Stress Analysis of Laminated HSDT Beams Considering Bending Extension Coupling

Abstract

This study demonstrates a mixed finite element formulation procedure for the bending and stress analyses of laminated composite beams. The finite element method is based on the Hellinger-Reissner variational principle, while the beam assumptions are based on the Higher Order Shear Deformation Theory (HSDT). Reddy’s shear function is employed for the beam theory where the beam is discretized by two-noded linear elements. The displacements and stress resultants are obtained directly at the nodes according to the proposed mixed formulation. The validation of current study is performed by comparison and convergence analyzes for various lamination cases under different boundary conditions.

Keywords

• Higher Order theory
• laminated composite beam
• Hellinger-Reissner
• mixed finite element formulation
• stress analysis

References

1. V.V. Vasiliev, E.V. Morozov, Advanced Mechanics of Composite Materials and Structural Elements, Elsevier, London, UNITED KINGDOM, 2013. http://ebookcentral.proquest.com/lib/itup/detail.action?docID=1221537 (accessed April 9, 2021).
2. İ. Çömez, U.N. Aribas, A. Kutlu, M.H. Omurtag, An Exact Elasticity Solution for Monoclinic Functionally Graded Beams, Arab J Sci Eng. 46 (2021) 5135–5155. https://doi.org/10.1007/s13369-021-05434-9.
3. M. Dorduncu, Peridynamic modeling of delaminations in laminated composite beams using refined zigzag theory, Theoretical and Applied Fracture Mechanics. 112 (2021) 102832. https://doi.org/10.1016/j.tafmec.2020.102832.
4. M. Dorduncu, Stress analysis of laminated composite beams using refined zigzag theory and peridynamic differential operator, Composite Structures. 218 (2019) 193–203. https://doi.org/10.1016/j.compstruct.2019.03.035.
5. L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, apud Marcum-Michaelem Bousquet, 1744.
6. J. Bernoulli, Curvatura laminae elasticae, Acta Eruditorum Lipsiae. 1694 (1964) 262–276.
7. B.A. Boley, On the Accuracy of the Bernoulli-Euler Theory for Beams of Variable Section, Journal of Applied Mechanics. 30 (1963) 373–378. https://doi.org/10.1115/1.3636564.
8. A.H. Modaress-Aval, F. Bakhtiari-Nejad, E.H. Dowell, H. Shahverdi, H. Rostami, D.A. Peters, Aeroelastic analysis of cantilever plates using Peters’ aerodynamic model, and the influence of choosing beam or plate theories as the structural model, Journal of Fluids and Structures. 96 (2020) 103010. https://doi.org/10.1016/j.jfluidstructs.2020.103010.

