Let the split ring in Problem 10.33 be rounded on the inside surface so as to apply a line load at the center of the ring. Determine the maximum principal stresses at the inner radius of the cylinder.

A closed-end steel cylinder (E = 200 GPa and v = 0.29) has an inside radius a = 2.00 m, wall thickness h = 10 mm, and hemispherical ends. Since the state of stress is different for cylinder and hemisphere, their radial displacements will be different. Show that the length L/2 [see Eq. (10.43)] is small compared to a so that the short length of the hemisphere can be considered another cylinder. Determine the shear force w in terms of internal pressure p, at the junction of the cylinder and hemisphere (assumed to be another cylinder). Note that the bending moment at the junction is zero because of symmetry. Determine the maximum bending StISS Gz2(bendingy in the cylinder, axial stress o,,, and circumferential stress 099 at the outside of the cylinder at the location where the maximum bending stress occurs, and the ratio of the maximum shear stress at that location to the maximum shear stress in the cylinder at a distance far from the junction.

Ans. w = 13.73p,3 Gzz(bending) = 29-17P13 O22 = 129.2p,; O99 = 174.6p,; ratio = 0.874 REFERENCES 439 REFERENCES Boresi, A. P. and Chong,.K. P. (1987). Elasticity in Engineering Mechanics. New York: Elsevier. Hetnyi, M. (1946). Beams on Elastic Foundations. Ann Arbor: Univ. of Michigan Press. Iyengar, K. T. S. R. and Ramu, S. A. (1979). Design Tables for Beams on Elastic Foundations and Related Problems. London: Elsevier Appl. Sci. Pub., Ltd. James, R. W. and Raba, G. W. (1991). Behavior of Welded Steel Water-Storage Tank. J. Structural Eng., 117, (1): 61-79. Jones, R. and Xenophontos, J. (1977). The Vlasov Foundation Model. Int. J. Mech. Sci., 19, (6): 317-324. Kerr, A. D. (1964). Elastic and Viscoelastic Foundation Models. J. Appl. Mech., Trans. ASME, 31, Series E (3): 491-498. Scott, R. F. (1981). Foundation Analysis. Englewood Cliffs, N. J. Prentice Hall. Selvadurai, A. P. S. (1979). Elastic Analysis of Soil Foundation Interaction. London: Elsevier Appl. Sci. Publ., Ltd.

Ting, B.-Y. (1982). Finite Beams on Elastic Foundation with Restraints. J. Structural Div., 108, (ST 3): 611-621. Vallabhan, C. V. G. and Das, Y. C. (1987). A Note on Elastic Foundations. IDR Report, Dept. of Civil Engr., Texas Tech. Univ., Lubbock, Texas. Vallabhan, C. V. G. and Das, Y. C. (1988). Parametric Study of Beams on Elastic Foundations. ASCE J. Engr. Mech., 114, (12): 2072-2082. Vlasov, V. Z. and Leontev, N. N. (1966). Beams, Plates and Shells on Elastic Foundations. Washington, D. C.: NASA TT F-357 (translated from Russian). Westergaard, H. M. (1948). New Formulas for Stresses in Concrete Pavements of Airfields Trans. Amer. Soc. Civil Engineers, 113: 425-444. 11 THE THICK-WALL CYLINDER 11.1 BASIC RELATIONS