Mathematical programming approaches for obtaining sharp collapse load bounds in interval limit analysis

Interval elastoplastic analysis of structures

“…In particular, the interval responses in Figs. 10, 12 and 14 attain the maximum load capacities of the structures involved, which coincide with the critical collapse load bound limits reported by the interval limit analysis approach[6]. This supports the accuracy of our interval elastoplastic algorithm.…”

“…Unfortunately, the computational effort to successfully process practical-size problems is often intractable. An alternative, though still computationally demanding approach, is to use a 0-1 programming formulation as was done for interval limit analysis [6]. The method we propose in the next section overcomes these computational difficulties; and, for large scale problems, a Monte Carlo approach, as we have used in this paper, appears to be reasonable for verifying to some extent the solutions obtained.…”

“…The elastic material relations between the stresses Q and the elastic strains e are given in (4), where S 2 R mÂm is the conventional (positive definite) elastic stiffness matrix. The assumed piecewise linear (PWL) holonomic elastoplastic constitutive laws are given in (5) and (6). An associative flow rule defines plastic strains p 2 R m as functions of plastic multipliers z 2 R m through a (constant) normality matrix N 2 R mÂy that collects all unit normal directions to the yield hyperplanes.…”

“…Instead, we attempted to (partially) validate the accuracy of the computed interval elastoplastic responses by using 200,000 Monte Carlo deterministic simulations for some selected load multipliers a. Additionally, we carried out interval limit analyses [6] to provide some reference results on the most maximum collapse load limit a up and the most minimum collapse limit a lo .…”

“…A well-known technique, and one that is used in the present study, to take into account the influences of uncertain data is an interval approach (or so-called ''convex'' model) [5][6][7] that assumes the uncertainties are nonprobabilistic but are described by a set of independent parameters lying explicitly within known continuous interval ranges. Typically, one would then compute, for any given load level, the two bound solutions (namely most favorable and least favorable) to some response variable of interest, thus enabling a more reliable assessment of the structural safety of the structure under consideration.…”