An energetic view on the limit analysis of normal bodies

Abstract

This note presents a limit analysis for normal materials based on energy minimization. The class of normal materials includes some of those used to model masonry structures, namely, no–tension materials and materials with bounded compressive strength; it also includes the Hencky plastic materials. Considering loads $\mathfrak {L}(\lambda )$ that depend affinely on the loading multiplier $\lambda \in \mathbb {R},$ we examine the infimum $I_0(\lambda )$ of the potential energy $I(\boldsymbol {u},\lambda )$ over the set of all admissible displacements $\boldsymbol {u}.$ Since $I_0(\lambda )$ is a concave function of $\lambda$, the set $\Lambda$ of all $\lambda$ with $I_0(\lambda )>-\infty$ is an interval. Each finite endpoint $\lambda _{\mathrm {c}} \in \mathbb {R}$ of $\Lambda$ is called a collapse multiplier, and we interpret the loads corresponding to $\lambda _{\mathrm {c}}$ as the loads at which the collapse of the structure occurs. We show that the standard definition of collapse based on the collapse mechanism does not capture all situations: the collapse mechanism is sufficient but not necessary for the collapse. We then examine the validity of the static and kinematic theorems of limit analysis under the present definition. We show that the static theorem holds unconditionally while the kinematic theorem holds for Hencky plastic materials and materials with bounded compressive strength. For no–tension materials it generally does not hold; a weaker version is given for this class of materials.