Volume 21 - Composites Part 6 pdf

Livermore, CA MSC- NASTRAN/DYTRAN MSC Software Corp., Costa Mesa, CA http://www.mechsolutions.com/products/patran/lammod.html VR&D-GENESIS Vanderplaats Research & Development, Inc., Colorado Springs, CO http://www.vrand.com/genesis_fact.htm Pre- and postprocessors PATRAN MSC Software Corp., Costa Mesa, CA http://www.mechsolutions.com/products/patran/patran2000.htm HyperMesh Altair Engineering, Inc., Troy, MI http://www.altair.com/ The pre- and postprocessors of most of the commercial codes listed in Table 2 support the key aspects related to the analysis of fibrous composite materials. These include: • Input of ply properties based on unidirectional as well as 2-D woven fiber architecture • Ply lay-ups • Vector orientations used to define ply orientation in space • Computation of 3-D effective properties • Computation of [A], [B], and [D] stiffness matrices for plate and shell elements • Recovery of strains and stresses in various coordinate systems, such as global axis, local element axis, laminate axis, and lamina axis • First ply failure based on either point stress/ strain (maximum strain/stress) or quadratic failure (Tsai- Wu, Hill, Hashin) criterion Some of the explicit analysis codes, such as LS-DYNA, MSC-DYTRAN, PAM-CRASH, and ABAQUS also provide progressive damage material models for ultimate failure load prediction. However, these computational progressive damage models have not been experimentally verified for a wide variety of structural applications. Based on the author's personal experience, ESI-SYSPLY is perhaps the most comprehensive and user-friendly pre- and postprocessor program currently available for composite FEA. However, in the current form, it does not have the interface with most widely used FEA solvers, such as NASTRAN and ABAQUS, thereby severely limiting its utility. Over the years pre- and postprocessing tools have been highly optimized for the FEA of metallic structures. These tools now need significant enhancement in their capabilities to accurately and efficiently analyze and design complex structures manufactured from advanced composite materials. Reference cited in this section 4. J.M. Whitney, Structural Analysis of Laminated Anisotropic Plates, Technomic, 1987 Finite Element Analysis Naveen Rastogi, Visteon Chassis Systems Numerical Examples The continuity of transverse stresses at the layer interfaces and the free-edge effects are unique aspects in the analysis of multilayered composite structures. Finite element analysis of two classical examples from the mechanics of composite materials illustrates these aspects of multilayered composite structures. The first example is a problem of transverse bending of a simply supported [0/90/0] T laminated plate, the benchmark solution of which was obtained by Pagano (Ref 30). The second numerical example is of a [0/90] s laminated plate under uniform extension (Pagano, Ref 31, 32), illustrating the free-edge effects in multilayered composite structures. Example 1: Transverse Bending of a Laminated Plate. A simply supported [0/90/0] T laminated plate is subjected to sinusoidal loading on the top surface. Laminated plates with two different aspect ratios are considered. For a/h= 4, the plate represents a thick multilayered structure. For a/h= 50, a thin-walled structure is represented. All layers are assumed to be of equal thickness. The material properties for the orthotropic lamina are (Ref 30): E 1 /E 2 = 25, E 2 =E 3 , G 12 /E 2 =G 13 /E 3 = 0.5, G 23 /E 2 = 0.2, ν 12 =ν 13 =ν 23 = 0.25. The origin of the right- handed coordinate system