CHAPTER Nonlinear Continuum Spectral Shell Element In Chapter 4, we presented a detailed derivation of HT-CS and HT-CS-X elements, the associated linear finite element formulation and several challenging benchmark problems to assess the performances of higher order elements. Based on the performances of HT-CS and HT-CS-X elements in Discriminating and Revealing Test Cases, it was found that HT-CS-X elements are more robust in handling a wide range of shell problems. Further, the element was also tested for its capability to handle stress resultants in challenging linear plate bending problems (Morleys skew plate and corner supported square plate). Having assessed the performance of HT-CS-X element in linear plate\shell analysis, in this chapter, we shall extend the linear finite element formulation of HT-CS-X elements to a nonlinear formulation that caters for large deflection problems. The performance of the developed nonlinear continuum shell element will be assessed in several geometric nonlinear shell problems. Moreover, its superiority over lower order elements in handling stresses in the nonlinear regime will be discussed. The large deflection analysis of shells has drawn the attention of many researchers due to its importance in engineering practice. There have been numerous research studies on the geometric nonlinear analysis of shells that undergo large deflections, moderate\finite rotations (see for example, Simo et 222 Nonlinear Continuum Spectral Shell Element al.,1989; Saleeb et al, 1990; Sze et al., 1999; Balah and Al-Ghamedy, 2002; Arciniega and Reddy, 2007). Most of the researchers presented the nonlinear load versus deflection response of shell structures computed from various kinds of shell elements that were developed to tackle nonlinear behaviour. The ability of finite elements in handling stresses in nonlinear regime has received relatively less attention. The accurate prediction of stress distributions in the nonlinear region is very crucial from a design point of view. Furthermore, the correct estimation of peak stresses and their localization in nonlinear region provides a sound basis to perform reliability and failure studies which decide the safety of a structural component. Achieving good and reliable levels of accuracy in the highly nonlinear range with lesser computational resources is certainly not possible with lower order finite elements. Although the nonlinear load versus deflection response of the structure may be traced accurately with coarser mesh designs of lower order finite elements, one may require very fine mesh designs in order to achieve good accuracy of stress values. Furthermore, the accuracy of stresses predicted by lower order elements may be highly erroneous in problems which involve steep stress gradients. Hence, we use higher order finite elements that have enriched shape functions and render many advantages over conventional lower order finite elements such as the accommodation of high aspect ratio elements, better prediction of stresses with coarser meshes, ability to handle steep stress gradients, less sensitivity to input data (locking mechanisms) and lesser computational resources as compared to lower order finite elements. In this chapter, we shall present the nonlinear finite element formulation of a continuum shell element that accounts for large deflections and moderate 223 Nonlinear Continuum Spectral Shell Element rotations of shell structures. The accuracy of the proposed element will be verified via several nonlinear benchmark problems and their superior performance over conventional lower order shell finite elements will be illustrated in the nonlinear stress analysis of the shell problems. This chapter has been organized into four main sections. The first section deals with the development of nonlinear finite element model for shells under the framework of Total Lagrangian approach followed by a description on nonlinear solution algorithms adopted in the present work. In the second section, we verify the performance of HT-CS-X elements in selected nonlinear shell benchmark problems. Following this, we will present the performance of HT-CS-X elements in handling stresses in nonlinear region much efficiently as compared to ABAQUS S8R lower order shell elements. The performances of HT-CS-X element and ABAQUS S8R element will be compared in the context of accuracy and distribution of stresses, relative ease of mesh designs and number of degrees of freedom to achieve a smooth stress variation. In the last section of this chapter, we will present a detailed nonlinear analysis of laminated composite hyperboloid shells which are challenging due to their negative value of Gaussian curvature and complex behaviour. 