Abstract—Geometrically nonlinear axisymmetric bending of a shallow spherical shell with a point support at the apex under linearly varying axisymmetric load was investigated numerically. The edge of the shell was assumed to be simply supported or clamped. The solution was obtained by the finite difference and the Newton- Raphson methods. The thickness of the shell was considered to be uniform and the material was assumed to be homogeneous and isotropic. Sensitivity analysis was made for two geometrical parameters. The accuracy of the algorithm was checked by comparing the deflection with the solution of point supported circular plates and good agreement was obtained. Keywords—Bending, nonlinear, plate, point support, shell. I. INTRODUCTION INCE plate and shell structures have been extensively used in various disciplines like aerospace, marine, civil, nuclear, and automotive engineering, a large number of studies on plates and shells have been published (e.g. [1]- [17]). The studies on the large deflection bending of shallow spherical shells have basically focused on conventional boundary conditions (i.e., simply supported or clamped along the edge). However, for design purposes or for increasing the load carrying capacity of the structure, point supports have frequently been used. In the current study a shallow spherical shell with a circular plan form supported by a point support at the apex under linearly varying axisymmetric load undergoing large deflection was investigated. The influence of the geometrical parameters on the deflection and on the bending moment was examined by sensitivity analysis. The effect of the boundary conditions was studied. The accuracy of the algorithm was checked by comparing the deflection of a circular plate with those available in the literature. II. FORMULATION The smallest radius of curvature of a shallow shell at every point is larger than the greatest lengths measured along the midsurface of the shell [17]. The depth of a shallow spherical shell is limited by the relation [7] given by η < 1/8. The geometrically nonlinear shallow spherical shell equations were used in the current study. The equilibrium equations, the stress-strain relations, and the strain-displacement relations presented by Huang [7] were reorganized and rearranged in M. Altekin is with Yildiz Technical University, Istanbul, 34220 Turkey (phone: 90-212-383-5148; fax: 90-212-383-5133; e-mail: altekin@ yildiz.edu.tr). R. F. Yükseler was with Yildiz Technical University, Istanbul, 34220 Turkey (retired; e-mail: yukseler@yildiz.edu.tr). terms of three displacement components ( ) w,u, β where w ′ β = , and three stress resultants ( ) r r r n ,q ,m given by 1 0 ′ ⎛ ⎞ ′ ′ = + + + − = ⎜ ⎟ ⎝ ⎠ r r r w L m rm D rq r νβ (1) ( ) 2 1 0 ′ = − + − = r r u L n rn Et r ν (2) ( ) 3 2 0 2 2 1 2 0 ′ ′ = + + + + ′ ′ + + + + = r r r r r r L q rq r n h n a Et h wu Et u wn rq r a β ν ν (3) 4 0 ′ = + + = r L m D D r ν β β (4) ( ) 2 5 2 2 1 2 2 1 0 ′ = + + − − + = r r L u h a u n Et r β β ν ν (5) 6 0 ′ = − = L w β (6) were obtained where ( ) 3 0 0 2 , () . 12 1 = = = − Et r D q q r q a ν (7) Substituting the nondimensional parameters given by , , , = = = w u a W U c t t t (8) , , , 2 = = = q h r Q E a a η ξ (9) 2 , , = = = r r r r r r n q m N Q M Et Et Et (10) into (1)-(6), the differential operators ( ) 1 2 3 4 5 6 L ,L ,L ,L ,L ,L are obtained as follows: ( ) ( ) 1 2 2 2 1 12 1 0 12 1 = + + − + − = − r r r dM dW L M d d c d c Q d c ξ ξ ξ ν ξ ν β ξ ξ ν (11) M. Altekin, R. F. Yükseler Axisymmetric Nonlinear Analysis of Point Supported Shallow Spherical Shells S World Academy of Science, Engineering and Technology International Journal of Mechanical and Mechatronics Engineering Vol:7, No:11, 2013 2180 International Scholarly and Scientific Research & Innovation 7(11) 2013 scholar.waset.org/1307-6892/17336 International Science Index, Mechanical and Mechatronics Engineering Vol:7, No:11, 2013 waset.org/Publication/17336