CHAP TER Higher Order Triangular Mindlin Plate Element The construction of successful triangular plate bending elements posed difficulties due to the requirement of inter-element continuity of normal slopes (Melosh, 1961; Irons and Draper, 1965). It has been observed that the formulations of C1 and C2 continuity plate bending elements based on the classical thin (Kirchhoff) plate theory led to either incompatible elements or they involved complicated formulation and programming. In the last few decades, several attempts have been made to develop simple and efficient plate bending elements using displacement models satisfying only C0 continuity requirement (Hinton and Pugh, 1977; Reddy, 1980). These models are based on the first-order shear deformation plate theory, which incorporates the effect of transverse shear deformation (Mindlin, 1951). The performance of these elements in representing stress resultants has been good enough for some of the common plate problems involving simply supported and clamped edges. But when the models are applied to plates with free edges, these elements fail to predict the stress resultants accurately. The reason for the failure of these displacement finite elements can be attributed to the use of lower-order displacement field that is inadequate for predicting the variation 15 Higher Order Triangular Mindlin Plate Element of stress resultants which are defined by higher order derivatives of the displacement field. The quest for a more robust finite element (that has the ability to predict stress resultants accurately and is free of numerical problems) prompted researchers to develop higher order finite elements, both as separate elements or in the so-called framework of the p-version of the finite elements (Babuska et al., 1981; Croce and Scapolla, 1992). A list of research works pertaining to the development and assessment of the p version finite element method has been presented in Chapter 1. Hence, we proceed to outline research studies pertaining to the development of higher order plate bending elements. Peano (1976) proposed new families of C0 and C1 interpolations over triangles which were complete up to any polynomial of degree p. A family of higher order sub-parametric quadrilateral bending elements with up to 25 nodes was developed by Cheung et al. (1980). Wang et al. (1984) formulated a family of triangular finite elements of degree p having C1 continuity and analyzed simply supported square and equilateral triangular plates subjected to a central point load and uniformly distributed load. Chan et al. (1986) presented the large deflection analysis of plates having irregular shapes such as skewed, trapezoidal and curved plates that were modeled by Cheung (1980). Rank et al. (1988) studied the accuracy of using p-version finite elements in predicting bending moments and shear forces of simply supported circular and rhombic plates, which were known to exhibit oscillations in shear force very near to the boundary even when polynomial degrees of and are used. A high precision shear deformable element for the analysis of laminated composite plates of different shapes was developed by Sheikh et al. (2002). In 16 Higher Order Triangular Mindlin Plate Element this element, a complete fourth-order polynomial was used to express the transverse displacement w while the in-plane displacements (u and v) and bending rotations were expressed as cubic polynomials. Xenophontos et al. (2003) studied Reissner-Mindlin plates with curved boundaries using a pversion MITC finite element method. They developed p-MITC quadrilateral elements to obtain the shear force variations in circular, clamped and simply supported plates. Pontaza and Reddy (2004, 2005) used least-squares formulation to develop plate and shell elements, where higher-order interpolation of the field variables was employed. Houmat (2005) applied the hp version of the finite element method to study the vibration of membranes using a polynomially enriched triangular element. Ribeiro (2006) studied the large amplitude, geometrically non- linear periodic vibrations of shear deformable composite laminated plates using a p- version, hierarchical finite element. Reddy and Arciniega (2006) and Arciniega and Reddy (2007) studied the bending and buckling of composite and functionally graded plates and shells under mechanical and thermal loading using shear deformable, quadrilateral C0 continuity elements having higher-order interpolation functions. The degrees of interpolation functions that were used for representing the field variables were varied from p = to p = 8, which resulted in quadrilateral elements having 25 and 81 nodes respectively (Q25 and Q81). These elements were shown to be free of shear as well as membrane locking. The buckling problem of ceramic- metal plates with simply supported edges