Multiobjective optimization of prestressed concrete structures

This paper presents a practical and efficient approach to the optimization of prestressed concrete structures if two or more (possibly conflicting) objectives must simultaneously be satisfied. The most relevant objective function is adopted as the primary criterion, and the other objective functions are transformed into constraints by imposing some lower and upper bounds on them. The single-objective optimization problem is then solved by the projected Lagrangian algorithm. Two numerical examples illustrate the application of the approach to the design of a posttensioned floor slab and a pretensioned highway bridge system for two conflicting objectives: minimum cost and minimum initial camber. The Pareto optima achieve a compromise between the two conflicting objectives and represent more rational solutions than those obtained by independent optimizing each objective function. INTRODUCTION In the last three decades, much work has been done in structural optimization, in addition to considerable developments in mathematical optimization (Templeman 1983; Levy and Lev 1987). However, most of the applications on structural optimization found in the literature deal with theoretical rather than real-world engineering problems (Cohn 1992). One of the major difficulties inherent in solving realistic engineering problems is the selection of a meaningful objective function that includes all relevant criteria with adequately chosen weighting factors for a given structural design problem. An explicit formulation of a single-objective function is not a straightforward task, as the selection of the weighting factors can be fairly subjective. This problem may be overcome by considering the multiobjective (multicriteria, or vector) optimization problem instead of the ill-defined single-objective optimization problem. Several approaches have been proposed in the literature to solve this type of problems: weighting objectives; constraint (or trade-off) approach; goal programming; minimax approach, among others (Osyczka 1984; Duckstein 1984; Eschenauer et al. 1990). The application of multiobjective optimization to structural problems is very limited (Koski 1984; Eschenauer et al. 1990) when compared to its applications in operations research and control theory. It is believed that multiobjective optimization will be used more often when optimization will be concerned with real structures (buildings, bridges, etc...) instead of theoretical models, the type of which are found in structural optimization books. Often structural design must satisfy several (possibly conflicting) objectives such as: minimum cost, maximum safety, minimum weight, minimum volume of materials (concrete, steel), minimum deflection (camber), and so forth. In such cases, the multiobjective optimization offers an alternative approach to single-objective optimization. This alternative is preferable because it simultaneously considers all competing design objectives and results in merit values that cannot be further improved without impairing some of the objectives. In the present paper, the constraint approach is used to transform the multiobjective optimization into a single-objective optimization problem. In general, this is a nonlinear programming problem that may be solved by a variety of available techniques. The GAMS-MINOS program based on the projected Lagrangian algorithm is used (Murtagh and Saunders 1978; Brook et al. 1988). Two numerical examples illustrate the application of the approach to the design of a posttensioned floor slab and a pretensioned highway bridge system for two conflicting objectives: minimum cost and minimum initial camber. The interest of the present paper is that it introduces the multiobjective concept to the optimization of prestressed concrete structures, which are characterized by a large number of constraints and occasionally conflicting objectives. The multiobjective optimization is a direct (rather than a trial-and-error) approach to design, with more than one solution satisfying all design requirements (constraints and objectives). The outcome is a more efficient and flexible design procedure with great potential for some practical structural applications, as demonstrated by the floor slab and bridge system. MULTIOBJECTIVE OPTIMIZATION PROBLEM AND PARETO OPTIMUM The multiobjective optimization problem may be formulated as follows. Determine a vector of design variables that satisfy the constraints and minimize (or maximize) a vector of objective functions. Mathematically, this can