5.1 Description of motion Consider a deformable body of known geometry, constitution and loading that occupies an initial configuration in which a particle X occupies the position X having Cartesian coordinates (X, Y, Z). After the application of loads, the body assumes a new position x in the deformed configuration having coordinates (x, y, z). The objective is to determine the final configuration of a body subjected to a total load say Pmax . A straightforward way of determining 224 Nonlinear Continuum Spectral Shell Element the final configuration from a known initial configuration is to assume that the total load Pmax is applied in increments so that the body occupies several intermediate configurations i (i =1,2,) prior to attaining the final configuration. The magnitude of load increments should be such that the computational procedure that is employed to trace the response of the body (such as the Newton Raphson method and the Arc- length method) is capable of predicting the deformed configuration at the end of each load step. In the determination of an intermediate configuration i, one may use any of the previously known configurations , , ,i-1 as the reference configuration R. If the initial configuration is used as the reference configuration with respect to which all quantities are measured, it is called the Total Lagrangian description. We consider three equilibrium configurations of the body namely, , and which correspond to three different loads. denotes the initial undeformed configuration, denotes the last known deformed configuration and denotes the current deformed configuration to be determined. It is assumed that all variables such as displacements, strains and stresses are known up to configuration . The objective is to develop a formulation to determine the displacements and stresses of the body in the deformed configuration . In the next section, we present the strain and stress measures employed in the Total Lagrangian formulation. A detailed derivation of relevant stress and strain measures for a Total Lagrangian approach can be seen in standard textbooks on nonlinear finite element formulation (Reddy, 2004). Hence we 225 Nonlinear Continuum Spectral Shell Element present the final equations that are necessary for the development of nonlinear finite element model. 5.1.1 Green strain tensor We adopt the Green-Lagrange strain tensor or simply referred to as the Green strain tensor to measure the deformation of a body. The Green strain tensor is symmetric and is expressed as follows: E= T F ì F - I = (C - I ) 2 ( ) (5.1) where C = F T ì F is called the right Cauchy-Green deformation tensor and F is the deformation gradient tensor defined as ộ ả 1x ờ ả 1x T ổ ảx ờả y F =ỗ ữ = ả x ố ảX ứ ờả z ờở ả x ả 1x ả 0y ả 1y ả 0y ả 1z ả 0y ả 1x ự ỳ ả 0z ỳ ả 1y ỳ ả 0z ỳ ỳ ả 1z ỳ ả z ỳỷ (5.2) The Green strain tensor can be written as EI J = ổ ả uI ả uJ ả uK ả uK ỗ + + ỗố ảX J ảX I ảX I ảX J ữữ ứ (5.3) where u I denotes the component of displacement. The subscripts I,J,K take the values of 1,2,3 ( u1 = u , u = v and u = w ). u, v denote the in-plane displacements and w denotes the transverse displacements. Likewise, X I denotes the components of Cartesian coordinates ( X = X , X = Y and X = Z ). 226 Nonlinear Continuum Spectral Shell Element The Green-strain components can be written as Ex x = Ey y Ez z 2 ả u ộổ ả u ổ ả v ổ ả w ự + ờỗ + + ữ ỗ ữ ỗ ữ ỳ ảX ờởố ảX ứ ố ảX ứ ố ảX ứ ỳỷ 2 ả v ộổ ả u ổ ả v ổ ả w ự = + ờỗ ữ +ỗ ữ +ỗ ữ ỳ ảY ờởố ảY ứ ố ảY ứ ố ảY ứ ỳỷ 2 ả w ộổ ả u ổ ả v ổ ả w ự = + ờỗ ữ +ỗ ữ +ỗ ữ ỳ ảZ ờởố ảZ ứ ố ảZ ứ ố ảZ ứ ỳỷ ổ ảu ảv ảu ảu ảv ảv ả w ả wử Ex y = ỗ + + + + ữ ố ảY ảX ảX ảY ảX ảY ảX ảY ứ ổ ảu ả w ảu ảu ảv ảv ả w ả wử Ex z = ỗ + + + + ữ ố ảZ ảX ảX ảZ ảX ảZ ảX ảZ ứ ổ ảv ả w ảu ảu ảv ảv ả w ả wử Ey z = ỗ + + + + ữ ố ảZ ảY ảY ảZ ảY ảZ ảY ảZ ứ (5.4) 5.1.2 Stress tensor The equation of equilibrium has to be derived for the deformed configuration of the body, i.e. at configuration . Since the geometry of the deformed configuration is unknown, the equations are written in terms of the known reference configuration . In doing so, it becomes necessary to introduce various measures of stress. These stress measures emerge when the elemental volumes and