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Recent numerical work has investigated incipient failure of yield stress materials under gravitational loading, for both the rectangular block and cylinder geometries [Chamberlain et al.; 2001, Int. J. Mech. Sci. 43(3):793-815, 2002, Int. J. Mech. Sci. 44(8):1779-1800]. While the rectangular block solution is exact, the cylinder solutions give lower bounds on the height of incipient failure. Consequently, we construct upper bound solutions for the height of incipient failure of a cylinder under gravitational loading. This closes the cylinder problem and quantifies the accuracy of the Haar-Karman hypothesis used in slip-line analysis. For completeness, we also give a simple lower bound solution for the cylinder, as well as upper and lower bound solutions for the two-dimensional rectangular block. These results have the advantage of being analytical, in contrast to the previous purely numerical results.

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