Reliability analysis for elastoplastic mechanical structures under stochastic uncertainty
Authors: Marti K. , Kaymaz I. 
Published: February 2006

Problems from plastic limit load or shakedown analysis and optimal plastic design are based on the convex yield criterion and the linear equilibrium equation for the generic stress (state) vector σ. Having to take into account, in practice, stochastic variations of the vector y = y(ω) of model parameters, e.g. yield stresses, external loadings, cost coefficients, etc., the basic stochastic plastic analysis or optimal plastic design problem must be replaced – in order to get robust optimal designs/load factors – by an appropriate deterministic substitute problem. For this purpose, the existence of a statically admissible (safe) stress state vector is described first by means of an explicit scalar state function s* = s* (y,x) depending on the parameter vector y and the design vector x. The state or performance function s* (y,x) is defined by the minimum value function of a convex or linear program based on the basic safety conditions of plasticity theory: A safe (stress) state exists then if and only if s* < 0, and a safe stress state cannot be guaranteed if and only if s* ≥ 0. Hence, the probability of survival can be represented by ps = P(s* (y(ω),x)<0).

Using FORM, the probability of survival is approximated then by the well‐known formula ps ∼ Φ ($\| {z_x^\ast } \|$|) where $\| {z_x^\ast } \|$ denotes the length of a so‐called β‐point, hence, a projection of the origin 0 to the failure domain (transformed to the space of normal distributed model parameters z(ω)=T(y(ω))). Moreover, Φ = Φ (t) denotes the distribution function of the standard N(0,1) normal distribution. Thus, the basic reliability condition, used e.g. in reliability‐based optimal plastic design or in limit load analysis problems, reads

with a prescribed minimum probability αs. While in general the computation of the projection $z_x^\ast $ is very difficult, in the present case of elastoplastic structures, by means of the state function s* = s* (y,x) this can be done very efficiently: Using the available necessary and sufficient optimality conditions for the convex or linear optimization problem representing the state function s* = s* (y,x), an explicit parameter optimization problem can be derived for the computation of a design point $z_x^\ast $. Simplifications are obtained in the standard case of piecewise linearization of the yield surfaces.

In addition, several different response surface methods including the standard response surface method are also applied to compute a β‐point $z_x^\ast $ in order to reduce the computational time as well as having more accurate results than the first order approximation methods by using the obtained response surface function with any simulation methods such as Monte Carlo Simulation. However, for the problems having a polygon type limit state function, the standard response surface methods can not approximate well enough. Thus, a response surface method based on the piecewise regression has been developed for such problems. Applications of the methods developed to several types of structures are presented for the examples given in this paper.