Development of higher order triangular element for accurate stress resultants in plated and shell structures 3

CHAPTER Enhanced Higher Order Triangular Reddy Plate Elements There have been several promising attempts to analyze the interlaminar stresses and stress resultants of laminated composite plates using conventional lower-order finite elements (Icardi, 1998; Kant and Swaminathan, 2000; Wanji and Zhen, 2008). However, the performance of these finite elements has not been satisfactory in predicting smooth and accurate variations of stresses, especially for laminated composite plates with free edges, clamped edges, reentrant corners and highly skewed geometry where regions of high stress gradients exist. Even though one uses a highly refined mesh to model plate problems with significant stress gradients, the accuracy of stresses obtained by using lower-order finite elements is rather poor. One of the plausible ways of overcoming such problems is to employ a finite element formulation having superior continuity of shape/interpolation functions. This is done by using a higher order p of polynomial basis function which results in the development of a p- version finite element method. Details regarding research related to pversion finite elements were presented in Chapter 1. In this chapter, we shall review research works on stress analysis of laminated composite plates. Various problems concerning laminated composite plates and shells were studied using the p-FEM viz., interlaminar 97 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ stress variation of laminated composites using three-node axisymmetric curved shell element (Liu and Surana, 1993) and quadrilateral element (Liu and Surana, 1995), bending and buckling problems of composite and functionally graded ceramic metal plates and shells (using quadrilateral C0 continuity elements) (Reddy and Arciniega, 2006; Arciniega and Reddy, 2007). Although the aforementioned studies cover a wide range of engineering problems, there has been limited research showing the performance of pversion elements in the stress analysis of laminated composite plate problems using triangular elements. In the context of stress-analysis of laminated composite structures which is very crucial to their design and performance, several specialized finite elements based on zig- zag and/or layerwise theories have evolved since 1935. A classic and comprehensive historical review on zig- zag theories was presented by Carrera (2003). The vast literature on finite elements developed to study interlaminar stresses in laminated composite plates have been validated for plates with simply supported edges. However, the ability of these elements in obtaining accurate stress resultants (especially the vanishing of shear forces and twisting moments along the free edge of a plate) and interlaminar stresses for a laminated composite plate with free edges and highly skewed geometry has not been tested. The influence of singularities in the form of point supports on the transverse shear stresses of symmetric and antisymmetric cross-ply laminated composite plates has also not been hitherto studied. The main objective of this chapter is to formulate a higher-order triangular element based on the third-order shear deformation theory of Reddy (1984) (TSD-R) and a layerwise theory (LT-R) of Reddy (2004), so that accurate 98 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ stress resultants and interlaminar stresses for laminated composite plates can be obtained for a wide range of thickness-to-length ratios (without any numerical problems such as shear locking) by the use of higher order shape functions. Finite elements based on layerwise theories have been applied by many researchers to capture accurate interlaminar and localized stresses in laminated composite structures. Carrera and Demasi (2002 a,b) have presented extensive formulations and numerical evaluations for multilayered plate elements encompassing the layerwise and equivalent single layer models. A quasi-recent displacement finite element model based on the layerwise theory of Reddy has been employed by Garỗóo et al. (2004) to study response details of adaptive structures composed of composite materials and piezoelectric inserts. While the aforementioned works by various researchers have been based on