differential quadrature method,” Eur. J. Mech.A/Solids, 60, pp. 166-182. [43] Ansari, R., Faghih Shojaei, M., and Gholami, R., 2016, “Size-dependent nonlinear mechanical behavior of third-order shear deformable functionally graded microbeams using the variational differential quadrature method,” Compos. Struct., 136, pp. 669-683. [44] Ansari, R., Faghih Shojaei, M., Shahabodini, A., and Bazdid-Vahdati, M., 2015, “Threedimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach,” Compos. Struct., 131, pp. 753-764. [45] Eremeyev, V. A., Lebedev, L. P., and Cloud, M. J., 2015, “The Rayleigh and Courant variational principles in the six-parameter shell theory,” Math. Mech. Solids, 20, pp. 806822. [46] Eremeyev, V. A., and Pietraszkiewicz, W., 2014, “Refined theories of plates and shells,” ZAMM, 94, pp. 5-6. [47] Altenbach, H., and Eremeyev, V. A., 2014, “Actual developments in the nonlinear shell theory-state of the art and new applications of the six-parameter shell theory,” In Shell structures: Theory and applications, 3, pp. 3-12. [48] Altenbach, H., and Eremeyev, V. A., 2011, “On the shell theory on the nanoscale with surface stresses,” Int. J. Eng. Sci., 49, pp. 1294-1301. [49] Eremeyev, V. A., Lebedev, L. P., and Altenbach, H., 2012, “Foundations of micropolar mechanics,” Springer Science & Business Media. 34 [50] Altenbach, H., and Eremeyev, V. A., 2008, “Analysis of the viscoelastic behavior of plates made of functionally graded materials,” ZAMM, 88, pp. 332-341. [51] Altenbach, H., and Eremeyev, V. A., 2009, “On the bending of viscoelastic plates made of polymer foams,” Acta Mech., 204, p. 137. [52] Altenbach, H., and Eremeyev, V. A., 2009, “On the time-dependent behavior of FGM plates,” Key Eng. Mater., 399, pp. 63-70. [53] Altenbach, H., and Eremeyev, V. A., 2011, “Mechanics of viscoelastic plates made of FGMs,” In Computational modelling and advanced simulations (pp. 33-48). Springer, Dordrecht. [54] Altenbach, H., and Eremeyev, V. A., 2010, “On the effective stiffness of plates made of hyperelastic materials with initial stresses,” Int. J. Non-Linear Mech., 45, pp. 976-981. [55] Eremeyev, V. A., and Pietraszkiewicz, W., 2004, “The nonlinear theory of elastic shells with phase transitions,” J. Elast., 74, pp. 67-86. [56] Eremeyev, V. A., and Pietraszkiewicz, W., 2011, “Thermomechanics of shells undergoing phase transition,” J. Mech. Phys. Solids, 59, pp. 1395-1412. [57] Tornabene, F., Viola, E., and Inman, D. J., 2009, “2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures,” J. Sound Vib., 328, pp.259-290. [58] Altenbach, J., Altenbach, H., and Eremeyev, V. A., 2010, “On generalized Cosserat-type theories of plates and shells: a short review and bibliography,” Arch. Appl. Mech., 80, pp. 73-92. [59] Altenbach, H., and Eremeyev, V. A., 2009, “On the linear theory of micropolar plates,” ZAMM, 89, pp. 242-256. [60] Della Corte, A., dell’Isola, F., Esposito, R., and Pulvirenti, M., 2017, “Equilibria of a clamped Euler beam (Elastica) with distributed load: large deformations,” Math. Model. Meth. Appl. Sci., 27, pp. 1391-1421. [61] dell’Isola, F., Giorgio, I., Pawlikowski, M., and Rizzi, N. L., 2016, “Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, 35 experimental and numerical examples of equilibrium,” Proc. Royal Soc. London A, 472, (2185). [62] Shu, C., 2000, “Differential quadrature and its application in engineering,” Springer, London. [63] Ansari, R., Shojaei, M. F., Rouhi, H., and Hosseinzadeh, M., 2015, “A novel variational numerical method for analyzing the free vibration of composite conical shells,” Appl. Math. Model., 39, pp. 2849-2860. 