was studied using two shear deformation theories, namely FSDT (first-order shear deformation theory) and TSDT (third-order shear deformation theory). This 17 Higher Order Triangular Mindlin Plate Element resulted in the elements having 405 and 567 degrees of freedom corresponding to FSDT and TSDT for the Q81 element. The aforementioned literature survey indicates that most researchers have validated their displacement-based plate and shell finite elements by considering plates with simply supported and clamped edges. The ability of the finite element in handling the more challenging free edge boundary condition has received little attention. At the free edge, the stress resultants should be zero. But a few studies have pointed out the variation of stress resultants in the vicinity of free edge. The aforementioned statement is illustrated in Figs. 2.1a and 2.1b which show the variations of twisting moment and transverse shear force for a corner supported isotropic, square, thick steel plate under uniformly distributed load. The plate problem was analysed using a 20ì20 mesh (ie. 400 elements) of 8-node serendipity element. It is evident from Fig. 2.1 that the values of transverse shear force and twisting moment show a marked deviation from the zero value at the free edge. Hence, conventional, displacement-based plate finite elements with low order interpolation are deficient in predicting the values of stress resultants, particularly when the plate has free edges. In this chapter, we formulate a higher-order, displacement-based, triangular plate element that has the capability to predict stress resultants accurately. By higher order, we refer to the degree of polynomial basis that is employed to derive the shape/interpolation functions associated with the field variables. The choice of a triangular shape renders greater versatility in accommodating plate shapes with angular corners and arbitrary shapes, when compared to rectangular elements. The triangular plate element is based on the well known 18 Higher Order Triangular Mindlin Plate Element Mindlin plate model which is a first-order shear deformation plate theory and it considers the displacement field as linear variations of midsurface transverse displacements. The accuracy and validity of the proposed element will be established by conducting convergence and comparison studies on the displacements and stress resultants for a variety of boundary conditions. In order to enhance the performance of plate element in predicting distributions of stress resultants accurately, the variation of field variable (generalized displacements) is represented by higher degree polynomial basis functions. We shall first present the finite element layout and derivation of shape functions for arbitrary degree p followed by a brief description on the formulation of finite element matrices based on the Mindlin plate theory. Next, the optimal value of p will be determined based on the performance of various finite element schemes in a set of examples that involve comparison of stress resultants. (a) 19 Higher Order Triangular Mindlin Plate Element (b) Fig. 2.1 Distribution of stress resultants for a corner supported, square plate obtained using ABAQUS S8R elements having 7437 d.o.f. (a) Normalized twisting moment M xy and (b) Normalized transverse shear force Q x 2.1 Finite element layout and derivation of s hape functions In the development of the triangular higher-order element, we adopt the nodal basis formulation. Its main advantage over the modal/hierarchical basis formulation is that the degrees of freedom (d.o.f) are associated with the value of solution at a specific location within the element. This feature enables a straightforward interpretation and visualization of the computed results in the vicinity of regions having high stress gradients. We consider a master isosceles, right angle, triangular element having a degree of polynomial p of the basis function. The polynomial basis function will be used to derive shape functions that define the variation of field variables (displacement, stresses etc.) inside a finite element. For a given degree p, the number of geometric nodes comprises vertex nodes, ( p - 1) 20 Higher Order Triangular Mindlin Plate Element nodes along each edge, and ( p - 1)( p - ) / bubble nodes in the interior of the triangular element. Table 2.1 gives the number and type of shape functions associated with a given degree of polynomial p. The polynomial basis functions can comprise of any set of complete polynomials such as homogenous polynomial expansions (whose terms are monomials all having the same total degree), orthogonal polynomials such as Legendre, Jacobi, Appel and Proriol polynomials (Pozrikidis, 2005). Although, the use of orthogonal polynomial expansions ensure a well conditioned nature of global stiffness