Plastic Hinges in a Beam

This talk focuses minimization of one-dimensional free discontinuity problem with second-order energy dependent on jump integrals but not on the cardinality of the discontinuity set. Related energies, describing loaded elastic–plastic beams, are not lower semicontinuous in BH (the space of displacements with second derivatives which are measures). 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 the Sobolev space, H2. Moreover, we can always select one minimizer whose number of plastic hinges does not exceed two 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. If the beam is under the action of a skew-symmetric load then the safe load condition is less stringent than in the general case.