is chosen at the corner of the middle surface of the plate, that is, 0 ≤x≤a, 0 ≤y≤b, and–(h/2) ≤z≤ (h/2) (see Fig. 6). Fig. 6 A simply supported [0/90/0] T laminate subjected to sinusoidal loading on the top surface This problem is analyzed by using the novel 3-D FEA tool SAVE, developed by the author (Ref 21, 22). For the laminated plate problem described previously, a quick comparison between 3-D elasticity solution of Pagano (Ref 30) and the 3-D structural analysis code SAVE is presented in Table 3 for various a/h ratios. The normalized quantities used in Table 3 are described as: , , , , , , where s=a/h and q 0 is the peak magnitude of the applied sinusoidal pressure load at the center of the laminated plate at the top surface. The results presented in Table 3 demonstrate the accuracy of SAVE analysis code in the 3-D analysis of multi- layered structures. Results from SAVE analysis can now be used as a basis to compare with the results obtained from commercial FEA codes. Table 3 Comparison between the results obtained for various a/h ratios for a [0/90/0] T simply supported laminate subjected to sinusoidal loading on the top surface Quantity a/h= 2 a/h= 4 a/h= 10 a/h= 50 a/h= 100 Source 1.4361.436 0.8010.801 0.5900.590 0.5410.541 0.5390.539 (a) (b) –0.937–0.938 –0.754–0.755 –0.590–0.590 –0.541–0.541 –0.539–0.539 (a) (b) 0.6680.669 0.5340.534 0.2840.285 0.1840.185 0.1810.181 (a) (b) –0.742–0.742 –0.556–0.556 –0.288–0.288 –0.185–0.185 –0.181–0.181 (a) (b) –0.0859– 0.0859 –0.0511– 0.0511 –0.0288– 0.0289 –0.0216– 0.0216 –0.0214– 0.0213 (a) (b) 0.07020.0702 0.05050.0505 0.02900.0289 0.02160.0216 0.02140.0213 (a) (b) 0.1640.164 0.2560.256 0.3570.357 0.3930.393 0.3950.395 (a) (b) 0.25910.2591 0.21720.2172 0.12280.1228 0.08420.0842 0.08280.0828 (a) (b) (a) (a) From the SAVE analysis (Ref 21, 22). (b) By Pagano (Ref 30) Next, the example problem is analyzed using commercial FEA codes such as ABAQUS (Ref 33), NASTRAN (Ref 34), and I-DEAS (Ref 35). The solid elements—C3D8 and C3D20 in ABAQUS, and CHEXA and CHEXA20 in NASTRAN—and linear and parabolic brick in I- DEAS are used in the analyses. Results from the commercial FEA codes and the SAVE analysis are compared in Table 4 for a/h= 4. The mesh description shown in Table 4 represents the number of elements that are used to represent each composite layer in the three orthogonal coordinate directions. For example, a 12 × 12 × 2 finite element mesh represents 12 solid elements in x- and y-direction each, and 2 solid elements in the z-direction, in every single composite layer. As is shown in Table 4, the numerical values of the six stress components as obtained from the 3-D FEA using solid elements with quadratic shape functions (parabolic in I-DEAS, CHEXA20 in NASTRAN, and C3D20 in ABAQUS) are within 5% of the exact values. Only the transverse shear stress component, τ yz , shows some significant difference (about 12 %) from the exact solution. However, as is shown in Table 4, the accuracy in the solution of this stress component is increased significantly by refining the FE mesh. In Table 4, compare the results obtained from I-DEAS and ABAQUS analyses with progressive mesh refinement (6 × 6 × 2, 12 × 12 × 2, and 20 × 20 × 4 meshes of parabolic solid elements). It is also worth noting that a sufficiently accurate solution to the problem being analyzed could be obtained by using parabolic brick elements in a coarse mesh (6 × 6 × 2 per layer) with 2916 DOF only. However, in spite of using a more refined mesh (12 × 12 × 2 per layer), solid elements with linear shape functions (linear brick in I-DEAS and C3D8 