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This book reports on the progress in the theoretical development and the numerical application of shakedown and limit analysis. For a long time the lack of appropriate numerical algorithms had been an obstacle to wide-spread industrial application of the powerful theorems of Direct Methods. This book demonstrates in a convincing manner how through recent achievements in the field of optimization and steadily increasing computing power this obstacle could be removed. This is illustrated by the presentation of a large variety of new developments such as new problem-tailored algorithms, limit states of structures in civil and mechanical engineering, limit states of materials, plastic topological design and pavement mechanics.