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Optimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min-max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min-max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.

Under operational conditions, a structure is usually subjected to uncertainties which may arise from fluctuation and scatter of external loads, environmental conditions, boundary conditions, and geometrical and material properties. However quite often, design uncertainties arise from incomplete knowledge and unpredictable nature of the load under operational conditions.

In the present study optimality conditions are derived for the optimal robust design of a beam subject to a transverse load which has unknown and known parts. Moreover uncertain moments and/or shear forces may be acting at the endpoints of the beam. In conventional design, it is common practice to neglect the load uncertainties when analyzing a structure and assessing the structural performance on the basis of a deterministic model. To compensate for performance variability caused by load variations, a safety factor is introduced magnitude of which correlates with the level of uncertainty with higher levels leading to larger safety factors. However, the safety factors specified may be either too conservative or too small to compensate for the lack of knowledge of operational loads. Efficiency and reliability of the structure can be improved by taking the load uncertainties into consideration in the design process leading to a design which is robust under load variations. This approach is equivalent to optimizing the design for worst case loading and leads to an optimization problem for the area function and to an antioptimization problem for the load function. The final design is robust in the sense that the sensitivity of the beam to load variations is substantially reduced. This is accomplished by maximizing its potential energy over loading while minimizing it with respect to its cross-sectional shape. Mathematically these results in a min-max optimal design problem can be studied by variational calculus.

Finding the worst case loading on a structure corresponds to an antioptimization problem the examples of which can be found [

An alternative strategy to treat the uncertainties is convex modeling in which the uncertainties belong to a convex set [

In the present work, a system of nonlinear differential equations is derived in terms of state, design and load variables using calculus of variations and the methods of Lagrange multipliers and slack variables. These expressions serve as the optimality and antioptimality conditions of the problem. The design variable is the cross-sectional shape of the beam and is subject to a volume constraint. The load variables comprise a combination of deterministic and uncertain transverse loads as well as uncertain moments and shear forces which may act at the boundaries. The only constraints on the unknown loads involve finite norms and an upper bound on the transverse load. The optimization method involves a minmax formulation where the objective is to minimize the compliance with respect to the cross-sectional shape and maximize it with respect to the unknown loads. The formulation ensures that the optimal designs correspond to the most unfavorable loading and, therefore, these designs are conservative for any other loading. The solution methodology is illustrated with several examples involving cases which allow the computation of closed-form solutions.

The differential equation governing the deflection of a variable cross-section beam subject to a combination of uncertain and deterministic loads, as shown in Figure

Beam diagram with external forces.

Physically the stiffness of a beam has a finite value and in practice the volume of the beam is specified leading to a design constraint which can be expressed mathematically as

The design optimization can be achieved by choosing a suitable performance index for the problem which serves as an objective functional of a minmax problem. In the present case a suitable objective functional is the potential energy of the beam which is given by

Find the cross-sectional area

The robust design problem constitutes an optimization problem with respect to the area function

The derivations of the optimality and antioptimality conditions are given next. In view of the presence of several constraints on the design problem, the Lagrange multiplier technique is implemented by introducing the Lagrangian given by

The variation of

The variation of

The theoretical framework developed in Sections

We consider a simply supported beam subject to an uncertain load,

Governing equation of the deflection of the beam given by (

For the computation of the optimal area function

It is noted that the number of unknowns equals the number of equations resulting in unique solutions. This aspect the method of solution will be illustrated in the next section by applying the technique to several problems of practical interest.

To assess the efficiency of the optimal designs, comparisons are made with uniform beams under uncertain loads for which

Let the beam be subjected to only the uncertain transverse load

Curves of optimal

Let the beam subject to only the uncertain transverse load

Curves of optimal

The optimality conditions were derived for robust shape design of beams subject to unknown loads with the moment of inertia related to the area function as

Analytical solutions were obtained for the special case under consideration and several example problems were solved involving various cases of loadings. Numerical results were given for the optimal area and antioptimal load functions. The efficiencies of the optimal designs were computed in terms of the maximum deflections of the optimal beam and the uniform beam under least favorable loading. It was shown that for the cases studied the design efficiency can exceed 70

The first author (IK) would like to acknowledge the financial support provided by American University of Sharjah and University of KwaZulu-Natal (UKZN) to enable him to spend a research leave at UKZN in 2006.