lower-order finite elements, the present study embarks on the application of a p- version triangular element for the stress analysis of laminated composite structures. To enhance the accuracy of interlaminar stresses, we shall adopt the equilibrium approach, wherein the higher-order continuity of the shape functions is exploited. The degree of polynomial approximation needed in the in-plane as well as out-of-plane directions (i.e. along the thickness of the laminate) for accurate solutions will also be investigated. The efficiency of the proposed element will be discussed in the context of the accuracy in computed stresses and stress resultants, computational efficiency, and ability to handle complex problems such as laminated composite plates with free edges, highly skewed geometry, and cutouts. 99 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ In the next section, we begin with a description of the third-order shear deformation plate theory of Reddy (TSD-R) and the associated higher order finite element formulation. 3.1 Third Orde r Shear Deformable Reddys Plate Theory (TSD-R) The third-order shear deformable laminated plate theory of Reddy (Phan and Reddy, 1985) is based on the same assumptions as used for the classical and Mindlin plate theories, except that the assumption on the straightness and normality of a transverse normal after deformation is relaxed by expanding the displacements u, v and w as cubic functions of the thickness coordinate. This theory assumes the following displacement field ổ u ( x , y , z ) = u ( x , y ) + zf x ( x , y ) + z ỗ - ố 3h ảw ửổ ữ ỗf x + ữ ảx ứ ứố ổ v ( x , y , z ) = v ( x , y ) + zf y ( x , y ) + z ỗ - ố 3h ảw ửổ ữ ỗỗ f y + ữữ ảy ứ ứố w( x, y, z ) = w0 ( x, y ) (3.1) The displacement field provides a quadratic variation of transverse shear strains (and hence stresses) and vanishing of transverse shear stresses at the top and bottom of a plate. Thus, there is no need to use shear correction factors in a third order shear deformable plate theory. The third order theory provides a slight increase in accuracy for in-plane stresses but a marked improvement in transverse shear stresses relative to MPT. Thus they are capable of handling both thick and thin plates effectively. The third-order theory contains lower order theories, including the classical plate theory and the first order shear deformation theory as special cases (Phan and Reddy, 1985). 100 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ The nonlinear strain-displacement relations in TSD-R may be expressed as follows: ỡe xx(1) ỹ ỡe xx(3) ỹ ỡe xx ỹ ỡe xx( ) ỹ ù (1) ù ù (3) ù ù ù ù (0) ù ớe yy ý = ớe yy ý + z ớe yy ý + z ớe yy ý ùg ù ùg ( ) ù ùg (1) ù ùg (3) ù ợ xy ỵ ợ xy ỵ ợ xy ỵ ợ xy ỵ (3.2 a) ỡg yz ỹ ỡg yz( ) ỹ ỡg yz( ) ỹ ý = (0) ý + z ( 2) ý ợg xz ỵ ợg xz ỵ ợg xz ỵ (3.2 b) where ỡ ảu ổ ảw0 + ỗ ữ ù ảx ố ảx ứ (0) ù ỡe xx ỹ ù ( ) ù ùù ảv0 ổ ảw0 ữữ + ỗỗ ớe yy ý = ùg ( ) ù ù ảy ố ảy ứ ợ xy ỵ ù ảu ảv0 ảw0 ảw0 ù ảy + ảx + ảx ảy ùợ ỹ ù ù ùù ý ù ù ù ùỵ (3.3 a) ỡ ảf x ỹ ù ù ảx ỡe xx(1) ỹ ù ù ù (1) ù ù ảf y ù ớe yy ý = ý ảy ùg (1) ù ù ù ợ xy ỵ ù ảf x ảf y ù ù ảy + ảx ù ợ ỵ (3.3 b) ỡ ỹ ảf x ả w0 + ù ù ảx ảx ỡe xx(3) ỹ ù ù ảf y ả w0 ù ( 3) ù ù ù + ớe yy ý = -C1 ý ảy ảy ùg (3) ù ù ù ợ xy ỵ ả w0 ù ù ảf x ảf y ù ảy + ảx + ảx ảy ù ợ ỵ (3.3 c) ỡg ợg (0) yz (0) xz ảw0 ỹ ỡ f + y ù ỹ ù ảy ùù = ý ý ỵ ùf + ảw0 ù x ảx ùỵ ợù (3.3 d) 101 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ ỡg ợg ( 2) yz ( 2) xz ảw0 ỹ ỡ f + y ỹ ùù ảy ùù ý = -C ý ỵ ùf x + ảw0 ù ùợ ảx ùỵ (3.3 e) C1 and C are constants that are expressed as follows: C1 = and 3h C = 3C1 . In the present formulation, we shall ignore the nonlinear strain ổ ảw ổ ảw terms namely ỗ ữ , ỗỗ ữữ ố ảx ứ ố ảy ứ ổ ảw ảw ữữ . , ỗỗ ố ảy ảx ứ The relation between stress resultants and strains in TSD-R can be given as: ỡ {N }ỹ ộ[A] [B ] [E ]ự ù ù ỳ ớ{M }ý = ờ[B ] [D ] [F ]ỳ ù {P} ù ờ[E ] [F ] [H ]ỳ ợ ỵ ở1444244ỷ4 99 { }ỹ { }ùý { }ùỵ ỡ e (0 ) ù (1) ớe ù e (3 ) ợ (3.4 a) 91 [C _ TSD _ R ] ỡ{Q}ỹ ộ [A]S ý= ợ{R}ỵ ở[D ]S where {N }= N xx {M }= {P}= M xx Pxx N yy M yy Pyy Pxy N xy M xy T T T [D]S ự [F ]S ỳỷ are the in-plane force resultants, are the in-plane moment resultants, and {R}= R y resultants, and {Q}= Q y (3.4 b) Qx T Rx T are the higher order stress are the transverse shear force resultants. The expressions for material stiffness coefficients are