matrix, the choice of polynomials have marginal influence on the accuracy of solution, especially, when the structure to be analyzed is linear and elastic. Herein, we adopt a simple basis function comprising of a complete polynomial of degree p, whose individual terms are monomials defined in terms of area coordinates L1 , L2 and L3 (Zienkiewicz, 1967). For instance, the polynomial basis for degree p = 1, consists of the following three terms of a complete linear polynomial a L1 + a L2 + a L3 (2.1) For p = 2, the polynomial basis consists of a L1 L2 + a L2 L3 + a L3 L1 + a L12 + a L22 + a L23 (2.2) Note that for a degree p, the total number of nodes corresponds to the number of terms contained in the complete polynomial expression. Thus a polynomial basis of degree p is formed by all possible pth order combinations of area coordinates. 21 Higher Order Triangular Mindlin Plate Element Table 2.1 Number of shape functions for degree of polynomials p Degree of polynomial p Vertex shape functions - Inte rior shape functions - Total numbe r of shape functions - 10 15 12 21 15 10 28 3 18 21 15 21 36 45 Edge shape functions Having established the complete polynomial basis, the shape functions can be derived as follows. Let y (r, s ) denotes the variation of any field quantity (say displacements, stresses) and is given as y (r , s ) = f1 f K f N -1 f N ỡ c1 ỹ ùc ù ùù ùù M ý = f {c} ùc ù ù N -1 ù ùợ c N ùỵ (2.3) f1 , f , K , f N denote the individual terms of the polynomial basis and c1 , c , ., c N -1 , c N denote the unknown coefficients. In terms of the shape functions, y (r, s ) can be expressed as y (r , s ) = Q1 Q K Q N -1 QN ỡ y1 ỹ ùy ù ùù ùù M ý = Q {y } ùy ù ù N -1 ù ợù y N ỵù (2.4) 22 Higher Order Triangular Mindlin Plate Element where Q i denote the shape functions and y i denote the value of field quantity at node i which is to be solved. We construct the Vandermonde matrix by substituting the nodal coordinates into the individual terms of the polynomial basis, i.e. f1 (r2 , s ) ộ f1 (r1 , s1 ) f (r , s ) f (r2 , s ) 1 Vf = K K ờf N -1 (r1 , s1 ) f N -1 (r2 , s ) ờở f N (r1 , s1 ) f N (r2 , s ) f1 (rN , s N ) f (rN , s N ) K K K K ự ỳ ỳ ỳ ỳ K f N -1 (rN , s N )ỳ K f N (rN , s N ) ỳỷ T (2.5) In order to determine the unknown coefficients ci , we invoke the cardinal interpolation condition (which states that the value of the shape function Q i at node i is unity whereas for the remaining nodes, the value is zero) [V ] { c } = {y } ị { c } = [V ] {y } -1 f f (2.6) By substituting Eq. (2.6) into Eq. (2.3), one obtains y (r , s ) = f {c } = f [V ] {y } -1 f (2.7) The shape functions [Q] are given by [Q] = [ ] f Vf -1 . (2.8) Based on the shape functions given in Eq. (2.8) which dictate the variation of displacement/stresses inside a finite element, one can develop a family of finite elements (say for example 2D plate elements and degenerated shell elements) which can be tailored to analyse any complex structure. Figure 2.2 shows 3D plot of some edge and interior (bubble) shape functions for various p values. 23 Higher Order Triangular Mindlin Plate Element p Edge shape functions Bubble shape functions Fig. 2.2 Plot of some edge and internal shape functions for various degrees of polynomial basis p 2.2 Mindlin plate theory In the Mindlin plate theory (MPT), also commonly referred to as the first order shear deformation plate theory, the Kirchhoff hypothesis is relaxed by assuming that the transverse normals not necessarily remain perpendicular to the midsurface after deformation. The inextensibility of transverse normals requires that w (transverse deflection) not be a function of the thickness coordinate, z. The displacement field of MPT is given by u ( x , y , z ) = u ( x , y ) + zf x ( x , y ) v ( x , y , z ) = v ( x , y ) + zf y ( x , y ) w( x, y, z ) = w0 ( x, y ) (2.9) 24 Higher Order Triangular Mindlin Plate Element ABAQUS S8R elements. It can be observed from Fig. 2.20 that Mesh A of ABAQUS S8R elements having 5310 d.o.f. completely fails to represent the behavior in the neighborhood of reentrant corner. As the mesh is progressively refined, the bending moment M x tend to get localized and the strength of singularity at the corner point is represented accurately. A simple mesh design of HT-M45 elements having nearly 17 times lesser d.o.fs as compared to ABAQUS S8R elements could furnish excellent distributions of bending moments in the vicinity of reentrant corner. The singularity problem at the corner point is taken care of automatically by the higher order element due to the presence of an adequate number of side nodes and the higher order shape functions. Hence, specially tailored mesh designs and complex singularity functions for modeling stress singularities at obtuse corners for plates are not necessary. This example also brings out the ability of HT-M45 elements in accommodating large aspect ratios due to the presence of geometric nodes distributed along the sides and the interior of the triangular domain The contour plots of bending moments and twisting moment obtained using Mesh D of HT-M45 show very smooth distributions of stress resultants even in the vicinity of the re-entrant corner (see Figs. 2.21 a-c). The singular behavior in moments at the re-entrant corner is found to be localized to a small region and does not influence the values of stress resultants elsewhere in the plate. 