in ABAQUS) do not provide an accurate solution. For the numerical problem analyzed here, solid elements with linear shape functions, also known as constant strain elements, can be erroneous in the bending stress values by as much as 30%. Table 4 Comparison among the results obtained for a/h= 4 from various 3-D analyses for a [0/90/0] T simply supported laminate subjected to sinusoidal loading on the top surface Quantity SAVE, 1 × 1 × 1,M= 6, 1,805 DOF ABAQUS, 20 × 20 × 4, C3D20, 61,200 DOF ABAQUS, 12 × 12 × 2, C3D20, 11,664 DOF NASTRAN, 12 × 12 × 2, CHEXA20, 11,664 DOF I-DEAS, 12 × 12 × 2 (parabolic), 11,664 DOF I-DEAS, 6 × 6 × 2 (parabolic), 2,916 DOF I-DEAS, 12 × 12 × 2 (linear), 3,024 DOF ABAQUS, 12 × 12 × 2, C3D8, 3,024 DOF 1.0 1.005 1.02 1.02 1.02 1.025 0.950 0.826 0.801 0.788 0.800 0.800 0.769 0.773 0.760 0.571 –0.754 –0.744 –0.750 –0.750 –0.725 –0.729 –0.716 –0.547 0.534 0.528 0.532 0.532 0.516 0.521 0.514 0.483 –0.556 –0.550 –0.554 –0.554 –0.539 –0.543 –0.537 –0.514 –0.0511 –0.0508 –0.0509 –0.0498 –0.0500 –0.0504 –0.0492 –0.0519 0.0505 0.0503 0.0503 0.0494 0.0496 0.0500 0.0487 0.0514 0.256 0.257 0.255 0.256 0.257 0.259 0.252 0.261 0.2172 0.2228 0.2400 0.2398 0.2408 0.2427 0.1743 0.1703 DOF, degrees of freedom; M, degree of polynomial The through-the-thickness distributions of six stress components, as shown in Fig. 7, demonstrate many unique features of the 3-D stress state in multilayered laminated composite structures. Note the jump in the in-plane normal stress components σ xx and σ yy at the layer interfaces (see Fig. 7a and b). In multilayered laminated structures, in-plane stresses σ xx , σ yy , and τ xy are discontinuous (hence, the in-plane strains xx , yy , and γ xy are continuous) at the layer interfaces. On the other hand, the transverse stresses σ zz , τ yz , and τ xz are continuous at the material interfaces, as shown in Fig. 7(c)–(e). However, the out-of-plane strains zz , γ yz , and γ xz are now discontinuous (or jump) at these interfaces. Fig. 7 Through-the-thickness distributions for a/h= 4. (a) σ xx . (b) σ yy . (c) σ zz . (d) τ xz . (e) τ yz . (f) τ xy . In the legend for these curves, M is the degree of the polynomial. Continuity of the transverse normal and shear stresses at the layer interface, is a unique and important aspect in the analysis of multilayered laminated structures. Table 5 presents the actual numerical values of transverse stress components σ zz , τ yz , and τ xz at the layer interfaces, thereby providing a deeper insight into the continuity of these interlaminar stresses. The notation “T” represents values computed at the interface approaching from the top; similarly, “B” represents values computed at the interface approaching from the bottom. While the SAVE analysis code is almost perfect in satisfying the continuity of interlaminar stresses at the interfaces, ABAQUS analysis with a very refined mesh (20 × 20 × 4 per layer with 61,200 DOF) is also reasonably good in achieving that goal. However, as the mesh size becomes coarser, the commercial FE analyses results tend to become more distinct as well as less accurate at the interface (refer to Table 5). Once again, in all the commercial FE analyses, the transverse shear stress component, τ yz , shows the most significant differences. As is shown in Table 5, the continuity of interlaminar stresses at the layer interfaces is the worst from the FEA with constant strain elements, thereby making them unsuitable for transverse bending analysis of multilayered composite structures. Table 5 Interlaminar stresses as obtained at the ply interfaces from various 