9. S.P. Timoshenko, LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars, (1921) 3.
10. W. Wagner, F. Gruttmann, A displacement method for the analysis of flexural shear stresses in thin-walled isotropic composite beams, Computers & Structures. 80 (2002) 1843–1851. https://doi.org/10.1016/S0045-7949(02)00223-7.
11. U. Lee, I. Jang, Spectral element model for axially loaded bending–shear–torsion coupled composite Timoshenko beams, Composite Structures. 92 (2010) 2860–2870. https://doi.org/10.1016/j.compstruct.2010.04.012.
12. J.N. Reddy, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics. 51 (1984) 745–752. https://doi.org/10.1115/1.3167719.
13. E. Reissner, On transverse bending of plates, including the effect of transverse shear deformation, International Journal of Solids and Structures. 11 (1974) 569–573.
14. K.P. Soldatos, A transverse shear deformation theory for homogeneous monoclinic plates, Acta Mechanica. 94 (1992) 195–220. https://doi.org/10.1007/BF01176650.
15. A.J.M. Ferreira, C.M.C. Roque, P.A.L.S. Martins, Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates, Composite Structures. 66 (2004) 287–293. https://doi.org/10.1016/j.compstruct.2004.04.050.
16. M.V.V.S. Murthy, D. Roy Mahapatra, K. Badarinarayana, S. Gopalakrishnan, A refined higher order finite element for asymmetric composite beams, Composite Structures. 67 (2005) 27–35. https://doi.org/10.1016/j.compstruct.2004.01.005.
17. E. Madenci, Yüksek Mertebe Kayma Deformasyon Teorisine Dayali Çapraz Tabakali Kompozit Plaklarin Karisik Sonlu Eleman Yöntemi İle Anali̇zi̇, Selcuk Universitesi, 2016.
18. K. Chandrashekhara, K.M. Bangera, Free vibration of composite beams using a refined shear flexible beam element, Computers & Structures. 43 (1992) 719–727. https://doi.org/10.1016/0045-7949(92)90514-Z.
19. D.K. Maiti, P.K. Sinha, Bending and free vibration analysis of shear deformable laminated composite beams by finite element method, Composite Structures. 29 (1994) 421–431.
20. S. Xiaoping, S. Liangxin, An improved simple higher-order theory for laminated composite plates, Computers & Structures. 50 (1994) 231–236. https://doi.org/10.1016/0045-7949(94)90298-4.
21. A.M. Zenkour, Transverse shear and normal deformation theory for bending analysis of laminated and sandwich elastic beams, Mech. of Adv. Mat. & Structures. 6 (1999) 267–283. https://doi.org/10.1080/107594199305566.
22. H. Matsunaga, Vibration and buckling of multilayered composite beams according to higher order deformation theories, Journal of Sound and Vibration. 246 (2001) 47–62. https://doi.org/10.1006/jsvi.2000.3627.
23. P. Subramanian, Dynamic analysis of laminated composite beams using higher order theories and finite elements, Composite Structures. 73 (2006) 342–353. https://doi.org/10.1016/j.compstruct.2005.02.002.
24. W. Zhen, C. Wanji, An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams, Composite Structures. 84 (2008) 337–349. https://doi.org/10.1016/j.compstruct.2007.10.005.
25. A. Ozutok, E. Madenci, Static analysis of laminated composite beams based on higher-order shear deformation theory by using mixed-type finite element method, Int. J. Mech. Sci. 130 (2017) 234–243. https://doi.org/10.1016/j.ijmecsci.2017.06.013.
26. A. Kutlu, M. Dorduncu, T. Rabczuk, A novel mixed finite element formulation based on the refined zigzag theory for the stress analysis of laminated composite plates, Composite Structures. 267 (2021) 113886. https://doi.org/10.1016/j.compstruct.2021.113886.
27. A. Kutlu, Mixed finite element formulation for bending of laminated beams using the refined zigzag theory, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications. 235 (2021) 1712–1722. https://doi.org/10.1177/14644207211018839.
28. A. Kutlu, G. Meschke, M.H. Omurtag, A new mixed finite-element approach for the elastoplastic analysis of Mindlin plates, J Eng Math. 99 (2016) 137–155. https://doi.org/10.1007/s10665-015-9825-7.
29. U.N. Aribas, M. Ermis, N. Eratli, M.H. Omurtag, The static and dynamic analyses of warping included composite exact conical helix by mixed FEM | Elsevier Enhanced Reader, Composites Part B: Engineering. 160 (2019) 285–297. https://doi.org/10.1016/j.compositesb.2018.10.018.
30. U.N. Aribas, M. Ermis, A. Kutlu, N. Eratli, M.H. Omurtag, Forced vibration analysis of composite-geometrically exact elliptical cone helices via mixed FEM, Mechanics of Advanced Materials and Structures. (2020) 1–19. https://doi.org/10.1080/15376494.2020.1824048.
31. U.N. Aribas, M. Ermis, M.H. Omurtag, The static and stress analyses of axially functionally graded exact super-elliptical beams via mixed FEM, Arch Appl Mech. 91 (2021) 4783–4796. https://doi.org/10.1007/s00419-021-02033-w.
32. D. Shao, S. Hu, Q. Wang, F. Pang, Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions, Composites Part B: Engineering. 108 (2017) 75–90. https://doi.org/10.1016/j.compositesb.2016.09.093.
33. A.S. Sayyad, Y.M. Ghugal, Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures. 171 (2017) 486–504. https://doi.org/10.1016/j.compstruct.2017.03.053.
34. Y. Bab, A. Kutlu, Mixed finite element formulation based on higher order theory for stress calculation of laminated composite beams, in: Proceedings 22. National Mechanics Congress, Adana, Turkey, September 06-10.
35. P. Shi, C. Dong, F. Sun, W. Liu, Q. Hu, A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis, Composite Structures. 204 (2018) 342–358. https://doi.org/10.1016/j.compstruct.2018.07.080.
36. R.M. Jones, Mechanics of composite materials, CRC Press, 1999.
37. E. Hellinger, Die Allgemeinen Ansätze der Mechanik der Kontinua, in: F. Klein, Conr. Müller (Eds.), Mechanik, Vieweg+Teubner Verlag, Wiesbaden, 1907: pp. 601–694. https://doi.org/10.1007/978-3-663-16028-1_9.
38. E. Reissner, On a Variational Theorem in Elasticity, Journal of Mathematics and Physics. 29 (1950) 90–95. https://doi.org/10.1002/sapm195029190.
39. N.J. Pagano, Exact solutions for rectangular bidirectional composites and sandwich plates, (n.d.) 15.
40. K.A. Hasim, Isogeometric static analysis of laminated composite plane beams by using refined zigzag theory, Composite Structures. 186 (2018) 365–374. https://doi.org/10.1016/j.compstruct.2017.12.033.
41. P. Vidal, O. Polit, A sine finite element using a zig-zag function for the analysis of laminated composite beams, Composites Part B: Engineering. 42 (2011) 1671–1682. https://doi.org/10.1016/j.compositesb.2011.03.012.
42. A.A. Khdeir, J.N. Reddy, An exact solution for the bending of thin and thick cross-ply laminated beams, Composite Structures. 37 (1997) 195–203.
43. S. Kapuria, P.C. Dumir, N.K. Jain, Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams, Composite Structures. 64 (2004) 317–327. https://doi.org/10.1016/j.compstruct.2003.08.013.