81 Higher Order Triangular Mindlin Plate Element (a) (b) Fig. 2.19 Variation of normalized bending moments at various sections of a simply supported square plate with a central square cutout 82 Higher Order Triangular Mindlin Plate Element Fig. 2.20 Enlarged view of normalized bending moment M x obtained using various mesh designs of ABAQUS and HT-M45 elements in the vicinity of reentrant corner O (see Fig. 2.7e). The bracketed values indicate the number of d.o.f. (a) (b) (c) Fig. 2.21 Contour plots of normalized stress resultants shown for one quadrant of a simply supported square plate with a square cutout (a) Bending moment M x qa (b) Bending moment M y qa (c) Twisting moment M xy qa 83 Higher Order Triangular Mindlin Plate Element Having established the superior performance of HT-M45 elements in various problems concerning isotropic plates, we shall now assess their ability in predicting interlaminar stresses in laminated composite plates. 2.3.9 Simply supported, symmetric cross-ply, laminated composite square plate Consider a symmetric (0 90 90 0 ) cross-ply laminated square plate of side length a and total thickness h. The material properties for the plate are E1 = 25E , E = E3 , G12 = G13 = 0.5 E , G23 = 0.2 E , and n 12 = n 23 = n 13 = 0.25 . Subscript denotes the direction parallel to the fibers and subscript the transverse direction. The shear correction factor is assumed to be 5/6. All lamina have equal thickness. The plate is subjected to a transverse sinusoidal load q( x, y ) = q sin (p x a ) sin (p y a ) . The problem at hand is to determine the deflections and stresses for this loaded plate. The deflection and stress values will be presented at the following pertinent locations. ã For transverse deflection - w is evaluated at (0.5 a,0.5 a ) ã For in-plane normal bending stress - s xx - is evaluated at (0.5 a,0.5 a,0.5 h ) ã For in-plane normal bending stress - s yy - is evaluated at (0.5 a,0.5 a,0.25 h ) ã For in-plane normal shear stress - s xy - is evaluated at (0,0,0.5 h ) ã For transverse shear stress - s xz - is evaluated at (0,0.5 a,0 ) ã For transverse shear stress - s yz - is evaluated at (0.5 a,0,0 ) 84 Higher Order Triangular Mindlin Plate Element The results for deflections and stresses will be presented in a nondimensional form defined by w E2 h w= ; q0 a s xx s xy h s h s xx h s xx h = ; s yy = ; s xy = ; s xz = xz ; 2 q0 a q0 a q0 a q0 a s yz = s yz h q0 a where h denotes the thickness of the laminated plate, and q denotes the maximum load value and a denotes the length of the square plate. The deflections and stresses obtained from HT-M45 elements for laminated plates with various thickness to length ratios h/a = 0.25, 0.1, 0.05, 0.01 are compared with the 3D elasticity solution of Pagano (referred to as ELS), exact Navier solutions based on the third-order shear deformation theory (TSDT), exact Navier solutions based on the first-order shear deformation theory (FSDT) and exact Navier solutions based on the Classical laminate plate theory (CLPT). Table 2.14 shows the convergence study for various quadrant mesh sizes 44, 55, 66, 77 and 88. Monotonic convergence of the results to the exact solution is observed for all cases. It can be seen that even 44 quadrant mesh size can furnish results that are very close to the exact Navier solutions. It is evident from Table. 2.15 that the maximum difference in the results for transverse shear stress s xz and computed using HT-M45 and Paganos 3D elasticity solutions is more than 20% when thickness-to- length ratio h/a = 0.25. Hence the present element cannot be used to model thick plates. As the plate becomes rather thin, say h/a = 0.01, the HT-M45 results converge to 85 Higher Order Triangular Mindlin Plate Element CLPT solutions without any shear locking problem. This shows the ability of the present element in predicting deflections and stresses of thin plates accurately. The results of transverse shear stresses computed using MPT agree very well with 3D elasticity solutions for h/a = 0.05 and 0.01. Table 2.15 Comparison of non-dimensionalised maximum deflections and stresses for simply supported (SS1) symmetric cross ply (0 90 90 0 ) square plates subjected to transverse sinusoidal load h/a M esh Size 44 66 0.25 HT-M45 55 s xx s s xy s xz s yz 1.7102 0.406 0.577 0.0308 0.138 0.264Đ 0.187 0.295Đ 