Research PapersNo Access

Isogeometric Stress, Vibration and Stability Analysis of In-Plane Laminated Composite Structures

S. Faroughi

Faculty of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

E-mail Address: sh.farughi@uut.ac.ir

Corresponding author.

Search for more papers by this author

E. Shafei

Faculty of Civil Engineering, Urmia University of Technology, Urmia, Iran

E-mail Address: e.shafei@uut.ac.ir

Search for more papers by this author

, and

D. Schillinger

Department of Civil, Environmental and Geo-Engineering, University of Minnesota, Twin Cities, USA

E-mail Address: dominik@umn.edu

Search for more papers by this author

https://doi.org/10.1142/S0219455418500700Cited by:7 (Source: Crossref)

Abstract

We present a computational study that develops isogeometric analysis based on higher-order smooth NURBS basis functions for the analysis of in-plane laminated composites. Focusing on the stress, vibration and stability analysis of angle-ply and cross-ply 2D structures, we compare the convergence of the strain energy error and selected stress components, eigen-frequencies and buckling loads according to overkill solutions. Our results clearly demonstrate that for in-plane laminated composite structures, isogeometric analysis is able to provide the same accuracy at a significantly reduced number of degrees of freedom with respect to standard $C^{0}$ finite elements. In particular, we observe that the smoothness of spline basis functions enables high-quality stress solutions, which are superior to the ones obtained with conventional finite elements.

Keywords:

Remember to check out the Most Cited Articles!
Remember to check out the structures