given by (A , B ij NL z k +1 ij , Dij , Eij , Fij , H ij )= ( ) ũ Q (1, z, z k ij ) , z , z , z dz , for i,j = 1,2,6,5 k =1 z k (3.5 a) (A , D ij ij NL z k +1 , Fij )S = ( ) ũ Q (1, z k ij ) , z dz , for i,j = 4,5 (3.5 b) k =1 z k 102 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ The expressions for Aij , Bij and Dij are given in Eqs. (2.12) in Chapter 2. The index NL denotes the number of lamination layers. The expressions for additional stiffness coefficients that are present in TSD-R are given as follows: Eij = NL (k ) ( ) Q ij z k4+1 - z k4 ; Fij = k =1 H ij = NL ( ) Q (z k ij k +1 - z k7 NL ( ) Q (z k ij k +1 ) - z k5 ; k =1 ) (3.5 c) k =1 The stiffnesses E ij , Fij and H ij involve fourth or higher powers of the thickness and hence contribute less to the thin laminate solutions. 3.2 Finite Ele ment implementation of HT-TSD-R45 In this section, we develop the higher order displacement finite element model of TSD-R, referred from hereon as HT-TSD-R45, based on the principle of virtual work. The displacement field requires only C0 continuity for all its variables. The number of variables to be interpolated in the finite element model is seven for the third-order shear deformation theory (TSD-R). Displacement finite element based on Eq. (3.3) requires C1 continuity because of the presence of first order derivatives of the transverse displacement in the weak form of TSD-R. In order to relax the continuity in finite element formulation, we introduce two auxiliary variables namely, y x and y y , which are defined as follows: yx = ảw0 ảw + fx ; y y = + f y ảx ảy Hence we have (u , v , w0 , f x , f y ,y x ,y y ) (3.6) as the generalized degrees of freedom in TSD-R. 103 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ The field variables can be expressed in terms of shape functions having degree p = (45 geometric nodes) as follows: 45 u ( x, y , z ) = u i Q i ( x, y ) i =1 45 v ( x, y , z ) = v i Q i ( x, y ) i =1 45 w0 ( x, y, z ) = wi Q i ( x, y ) i =1 45 f x ( x, y, z ) = f xi Q i ( x, y ) i =1 45 f y ( x, y, z ) = f yi Q i ( x, y ) i =1 45 y x ( x, y, z ) = ồy xi Q i ( x, y ) i =1 45 y y ( x, y, z ) = ồy yi Q i ( x, y ) (3.7) i =1 The derivation of shape functions Q i for a 45 noded triangular element has been discussed in Chapter 2. Hence we proceed to give the finite element matrices involved in TSD-R formulation. The strain-displacement relationship for TSD-R can be written as: {e }TSD - R = [B ]TSD - R {d i } (3.8) where {d i }= u 0i v0i w0i f xi f yi y xi y yi T , i =1 to 45 for the 45- noded element. The strain-displacement matrix [B ] TSD - R can be expressed in terms of shape functions N i as: 104 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ ộổ ảQi ờỗ ảx ữ ứ ờố ờ ờổỗ ảQi ửữ ờỗố ảy ữứ ờ ờ ờ ờ [Bi ] TSD - R = ờờ ờ ờ ờ ờ ờ ờ ờ ờở ổ ảQi ỗ ữ ố ảx ứ ổ ảQi ỗỗ ữữ ố ảy ứ ổ ảQi ỗỗ ữữ ả y ố ứ ổ ảQi ỗ ữ ố ảx ứ ổ ảQ - C1 ỗ i ữ ố ảx ứ ổ ảQ - C1 ỗỗ i ữữ ố ảy ứ Qi Qi - C2 Q i ổ ảQi ỗỗ ữữ ố ảy ứ ổ ảQi ỗ ữ ố ảx ứ ổ ảQi ỗỗ ữữ ố ảy ứ ổ ảQi ỗ ữ ố ảx ứ 0 0 ự ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ổ ảQi ửỳ ữữỳ - C1 ỗỗ ố ảy ứỳ ổ ảQ ửỳ - C1 ỗ i ữỳ ố ảx ứỳ ỳ ỳ ỳ ỳ ỳ - C2 Qi ỳỳ ỳỷ i 0 (3.9) Having formulated the strain displacement matrix, the element stiffness matrix [K e ] and the load is obtained as shown in Eqs. (2.18) and (2.20) of Chapter 2. The resulting element stiffness matrix has an order of 315. This may be reduced to 168 degrees of freedom by eliminating the degrees of freedom of the internal nodes through the static condensation technique. Likewise the stiffness matrices computed for all the elements are assembled together to form the final structural stiffness matrix. The boundary conditions are imposed and the system of equations is solved for the unknown nodal 105 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ displacements. Once the nodal displacements are known, the stress resultants at any point within the element are evaluated using Eq. (2.26) (see Chapter 2). The appropriate expressions for strains are substituted from Eq. (3.3) into Eq. (2.26) to obtain stresses. As laminated composite materials undergo the transition from secondary structural components to primary critical structural components, it becomes important to perform a highly accurate assessment of localized regions where damage initiation is likely to occur. MPT and TSD-R