1.7102 0.406 0.577 0.0308 0.139 0.266Đ 0.190 0.290Đ 1.7102 0.406 0.577 0.0308 0.139 0.192 yy 0.267 Exact ELS Exact TSDT Exact FSDT 44 66 HT-M45 55 0.288Đ 0.406 0.577 0.0308 0.139 0.267Đ 0.193 0.286Đ 1.7102 0.406 0.577 0.0308 0.139 0.268Đ 0.194 0.285Đ 1.954 0.720 0.663 0.0467 0.219 0.292 1.894 0.665 0.632 0.044 0.206 0.231Đ 0.239 0.298Đ 1.710 0.406 0.576 0.0308 0.140 0.269Đ 0.196 0.280Đ 0.663 0.499 0.362 0.0241 0.165 0.314Đ 0.120 0.195Đ 0.663 0.499 0.362 0.0241 0.165 0.315Đ 0.123 0.191Đ 0.663 0.499 0.361 0.0241 0.166 0.316Đ 0.125 0.188Đ 0.663 0.499 0.361 0.0241 0.166 0.317Đ 0.126 0.187Đ 0.663 0.499 0.361 0.0241 0.166 0.127 77 88 Đ 1.7102 77 88 0.2 w 86 Higher Order Triangular Mindlin Plate Element Exact ELS Exact TSDT Exact FSDT 44 66 HT-M45 55 77 ELS Exact TSDT Exact FSDT 44 55 66 HT-M45 0.01 Exact 0.559 0.401 0.0275 0.301 0.196 0.715 0.546 0.389 0.0268 0.307 0.192 0.264Đ 0.153Đ 0.663 0.499 0.361 0.0241 0.318 0.167Đ 0.181 0.130Đ 0.491 0.528 0.296 0.0221 0.173 0.099 0.329Đ 0.165Đ 0.491 0.528 0.296 0.0221 0.174 0.331Đ 0.103 0.161Đ 0.491 0.528 0.296 0.0221 0.174 0.331Đ 0.105 0.158Đ 0.491 0.528 0.296 0.0221 0.174 0.106 0.332Đ 0.156Đ 0.491 0.528 0.296 0.0221 0.174 0.332Đ 0.106 0.155Đ 0.517 0.543 0.308 0.0230 0.328 0.156 0.506 0.539 0.304 0.0228 0.330 0.154 0.282Đ 0.123Đ 0.491 0.527 0.296 0.0221 0.175 0.333Đ 0.109 0.150Đ 0.434 0.538 0.271 0.0213 0.176 0.338Đ 0.092 0.153Đ 0.434 0.538 0.271 0.0213 0.177 0.339Đ 0.095 0.149Đ 0.434 0.538 0.27 0.0213 0.177 0.339Đ 0.097 0.146Đ 0.434 0.538 0.27 0.0213 0.178 0.339Đ 0.098 0.144Đ 0.434 0.538 0.27 0.0213 0.178 0.339Đ 0.099 0.143Đ 0.438 0.539 0.276 0.0216 0.337 0.141 0.434 0.539 0.271 0.0213 0.290 0.112 0.339Đ 0.139Đ 77 88 Exact ELS Exact TDST 0.185Đ 0.743 0.05 88 0.317Đ 87 Higher Order Triangular Mindlin Plate Element Exact FSDT Exact CLPT 0.434 0.538 0.27 0.0213 0.178 0.339Đ 0.101 0.139Đ 0.431 0.539 0.269 0.0213 0.339 0.138 Note: Đ denotes the values of transverse shear stresses obtained from equilibrium equations 2.3.10 Antisymmetric cross-ply square plates with different support conditions Next, we shall verify the performance of HT-M45 elements for antisymmetric cross-ply plates having various support conditions. The material properties are considered to be the same as defined in the previous example. The thicknessto-length ratios considered are: h/a=0.2,0.1. We will consider two antisymmetric lamination schemes namely, (0 90 ) and (0 ) 90 K 10 layers. Owing to symmetry, we consider half plate model shown in Fig.2.22 (i.e. shaded portion of the plate) for the analyses of SC, FS and FC boundary conditions. The notations SC, FS and FC are explained below. Referring to Fig. 2.22a, ã SC denotes simply supported edge along x = and clamped edge along x=a ã FS denotes free edge along x = and simply supported edge along x = a ã FC denotes free edge along x = and clamped edge along x = a The edges y = -a/2 and y = a/2 are simply supported. The plate is subjected to transverse sinusoidal load described as q( x, y ) = q sin (p x a ) cos (p y a ) . The deflections and stresses for half plate model are computed at the following locations: 88 Higher Order Triangular Mindlin Plate Element ã w is evaluated at (a/2,0) ã s xx and s yy are evaluated at (a 2,0, - h ) and (a 2,0, h ) respectively ã s yz is evaluated at (a 2, a ,0 ) We use quarter plate models (shaded portion) for laminated plates having CC and FF boundary conditions. Referring to Fig. 2.22b, we have CC denoting clamped boundary edges along x = - a/2 and x = a/2, and FF denoting free edges along x = - a/2 and x = a/2. The plate is subjected to transverse sinusoidal load described as q( x, y ) = q cos (p x a ) cos (p y a ) . The deflections and stresses for quarter plate model are computed at the following locations: ã w is evaluated at (0,0) ã s xx and s yy are evaluated at (0,0, - h ) and (0,0, h 2) respectively ã s yz is evaluated at (0, a ,0 ) The results for deflection and stresses are shown in nondimensional form as follows: w= w E2 h 10 ; q0 a s yz = s yz h q0 a s xx = s xx h s xx h 10 s = 10 ; ; yy q0 a q0 a 10 The results obtained for this example case are compared with exact Levy or Navier solutions Table 2.16 contains the results of deflection and stresses at critical locations obtained by different methods for antisymmetric plates having various boundary conditions. 