3-D analyses of a [0/90/0] T simply supported laminated plate (a/h= 4) subjected to sinusoidal loading on the top surface Quantity SAVE, 1 × 1 × 1 mesh (M= 6), 1,805 DOF ABAQUS, 20 × 20 × 4, C3D20, 61,200 DOF ABAQUS, 12 × 12 × 2, C3D20, 11,664 DOF NASTRAN, 12 × 12 × 2, CHEXA20, 11,664 DOF I-DEAS, 12 × 12 × 2 (parabolic), 11,664 DOF I-DEAS, 6 × 6 × 2 (parabolic), 2,916 DOF I-DEAS, 12 × 12 × 2 (linear), 3,024 DOF ABAQUS, 12 × 12 × 2, C3D8, 3,024 DOF TB 0.2690.269 0.2670.270 0.2580.265 0.2580.265 0.2640.269 0.2660.271 0.3820.182 0.3890.099 TB 0.25750.2575 0.25750.2625 0.25750.2725 0.25750.2725 0.25750.2750 0.26000.2770 0.25330.2693 0.26130.2730 TB 0.07580.0758 0.08280.0763 0.10100.0778 0.10050.0775 0.10100.0780 0.09670.0786 0.17090.0645 0.16430.0665 TB 0.7180.718 0.7190.721 0.7190.725 0.7190.725 0.7220.726 0.7280.732 0.8080.605 0.8810.582 TB 0.25250.2525 0.25750.2525 0.27000.2525 0.27000.2525 0.27000.2525 0.27330.2548 0.27250.2508 0.26800.2618 TB 0.08930.0893 0.08980.0963 0.09100.1145 0.09100.1140 0.09150.1145 0.09220.1155 0.07500.1778 0.07380.1786 DOF, degrees of freedom; T, top; B, bottom; M, degree of polynomial In general, a 3-D analysis using discrete layer- by-layer representation of the laminate can be performed with reasonable accuracy, using solid elements with quadratic shape functions in any of the commercial FE codes evaluated here. However, due to limitations in the available computational resources, many times it may not be possible to discretize a complex, practical structure completely with 3-D solid elements. In addition, most of the real-life structures are thin- walled, so as to justify the use of 2-D shell elements in their analyses. However, the limitations, or bounds, of using 2-D shell elements to accurately analyze multi-layered composite structures needs to be well understood. The thick laminated plate problem (a/h= 4) described previously is now analyzed using 2-D shell elements available in the commercial FE codes, namely, ABAQUS (S4R), NASTRAN (CQUAD4), I-DEAS (linear shell), and MECHANICA (p-type shell) (Ref 36). The type of shell element used in each analysis is mentioned in parentheses. The stress solutions obtained from various 2-D shell analyses are compared with the exact 3-D solution, as shown in Table 6. The 2-D shell analysis does not provide the transverse normal stress component, σ zz . Note the very high numerical discrepancy among the stress values as obtained from exact 3-D solution and various 2-D analyses using shell elements. The largest discrepancy is in the magnitude of stress in the fiber direction in 0° layer, where the stress values from the 2-D analysis are almost 50% lower than the exact 3-D values. At the same time, the similarity among the 2-D analyses solutions is remarkable. Except for the values of transverse shear stress, τ yz , as obtained from MECHANICA, the numerical results for the stresses obtained from the 2-D shell analyses are essentially the same. A systematic attempt was made to check the convergence of the 2-D solutions by increasing the order of shell elements (e.g., S4R to S8R in ABAQUS, CQUAD4 to CQUAD8 in NASTRAN, and linear shell to parabolic shell in I-DEAS), as well as by refining the FE mesh in the model. No further improvement in the numerical solution of the problem was observed. Table 6 Comparison among the results obtained for a/h= 4 from various 2-D analyses for a [0/90/0] T simply supported laminate subjected to sinusoidal loading on the top surface Quantity SAVE (3-D), 1 × 1 × 3 mesh (M= 6) 1,805 DOF I-DEAS (2-D), 24 × 24 mesh (linear shell), 3,553 DOF ABAQUS (2- D), 