We focus on a variational model of an elastic-plastic beam which is clamped at both endpoints and subject to a transverse L^1 load: the mathematical formulation is a 1D free discontinuity problem with second-order energy dependent on gradient-jump integrals but not on the cardinality of gradient-discontinuity set. The related energy is not lower semicontinuous in BH; moreover the relaxed energy is finite also when second derivatives have Cantor part. Nevertheless we show that if a safe load condition is fulfillled, then minimizers exist and they actually belong 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^2. 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.