89 Higher Order Triangular Mindlin Plate Element (a) (b) Fig. 2.22 (a) Half-plate model of a square antisymmetric cross-ply plate having various boundary conditions (b) Quarter plate model of a square antisymmetric cross-ply plate simply supported along edges and having the other two edges as clamped or free It can be seen that for the same thickness-to- length ratio of the laminated plate, as the number of layers k is increased, the solutions obtained using HTM45 are close to exact FSDT solution for most of the boundary conditions having a maximum relative difference of about 1.5%. The results of HT-M45 are seen to have a maximum difference of 18% for h/a=0.2 and k = 10. For laminated plates simply supported on two opposite sides and having free edges on the other two side (FF), when k =2 and thickness-to- length ratio h/a=0.2 and 0.1, the results obtained for transverse shear stress by HT-M45 and exact solution based on TSDT differ by a maximum value of 20%. The results obtained agree very well with exact FSDT solutions for all the boundary conditions considered. It can be noted from Tables 2.16 and 2.17 that results of in-plane normal stresses obtained from HT-M45 agree well with exact TSDT solutions even when h/a = 0.2 and 0.1. The results for transverse shear stresses are particularly seen to show a marked difference with respect to exact 90 Higher Order Triangular Mindlin Plate Element TSDT solutions for boundary conditions and h/a ratios considered herein. This clearly point out the need for resorting to a refined or a higher order shear deformable laminated theory that will be able to provide accurate results of interlaminar stresses especially the transverse shear stresses for thick and thin plates. Table 2.16 Comparison of non-dimensionalized deflections and stresses of antisymmetric cross ply square laminated plate simply supported on two opposite sides and having the following boundary conditions on the other two sides k h/a CC SC w s xx s w s xx s HT-M 45 1.449 5.339 6.035 2.287 1.257 3.911 5.154 1.936 Exact TSDT Exact FSDT 1.333 1.477 6.816 5.338 6.725 6.034 2.543 2.297 1.088 1.257 5.679 3.911 5.505 5.153 2.095 1.958 HT-M 45 0.866 5.495 5.109 1.983 0.657 4.451 3.8 1.500 Exact TSDT 0.848 5.91 5.219 2.29 0.617 4.952 3.803 1.725 Exact FSDT 0.883 5.494 5.109 1.993 0.656 4.45 3.799 1.523 HT-M 45 1.025 3.708 4.629 2.490 0.945 2.275 4.213 2.225 Exact TSDT Exact FSDT 1.001 1.045 5.196 3.707 5.635 4.628 2.974 2.498 0.879 0.945 4.025 2.275 4.963 4.212 2.601 2.248 HT-M 45 0.471 3.643 3.905 2.117 0.385 2.692 3.136 1.685 Exact TSDT Exact FSDT 0.473 0.480 4.066 3.642 4.11 3.904 2.622 2.126 0.375 0.385 3.193 2.692 3.803 3.135 2.083 1.708 yy s yz yy s yz 0.2 0.1 0.2 10 0.1 k Number of lamination layers. ; k = for (0 90 ) lamination scheme; k = 10 for (0 90 K) 10 layers 91 Higher Order Triangular Mindlin Plate Element Table 2.17 Comparison of non-dimensionalized deflections and stresses of antisymmetric cross ply square laminated plate simply supported on two opposite sides and having the following boundary conditions on the other two sides(continued) k h/a FF w s xx s FS yy s yz w s xx s FC yy s yz w s xx s yy s yz HT-M45 2.779 2.467 11.913 3.8801 2.335 4.432 9.849 3.382 1.898 2.436 8.048 2.739 Exact TSDT Exact FSDT 2.624 2.777 3.171 13.551 2.469 11.907 4.457 3.901 2.211 2.335 5.349 11.32 4.43 9.848 3.893 3.39 1.733 1.897 3.727 2.434 8.919 8.047 3.048 2.748 HT-M45 2.03 2.44 11.89 3.8614 1.688 4.438 9.849 3.374 1.223 2.792 7.151 2.44 Exact TSDT Exact FSDT 1.992 2.028 2.624 12.295 2.442 11.884 4.489 3.882 1.658 1.687 4.669 10.22 4.435 9.847 3.927 3.383 1.184 1.223 3.158 2.79 7.314 7.15 2.805 2.449 HT-M45 1.664 1.71 7.5859 3.8628 1.46 2.959 6.591 3.429 1.258 1.344 5.706 2.941 Exact TSDT Exact FSDT 1.651 1.663 2.482 1.712 9.454 7.583 4.784 3.883 1.45 1.46 3.946 8.252 2.957 6.59 4.234 3.437 1.214 1.258 2.608 1.343 6.934 5.706 3.535 2.951 HT-M45 9.154 1.721 7.5365 3.8323 0.8 2.969 6.567 3.413 0.612 1.595 5.03 2.596 Exact TSDT Exact FSDT 0.916 0.915 1.924 1.723 0.801 0.8 3.221 6.987 2.968 6.566 4.275 3.421 0.607 0.612 1.954 1.594 5.299 5.029 3.225 2.605 0.2 0.1 0.2 10 0.1 8.005 7.533 4.814 3.853 92 Higher Order Triangular Mindlin Plate Element 2.4 Conclusions In this chapter a higher-order triangular plate element based on the Mindlin plate theory is developed to accurately determine the stress resultants. Two displacement based plate finite elements having degree of polynomial p=5 and p=8 were developed. The performances of the two elements in achieving accuracy of shear force and twisting moment along the free edges of isotropic plates were discussed. Based on the results, it was established that p=8 basis perform very well even for challenging problems with much less d.o.f as compared to p=5 and other lower order finite elements. Conventional finite plate/shell elements used in commercial software are based on lower-order polynomial expansions and they were shown to give inaccurate distributions of the stress resultants, although they yield satisfactory results for deflections. The reason is that stress resultants involve higher-order derivatives of the displacements. Therefore, in displacement formulations using conventional lower-order approximations, the accuracy of stress resultants is not good as compared to that obtained from higher-order approximations of the displacements. The higher-order elements