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A new algorithm to solve contact body problems is presented. The contact boundaries are divided into finite elements, and contact element matrices and vectors are computed using a modified variational principle. The contact conditions are incorporated into the total functional using the Lagrange multiplier method. This algorithm considers surface contacts rather than point contacts, and computes contact tractions directly. The Coulomb friction law is used as the friction condition. A transformation matrix based on static equilibrium is developed to solve extensive sliding problems. This algorithm is extended to the elasto‐plastic analysis and several example problems are presented to demonstrate the algorithm.