which may be called as equivalent single layer theories often prove adequate for modeling secondary structures which may be relatively thin compared to primary structures. Hence, even the estimation of global deflections, buckling loads etc. demands a refined laminate theory that can account for through thickness effects. Furthermore, a refined laminate theory can enable accurate assessment of 3D state of stress at the ply level. Such accurate assessment of stresses may be very important to study potential damage initiation. This is certainly not possible with MPT or TSD-R theories. Hence we shall develop a higher order layerwise plate element based on Reddys layerwise plate theory that contains full 3D kinematics and constitutive relations. Such an element enables accurate prediction of interlaminar stresses for very thick plates to moderately thick plates. 3.3 Layerwise plate theory of Reddy (LT-R) In contrast to equivalent single layer theories, layerwise theories are developed to by assuming that the displacement fields are continuous through the laminate thickness, whereas their derivatives with respect to the thickness 106 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ Based on the aforementioned numerical examples, it can be said that HTLT-R45 elements can be specifically employed to study interlaminar stresses for thick plates having thickness-to-length ratio h/a > 0.1. HT- TSD-R45 elements can be well suited to determine interlaminar stresses of moderately thick plates to very thin plates 0.01 Ê h a Ê 0.1 . Having assessed the behavior of HT-TSD-R45 elements in predicting interlaminar stresses in laminated composite plates, we shall now present two examples that involve circular cutouts to demonstrate their capability in tackling stresses and stress resultants in the vicinity of cutouts. 3.5.6 Stress analysis of a symmetric cross-ply plate with a hole under uniaxial te nsion We present the stress analysis of a symmetric cross-ply (0 90 90 0 ) plate with a hole subjected to uniaxial tension s . The aim of this example is to compare the variations of circumferential stresses predicted by HT-TSDR45 elements against various specialized finite elements that were developed by researchers for determining stress concentration around holes in laminated plates. The dimensions of the laminated plate with a hole are shown in Fig. 3.3d. The material E1 = 25106 psi, E2 = 3106 properties psi, of the laminated plate are: G12 = G13 = G 23 = 0.5 10 psi, n 12 = 0.336 . The thickness to length ratio h/a = 0.05. Owing to symmetry in geometry and loading conditions, we consider only one quarter of the plate and plot the circumferential stresses along the cutout edge. We employ MeshD of HT-TSD-R45 shown in Fig. 3.16d. The results are compared with two 158 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ finite elements that were developed by Nishioka and Atluri (1982) and Huang et al. (1994). Figure 3.17 shows the plot of circumferential stresses along the centerline of the cutout edge corresponding to the 00 and 900 layers. The results of circumferential stresses predicted by HT-TSD-R45 in Fig. 3.17 compare very well with the specialized finite elements tailored to handle stress concentration problems. A recent study by Zhen and Wanji (2009) shows the performance of a nonconforming BCIZ element in handling stress concentration problems in laminated composite plates with holes similar to the present numerical example. The nonconforming BCIZ element proposed by Zhen and Wanji (3 noded triangle having 26 dof per node) requires twice the number of degrees of freedom as compared to the present HT- TSD-R45 to achieve the same level of accuracy. The superior quality of higher-order shape functions obviates the need for inclusion of any additional functions to deal with free edge stresses. Thus the present higher-order element can be easily extended to study problems in fracture mechanics to yield results with better accuracy and lesser computational effort. (a) (b) 159 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (c) (d) Fig. 3.16 Finite element mesh of one quarter of a simply supported square plate with a circular cutout (a) Mesh A- ABAQUS (b) Mesh B ABAQUS (c) Mesh C ABAQUS (d) Mesh D HT-TSD-R45 (a) 160 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (b) Fig. 3.17 Comparison