presented herein predict very accurate distributions of stress resultants even for plates with point supports and also for rhombic plates. The relatively accurate prediction of stress resultants with lesser d.o.f and lesser computation time ( ằ 2.5 on a computer having a processor speed of 2.4 GHz inclusive of post processing computations) as compared to ABAQUS make the 45- node higher-order triangular finite element superior to conventional lower-order finite elements available in commercial programs. The astonishing result of this study is that the present higher order element is 93 Higher Order Triangular Mindlin Plate Element nearly 25 times more efficient than ABAQUS S8R elements with respect to the d.o.f used to achieve a specified degree of accuracy. However, the comparison of efficiency in terms of d.o.fs/computation time may not be fair because the present higher order elements and lower order elements in ABAQUS have been implemented in different software configurations. For example, we usually specify the points/edges in a structure that are to be finely meshed and let ABAQUS software the automated meshing for the problem which may not be as efficient as manual meshing. On the other hand, since the higher order elements were programmed using MATLAB, we could allow for a more flexible meshing. However the results reported herein indicate that one can definitely achieve very good accuracy with fewer higher order elements than a large number of lower order elements. This study has also brought out the pitfalls associated with the case of skew plates modeled using lower order finite elements. The present higher-order triangular element HT-M45 performs well even for skew plates with b = 750 and conforms well to the bending results obtained from the finest adaptive mesh of ABAQUS. HT-M45 elements exhibit a faster rate of convergence. Solutions within a reasonable accuracy of 5% can be obtained even for coarser meshes. This makes the preparation of input data for the problem very easy as compared to a finer mesh involving more number of lower order finite elements. The stress resultant study affirms the excellent behaviour of HT-M45 elements against conventional lower-order elements. Although the use of higher-order approximations results in greater number of d.o.f per node, the computational cost can be compromised with the level of accuracy obtained 94 Higher Order Triangular Mindlin Plate Element by modeling the structure with fewer higher-order elements compared to the conventional elements. The investigation of L-shaped plate with exemplifies the extent to which the region prone to singularity in stress resultants is being tackled. Although a number of research papers deal with new finite element formulations for plates, none have presented the actual variations and accuracy of stress resultants especially twisting moments and shear forces in the presence of free edges and singularities. Plates with point supports are known to exhibit severe stress concentrations at the supports and their prediction by commercial finite element codes (e.g., ANSYS and ABAQUS) contain oscillations. The aforementioned plate problems were selected as they have free edges and the results presented herein may serve as benchmark solutions for assessing the performance of other plate bending elements in providing accurate stress distributions. Furthermore, the triangular shape employed in formulating the element enables, relatively easier than the quadrilateral elements, in accurately capturing the structural behavior near sharp corners. This has been demonstrated by studying the distributions of stress resultants in a triangular isosceles triangular plate. Although HT-M45 elements show excellent performance in isotropic and laminated composite plates having thickness-to-length ratio (h / a ) < 0.1 , they fail to model interlaminar transverse shear stresses of thick plates accurately. Although the in-plane normal and shear stresses are reasonably accurate for moderately thick plates, the results of interlaminar transverse shear stresses are erroneous. Hence it is important to represent the variation of displacement field across the thickness of lamina by a higher order polynomial. Such a modification in displacement field together with an enriched set of higher 95 Higher Order Triangular Mindlin Plate Element order shape functions can efficiently tackle interlaminar stresses for laminated composite plates encompassing a wide range of h/a ratios (ranging from 0.2 to 0.001). In view of capturing the interlaminar stresses in laminated composite plates accurately, the present version of HT-M45 will be further enhanced to incorporate higher order variations of through thickness displacements. This is accomplished by employing a third order shear deformation theory of Reddy (TST-R) and a layerwise plate theory of Reddy (LT-R) which will be discussed in the ensuing chapter. 