24 × 24 mesh, S4R, 3,553 DOF MECHANICA, 2-D;p= 6 NASTRAN (2-D), 24 × 24 mesh, CQUAD4, 3,553 DOF 0.801 0.397 0.397 0.398 0.396 –0.754 –0.397 –0.397 –0.398 –0.396 0.534 0.592 0.592 0.592 0.592 –0.556 –0.592 –0.592 –0.592 –0.592 –0.0511 –0.0429 –0.0429 –0.0429 –0.0428 0.0505 0.0429 0.0429 0.0429 0.0428 0.256 0.310 0.310 0.310 0.310 0.2172 0.2750 0.2750 0.0675 0.2750 DOF, degrees of freedom; M or p, degree of polynomial Further insight into this subject is gained by analyzing the laminated plate problem described previously with a/h= 50. The same 2-D shell elements and FE mesh are used during the analysis. Numerical results from a typical 2-D shell analysis using SDRC I-DEAS and the 3-D exact analysis SAVE are presented both in tabular form (Table 7) and as plots of stress distributions through the thickness of the laminate (Fig. 8). Except for the transverse shear stress component, τ yz , numerical solutions obtained from the 2-D and 3-D analyses for this problem compare very well with each other. The numerical examples discussed here emphasize the need to understand the limits of 2-D shell elements in the analysis of anisotropic, multi-layered composite structures. Table 7 Comparison among the results obtained for a/h= 50 from various 2-D analyses for a [0/90/0] T simply supported laminate subjected to sinusoidal loading on the top surface Quantity SAVE (3-D), 1 × 1 × 3 mesh, (M= 6) 1,805 DOF I-DEAS (2-D), 24 × 24 mesh (linear shell), 3,553 DOF ABAQUS (2-D), 24 × 24 mesh, S4R, 3,553 DOF MECHANICA, 2-D;p= 6 NASTRAN, 24 × 24 mesh, CQUAD4, 3,553 DOF 0.541 0.536 0.536 0.536 0.536 –0.541 –0.536 –0.536 –0.536 –0.536 0.184 0.184 0.184 0.183 0.184 –0.185 –0.184 –0.184 –0.183 –0.184 –0.0216 –0.0215 –0.0215 –0.0215 –0.0215 0.0216 0.0215 0.0215 0.0215 0.0215 0.393 0.394 0.392 0.388 0.392 0.0842 0.104 0.104 0.025 0.104 DOF, degrees of freedom; M or p, degree of polynomial Fig. 8 Comparisons between 2-D and 3-D solutions for a/h= 50. (a) σ xx . (b) σ yy . (c) τ yz . (d) τ xz . (e) τ xy . In the legend for these curves, M is the degree of the polynomial. Example 2: Uniaxial Extension of a Laminated Plate. This example focuses on the free- edge effects in a [0/90] s laminated plate subjected to uniaxial extension (see Fig. 9). Pagano (Ref 31) presented the closed-form solution to this classical problem in 1974. Later on, Pagano and Soni (Ref 32) also solved this problem using a global- local variational model. Here, the results from the SAVE FE analysis program, performed using a 1 × 20 × 12 mesh of variable- order rectangular solid elements (Ref 21, 22), are presented. Fig. 9 A [0/90] S laminate subjected to uniform axial extension Due to the symmetry of geometrical and materials properties and the applied loading, only one-eighth of the configuration of the [0/90] s laminated plate (a=b= 4 h; see Fig. 9) need be analyzed. The uniform extension of the laminated plate is achieved by applying a uniform axial displacement at the ends x=a. For the purpose of analyses, the following lamina elastic constants are taken (Ref 31): E 1 = 138 GPa (20 × 10 6 psi) E 2 =E 3 = 14.5 GPa (2.1 × 10 6 psi) G 12 =G 13 =G 23 = 5.9 GPa (0.85 × 10 6 psi) ν 12 =ν 13 =ν 23 = 0.21 The distributions of interlaminar stresses as obtained from the analysis are plotted along the y-direction at x/a= 0.5. All the stress components are normalized by the applied axial strain. The distributions of the transverse normal stress, σ zz , as obtained from the analysis at the midsurface of the [0/90] s laminate in the 90° layer, are shown in Fig. 10. The normalized peak value of 2.0 GPa (0.29 × 10 6 psi) for