be staled as follows (Koski 1984): |Mathematical Expression Omitted~ where f = vector of objective functions; |f.sub.1~ = component objective functions (i = 1, 2,..., m); x = |(|x.sub.1~|x.sub.2~... |x.sub.n~).sup.T~ = design variable vector; |Omega~ = feasible set to which x belongs and is a subset of |R.sup.n~ |Mathematical Expression Omitted~ Since in multiobjective optimization problems some objective functions have to be minimized and others maximized, it is convenient to convert all problems into equivalent minimization problems. In general, there is no single optimal (or superior) solution that simultaneously yields a minimum for all m objective functions. A new concept, the Pareto optimum (noninferior, nondominated, or efficient solution) (Zadeh 1963; Carmichael 1980), is introduced as a solution to the multiobjective optimization problem. A vector |x.sup.*~ is a Pareto optimum for problem (1) if and only if there exists no x |is an element of~ |Omega~ such that |f.sub.i~(x) |is less than or equal to~ |f.sub.i~(|x.sup.*~, for i = 1, 2,...,m with |f.sub.j~(x) |is less than~ |f.sub.j~(|x.sup.*~) for at least one j. In other words, |x.sup.*~ is a Pareto optimum if there is no feasible solution x which may yield a decrease of some objective function without causing a simultaneous increase of at least another objective function. SOLUTION OF MULTIOBJECTIVE OPTIMIZATION PROBLEM: |epsilon~-CONSTRAINT APPROACH Several approaches, already applied in operations research and control theory, have been proposed in the literature for the solution of multiobjective optimization problems. Among these, the |epsilon~-constraint (trade-off) approach seems to gain wide acceptance because of its practicality and rationality, when compared to the weighting objectives approach. Indeed, in the weighting objectives approach, the major difficulty is the a-priori choice of the weighting factors of various objective functions. This is not obvious, especially in problems with several conflicting objectives (e.g. cost versus safety and/or serviceability). Furthermore, the optimal design obtained using such an approach may drastically change with varying weighting factors. Hence, the only way of using this approach is to carry out the single objective optimization for a large number of weighting factor values and combinations. A large set of Pareto optima may thus be obtained, from which the best solution may (subjectively) be selected. This approach is regarded as fairly primitive (Waltz 1967). The |epsilon~-constraint approach is based on minimization of one (the primary) objective function and considering the other objectives as constraints bound by some allowable levels ||epsilon~.sub.i~. Hence, a single objective minimization is carried out for the most relevant objective function |f.sub.1~ subject to additional constraints on the other objective functions. The levels ||epsilon~.sub.i~ are then altered to generate the entire Pareto optima set. In the design of prestressed concrete structures, the total structure cost minimization may be considered as the most relevant objective. However, in structural problems where the selection of the primary objective is not obvious, the above procedure is repeated for all objective functions (|f.sub.2~, |f.sub.3~,...,|f.sub.m~), and results in a much larger Pareto optima set. Assuming that minimizing |f.sub.i~(x) is the primary objective function, the multiobjective optimization problem may be formulated as follows: Minimize |f.sub.i~(x) (3a) Such that: |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ where m = number of objective functions; |n.sub.i~ = number of inequality constraints; and |n.sub.e~ = number of equality constraints. To get adequate ||epsilon~.sub.j~ values, single-objective optimizations are carried out for each objective function in turn by using mathematical programming techniques. For each objective function |f.sub.j~ (j = 1, 2,...,m), there is an optimal design vector |Mathematical Expression Omitted~ for which |f.sub.j~(|x.sup.*.sub.j~) is a minimum. Let |Mathematical Expression Omitted~ be the lower bound on ||epsilon~.sub.j~, i.e. |Mathematical Expression Omitted~ and |f.sub.j~(|x.sup.