of circumferential stresses along the cutout edge of a symmetric (0/90/90/0) cross-ply square plate subjected to uniaxial tension (a) variation along 00 ply and (b) variation along 900 ply 3.5.7 Symmetric cross-ply laminated composite plates with cutouts As a concluding example, we assess the performance of higher-order triangular element in tackling practical stress gradient problems. We consider a square (90 0 0 90 ) simply supported symmetric cross-ply laminated plate which was studied by Kapania et al. (1997). The plate has dimensions as shown in Fig. 3.3 e and is composed of four orthotropic layers of equal thicknesses (t = 0.05). Each layer models a G30-500/5208 Graphite-Epoxy fiber/matrix composite having the following material properties: E1 = 22.04 10 psi, E = 1.68 10 psi, G12 = 0.81 10 psi, G13 = 0.4562 10 psi, G 23 = 0.4562 10 psi and n 12 = 0.286 . The plate is subjected to a uniformly distributed load q = 0.1 psi. This example demonstrates the effectiveness of higher order triangular element HT- 161 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ TSD-R45 in plates with cutouts. In this section, we shall present a detailed stress analysis in the vicinity of the cutout and compare the stress resultant distributions obtained with those of ABAQUS finite element models. The details of mesh designs and corresponding d.o.f. adopted in ABAQUS are given in Fig. 3.16 and Table 3.4, respectively. Further, we shall compare the distribution of bending moments obtained near the cutout region with a global/local model proposed by Kapania et al. (1997) who used a two-step approach in estimating stresses near circular and elliptical cutouts. As a first step, a global analysis was performed based on the Ritz method by incorporating a perturbation function to match the circular cutout shape. The solution obtained using the global approach was used as a means to obtain geometric boundary conditions for the local region (i.e., vicinity of the cutout). The local model was then analyzed using finite element method. Figures 3.18 and 3.19 present the variation of bending moments M x and M y obtained using different finite element models along Edge A as shown in Fig. 3.3e. It can be observed from Figs. 3.18a and 3.19a, that the overall variations of bending moments M x and M y obtained using HT-TSD-R45 are in excellent agreement with Mesh A of ABAQUS S8R elements having only 8244 d.o.f. Figures 3.18b and 3.19b present the variations of bending moments near the edge of the cutout which is very crucial from failure and a design point of view. It can be inferred from Figs. 3.18b and 3.19b that MeshA of ABAQUS and the global- local model fail to give a correct estimation of peak stress near the cutout edge. In spite of its computational efficiency, the global- local model is not able to capture the peak stresses accurately. 162 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ Although the plot of bending moment M x , obtained using Mesh-D (10444 dof) of HT-TSD-R45 and Mesh-C (203,334 d.o.f) of ABAQUS S8R elements, exhibit minor deviation, the value of maximum bending moment M x differs by 8%. In order to verify the location and magnitude of this peak bending moment, we refine the mesh further in the vicinity of the cutout. The magnitude of peak bending moment obtained using a refined mesh of HTTSD-R45 having 35308 d.o.f. differed by less than 1% from that of HT-TSDR45 with 10444 d.o.f. This shows that the present higher-order element is capable of predicting accurate values of peak stresses near the edges of cutouts. Furthermore, the results obtained using Mesh-C with more d.o.f. not only fails to capture the peak bending moments but the moment distribution is seen to oscillate in the vicinity of the cutout. Therefore it fails to converge to the exact value of peak bending moment despite its demand for more degrees of freedom and computational time. A similar trend is observed in the variation of bending moment M y along Edge A of the laminated cross- ply plate. 