Minimizing the expected total cost of a structure, including maintenance and construction is a difficult problem because of the presence in the objective function of the yearly failure rates, which have to be calculated by an optimization problem per each failure mode. In this paper, a new method for the design of structures that minimizes the total expected costs of the structure during its lifetime based on Benders’ decomposition is presented. In addition, some tools for sensitivity analysis are introduced, which make it possible to determine how the cost and yearly failure rates of the optimal solution are affected by small changes in the input data values. The proposed method is illustrated by its application to the design of a composite breakwater under breaking and non-breaking wave conditions.
Engineering design of structural elements is a complicated and highly iterative process that usually requires an extensive experience. Iterations consist of a trial-and-error selection of the design variables or parameters, together with a check of the safety and functionality constraints, until reasonable structures, in terms of cost and safety, are obtained. Since maintenance and repair take place during the service lifetime of the structure, the associated costs must be added to construction costs. The objective of the design is to verify that the structure satisfies the project requirements during its lifetime in terms of acceptable failure rates and cost (see Losada [1] and ROM [2]).
Since repair depends on the modes of failure and their frequencies, these must be defined. Each mode of failure m is defined by its corresponding limit state equation as, for example:
equation(1)
View the MathML sourcegm(x1,x2,…,xn)=hsm(x1,x2,…,xn)-hfm(x1,x2,…,xn);m∈M,
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where (x1, x2, …, xn) refer to the values of the variables involved, gm(x1, x2, …, xn) is the safety margin and hsm(x1, x2, …, xn) and hfm(x1, x2, …, xn) are two opposing magnitudes (such as stabilizing and mobilizing forces, strengths and stresses, etc.) that tend to avoid and produce the associated mode of failure, respectively, and M is the set of all failure modes.
The failure occurs when the critical variables satisfy gm ⩽ 0. With the consideration of all extreme events (see [3] and [4]) that may occur in the reference period, the different failure rates for all failure modes can be estimated.
Over the last few years, design methods have been improved by applying optimization techniques. The main advantage is that these techniques lead to optimal design and automation. Designer’s concerns are only the constraints to be imposed on the problem and the objective function.
Some authors consider the construction cost [5], [6], [7], [8] and [9] or the total cost (construction, maintenance and repairs) as the design criteria [10], [11], [12], [13] and [14]. The main problem of including repair and maintenance cost is that in such a case the cost function includes yearly failure rates, the calculation of which implies solving as many optimization problems as failure modes. Thus, use of optimization programs is not straightforward.
In addition to requiring optimal solutions to problems, some interest is shown by people in knowing how sensitive are the solutions to data values. A sensitivity analysis provides excellent information on the extent to which a small change in the parameters or assumptions (data) modifies the resulting design (geometric dimensions, costs, reliabilities, etc.).
The aims of this paper are: (a) to present a decomposition design method that permits solving the total cost minimization problem and (b) to provide tools to perform a sensitivity analysis.
The paper is structured as follows. In Section 2, the proposed method for optimal design based on Benders’ decomposition is presented. In Section 3, a technique for performing a sensitivity analysis is explained. Section 4 illustrates the proposed method by an example dealing with the design of a composite breakwater. Section 5 is devoted to the discussion of the statistical assumptions. Section 6 presents a numerical example. Finally, Section 7 gives some conclusions.
The methodology presented in this paper provides a rational and systematic procedure for automatic and optimal design of engineering works. The engineer is capable of obtaining optimal yearly failure rates for the different modes of failure, so that the choice of the safety level for which the structure has to be designed taking into account the different consequences of a complete of partial failure depending on the structure and the environment is carried out. In addition, a sensitivity analysis can be easily performed by transforming the input parameters into auxiliary variables, which are set to their associated actual values. The provided example illustrates how this procedure can be applied and proves that it is practical and useful.
Some additional advantages of the proposed method are:
1.
The method allows an easy connection with optimization frameworks.
2.
The responsibility for iterative methods is given to the optimization software.
3.
The reliability analysis takes full advantage of the optimization packages, which allows the solution of huge problems without the need of being an expert in optimization techniques.
4.
Sensitivity values with respect to the target reliability levels are given, without additional cost, by the values of the dual problem.
5.
It can be applied to different types of problems such as linear, non-linear, mixed-integer problems. The designer needs just to choose the adequate optimization algorithm.