this stress component, which occurs at the free edge y/b= 1, compares very well with those obtained by Pagano (Ref 31) and Pagano and Soni (Ref 32). Next, the distributions of the normalized transverse normal stress component, σ zz , as obtained from the analysis at the interface of the 0/90 layers, are shown in Fig. 11. Note that in Fig. 11 the numerical results from both 0° layer and 90° layer are plotted separately but appear superimposed. As is shown in Fig. 11, the continuity of the transverse normal stress component, σ zz , is satisfied extremely well. Similar observations are made regarding the distributions of the transverse shear stress component, τ yz , as shown in Fig. 12. Note the high gradients of interlaminar stresses that occur in the vicinity of the free-edge at y/b= 1 (see Fig. 11 and 12). These stresses are normally the primary cause of delamination failure in multilayered laminated structures. Fig. 10 Interlaminar normal stress, σ zz , at the midsurface Fig. 11 Interlaminar normal stress, σ zz , at the 0/90 interface Fig. 12 Interlaminar shear stress, τ yz , at the 0/90 interface References cited in this section N. Rastogi, “Variable-Order Solid Elements for Three-Dimensional Linear Elastic Structural Analysis,” AIAA Paper 99-1410, Proc. of the American Institute of Aeronautics and Astronautics/American Society of Mechanical Engineers/American Society of Civil Engineers/ American Helicopter Society/ American Society of Composites 40th Structures, Structural Dynamics and Materials Conference, 12– 16 April 1999 (St. Louis, MO) 22. N. Rastogi, “Three-Dimensional Analysis of Composite Structures Using Variable-Order Solid Elements,” AIAA Paper 99-1226, Proc. of the American Institute of Aeronautics and Astronautics/American Society of Mechanical Engineers/American Society of Civil Engineers/AHS/ASC 40th Structures, Structural Dynamics and Materials Conference, 12–16 April 1999 (St. Louis, MO) 30. N.J. Pagano, Exact Solutions for Bi-Directional Composites and Sandwich Plates, J. Compos. Mater., Vol 4, 1970, p 20–34 31. N.J. Pagano, On the Calculation of Interlaminar Stresses in Composite Laminate, J. Compos. Mater., Vol 8, 1974, p 65–77 32. N.J. Pagano and S.R Soni, “Models for Studying Free-Edge Effects,”Interlaminar Response of Composite Materials, Composite Materials Series, Vol 5, N.J. Pagano, Ed., Elsevier Science Publishing Company, Inc., New York, NY, 1989, p 1–68 33. ABAQUS/standard version 5.8, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI [...]... Helicopter Society/American Society of Composites Structures, Structural Dynamics and Materials Conference, 20–23 April 1998 (Long Beach, CA) 20 R.A Naik, “Analysis of Woven and Braided Fabric Reinforced Composites, ” NASA- CR-194930, June 1994 21 N Rastogi, “Variable-Order Solid Elements for Three-Dimensional Linear Elastic Structural Analysis,” AIAA Paper 9 9-1 410, Proc of the American Institute of... Structures, Taylor & Francis, 1993 26 E.R Johnson and N Rastogi, “Load Transfer in the Stiffener-to-Skin Joints of a Pressurized Fuselage,” NASA-CR-19 861 0, May 1995 27 M.B Woodson, E.R Johnson, and R.T Haftka, “A Vlasov Theory for Laminated Circular Open Beams with Thin-Walled Open Sections,” AIAA Paper 9 3-1 61 9, Proc of the American Institute of Aeronautics and Astronautics/American Society of Mechanical... Society of Composites 34th Structures, Structural Dynamics and Materials Conference (LaJolla, CA), 1993 28 V.Z Vlasov, Thin-Walled Elastic Beams, National Science Foundation, 1 961 29 N.R Bauld and L Tzeng, A Vlasov Theory for Fiber-Reinforced Beams with Thin- Walled