163 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) Fig. 3.18 Variation of bending moment M x for a simply supported square plate with a central circular cutout (a) M x along Edge A (b) M x in the vicinity of cutout. Bracketed values indicate number of degrees of freedom 164 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) Fig. 3.19 Variation of bending moment M y for a simply supported square plate with a central circular cutout (a) M y along Edge A (b) M y in the vicinity of cutout. Bracketed values indicate number of degrees of freedom In order to reveal the influence of cutout on the stress distributions in the laminated plate, we present contour plots of stress resultants M x , M y , M xy and 165 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ Q x for one quarter of the laminated composite plate. Figures 3.20-3.22 show the contour plots of stress resultants obtained for Mesh C of ABAQUS that has a highly refined mesh with more number of nodes distributed in the vicinity of the cutout. The enlarged views of stress distributions around the cutout edge are shown in Fig. 3.23. By comparing the contour plots of stress resultants, we observe that HT- TSD-R45 with 10444 d.o.f. performs on par with Mesh C of ABAQUS (203,334 d.o.f.) which has a highly refined mesh near the cutout edge in capturing the stress gradients accurately. On the other hand, meshes A and B of ABAQUS exhibit significant stress gradients in the vicinity of the cutout, thereby yielding incorrect values of peak stresses. The contour plots of twisting moment M xy and transverse shear force Q x display substantial irregularities in their distributions. The key point to note in this example is that an accurate prediction of peak stresses in the neighborhood of a cutout that is prone to high stress gradients, demands the finite element discretization to have elements that have polynomially enriched basis functions rather than densely clustered lower-order finite elements. For an easy comparison of computational efficiency, Table 3.4 presents the degrees of freedom corresponding to various finite element meshes. 166 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) Fig. 3.20 Contour plots of normalised bending moment M x for a simply supported square plate with a central circular cutout (a) Mesh D of HT-TSD-R45 (b) Mesh C of ABAQUS S8R elements 167 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) Fig. 3.21 Contour plots of normalised twisting moment M xy for a simply supported square plate with a central circular cutout (a) Mesh D of HT-TSD-R45 (b) Mesh C of ABAQUS S8R elements 168 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) Fig. 3.22 Contour plots of normalised transverse shear force Q x for a simply supported square plate with a central circular cutout (a) Mesh D of HT-TSD-R45 (b) Mesh C of ABAQUS S8R elements 169 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ (a) (b) (c) (d) (e) (f) (g) (h) Fig. 3.23 Enlarged view of stress resultants M xy (figures. (a)-(d)) and Q x (figures. (e)-(h)) in the vicinity of cutout obtained using the following mesh designs. (a), (e) Mesh D of HT-TSD-R45 ; (b), (f) Mesh A of ABAQUS ; (c),(g) Mesh B of ABAQUS ; (d),(h) Mesh C of ABAQUS 170 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ Table 3.4 Mesh type and degrees of freedom adopted for the analysis of laminated composite plates using higher order triangular elements and ABAQUS No. of elements 128 Degrees of freedom 10368 400 7686 1600 29766 100100 Đ 10000 182406 Mesh A 429 8244 Mesh B 1787 33168 Mesh C 10876 203334 HT-TSD-R45 Mesh D 132 10444 GlobalLocal Method (1997) 6199 Example case Corner supported cross-ply laminated plate Model Mesh Design HT-TSD-R45 88 Đ 2020 Đ ABAQUS S8R ABAQUS S8R Simply supported crossply laminated plate with circular cutout 4040 Đ Note: Đ denotes refined mesh near the location of point support 3.6 Conclusions In this chapter, higher-order triangular plate elements based on the third-order shear deformation theory and a layerwise theory with variable interpolation through the laminate thickness are developed for the determination of stresses in laminated composite plates. The polynomial degree of the element shape functions is increased uniformly over the entire finite element mesh in order to improve the accuracy of solution rather than the traditional approach of mesh refinement using lower-order finite elements. The credibility of the developed higher-order finite elements to predict, especially the transverse shear stresses through the thickness of the laminated composite plate, was emphasized. 