areas are transformed from the deformed configuration to the undeformed configuration. In the Total Lagrangian approach we use the second Piola-Kirchhoff stress tensor denoted as S. The second Piola-Kirchhoff stress S can be expressed in terms of Cauchy stress tensor s (which is defined to be the current force per unit deformed area) by the following transformation S = JF -1 ì s ì F -T (5.5) 227 Nonlinear Continuum Spectral Shell Element where, J denotes the determinant of the deformation gradient tensor F . The second Piola-Kirchhoff stress tensor S, gives the transformed current force per unit undeformed area. The stress tensor S is symmetric whenever the Cauchy stress tensor s is symmetric. For details on the transformation of various measures one may refer to books by Bathe (1996) and Reddy (2004). Having mentioned about the strain and stress measures, it can be shown that the rate of internal work done in a continuous medium in the current configuration can be expressed as (Reddy 2004): W= S : E& dV Vũ (5.6) Thus the second Piola-Kirchhoff tensor S is the work conjugate to the rate of the Green-Lagrange strain tensor E& . The following notations are used in this chapter. A left superscript on a quantity denotes the configuration in which the quantity occurs and a left subscript denotes the configuration with respect to which the quantity is measured. For example, i j H refers to a quantity H (say displacements, stresses) that occurs in configuration i but is measured in configuration j. When the quantity is measured in the same configuration, the left subscript is omitted. The left superscript will be omitted for all incremental quantities that occur between configurations and . The right subscript refers to the components of Cartesian coordinate system. When the body deforms under the action of externally applied loads, a particle X occupying position (X, Y, Z) in configuration moves to a new position x having coordinates (x, y, z) in configuration . The components of particle X can be written as ( x = x, y , z ) and that of x can be written 228 Nonlinear Continuum Spectral Shell Element as x = ( x, y, z ) . The total displacements of a particle X in the two configurations and can be written as: i u = 1xi - xi , (i =1,2,3) (5.7 a) u i = xi - xi , (i =1,2,3) (5.7 b) The displacement increment of a point from configuration to is u i = 02 u i - 01u i , (i =1,2,3) (5.8) 5.1.3 Green Strain tensor and stress tensor for various configurations The components of Green strain tensor in configurations and are given in terms of displacements as: 1 ổỗ ả 01u i ả u j ả 01u k ả 01u k ửữ Eij = + + ỗố ả x j ả xi ả xi ả x j ữứ (5.9 a) Eij = ổỗ ả 02 u i ả u j ả 02 u k ả 02 u k ửữ + + ỗố ả x j ả xi ả xi ả x j ữứ (5.9 b) The incremental Green-Lagrange strain components e ij which are obtained in moving from configuration to are given as e i j = ei j + h i j (5.10) where, ei j are linear components of strain increment tensor expressed as ei j = ổ ả u i ả u j ả 01u k ả u k ả u k ả 01u k ỗ ữ + + + 0 ỗ ả 0x ữ ả x ả x ả x ả x ả x j i i j i j ố ứ (5.11) The nonlinear components 0h i j are given by hi j = ả uk ả uk ả xi ả x j (5.12) 229 Nonlinear Continuum Spectral Shell Element For geometrically linear analysis, only two configurations = and are involved. Thus (ảu k ) u i = and ( u i = u i . The terms involving products of ) ả xi and ảu k ả x j are small and hence are neglected. Consequently, the linear components of strain increment tensor ei j become the same as the components of the Green- Lagrange strain tensor Eij and both reduce to infinitesimal strain components ei j = ổ ả ui ả u j ỗ + ỗ ả 0x ả xi j ố ữ ữ ứ (5.13) The second Piola-Kirchhoff stress tensor components in configurations and are denoted by S i j and S i j respectively. They are related by the following equation where, S i j = 01S i j + S i j (5.14) S i j are the components of the Kirchhoff stress increment tensor and are given by: S i j = C ijkl e k l (5.15) C ijkl denotes the incremental constitutive tensor with respect to configuration . In the present work, since we deal with geometric nonlinearity, the components of the constitutive matrix are the same as that obtained for a linear analysis. 