*.sub.i~) be the upper bound on ||epsilon~.sub.j~, i.e. |Mathematical Expression Omitted~ Thus |Mathematical Expression Omitted~ By expressing the Kuhn-Tucker conditions for problem (3), it is easily confirmed that the constraints |f.sub.j~(x) |is less than or equal to~ ||epsilon~.sub.j~ are active for the Pareto optimal solutions (Carmichael 1980). SOLUTION OF SINGLE OBJECTIVE OPTIMIZATION PROBLEM: PROJECTED LAGRANGIAN ALGORITHM After the multiobjective optimization is transformed into an equivalent single-objective optimization problem by the |epsilon~-constraint approach, the latter may be solved using mathematical programming techniques. As generally the prestressed concrete optimization problem is nonlinear, any nonlinear programming method may be used (Gill et al. 1981; Morris 1982). In the present paper, the projected Lagrangian approach, implemented in the MINOS program (Murtagh and Saunders 1977, 1978, 1980) is used. This program is part of a general-optimization software GAMS (Brook et al. 1988) and has been successfully used in optimal bridge design (Lounis and Cohn 1992). Since detailed presentations of the algorithm may be found elsewhere (Murtagh and Saunders 1977, 1978, 1980; Gill et al. 1981; Murtagh 1981, Brook et al. 1988), only a brief description is given here. The projected Lagrangian method is based on a method that involves a sequence of major iterations. Each iteration requires the solution (by the reduced gradient method) of a linearly constrained subproblem where the nonlinearities are confined to the objective function only. The nonlinear programming problem may be formulated as follows: Minimize f(x) (6a) Such that:g(x) |is less than or equal to~ 0 (6b) h(x) = Ax + B |is less than or equal to~ 0 (6c) |x.sup.l~ |is less than or equal to~ x |is less than or equal to~ |x.sup.u~ (6d) where g and h = vectors of the nonlinear and linear constraints, respectively; A and B = constant matrix and vector, respectively; and x, |x.sup.l~ and |x.sup.u~ = vectors of design variables and corresponding lower and upper bounds. At the start of each iteration, the nonlinear constraints |g.sub.j~ (j = 1, 2,...,|n.sub.n~) are linearized at the current point |x.sub.k~ using first-order Taylor's series expansions, i.e. |Mathematical Expression Omitted~ where |n.sub.n~ = number of nonlinear constraints; and J = Jacobian matrix. At each major iteration, the original nonlinearly constrained problem is transformed into a linearly constrained problem, which is then solved by the reduced gradient algorithm Minimize |Mathematical Expression Omitted~ Such that: |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ where ||lambda~.sub.k~ = Lagrange multiplier estimates at kth iteration; C and D = constant matrix and vector, respectively; and |rho~ = a positive penalty parameter. The new objective function, (8a) is called an augmented Lagrangian function, and includes the original objective function and a term involving Lagrange multiplier estimates ||lambda~.sub.k~. The quadratic term |Mathematical Expression Omitted~ is a penalty function added for improved convergence (especially when the initial design point is poorly chosen) and dropped near the optimum. The linearly constrained problem, (8a)-(8c) is solved by MINOS (Murtagh and Saunders 1978) as follows. Assume that at the current iterate |x.sub.k~ there are r active constraints; the objective is to find a search direction |delta~x that is feasible and usable for the new design point |x.sub.k+1~ |x.sub.k+1~ = |x.sub.k~ + |delta~x Since the objective function may be nonlinear, we cannot assume that all variables will be basic at an optimum solution. The notion of superbasic variables is introduced along with the partition of the set of active constraints |Mathematical Expression Omitted~ where B, S, and N = components of matrix C associated with the basic, superbasic, and nonbasic variables |x.sub.B~, |x.sub.S~, and |x.sub.N~, respectively. Feasible Direction The direction |delta~x must be feasible, i.e. the new design point |x.sub.k+1~ must satisfy the constraints (8b) and (8c). By substituting for |x.sub.k+1~ in (10) we get |Mathematical Expression Omitted~ Hence, the search direction |delta~x should be orthogonal to the gradients of the active constraints. Usable Direction Moreover, the feasible direction |delta~x must be usable, i.e. a direction of descent of the objective function |Mathematical Expression Omitted~ In (12), the vectors g (gradient vector of the objective function) and |delta~x have been partitioned according to the partitioning of x. Eq. (12)indicates that the gradient at |x.sub.k+1~ is orthogonal to the surface of the active constraints and therefore may be expressed as a linear combination of their normals. Parameters |lambda~ and |mu~ are the Lagrange multipliers and H is the Hessian matrix. NUMERICAL EXAMPLES Example 1: Multiobjective Optimal Design of a Posttensioned Slab The posttensioned, simply supported slab with the geometry, loading, cross-section, and