Open Cross-Sections, Int J Solids Struct., Vol 20 (No 3), 1984, p 277–297 30 N.J Pagano, Exact Solutions for Bi-Directional Composites. .. performed with V-Lab The Material and Laminate Labs provide the standard capabilities of a composites program; the Joint Lab is an added analysis type Using the Joint Lab can be a bit confusing and so is one case where it pays to first read the Help file The analysis starts by defining joint materials The module can examine composite-to- composite or composite-tometal joints For composite-to-composite joints,... Think Composites software package, developed by Dr Stephen Tsai, includes the Mic- Mac spreadsheet, GenLam, and LamRank Information about the package, plus a “lite” version of Mic-Mac, is available from http:// www.thinkComp.com That site also includes a downloadable version of Theory of Composites Design (Ref 5) in PDF format Documentation for the Think Composites software is in the textbook Mic-Mac... /www.mecheng.asme.org): A large collection of engineering shareware and freeware Browse by category or search by keyword E -Composites. com (http://www.e -composites. com/software_store.htm): On-line composite software store divided into two categories: Analysis and Design, and Manufacturing ER-Online (http://www.er-online.co.uk/software.htm) Brief reviews of engineering programs with links to publisher home pages Lycos... References cited in this section 6 Alibre.com, Alibre Design, http://www.alibre.com 7 Engineering-e.com, e.visualNastran 4D, http:// www.engineering-e.com/computing/ Computer Programs Barry J Berenberg, Caldera Composites References 1 W.C Young and R.G Budynas, Roark's Formulas for Stress and Strain, 7th ed., McGraw- Hill, 2001 2 Composite Materials Handbook, MIL- HDBK-17, U.S Army Research Laboratory,... special types of laminates can be defined: θ-laminates and p-laminates Theta-laminates can have any number of plies specified as a variable angle, or “θ.” In other words, θ-laminates define a family of laminates, such as [0/±θ / 90]S When running a basic laminate analysis, θ is automatically varied over a range, similar to a multiple-angle analysis For p-laminates, the total thickness of the laminate... Engineers/American Society of Civil Engineers/ American Helicopter Society/ American Society of Composites 40th Structures, Structural Dynamics and Materials Conference, 12– 16 April 1999 (St Louis, MO) 22 N Rastogi, “Three-Dimensional Analysis of Composite Structures Using Variable-Order Solid Elements,” AIAA Paper 9 9-1 2 26, Proc of the American Institute of Aeronautics and Astronautics/American Society of Mechanical... stability; thin- and thick-wall pressure vessels) Notched laminates; ply drop-offs; free-edge effects (builtin finiteelement model) Design Thermal curvature; laminate surveys (tabular form of carpet plots); laminate parametric analysis (family studies) Micromechanics: plot constants versus volume/ weight fraction, fiber direction; multiple materials.Laminates: carpet plots; family studies; θ-laminates (variable . 0.5 36 0.5 36 0.5 36 0.5 36 –0.541 –0.5 36 –0.5 36 –0.5 36 –0.5 36 0.184 0.184 0.184 0.183 0.184 –0.185 –0.184 –0.184 –0.183 –0.184 –0. 02 16 –0. 0215 –0. 0215 –0. 0215 –0. 0215 . × 12 × 2, C3D20, 11 ,66 4 DOF NASTRAN, 12 × 12 × 2, CHEXA20, 11 ,66 4 DOF I-DEAS, 12 × 12 × 2 (parabolic), 11 ,66 4 DOF I-DEAS, 6 × 6 × 2 (parabolic), 2,9 16 DOF I-DEAS, 12 × 12 × 2 (linear),. C3D20, 61 ,200 DOF ABAQUS, 12 × 12 × 2, C3D20, 11 ,66 4 DOF NASTRAN, 12 × 12 × 2, CHEXA20, 11 ,66 4 DOF I-DEAS, 12 × 12 × 2 (parabolic), 11 ,66 4 DOF I-DEAS, 6 × 6 × 2 (parabolic), 2,916