171 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ Although exact solutions for symmetric and antisymmetric laminated composite plates with regular geometry are readily available, there has been less attention to the problem of skew laminated composite plates subjected to transverse load. The present work has brought out the significance of employing polynomial shape functions (for in-plane as well as transverse displacements across the thickness of the lamina) in predicting interlaminar stresses in skew laminated composite plates with various skew angles. While a great deal of research on interlaminar stresses in laminated composites focus on structures having simply supported and clamped boundary conditions, problems of interlaminar transverse shear stresses for structures having point supports and free edges have not received much attention. This study has brought forth the shortcomings existing in the formulation of conventional lower-order finite elements that have been implemented in ABAQUS and its influence on the accuracy of computed interlaminar transverse shear stresses for antisymmetric laminated composite structures. Finally, a detailed stress analysis and visualization of computed stresses in laminated composite plates with cutouts are presented, with the view to establish the accuracy and computational efficiency of higher-order triangular elements. The present higher-order triangular finite elements were shown to possess excellent computational capabilities (nearly 20 times more efficient in terms of degrees of freedom necessary to achieve a specified accuracy) as compared to conventional lower-order finite elements. Furthermore, the presence of higher-order shape functions accommodates a wide range of deformations and sidesteps the use of additional functions/adjustments to 172 Enhanced Higher Order Triangular Reddy Plate Elements _______________________________________________________________ handle high stress gradient problems and free edge boundary conditions. The present work has been the first one to present a detailed stress assessment with visualization of a higher-order triangular element based on the nodal basis approach. The comprehensive results reported in this work indicate that the present higher-order triangular element can be a potential element implemented in element library of any commercial finite element software. For problems involving linear analysis, the nodal distribution has negligible effect on the accuracy of numerical solution. When the nonlinear behavior of a system has to be studied, the distribution of nodes inside a finite element plays a significant role in the accuracy and convergence of a solution. This prompts one to employ spectral finite elements that were originally applied to solve problems in computational fluid mechanics and magnetostatics (Sherwin and Karniadakis,1995). Hence, in the next chapter, we shall establish the best layout for the 45 geometric nodes for robustness of the element to handle linear and geometrically nonlinear problems. In the subsequent chapters, we shall also present a comparative study of the performances of finite elements with respect to their nodal distributions in various plate and shell problems. 173 [...]... 0.0 234 0.0 234 0 .38 50 0 .38 26 0 .37 71 0.0 938 0.0919 0.0944 LQ1-22 LQ1 -33 LQ1-44 0. 434 9 0. 434 9 0. 434 9 0. 539 9 0. 539 6 0. 539 5 0.1811 0.1809 0.1808 0.0214 0.0214 0.0214 0 .38 26 0 .39 04 0 .39 30 0.0587 0.0754 0.0826 LC1-22 LC1 -33 LC1-44 0. 434 9 0. 434 9 0. 434 9 0. 539 9 0. 539 6 0. 539 5 0.1811 0.1809 0.1808 0.0214 0.0214 0.0214 0 .38 14 0 .38 93 0 .39 21 0.0541 0.0690 0.0754 44 66 8 8 0. 434 5 0. 434 5 0. 434 5 0. 539 4 0. 539 4 0. 539 4 0.1806... presented herein are obtained by using an 88 mesh of HT-TSD-R45 and a 33 mesh of HT-LT-R45 Figures 3. 4 and 3. 5 show the variations of in- plane and transverse shear stresses across the thickness of the laminate for various h/a ratios It can be seen from Fig 3. 5 that the through thickness variation of in- plane and transverse shear stresses obtained using a 33 mesh of HT-LT-R45 with cubic interpolation... layouts of the two higher order triangular plate elements HT-TSD-R45 and HT-LT-R45 are given in Fig 3. 2 119 Enhanced Higher Order Triangular Reddy Plate Elements _ (a) (b) Fig 3. 2 Layouts of (a) HT-TSD-R45 (b) HT-LT-R45 3. 