A variational principle for plastic hinges in a beam

We focus the minimization of 1D free discontinuity problem with second order energy dependent on jump integrals but not on the cardinality of the discontinuity set, in the framework of L∞ load. The related energies are not lower semi continuous in BH. Nevertheless we show that if a safe load condition is fulfilled then minimizers exist and they belong actually to SBH, say their second derivative has no Cantor part. If in addition a stronger condition on load is fulfilled then minimizer is unique and belongs to H. Moreover we can always select one minimizer whose number of plastic hinges does not exceed 2 and is the limit of minimizers of penalized problems. When the load stays in the gap between safe load and regularity condition then minimizers with hinges are allowed; if in addition the load is symmetric and strictly positive then there is uniqueness of minimizer, the hinges of such minimizer are exactly two and they are located at the endpoints.