We focus on a variational model of an elastic-plastic beam which is clamped at both endpoints and subject to a transverse L(infinity) 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 fulfilled, 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.
A Variational Principle For Plastic Hinges In A Beam
PERCIVALE, DANILO;
2009-01-01
Abstract
We focus on a variational model of an elastic-plastic beam which is clamped at both endpoints and subject to a transverse L(infinity) 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 fulfilled, 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.