36 (a) (b) 1 -u 1 (tip) - Present 0 0.9 u3 (tip) - Present -u 1 (tip) - [10] u3 (tip) - [10] 0 0.8 -u 1 (tip) - uni. load u3 (tip) - uni. load 0 0.7 0 0.6 0 0.5 0 0.4 0 0.3 0 0.2 0 0.1 0 0 1 2 3 4 5 deflectio on (c) 37 6 7 8 9 (d) (e) Figure 1: (a) Cantilev ver subjecteed to end shear force, (b b) Cantileveer subjected d to unifoormly distribbuted load (c) ( Load–deefection curv ves, (d) Defformed connfiguration for f the maaximum endd shear forcce, (e) Deforrmed config guration forr the maxim mum uniform mly diistributed lo oad 38 (a) 1 0 0.9 0 0.8 0 0.7 0 0.6 0 0.5 0 0.4 -u 1 (tip)) - Present 0 0.3 u3 (tip) - Present 0 0.2 -u 1 (tip)) - [10] 0 0.1 0 u3 (tip) - [10] 0 3 6 9 deflectioon (b) 39 12 15 (c) Figurre 2: (a) Canntilever sub bjected to ennd bending moment, (b b) Load–deffection curv ves, (c) D Deformed co onfigurationn for the maaximum ben nding momeent 40 (a) 1 0 0.9 u 3 (A) - Preseent 0 0.8 u 3 (A) - [10] u 3 (B) - Preseent u 3 (B) - [10] 0 0.7 0 0.6 0 0.5 0 0.4 0 0.3 0 0.2 0 0.1 0 0 2 4 6 8 10 12 14 16 18 deflectioon (b) (c) Figgure 3: (a) Slit S annularr plate underr the line fo orce , (b) Load–defect L tion curves,, (c) Deformed configuraation for thee maximum line force 41 (a) 1 0 0.9 u 1 (A) - Pressent 0 0.8 u 1 (A) - [10]] -u 2 (B) - Present -u 2 (B) - [10]] 0 0.7 0 0.6 0 0.5 0 0.4 0 0.3 0 0.2 0 0.1 0 0 1 2 3 4 5 deflectio on (b) 42 6 7 8 9 (c) mispherical shell subjeected to alterrnating radiial forces, (bb) Load–defection Figure 4: (a) Hem cuurves, (c) Deformed coonfiguration n for the max ximum forcces 43 (a) 1 0 0.9 u 2 (A) , Pressent 0 0.8 -u 3 (C) , Pressent -u 3 (B) , Pressent 0 0.7 u 2 (A) , [10]] 0 0.6 -u 3 (C) , [10]] -u 3 (B) , [10]] 0 0.5 0 0.4 0 0.3 0 0.2 0 0.1 0 0 0.5 5 1 1.5 2 2.5 deflectioon (b) 44 3 3 3.5 4 4.5 5 (c) Figu ure 5: (a) Open-end cyllindrical sheell under raadial pulling g forces, (b)) Load–defeection c curves, (c) Deformed D coonfiguration n for the maaximum loaad 45 (a) (b) 46 (c) Figuree 6: (a) Sem mi-cylindrical shell undder an end pinching p force, (b) Loadd–defection n curve, (c) Defo ormed confiiguration forr the maxim mum load 47 load parameter (a) (b) 48 (c) Figuree 7: (a) Cyliindrical shelll under raddial pinching g forces resttrained withh rigid diaph hragms, (b) Load–ddefection cu urves, (c) Deeformed con nfiguration for the maxximum load d 49 (a) 1 -u 2 (A) - Pressent 0 0.9 -u 2 (A) - [10]] 0 0.8 0 0.7 0 0.6 0 0.5 0 0.4 0 0.3 0 0.2 0 0.1 0 0 5 10 15 deflectioon (b) 50 20 255 30 (c) Figgure 8: (a) Hinged H cylindrical rooff under a ceentral pinchiing force, (bb) Load–defection c curve, (c) Deformed D coonfiguration n for the maaximum loadd 51 (a) (b) 52 16 14 12 10 8 6 4 HSDT, 2TS, 2TS-NL [33] ANSYS-3D [33] 6-parameter 2 0 0 3 6 9 P 10 12 15 4 (c) 0 HSDT, 2TS [33] ANSYS-3D [33] 2TS-NL [33] 6-parameter -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0 3 6 9 P (d) 53 10 4 12 15 10-3 -1 -2 -3 -4 -5 -6 -7 ANSYS-3D [33] 2TS-NL [33] HSDT & 2TS [33] 6-parameter -8 -9 -10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/L (e) Figure 9: (a) Simply-supported square plate subjected to uniformly distributed load, (b) Deformed configuration for the maximum load, (c) Non-dimensional maximum transverse deflection curve at the center of the plate versus the non-dimensional uniformly distributed load, (d) Maximum rotation versus the non-dimensional uniformly distributed load, (e) Non-dimensional thickness deformation (Δ ⁄ ) at 1428.6 54 ⁄2 for the non-dimensional pressure Table 1: Comparison of the number of load increments (NINC) and the number of iterations (NITER) required to obtain the ultimate solution for the cantilever subjected to end shear force Ref. [10] 8 1 S4R elements 16 present 1 S4R elements 15 5 nodes NINC 15 15 15 NITER 78 80 57 Table 2: Comparison of the number of load increments (NINC) and the number of iterations (NITER) required to obtain the ultimate solution for the cantilever subjected to end bending moment Ref. [10] 8 1 S4R elements 16 present 1 S4R elements 15 5 nodes NINC 125 125 103 NITER 715 714 394 55 Highlights A new approach is proposed for the nonlinear large deformation analysis of shells based on the six-parameter shell theory. The proposed method is implemented in the context of weak form formulation and can be easily used for various geometries. Easy implementation, general formulation, absence of locking problem, accuracy and low computational cost are the main advantages of proposed approach. 56

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