96 [...]... 2 q cos 2 q + Q 12 sin 4 q + cos 4 q ) Q 22 = Q11 sin 4 q + 2( Q 12 + 2Q66 )sin 2 q cos 2 q + Q 22 cos 4 q Q 16 = (Q11 - Q 12 - 2Q66 )sin q cos 3 q + (Q 12 - Q 22 + 2Q66 ) cos q sin 3 q Q 26 = (Q11 - Q 12 - 2Q66 ) cos q sin 3 q + (Q 12 - Q 22 + 2Q66 )sin q cos 3 q ( Q 66 = (Q11 + Q 22 - 2Q 12 - 2Q66 )sin 2 q cos 2 q + Q66 sin 4 q + cos 4 q ) Q 44 =Q 44 cos 2 q +Q 55 sin 2 q Q 45 = (Q 55 -Q 44 ) cos q sin q Q 55... basis for p = 8 is given by 8 ộ L1 ờ 6 2 ờ L1 L2 [S 45 ] = ờ L5 L3 ờ 5 22 3 ờ L1 L2 L3 ờ L4 L3 L ở 1 2 3 L8 2 2 6 L1 L2 L8 3 6 2 L2 L3 7 L1 L2 L6 L2 3 2 L7 L1 2 6 2 L3 L1 L7 L3 2 6 2 L1 L3 L7 L2 3 5 3 L1 L2 L7 L1 3 5 3 L2 L1 3 L5 L1 3 5 L1 L2 L2 3 5 L1 L3 3 2 L5 L1 L3 2 4 L1 L4 2 L5 L2 L1 2 3 L4 L4 2 3 2 L5 L1 L2 3 4 L4 L1 3 L5 L2 L1 3 2 6 L1 L2 L3 4 L1 L2 L2 2 3 L6 L1L3 2 2 L4 L1 L2 2 3 4 L1 L3 L2 3... 39 Higher Order Triangular Mindlin Plate Element The polynomial basis function for p = 5 is given as 5 ộ L1 ờ 4 [S 21 ] = ờ L3 L1 3 ờ L1 L2 ở 3 L5 2 4 L1 L 3 2 L1 L2 L3 2 L5 3 3 L1 L2 2 2 2 L3 L2 L1 4 L1 L 2 2 L1 L2 3 2 2 L1 L3 L2 L4 L1 2 L3 L2 2 3 3 L1 L2 L3 L4 L 3 2 L2 L3 2 3 L1L3 L3 2 L4 L 2 ự 3 2 ỳ L3 L1 ỳ 3 L1L2 L3 ỳ 3ỷ (2. 29) The polynomial basis function [S 21 ] with p = 5 contains 21 terms and. .. the individual terms within the matrix (ả 2 Q ảR 2 ) can be expressed in terms of area coordinates as: Q rr = Q rs = ả 2Q ả 2Q ả 2Q -2 + 2 2 ảL1 ảL1 ảL2 ảL2 ả 2Q ả 2Q ả 2Q ả 2Q + = Q sr 2 ảL1 ảL1 ảL2 ảL1 ảL3 ảL2 ảL3 Q ss = ả 2Q ả 2Q ả 2Q -2 + 2 2 ảL1 ảL1 ảL3 ảL3 (2. 41 a) (2. 41 b) (2. 41 c) where L1 , L2 , L3 denote the area coordinates of a triangular element Having obtained the matrix given by Eq (2. 35),... resulting finite element has 21 geometric nodes inside its domain In the subsequent sections of this chapter, we shall examine the performances of finite elements having p = 5 and p = 8 in selected plate bending problems The layouts of the two finite elements are shown in Fig 2. 6 11 17 16 18 10 12 19 13 14 20 9 20 15 35 36 34 37 44 33 38 45 43 32 39 40 41 42 31 25 26 27 28 29 21 14 21 19 16 17 15 1 8 2. .. 43 Higher Order Triangular Mindlin Plate Element ổ ả 2Q ả 2Q ả2x 2 y ử T = J -1 ỗ 2 - Q x 2 - Q y 2 ữ J -1 ỗ ảR ảX 2 ảR ảR ữ ố ứ (2. 39) Equation (2. 39) indicates the influence element geometry in (ả 2 Q ảX 2 ) term which could only be neglected for slightly distorted elements Since we use a regular mesh, Eq (2. 39) reduces to ổ ả 2Q ử T ả 2Q = J -1 ỗ 2 ữ J -1 ỗ ảR ữ ảX 2 ố ứ (2. 40) For a triangular element, ... 2 (a) 23 7 18 4 3 22 24 5 6 1 2 3 13 12 11 10 4 5 6 30 7 8 9 (b) Fig 2. 6 (a) 21 -node element (b) 45-node element 2. 2.3 Derivation of transverse shear stresses from equilibrium equations Before the presentation of results obtained by using the developed higherorder triangular plate elements, it is necessary to explain how we intend to determine the transverse shear stresses accurately for laminated... L1 2 3 3 L4 L1 L3 2 L4 L3 L1 3 2 3 L4 L1 L2 3 3 L1 L3 L2 2 3 2 L1 L3 L3 2 3 7 L1 L3 ự 3 5 ỳ L2 L3 ỳ L6 L1L2 ỳ 3 ỳ 2 L4 L1 L2 ỳ 3 2 3 L2 L1 L3 ỳ 2 3ỷ (2. 28) [S 45 ] denotes the complete 8th order polynomial with 45 terms Note that p = 5 also gives reasonably good results in the aforementioned examples p = 8 contains nodes of p = 5 and hence it can be said that the location of nodal points inside a finite... ả2x ả 2 x ờ ảr 2 =ờ 2 ảR 2 ờ ả x ờ ảr ảs ở ả2x ự ỳ ảs ảr ỳ ộ xrr = ả 2 x ỳ ờ xsr ở ảs 2 ỳ ỷ xrs ự xss ỳ ỷ ; ộ 2 y ờ ả 2 y ờ ảr 2 = 2 ảR 2 ờ ả y ờ ảr ảs ở 2 y ự ỳ ảs ảr ỳ ộ y rr =ờ ả 2 y ỳ ở y sr ỳ ảs 2 ỷ y rs ự y ss ỳ ỷ (2. 37) Therefore we have, ả 2Q ả 2Q T ả2x 2 y =J J + Qx 2 + Q y 2 ảR 2 ảX 2 ảR ảR (2. 38) The second derivatives of shape functions N with respect to Cartesian coordinates can be resolved... of selecting this example is to demonstrate the capability of various finite elements schemes in tackling stress resultants especially transverse shear force distribution in the vicinity of the free edge of the plate Such problems are 35 Higher Order Triangular Mindlin Plate Element typical of Very Large Floating Structures (Wang et al., 20 08) where the edges are free and the accurate computation of . +++= ( ) ( ) qqqq 44 12 22 6 622 11 12 cossincossin4 ++-+= QQQQQ ( ) qqqq 4 22 22 66 12 4 11 22 coscossin22sin QQQQQ +++= ( ) ( ) qqqq 3 6 622 12 3 66 121 1 16 sincos2cossin2 QQQQQQQ +-+ = . ) qqqq 3 6 622 12 3 66 121 1 26 cossin2sincos2 QQQQQQQ +-+ = ( ) ( ) qqqq 44 66 22 66 122 211 66 cossincossin 22 ++ += QQQQQQ qq 2 55 2 44 44 sincos QQQ += ( ) qq sincos 4455 45 QQQ. ) T NNNNN NNNNN NN NN srsrsr srsrsr srsrsr srsrsr V ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é = ,,, ,,, ,,, ,,, 22 11 122 1111 22 221 12 122 1111 fff fff fff fff f K K KKKK K K (2. 5) In order to determine the unknown coefficients i c , we invoke the cardinal interpolation condition