References

1. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis (CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida, 33431, 2004). Crossref, Google Scholar
2. Y. Wang, K. Lam and G. Liu, A strip element method for the transient analysis of symmetric laminated plates, Int. J. Solids Struct. 38 (2) (2001) 241–259. Crossref, Web of Science, Google Scholar
3. K. Liew, Y. Huang and J. Reddy, Vibration analysis of symmetrically laminated plates based on fsdt using the moving least squares differential quadrature method, Comput. Methods Appl. Mech. Eng. 192 (19) (2003) 2203–2222. Crossref, Web of Science, Google Scholar
4. K. Liew and X. Chen, Mesh-free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in-plane point loads, Int. J. Numer. Methods Eng. 60 (11) (2004) 1861–1877. Crossref, Web of Science, Google Scholar
5. A. Ferreira, C. Roque and P. Martins, Analysis of composite plates using higher-order shear deformation theory and a finite point formulation based on the multiquadric radial basis function method, Compos. B Eng. 34 (7) (2003) 627–636. Crossref, Web of Science, Google Scholar
6. A. Ferreira, A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates, Compos. Struct. 59 (3) (2003) 385–392. Crossref, Web of Science, Google Scholar
7. J. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, Int. J. Eng. Sci. 48 (11) (2010) 1507–1518. Crossref, Web of Science, Google Scholar
8. A. Khdeir and J. Reddy, Free vibration of cross-ply laminated beams with arbitrary boundary conditions, Int. J. Eng. Sci. 32 (12) (1994) 1971–1980. Crossref, Web of Science, Google Scholar
9. A. Khdeir and J. Redd, Buckling of cross-ply laminated beams with arbitrary boundary conditions, Compos. Struct. 37 (1) (1997) 1–3. Crossref, Web of Science, Google Scholar
10. S. Soni and C. A. Rao, Vibrations of orthotropic rectangular plates under inplane forces, Comput. Struct. 4 (5) (1974) 1105–1115. Crossref, Google Scholar
11. H. Matsunaga, Vibration of cross-ply laminated composite plates subjected to initial in-plane stresses, Thin Wall. Struct. 40 (7) (2002) 557–571. Crossref, Web of Science, Google Scholar
12. T. J. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Eng. 194 (39) (2005) 4135–4195. Crossref, Web of Science, Google Scholar
13. L. Piegl and W. Tiller, The NURBS Book (Springer Science & Business Media, Springer-Verlag Berlin, Heidelberg, 2012). Google Scholar
14. J. A. Cottrell, T. J. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA (John Wiley & Sons, New York, 2009). Crossref, Google Scholar
15. J. Kiendl, K.-U. Bletzinger, J. Linhard and R. Wüchner, Isogeometric shell analysis with Kirchhoff–Love elements, Comput. Methods Appl. Mech. Eng. 198 (49) (2009) 3902–3914. Crossref, Web of Science, Google Scholar
16. D. Benson, Y. Bazilevs, M. C. Hsu and T. Hughes, Isogeometric shell analysis: The Reissner–Mindlin shell, Comput. Methods Appl. Mech. Eng. 199 (5) (2010) 276–289. Crossref, Web of Science, Google Scholar
17. R. Echter, B. Oesterle and M. Bischoff, A hierarchic family of isogeometric shell finite elements, Comput. Methods Appl. Mech. Eng. 254 (2013) 170–180. Crossref, Web of Science, Google Scholar
18. Y. Guo and M. Ruess, Nitsche’s method for a coupling of isogeometric thin shells and blended shell structures, Comput. Methods Appl. Mech. Eng. 284 (2015) 881–905. Crossref, Web of Science, Google Scholar
19. Y. Guo, M. Ruess and D. Schillinger, A parameter-free variational coupling approach for trimmed isogeometric thin shells, Comput. Mech. 59(4) (2016) 693–715. https://doi.org/10.1007/s00466–016–1368–x Crossref, Web of Science, Google Scholar
20. A. Sankar, S. Natarajan, T. Merzouki and M. Ganapathi, Nonlinear dynamic thermal buckling of sandwich spherical and conical shells with cnt reinforced facesheets, Int. J. Struct. Stab. Dyn. (2016) 1750100. Online ready, https://doi.org/10.1142/S0219455417501000. Web of Science, Google Scholar
21. M. A. Scott, R. N. Simpson, J. A. Evans, S. Lipton, S. P. Bordas, T. J. Hughes and T. W. Sederberg, Isogeometric boundary element analysis using unstructured T-splines, Comput. Methods Appl. Mech. Eng. 254 (2013) 197–221. Crossref, Web of Science, Google Scholar
22. R. N. Simpson, M. A. Scott, M. Taus, D. C. Thomas and H. Lian, Acoustic isogeometric boundary element analysis, Comput. Methods Appl. Mech. Eng. 269 (2014) 265–290. Crossref, Web of Science, Google Scholar
