Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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820 YING LI AND CHANG-JUN CHENG [Bîrsan 2003] M. Bîrsan, “A bending theory of porous thermoelastic plates”, J. Therm. Stresses 26:1 (2003), 67–90. [Cheng et al. 2006] C.-J. Cheng, D.-F. Sheng, and J.-J. Li, “Quasi-static analysis for viscoelastic Timoshenko beams with damage”, Appl. Math. Mech. 27:3 (2006), 295–304. [Chirit,ă and Ciarletta 2008] S. Chirit,ă and M. Ciarletta, “On the structural stability of thermoelastic model of porous media”, Math. Methods Appl. Sci. 31:1 (2008), 19–34. [Chirit,ă and Scalia 2001] S. Chirit,ă and A. Scalia, “On the spatial and temporal behavior in linear thermoelasticity of materials with voids”, J. Therm. Stresses 24:5 (2001), 433–455. [Ciarletta et al. 2007] M. Ciarletta, B. Straughan, and V. Zampoli, “Thermo-poroacoustic acceleration waves in elastic materials with voids without energy dissipation”, Int. J. Eng. Sci. 45:9 (2007), 736–743. [Cicco and Diaco 2002] S. D. Cicco and M. Diaco, “A theory of thermoelastic materials with voids without energy dissipation”, J. Therm. Stresses 25:5 (2002), 493–503. [Cowin and Nunziato 1983] S. C. Cowin and J. W. Nunziato, “Linear elastic materials with voids”, J. Elasticity 13:2 (1983), 125–147. [Iesan 1986] D. Iesan, “A theory of thermoelastic materials with voids”, Acta Mech. 60:1-2 (1986), 67–89. [Kumar and Rani 2005] R. Kumar and L. Rani, “Interaction due to mechanical and thermal sources in thermoelastic half-space with voids”, J. Vib. Control 11:4 (2005), 499–517. [Luo et al. 2002] E. Luo, J.-S. Kuang, W.-J. Huang, and Z.-G. Luo, “Unconventional Hamilton-type variational principles for nonlinear coupled thermoelastodynamics”, Sci. China, A 45:6 (2002), 783–794. [Puri and Cowin 1985] P. Puri and S. C. Cowin, “Plane waves in linear elastic materials with voids”, J. Elasticity 15:2 (1985), 167–183. [Scalia et al. 2004] A. Scalia, A. Pompei, and S. Chirit,ă, “On the behavior of steady time-harmonic oscillations in thermoelastic materials with voids”, J. Therm. Stresses 27:3 (2004), 209–226. [Sharma et al. 2008] P. K. Sharma, D. Kaur, and J. N. Sharma, “Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids”, Int. J. Solids Struct. 45:18-19 (2008), 5049–5058. [Sheng and Cheng 2004] D.-F. Sheng and C.-J. Cheng, “Dynamical behaviors of nonlinear viscoelastic thick plates with damage”, Int. J. Solids Struct. 41:26 (2004), 7287–7308. [Singh 2007] B. Singh, “Wave propagation in a generalized thermoelastic material with voids”, Appl. Math. Comput. 189:1 (2007), 698–709. [Xi 2007] B. Q. Xi, “Analysis of the features of porous materials”, Science and technology information 23 (2007), 316. In Chinese. [Yu et al. 1996] T. X. Yu, J. L. Yang, S. R. Reid, and C. D. Austin, “Dynamic behaviour of elastic-plastic free-free beams subjected to impulsive loading”, Int. J. Solids Struct. 33:18 (1996), 2659–2680. [Zhang 2007] N.-H. Zhang, “Thermo-viscoelastic bending of functionally graded beam under thermal load”, Chinese Quarterly of Mechanics 28:2 (2007), 240–245. in Chinese. Received 20 Dec 2009. Revised 11 Apr 2010. Accepted 16 Apr 2010. YING LI: liying982@hotmail.com Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China CHANG-JUN CHENG: chjcheng@mail.shu.edu.cn Shanghai Institute of Applied Mathematics and Mechanics, Department of Mechanics, Shanghai University, Shanghai 200072, China
JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 5, No. 5, 2010 DYNAMIC STIFFNESS VIBRATION ANALYSIS OF THICK SPHERICAL SHELL SEGMENTS WITH VARIABLE THICKNESS ELIA EFRAIM AND MOSHE EISENBERGER A dynamic stiffness method is presented for determining the free vibration frequencies and mode shapes of thick spherical shell segments with variable thickness and different boundary conditions. The analysis uses the equations of the two-dimensional theory of elasticity, in which the effects of both transverse shear stresses and rotary inertia are accounted for. The displacement components are taken to be sinusoidal in time, periodic in the circumferential direction, constant through the thickness, and solved exactly in the meridional direction using the exact element method. The shape functions are derived from the exact solutions for the system of the differential equation of motion with variable coefficients. The dynamic stiffness matrix is derived from the exact shape functions and their derivatives. Highprecision numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and for several combinations of boundary conditions. Comparison is made with results of published research and with two- and three-dimensional finite element analyses. 1. Introduction Spherical shells are extensively used in civil, mechanical, aircraft, and naval structures. The free vibration of solid and hollow spheres has been a subject of study for more than a century. Historical reviews of the research into the vibrations of spherical shell are given in [Leissa 1973; Kang and Leissa 2000; Qatu 2002]. For segmented spherical shells very few studies can be found. Gautham and Ganesan [1992] used finite elements to study the free vibration analysis of open spherical shells, based on a thick (two-dimensional) shell theory. A thick shell finite element was derived and vibration frequencies were obtained for spherical caps with and without center cutout having simply supported or clamped boundary conditions. Lim et al. [1996] analyzed spherical shells with variable thickness using two-dimensional shell theory and the Ritz method, and the results were compared with finite element and experimental ones. For spherical shell segments based on three-dimensional analysis, Kang and Leissa [2000] used the Ritz method to obtain accurate frequencies for thick spherical shell segments of uniform or varying thickness. Their method does not yield exact solutions, but with proper use of displacement components in the form of algebraic polynomials, one is able to obtain frequency upper bounds, that are as close to the exact values as desired. Corrected results for the test cases in that paper appeared subsequently in [Kang and Leissa 2006]. In this paper the equations of motion for a thick spherical shell segment with variable thickness are derived. Then, these are solved for the dynamic stiffness matrix of the segment, and assembled for a complete structure. Keywords: vibrations, thick shell, spherical shell, variable thickness, dynamic stiffness, exact element method. Support given this research to the first author by the Technion – Israel Institute of Technology is gratefully acknowledged. 821
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