5.1.4 Total Lagrangian Formulation Having defined the necessary terms involved in the Total Lagrangian formulation, we now present the final equations of equilibrium. The equations of Lagrangian incremental description of motion for the displacement based 230 Nonlinear Continuum Spectral Shell Element finite element model considered herein are derived from the principle of virtual displacements. The detailed derivation of the equations of equilibrium can be found in the book by Reddy (2004). The weak form of the equilibrium equation that is suited for the development of displacement finite element model based on the Total Lagrangian formulation is given to be: ũ V C i j k l ek l d ( ei j ) d 0V + ũ V S i j d ( h i j ) d 0V = d ( 02 R) - d ( 01R) (5.16) where d ( 01 R ) = ũ 01 S i j d ( ei j ) d 0V is the equivalent nodal force vector and d ( R) = ũ V f i d u i d 0V + ũ 02 t i d u i d 0S is the externally applied load vector V S (sum of body force and traction force). The total stress components 01 S i j are evaluated using the following constitutive relation S i j = C i j k l 01E k l (5.17) where, 01Ek l are the Green-Lagrange strain components described in Eq. (5.4). 5.2 Finite Ele ment Model Continuum Shell Element The equilibrium equation that is required for the development of nonlinear displacement based degenerated shell finite element model for a solid continuum is given in Eq. (5.16). In order to derive the finite element equations for a shell element, the first step is to select appropriate interpolation (shape) functions for the displacement field and geometry. The coordinates and displacements are interpolated using the isoparametric concept which involves the same interpolation functions. This is done to ensure the displacement compatibility across element boundaries is preserved at all configurations 231 Nonlinear Continuum Spectral Shell Element HT-CS-X elements in handling deflections as well as stress distributions in the nonlinear regime for a shell having negative Gaussian curvature. 5.6.1 Hype rboloid shell clamped along both the edges We consider a hyperboloid shell subjected to a smoothly varying surface pressure q = q cos (2q ) that acts normal to the surface of the shell as shown in Fig. 5.30 and 5.31. The hyperboloid shell has an assumed length a = 40 in., radius at the throat region R_b = 7.5 in., radius at the top R_t = 15 in. and thickness h = 0.04 in. The hyperbolic curve is generated using the following equation R0 ( y ) = R _ b + ( y C ) , where C = 20 and R0 denotes the base radius at a specific point along the length of the hyperboloid shell. The material properties assumed are as follows: Youngs modulii E1 = 40 10 psi , E = E = 10 psi Shear modulii G12 = G13 = G 23 = 0.6 10 psi , and Poisson ratios n 12 = n 13 = n 23 = 0.25 . Subscript denotes material properties of the laminated shell in the circumferential direction and subscript denotes material properties in the meridional direction. The hyperboloid shell is clamped along both the edges and is highlighted in Fig. 5.30. 277 Nonlinear Continuum Spectral Shell Element Fig. 5.30 Geometry of a hyperboloid shell clamped along both the edges Fig. 5.31 Top view of a hyperboloid shell showing the distribution of periodic load along the circumference A similar example was considered by Basar et al. (1993), Wagner and Gruttmann (1994), Balah and Al- Ghamedy (2002) and Arciniega and Reddy (2007) who studied the nonlinear behaviour of hyperboloid shells undergoing finite rotations. They presented the nonlinear load versus deflection response of a symmetric hyperboloid shell subjected to inward and outward point loads. Herein, we study the nonlinear behaviour of the hyperboloid shell under the action of varying surface pressure along the circumferential direction and go a 278 Nonlinear Continuum Spectral Shell Element step ahead to present the stress distributions inside the shell at various load steps. In this example, we employ a 33 mesh of HT-CS-X elements having 964 d.o.fs to obtain the nonlinear response of an orthotropic and a symmetric cross-ply (0 90 90 0 ) hyperboloid shell. The nonlinear load versus deflection responses of the two shells are given in Fig. 5.32. The hyperboloid shell having a symmetric (0 90 90 0 ) lamination scheme is observed to have a larger load resisting capability as compared to an orthotropic shell due to the specific arrangement of layers that have their fibres oriented perpendicular to each other. The symmetric (0 90 90 0 ) hyperboloid shell will have a larger bending stiffness about an axis perpendicular to the fibre orientation as compared to an orthotropic shell. For the same value of thickness h, since the 0 layer is located farther from the shells midsurface, bending stiffness is high as compared to an orthotropic shell. It can be seen that the maximum load intensity say q max is three times higher for a (0 90 90 0 ) laminated composite shell as compared to an orthotropic shell for nearly the same amount of radial deflections considered at point O. Figures 5.33 and 5.34 show the deformed configuration at a final load value q max = ksi and q max = ksi corresponding to the orthotropic and (0 ) 90 90 0 laminated composite hyperboloid shells respectively. 