23. L. Heltai, J. Kiendl, A. DeSimone and A. Reali, A natural framework for isogeometric fluid–structure interaction based on bem–shell coupling, Comput. Methods Appl. Mech. Eng. 316(1) (2017) 522–546. Crossref, Web of Science, Google Scholar
24. H. Gómez, V. M. Calo, Y. Bazilevs and T. J. Hughes, Isogeometric analysis of the Cahn–Hilliard phase-field model, Comput. Methods Appl. Mech. Eng. 197 (49) (2008) 4333–4352. Crossref, Web of Science, Google Scholar
25. G. Vilanova, I. Colominas and H. Gomez, Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis, Int. J. Numer. Methods Biomed. Eng. 29 (10) (2013) 1015–1037. Crossref, Web of Science, Google Scholar
26. D. Schillinger, M. J. Borden and H. K. Stolarski, Isogeometric collocation for phase-field fracture models, Comput. Methods Appl. Mech. Eng. 284 (2015) 583–610. Crossref, Web of Science, Google Scholar
27. Y. Zhao, P. Stein and B. X. Xu, Isogeometric analysis of mechanically coupled Cahn–Hilliard phase segregation in hyperelastic electrodes of li-ion batteries, Comput. Methods Appl. Mech. Eng. 297 (2015) 325–347. Crossref, Web of Science, Google Scholar
28. M. J. Borden, T. J. Hughes, C. M. Landis, A. Anvari and I. J. Lee, A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects, Comput. Methods Appl. Mech. Eng. 312 (2016) 130–166. Crossref, Web of Science, Google Scholar
29. A. S. Sayyad and Y. M. Ghugal, On the buckling of isotropic, transversely isotropic and laminated composite rectangular plates, Int. J. Struct. Stab. Dyn. 14 (7) (2014) 1450020. Link, Web of Science, Google Scholar
30. M. Biswal, S. Sahu and A. Asha, Dynamic stability of woven fiber laminated composite shallow shells in hygrothermal environment, Int. J. Struct. Stab. Dyn. (2016) 1750084. Online ready, https://doi.org/10.1142/S0219455417500845. Web of Science, Google Scholar
31. R. Al-Masri and H. A. Rasheed, Analytical and finite element buckling solutions of fixed–fixed anisotropic laminated composite columns under axial compression, Int. J. Struct. Stab. Dyn. (2017) 1750103. Online ready, https://doi.org/10.1142/S0219455417501036. Link, Web of Science, Google Scholar
32. N. George, P. Jeyaraj and S. Murigendrappa, Buckling and free vibration of nonuniformly heated functionally graded carbon nanotube reinforced polymer composite plate, Int. J. Struct. Stab. Dyn. 17(6) (2017) 1750064. Link, Web of Science, Google Scholar
33. H. Wu, S. Kitipornchai and J. Yang, Free vibration and buckling analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets, Int. J. Struct. Stab. Dyn. 15 (7) (2015) 1540011. Link, Web of Science, Google Scholar
34. T. Rajanna, S. Banerjee, Y. M. Desai and D. Prabhakaran, Effect of reinforced cutouts and ply-orientations on buckling behavior of composite panels subjected to non-uniform edge loads, Int. J. Struct. Stab. Dyn. (2017) 1850058. Web of Science, Google Scholar
35. F. Auricchio, L. B. Da Veiga, C. Lovadina and A. Reali, The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: Mixed FEMs versus NURBS-based approximations, Comput. Methods Appl. Mech. Eng. 199 (5) (2010) 314–323. Crossref, Web of Science, Google Scholar
36. F. Auricchio, L. B. da Veiga, J. Kiendl, C. Lovadina and A. Reali, Locking-free isogeometric collocation methods for spatial Timoshenko rods, Comput. Methods Appl. Mech. Eng. 263 (2013) 113–126. Crossref, Web of Science, Google Scholar
37. D. Schillinger, J. A. Evans, A. Reali, M. A. Scott and T. J. Hughes, Isogeometric collocation: Cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Eng. 267 (2013) 170–232. Crossref, Web of Science, Google Scholar
38. A. Reali and H. Gomez, An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates, Comput. Methods Appl. Mech. Eng. 284 (2015) 623–636. Crossref, Web of Science, Google Scholar
39. M. C. Hsu and Y. Bazilevs, Fluid–structure interaction modeling of wind turbines: Simulating the full machine, Comput. Mech. 50(6) (2012) 821–833. Crossref, Web of Science, Google Scholar
40. Y. Bazilevs, M. C. Hsu and M. Scott, Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines, Comput. Methods Appl. Mech. Eng. 249 (2012) 28–41. Crossref, Web of Science, Google Scholar
41. D. Kamensky, M. C. Hsu, D. Schillinger, J. A. Evans, A. Aggarwal, Y. Bazilevs, M. S. Sacks and T. J. Hughes, An immersogeometric variational framework for fluid–structure interaction: Application to bioprosthetic heart valves, Comput. Methods Appl. Mech. Eng. 284 (2015) 1005–1053. Crossref, Web of Science, Google Scholar