5 Numerical Examples The performance of the higher- order triangular plate elements HT- TSD-R45 and HT-LT-R45 in predicting deflections and stresses of laminated... displacement formulation are capable of predicting better results as compared to their counterparts namely LD3 (LM3) and ED3 (EMC3) HT-LT-R45 (LC1) element stands ahead of LM4 and LD4 in the hierarchy of finite elements proposed by Carrera and Demasi (2002b) 1 23 Enhanced Higher Order Triangular Reddy Plate Elements _ ( ) Table 3. 1 Convergence and comparison of deflection and stresses... 0.0296 0 .36 26 0 .37 24 0.1249 0.16 23 LM 3 LD3 0.7528 0.7528 0.5801 0.5801 0.2797 0.2797 0.0296 0.0296 0 .36 26 0 .37 24 0.1249 0.16 23 EMZC3 0.7 634 0.5856 0.2829 0.00 83 0 .38 84 0.1270 EM C4 EM C3 0. 731 3 0. 732 3 0.5787 0.58 03 0.2708 0.2710 0.0084 0.0085 0 .31 18 0.2687 0.1259 0.1450 EDZ3 0.7 634 0.5856 0.2 834 0.00 83 0 .38 79 0.1491 ED4 ED3 0.7268 0.7246 0.5776 0.57 83 0.2694 0.2687 0.00 83 0.0085 0.2948 0.2849 0.1464... 0.0 234 0 .38 09 0 .38 40 0 .38 50 0.0877 0.0988 0.1015 LC1-22 LC1 -33 LC1-44 0.5159 0.5166 0.5166 0.55 43 0.5 532 0.5528 0.2094 0.20 93 0.2092 0.0 235 0.0 234 0.0 234 0 .38 18 0 .38 26 0 .38 46 0.0867 0.0920 0.0 934 44 66 0.5060 0.5060 0.2050 0.2050 0.0 231 0.0 231 0.5060 0.2050 0.0 231 0 .38 52 0 .38 66 0 .38 71 0.0895 0.0912 88 0.5510 0.5509 0.5509 0.5142 0.5200 0.5194 0.5520 0.5457 0.5826 0.2100 0.2127 0.2041 0.0 234 0.0 234 0.0 234 ... stresses at critical points of a simply supported 0 0 90 0 0 0 square laminate under transverse double sinusoidal load w (0,0,0 ) hử ổ s xx ỗ 0,0, ữ 2ứ ố LQ1-22 LQ1 -33 LQ1-44 2.0862 2.0 935 2.0 935 0.7916 0.79 03 0.7901 0. 533 9 0. 532 0 0. 531 1 0.0505 0.0505 0.0506 ổa ử ỗ ,0,0 ữ 2 ố ứ 0.2524 0.2 530 0.2 532 LC1-22 LC1 -33 LC1-44 2.0955 1.9927 1.9927 0.8 036 0.8016 0.8014 0. 533 8 0. 533 8 0. 533 6 0.0516 0.0511 0.0511... for integrating the stiffness and the load matrix of the present reference triangular element by a simple affine transformation The transverse shear stresses are determined from 3D equilibrium equations in a manner similar to that presented in Chapter 2 (see Eqns 2.28 to 2 .39 ) Thus we exploit the higher order continuity of the shape functions to determine accurate interlaminar transverse shear stresses... obtained using the higher- order triangular elements are seen to agree well with the solutions obtained by Kant and Pandya (1988) The transverse shear stresses obtained using both the higher- order elements show excellent agreement with the exact solution, whereas results of Kant and Pandya (1988) show substantial deviation from the exact solution Thus HT-TSD-R45 elements yield accurate interlaminar stresses... stresses for thin and moderately thick plates HT-LT-R45 elements with cubic interpolation are well suited for interlaminar stress analysis of thick laminated composite plates 129 Enhanced Higher Order Triangular Reddy Plate Elements _ (a) (b) 130 Enhanced Higher Order Triangular Reddy Plate Elements _ (c) Fig 3. 4 Variation of normalized stresses . Eq. (3. 3) requires C 1 continuity because of the presence of first order derivatives of the transverse displacement in the weak form of TSD-R. In order to relax the continuity in finite element. stress resultants given in Eqs. (3. 19) to (3. 23) corresponding to I th numerical plane can be represented in matrix form as 18 88 6 636 2616 44454445 45554555 44454445 45554555 33 332 3 13 26 232 212 16 131 211 18 0000 ~ 0000 0000 0000 0000 0000 ~ ~~ 0000 ~ 0000 ~ ~ ~ ~ ù ù ù ù ù ù ù ỵ ù ù ù ù ù ù ù ý ỹ ù ù ù ù ù ù ù ợ ù ù ù ù ù ù ù ớ ỡ ả ả ả ả ả ả + ả ả ả ả ả ả ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỷ ự ờ ờ ờ ờ ờ ờ ờ ờ ờ ờ ờ ở ộ = ù ù ù ù ù ỵ ù ù ù ù ù ý ỹ ù ù ù ù ù ợ ù ù ù ù ù ớ ỡ y W x W V U x V y U W x V x U AAAA DDBB DDBB BBAA BBAA AAAA AAAA AAAA N Q Q Q Q Q N N J J J J JJ J J J JIJIJIJI JIJIJIJI JIJIJIJI JIJIJIJI JIJIJIJI IJIJIJIJ JIJIJIJI JIJIJIJI I xy I y I x I y I x I z I yy I xx . I F . The stresses in the k th layer can be computed from the 3D stress strain equations. For the k th orthotropic lamina, we have )( )( 6 636 2616 5545 4544 36 333 231 26 232 221 16 131 211 )( 00 0000 0000 00 00 00 k xy xz yz zz yy xx k k xy xz yz zz yy xx QQQQ QQ QQ QQQQ QQQQ QQQQ ù ù ù ù ỵ ù ù ù ù ý ỹ ù ù ù ù ợ ù ù ù ù ớ ỡ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỳ ỷ ự ờ ờ ờ ờ ờ ờ ờ ờ ở ộ = ù ù ù ù ỵ ù ù ù ù ý ỹ ù ù ù ù ợ ù ù ù ù ớ ỡ g g g e e e s s s s s s