279 Nonlinear Continuum Spectral Shell Element Fig. 5.32 Nonlinear load versus radial deflection response at point O of a hyperboloid shell clamped at both ends and subjected to varying periodic surface load Initial undeformed configuration Deformed final configuration Fig. 5.33 Deformation of an orthotropic hyperboloid shell clamped along two edges and subjected to periodic surface pressure 280 Nonlinear Continuum Spectral Shell Element Initial undeformed configuration Deformed final configuration Fig. 5.34 Deformation of a symmetric cross-ply (0 90 90 0 ) hyperboloid shell clamped along two edges and subjected to periodic surface pressure Stress distributions in an orthotropic hyperboloid shell The stress distributions at the top and bottom face of the orthotropic hyperboloid shell at the end of the initial and final load steps (see Figs 5.37 to 5.39). For an orthotropic hyperboloid shell, we observe that for the given periodic load variation q = q cos (2q ) , the maximum circumferential stresses occur along the arc OC (see Fig. 5.30) at a circumferential angle of q = 0 and q = 90 . The meridional stresses attain peak values in the vicinity of the clamped edge and they are seen to have the greatest magnitudes at q = 0 and q = 90 . The plot of stress variations in an orthotropic hyperboloid shell indicates a redistribution of circumferential and meridional stresses in the final load step (see Figs. 5.35 and 5.36). This is further justified by a visual inspection of stress distributions shown in Figs 5.37 to 5.39. However, there is negligible redistribution of in-plane shear stresses within the hyperboloid shell. 281 Nonlinear Continuum Spectral Shell Element Fig. 5.35 Variation of circumferential stresses along OC (see Fig. 5.30) of an orthotropic hyperboloid shell at q = 0.15 ksi and q = ksi Fig. 5.36 Variation of meridional stresses along OD (see Fig. 5.30) of an orthotropic hyperboloid shell at q = 0.15 ksi and q = ksi 282 Nonlinear Continuum Spectral Shell Element s qq at q = ksi Bottom face of shell Top face of shell s qq at q = 0.15 ksi Fig. 5.37 Distributions of circumferential stresses s qq in clamped orthotropic hyperboloid shell subjected to varying periodic surface pressure 283 Nonlinear Continuum Spectral Shell Element s yy at q = ksi Bottom face of shell Top face of shell s yy at q = 0.15 ksi Fig. 5.38 Distributions of meridional stresses s yy in a clamped orthotropic hyperboloid shell subjected to varying periodic surface pressure 284 Nonlinear Continuum Spectral Shell Element In-plane shear stress at q = ksi Bottom face of shell Top face of shell In-plane shear stress at q = 0.15 ksi Fig. 5.39 Distributions of in-plane shear stresses in a clamped orthotropic hyperboloid shell subjected to varying periodic surface pressure 285 Nonlinear Continuum Spectral Shell Element Stress distributions in a symmetric (0 90 90 0 ) laminated composite hyperboloid shell Next, we present the stress distributions at the top and bottom face of the symmetric cross-ply (0 90 90 0 ) hyperboloid shell at the end of the initial and final load steps (see Figs 5.42 to 5.44). For a (0 90 90 0 ) hyperboloid shell, we observe that for the given periodic load variation q = q cos (2q ) , the maximum compressive circumferential stress occurs along the arc OC (see Fig. 5.30) at a circumferential angle of q = 58.5 and decreases monotonically towards the clamped edges. The meridional stresses are high in the vicinity of the clamped edge and they are seen to have the greatest magnitudes at q = 0 and q = 90 (see Fig. 5.43). The plot of stress variations in an orthotropic hyperboloid shell indicates a marked redistribution of circumferential and meridional stresses in the final load step (see Fig. 5.40 and 5.41). This can be further verified by visual inspection of stress distributions shown in Figs 5.42 to 5.44. Fig. 5.40 Variation of circumferential stresses along OC (see Fig. 5.30) of a (0 90 90 0 ) laminated composite hyperboloid shell at q = 0.3 ksi and q = ksi 286 