42. Y. Bazilevs, V. M. Calo, T. J. Hughes and Y. Zhang, Isogeometric fluid-structure interaction: Theory, algorithms, and computations, Comput. Mech. 43 (1) (2008) 3–37. Crossref, Web of Science, Google Scholar
43. M. J. Borden, M. A. Scott, J. A. Evans and T. J. Hughes, Isogeometric finite element data structures based on Bézier extraction of NURBS, Int. J. Numer. Methods Eng. 87 (1–5) (2011) 15–47. Crossref, Web of Science, Google Scholar
44. D. Schillinger, P. K. Ruthala and L. H. Nguyen, Lagrange extraction and projection for NURBS basis functions: A direct link between isogeometric and standard nodal finite element formulations, Int. J. Numer. Meth. Eng. 108(6) (2016) 515–534. Crossref, Web of Science, Google Scholar
45. C. H. Thai, H. Nguyen-Xuan, N. Nguyen-Thanh, T. Le, T. Nguyen-Thoi and T. Rabczuk, Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using nurbs-based isogeometric approach, Int. J. Numer. Methods Eng. 91 (6) (2012) 571–603. Crossref, Web of Science, Google Scholar
46. S. Shojaee, N. Valizadeh, E. Izadpanah, T. Bui and T.-V. Vu, Free vibration and buckling analysis of laminated composite plates using the NURBS-based isogeometric finite element method, Compos. Struct. 94 (5) (2012) 1677–1693. Crossref, Web of Science, Google Scholar
47. S. J. Lee and K. S. Park, Vibrations of Timoshenko beams with isogeometric approach, Appl. Math. Model. 37 (22) (2013) 9174–9190. Crossref, Web of Science, Google Scholar
48. S. Raknes, X. Deng, Y. Bazilevs, D. Benson, K. Mathisen and T. Kvamsdal, Isogeometric rotation-free bending-stabilized cables: Statics, dynamics, bending strips and coupling with shells, Comput. Methods Appl. Mech. Eng. 263 (2013) 127–143. Crossref, Web of Science, Google Scholar
49. J. Cottrell, T. Hughes and A. Reali, Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Eng. 196 (41) (2007) 4160–4183. Crossref, Web of Science, Google Scholar
50. T. J. Hughes, J. A. Evans and A. Reali, Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems, Comput. Methods Appl. Mech. Eng. 272 (2014) 290–320. Crossref, Web of Science, Google Scholar
51. A. T. Luu, N. I. Kim and J. Lee, NURBS-based isogeometric vibration analysis of generally laminated deep curved beams with variable curvature, Compos. Struct. 119 (2015) 150–165. Crossref, Web of Science, Google Scholar
52. K. Luo, C. Liu, Q. Tian and H. Hu, An efficient model reduction method for buckling analyses of thin shells based on IGA, Comput. Methods Appl. Mech. Eng. 309 (2016) 243–268. Crossref, Web of Science, Google Scholar
53. L. B. Nguyen, C. H. Thai and H. Nguyen-Xuan, A generalized unconstrained theory and isogeometric finite element analysis based on Bézier extraction for laminated composite plates, Eng. Comput. 32 (3) (2016) 457–475. Crossref, Web of Science, Google Scholar
54. G. E. Farin, Curves and Surfaces for CAGD: A Practical Guide (Morgan Kaufmann, 2002). Google Scholar
55. T. J. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (Courier Corporation, Englewood Cliffs, N.J., Prentice-Hall, 2012). Google Scholar
56. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method: Solid Mechanics (Butterworth-Heinemann, Linacre House, Jordan Hills, Oxford, OX2-8DP, 2000). Google Scholar
57. D. Schillinger, S. J. Hossain and T. J. Hughes, Reduced Bézier element quadrature rules for quadratic and cubic splines in isogeometric analysis, Comput. Methods Appl. Mech. Eng. 277 (2014) 1–45. Crossref, Web of Science, Google Scholar
58. M. Bartoň and V. Calo, Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis, Comput. Methods Appl. Mech. Eng. 305 (2016) 217–240. Crossref, Web of Science, Google Scholar
59. R. R. Hiemstra, F. Calabrò, D. Schillinger and T. J. Hughes, Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis, Comput. Methods Appl. Mech. Eng. 316(1) (2017) 966–1004. Crossref, Web of Science, Google Scholar
60. M. Hajianmaleki and M. S. Qatu, Static and vibration analyses of thick, generally laminated deep curved beams with different boundary conditions, Compos. B Eng. 43 (4) (2012) 1767–1775. Crossref, Web of Science, Google Scholar
61. J. O. Hallquist et al., LS-DYNA theory manual, Livermore Software Technology Corporation 3 (2006) 25–31. Google Scholar
62. D. Schillinger, L. Dede, M. A. Scott, J. A. Evans, M. J. Borden, E. Rank and T. J. Hughes, An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Methods Appl. Mech. Eng. 249 (2012) 116–150. Crossref, Web of Science, Google Scholar