Keywords: elastoplastic structures under stochastic uncertainty, representation of state/performance functions by minimum values of convex/linear programs, probability of survival/failure, first order reliability method (FORM), computation of design points by explicit programs.

Problems from plastic limit load or shakedown analysis and optimal plastic design [1,2] are based [3] on the convex yield criterion and the linear equilibrium equation for the generic stress (state) vector . Having to take into account, in practice, stochastic variations of the vector 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 depending on the parameter vector and the design vector . The state or performance function is defined [4] by the minimum value function of a convex or linear program

s.t.

based on the basic safety conditions of plasticity theory: Here, (46) represents the equilibrium equation involving the equilibrium matrix . Furthermore, (47) results from the yield condition described by means of the distance functionals of the feasible domains . Moreover, is the diagonal matrix containing the material strength parameters at the -th reference point , of the structure. A safe (stress) state exists then if and only if , and a safe stress state cannot be guaranteed if and only if . Hence, the probability of survival can be represented by

Using FORM, the probability of survival is approximated then by the well-known formula

where denotes the length of a so-called beta - or "design" point, hence, a projection of the origin 0 to the failure domain (transformed to the space of normal distributed model parameters . Moreover, denotes the distribution function of the standard 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 .

While in general the computation of the projection is very difficult, in the present case of elastoplastic structures, by means of the state function 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 , an explicit parameter optimization problem can be derived for the computation of a design point . Simplifications are obtained in the standard case of piecewise linearization of the feasible domains , or their surfaces (yield surfaces). Moreover, the (global) equilibrium matrix may be determined for different types of structures (plane and spatial trusses and frames) by a special FORTRAN module CMG.

The new method is an efficient and exact alternative to the existing approximation techniques in this area, such as Response Surface Methods (RSM). The use of the present direct reliability method in Reliability-Based Design Optimization (RBDO) of elastoplastic structure is discussed.

1

D.M. Frangopol, "Reliability-Based optimum structural design", In: Probabilistic Structural Mechanics Handbook, ed. by C. Sundarajan, 352-387, Chapman and Hall, New York, 1995.

2

M. Gasser, G.I. Schuëller, "Some basic principles in reliability-based optimization (RBO) of structures and mechanical components", In: Stochastic programming methods and technical applications, ed. by K. Marti and P. Kall, Lecture Notes in Economics and Mathematical Systems, 458, 80-103, Springer-Verlag, Berlin, 1998.

3

J.A. Kemenjarzh, "Limit Analysis of Solids and Structures", CRC Press, Boca Raton [etc.], 1996.

4

K. Marti, "Stochastic optimization methods in optimal engineering design under stochastic uncertainty", ZAMM 83, No. 11, 1-18, 2003. doi:10.1002/zamm.200310072

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