Nonlinear Continuum Spectral Shell Element Fig. 5.41 Variation of meridional stresses along OD (see Fig. 5.30) of a (0 90 90 0 ) laminated composite hyperboloid shell at q = 0.3 ksi and q = ksi 287 Nonlinear Continuum Spectral Shell Element s qq at q = ksi Bottom face of shell Top face of shell s qq at q = 0.3 ksi ( ) Fig. 5.42 Distributions of circumferential stresses s qq in a clamped 0 90 90 0 hyperboloid shell subjected to varying periodic surface pressure 288 Nonlinear Continuum Spectral Shell Element s yy at q = ksi Bottom face of shell Top face of shell s yy at q = 0.3 ksi ( ) Fig. 5.43 Distributions of meridional stresses s yy in a clamped 0 90 90 0 hyperboloid shell subjected to varying periodic surface pressure 289 Nonlinear Continuum Spectral Shell Element In-plane shear stress at q = ksi \ [ ] Bottom face of shell Top face of shell In-plane shear stress at q = 0.3 ksi ( ) Fig. 5.44 Distributions of in-plane shear stresses in a clamped 0 90 90 0 hyperboloid shell subjected to varying periodic surface pressure 290 Nonlinear Continuum Spectral Shell Element 5.7 Conclusions In this chapter, a nonlinear continuum spectral triangular shell element called HT-CS-X is developed under the framework of the total Lagrangian approach. The accuracy of the developed element is verified in selected nonlinear shell benchmark problems. A simple coarse mesh design of HT-CS-X elements together with the arc length method is able to trace complex nonlinear load versus deflection response of shells having multiple limit points. Further the ability of HT-CS-X elements in predicting better stress distributions in shells having singularities are demonstrated by cylindrical shell problems subjected to patch loads (where the sudden change in magnitude of loading introduces singularity) and multiple point loads on its surface. By using these sample examples, we have demonstrated that HT-CS-X element is able to ã Predict stress resultants of shell structures accurately. Note that the stress resultants are usually poorly estimated by lower order displacement-based h-version finite elements such as ABAQUS S8R elements, as evidenced from the presence of oscillations in stress distributions. HT-CS-X elements can handle steep stress gradients in problems involving stress singularities such as those present in proximity to sudden load variations and point loads ã Allow for easy mesh design because of its triangular shape and high tolerance for shape distortion (large aspect ratios). ã Save enormous storage/memory space and computational time since it requires considerably less d.o.f. for accurate solutions. It is worth noting that one needs only a 2GB RAM desktop computer to run the triangular higher order shell element computer code. The problems 291 Nonlinear Continuum Spectral Shell Element solved herein only took less than 15 seconds for each iteration. This can be further reduced by adopting robust solution techniques and parallel computing. As a step further, the nonlinear behaviour of an orthotropic and a symmetric cross-ply (0 90 90 0 ) laminated hyperboloid shell is studied. A detailed plot of stress distributions at the top and bottom faces of the hyperboloid shell are presented at the end of initial and final load steps. The contour plots of stress distributions clearly show the redistribution of stresses in the nonlinear regime which are very useful to carry out further detailed analyses involving material nonlinearities and failure prediction. The aforesaid contour plots of stresses were obtained with a very simple mesh design of HTCS-X elements involving 964 d.o.f. Achieving smooth distributions of stresses for such a complicated shell structure by using elements with lesser d.o.fs as compared to HT-CS-X elements is certainly not possible with lower order shell elements. In view of the aforementioned advantages of this triangular higher order shell element HT-CS-X, one could apply it successfully to solve many complicated problems in structural mechanics. 292 [...]