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The stresses are discussed to which the riding rings of rotary kilns are subjected. These stresses are caused partly by forces acting on the rings and partly by thermal gradients. By nature the stresses are static and dynamic. Particular attention is paid to the Hertzian pressure which must be regarded as the most significant load factor because, apart from surface stresses, other stresses are produced slightly below the surface. These stresses and the corresponding strengths were investigated statistically, based on 125 riding rings, the results being illustrated graphically. Referring to the fracture pattern of a spectacular case of riding ring breakage, the factors that caused it and the mechanism of failure are explained. Hints are given on how to dimension riding rings reasonably.