Abstract
Semi-analytical elasticity solutions for bending of angle-ply laminates in cylindrical bending are presented using the state-space-based differential quadrature method (SSDQM). Partial differential state equation is derived from the basic equations of elasticity based on the state space concept. Then, the differential quadrature (DQ) technique is introduced to discretize the longitudinal domain of the plate so that a series of ordinary differential state equations are obtained at the discrete points. Meanwhile, the edge constrained conditions are handled directly using the stress and displacement components without the Saint-Venant principle. The thickness domain is solved analytically based on the state space formalism along with the continuity conditions at interfaces. The present method is validated by comparing the results to the exact solutions of Pagano’s problem. Numerical results for fully clamped thick laminates are presented, and the influences of ply angle on stress distributions are discussed.
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References
Bellman, R., Kashef, B.G., Casti, J., 1972. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics, 10(1):40–52. [doi:10.1016/0021-9991(72)90089-7]
Bert, C.W., Malik, M., 1996. Differential quadrature method in computational mechanics: A review. Applied Mechanics Review, 49(1):1–28.
Bian, Z.G., Chen, W.Q., Lim, C.W., Zhang, N., 2005. Analytical solutions for single-and multi-span functionally graded plates in cylindrical bending. International Journal of Solids and Structures, 42(24–25):6433–6456. [doi:10.1016/j.ijsolstr.2005.04.032]
Chen, W.Q., Lee, K.Y., 2004a. On free vibration of cross-ply laminates in cylindrical bending. Journal of Sound and Vibration, 273(3):667–676. [doi:10.1016/j.jsv.2003.08.003]
Chen, W.Q., Lee, K.Y., 2004b. Three-dimensional exact analysis of angle-ply laminates in cylindrical bending with interfacial damage via state-space method. Composite Structures, 64(3–4):275–283. [doi:10.1016/j.compstruct.2003.08.010]
Chen, W.Q., Bian, Z.G., Ding, H.J., 2003a. Three-dimensional analysis of a thick FGM rectangular plate in thermal environment. Journal of Zhejiang University SCIENCE, 4(1):1–7.
Chen, W.Q., Lv, C.F., Bian, Z.G., 2003b. Elasticity solution for free vibration of laminated beams. Composite Structures, 62(1):75–82. [doi:10.1016/S0263-8223(03)00086-2]
Chen, W.Q., Lv, C.F., Bian, Z.G., 2004a. Free vibration analysis of generally laminated beams via state-space-based differential quadrature. Composite Structures, 63(3–4):417–425. [doi:10.1016/S0263-8223(03)00190-9]
Chen, W.Q., Ying, J., Cai, J.B., Ye, G.R., 2004b. Benchmark solution of imperfect angle-ply laminated rectangular plates in cylindrical bending with surface piezoelectric layers as actuator and sensor. Computers and Structures, 82(22):1773–1784. [doi:10.1016/j.compstruc.2004.05.011]
Lü, C.F., Huang, Z.Y., Chen, W.Q., 2007. Semi-analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending. Journal of Sound and Vibration, 304(3–5):987–995. [doi:10.1016/j.jsv.2007.03.023]
Messina, A., 2001. Two generalized higher order theories in free vibration studies of multi-layered plates. Journal of Sound and Vibration, 242(1):125–150. [doi:10.1006/jsvi.2000.3364]
Messina, A., Soldatos, K.P., 2002. A general vibration model of angle-ply laminated plates that accounts for the continuity of interlaminar stresses. International Journal of Solids and Structures, 39(3):617–635. [doi:10.1016/S0020-7683(01)00169-X]
Pagano, N.J., 1969. Exact solutions for composite laminates in cylindrical bending. Journal of Composite Materials, 3(3):398–411. [doi:10.1177/002199836900300304]
Pagano, N.J., 1970. Influence of shear coupling in cylindrical bending of anisotropic laminates. Journal of Composite Materials, 4(3):330–343. [doi:10.1177/002199837000400305]
Pu, J.P., Zheng, J.J., 2006. Structural dynamic responses analysis applying differential quadrature method. Journal of Zhejiang University SCIENCE A, 7(11):1831–1838. [doi:10.1631/jzus.2006.A1831]
Shu, C., 2000. Differential Quadrature and Its Application in Engineering. Springer-Verlag, London.
Shu, C., Richards, B.E., 1992. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 15(7):791–798. [10.1002/fld.1650150704]
Shu, X.P., Soldatos, K.P., 2000. Cylindrical bending of angle-ply laminates subjected to different sets of edge boundary conditions. International Journal of Solids and Structures, 37(31):4289–4307. [doi:10.1016/S0020-7683(99)00144-4]
Vel, S.S., Batra, R.C., 2000. The generalized plane strain deformations of thick anisotropic composite laminated plates. International Journal of Solids and Structures, 37(5):715–733. [doi:10.1016/S0020-7683(99)00040-2]
Vinson, J.R., Sierakowski, R.L., 2002. The Behavior of Structures Composed of Composite Materials. Kluwer, London.
Wang, H.Z., Chen, Y.M., Huang, B., 2003. Computation of one-dimensional consolidation of double layered ground using differential quadrature method. Journal of Zhejiang University SCIENCE, 4(2):195–201.
Yan, W., Chen, W.Q., 2004. Time-dependent response of laminated isotropic strips with viscoelastic interfaces. Journal of Zhejiang University SCIENCE, 5(11):1318–1321. [doi:10.1631/jzus.2004.1318]
Yan, W., Ying, J., Chen, W.Q., 2007. The behavior of angle-ply laminated cylindrical shells with viscoelastic interfaces in cylindrical bending. Composite Structures, 78(4):551–559. [doi:10.1016/j.compstruct.2005.11.017]
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Project supported by the National Natural Science Foundation of China (No. 10432030), the China Postdoctoral Science Foundation (No. 20060401071) and the Program for New Century Excellent Talent in University of China (No. NCET-05-0510)
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Lü, Cf., Lim, C.W. & Xu, F. Stress analysis of anisotropic thick laminates in cylindrical bending using a semi-analytical approach. J. Zhejiang Univ. - Sci. A 8, 1740–1745 (2007). https://doi.org/10.1631/jzus.2007.A1740
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DOI: https://doi.org/10.1631/jzus.2007.A1740
Key words
- Semi-analytical elasticity solution
- State-space-based differential quadrature method (SSDQM)
- Angle-ply laminates
- Cylindrical bending