... nonlinear response in 16 equal load steps Fig 5.5 Geometry of a cylindrical shell supported on rigid diaphragms and subjected to point loads located on diametrically opposite sides 252 Nonlinear Continuum Spectral Shell Element Figure 5 .6 shows a comparison of nonlinear load versus transverse deflection at point O obtained using a 33 mesh of HT-CS-X elements, 44 mesh of NSQ9 and 161 6 mesh of HMSH5 elements... thickness of the shell is small The deformed configuration of the cylindrical shell panel at the initial and final load steps are shown in Fig 5.4 250 Nonlinear Continuum Spectral Shell Element Fig 5.2 Geometry of clamped cylindrical shell panel Fig 5.3 Nonlinear load versus deflection response at point O of a clamped cylindrical shell panel subjected to uniform surface pressure Initial undeformed configuration... allowing for the conversion of the 3D shell model into a 2D model Moreover, the strains are assumed to be small The layout of HT-CS-X is shown in Fig 5.1 with Lobatto nodal distribution in the element In Fig 5.1, r and s denote the curvilinear coordinates of the element and t denotes the coordinate in the thickness direction The global Cartesian coordinates (x, y, z) of a point on the element are defined... the initial and final load steps Fig 5 .6 Nonlinear load versus deflection response at point O of a cylindrical shell supported by rigid diaphragms and subjected to two diametrically opposite point loads 253 Nonlinear Continuum Spectral Shell Element Initial undeformed configuration Deformed final configuration Fig 5.7 Deformation of a cylindrical shell subjected to two diametrically opposite point... deflection at point O obtained using a 33 mesh of HT-CS-X elements will be compared with the following three types of finite elements: 249 Nonlinear Continuum Spectral Shell Element 1 HMSH5 which is a geometric nonlinear five noded hybrid strain element developed by Saleeb et al (1990) 2 CSH9 which is a geometric nonlinear nine noded quasi conforming shell element developed by Guan and Tang (1995)...Nonlinear Continuum Spectral Shell Element The degenerated shell element is deduced from the 3D continuum element by imposing two kinematic constraints, i.e (i) the straight line normals to the midsurface before deformation remain straight but not necessarily normal after deformation, allowing for the effect of transverse shear deformation and (ii) transverse normal strain /stress components... 161 6 mesh of HMSH5 elements The maximum value of transverse deflection (at point O) computed at the end of the last load step Pmax using HT-CS-X and HMSH5 are exactly the same The maximum transverse deflection w obtained at Pmax using HT-CS-X and HMSH5 elements differ by 6% with respect to the results obtained from NSQ9 elements Figure 5 .6 shows the undeformed and deformed configuration of the cylinder... that involve multiple limit points 255 Nonlinear Continuum Spectral Shell Element Isotropic cylindrical shell panel Figures 5.9,5.10 and 5.11 present the nonlinear load versus deflection response of an isotropic cylindrical shell panel having thicknesses h = 25.4 mm, 12.5 mm and 6. 35 mm respectively The results of maximum transverse deflection w at point O are obtained for a 33 mesh of HT-CS-X elements... (qm ) (5 .68 ) Further details regarding the stepwise implementation of the arc-length method is discussed by Reddy (2004) Having discussed about the nonlinear finite element formulation of a degenerated continuum shell element and two efficient nonlinear solution procedures that are adopted in the present work, the next section begins with the assessment of HT-CS-X elements in several nonlinear shell benchmark... shown in Fig 5.5 The convergence behaviour of HT-CS-X elements was assessed in a similar problem (see Section 4.3.1 of Chapter 4) Herein we assess the performance of HT-CS-X elements in predicting the nonlinear behaviour of the aforesaid shell under the action of point loads The cylindrical shell has a length a = 200 mm, radius R = 100 mm and thickness h = 1 mm It should be noted that ratio of thickness-to- . ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é ¢¢ ¢¢ ¢¢¢ ¢¢¢ ¢¢¢ = ¢ 5545 4544 66 261 6 262 212 161 211 )(0 000 000 00 00 00 ][ CC CC CCC CCC CCC C k (5.42) where )()( 44 2 55 2 55 44554555 2 44 2 44 66 222 122211 22 66 66 12 22 22 2 11 2 26 22 4 66 12 22 11 4 22 66 12 22 22 2 11 2 16 12 44 66 2212 22 12 22 4 66 12 22 11 4 11 sin,cos )(, )()2( )]2)((][ )2(2 )]2)(([ )()4( )2(2 kk nm QnQmC QQmnCQnQmC QnmQQQnmC QQnmQmQnmnC QmQQnmQnC QQnmQnQmmnC QnmQQQnmC QnQQnmQmC qq. the performance of HT-CS-X elements in handling stresses in nonlinear region much efficiently as compared to ABAQUS S8R lower order shell elements. The performances of HT-CS-X element and ABAQUS. several challenging benchmark problems to assess the performances of higher order elements. Based on the performances of HT-CS and HT-CS-X elements in ‘Discriminating and Revealing Test Cases’,