tendon layout in Fig. 1 is to be designed for two objective functions: minimum cost and minimum initial camber. The constraints include all serviceability and ultimate limit slate requirements of the ACI1989 code ('Building' 1989). The design variables are the slab depth h, prestressing force P, net reinforcement index |omega~, and tendon eccentricity at midspan e. In addition to its own weight, the slab carries a superimposed deal load (partitions) |w.sub.SD~ = 1.3 kN/|m.sup.2~ and a live load |w.sub.L~ = 2.4 kN/|m.sup.2~. The unbonded tendons are of stress-relieved type with |f.sub.pu~ = 1,860 MPa, |f.sub.py~ = 1,580 MPa and an effective prestress at service |f.sub.se~ = 1,116 MPa. The concrete has |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |E.sub.c~ = 27,000 MPa, and |E.sub.ci~ = 25,000 MPa. The allowable stresses are ('Building' 1989) tension at transfer, 1.3 MPa; compression at transfer, 15 MPa; tension at service, 2.7 MPa; and compression at service, 13.5 MPa. Prestress losses of 15% between transfer and service are assumed. The primary-objective function is the minimization of the total slab cost-per-unit area, which may be stated as (Goble and Lapay 1971; Naaman 1976) |f.sub.1~(P, h, e) = |c.sub.c~h + |c.sub.p~Wp($/|m.sup.2~) (13) where |c.sub.c~ and |c.sub.p~ = unit costs of concrete and prestressing steel per unit volume and weight, respectively; and |W.sub.p~ = weight of prestressing per unit slab area. In this paper, we assume |c.sub.c~ = $80/|m.sup.3~ and |c.sub.p~ = $4/kg. The secondary objective function is the minimization of the initial camber due to prestressing and own weight of the slab before any superimposed dead or live load is applied. It should be pointed out that in most structural-concrete codes, no limit is specified on the initial camber of prestressed concrete structures, even though this is an important serviceability criterion. It is assumed that the initial camber is within adequate limits if the transfer stresses constraints arc satisfied. It is the merit of the multiobjective optimization that minimization of the camber may become an explicit objective function in the optimization process. The initial camber may be expressed as |Mathematical Expression Omitted~ where |w.sub.g~ and |w.sub.p~ = own weight of the slab and the equivalent load due to prestressing at transfer, respectively; |I.sub.g~ = moment of inertia of gross concrete section, ||delta~.sub.p~ and ||delta~.sub.g~ = camber and deflection due to prestressing and slab own weight, respectively. |Mathematical Expression Omitted~ The multiobjective optimization problem for this slab may be formulated as follows: Minimize |f.sub.1~(P, h, e) = |c.sub.c~h + |c.sub.p~|W.sub.p~ and |Mathematical Expression Omitted~ Such that: |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ 160 |is less than or equal to~ h |is less than or equal to~ 350 (16h) where |M.sub.g~, |M.sub.SD~, |M.sub.L~ and |M.sub.t~ = moments due to the slab own weight, superimposed dead load, live load, and total service load, respectively. Eqs. (16c) and (16d) represent the transfer and service stresses constraints, respectively. Eq. (16e) represents the ultimate limit state (ULS) flexural strength constraint. Eqs. (16f), (16g), and (16h) are the limits on minimum concrete cover, maximum net reinforcement index, and slab depth, respectively. Using the |epsilon~-constraint approach with the minimum cost as the primary objective, we transform the multiobjective optimization problem into a single-objective optimization problem by changing the minimum initial camber objective (16b) into a constraint. The bounds on ||epsilon~.sub.2~ will be determined by using (4a) and (4b). Minimization of |f.sub.1~ (total slab cost) alone yields the following optimal design vector: |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. The ULS flexural strength constraint (16e) is active. At this point (|Mathematical Expression Omitted~), by using (14), the initial camber is |Mathematical Expression Omitted~. Minimization of |f.sub.2~ (initial camber) alone yields the following optimal design vector: |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. Here also the strength constraint (16e)is active. From (4a) and (4b) we get the following values for the lower and upper bounds on ||epsilon~.sub.2~, respectively: |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ therefore 0.57 mm |is less than or equal to~ ||epsilon~.sub.2~ |is less than or equal to~ 5.73 mm (17c) Thus, the set of Pareto optima is obtained by varying ||epsilon~.sub.2~ in the preceding interval and the results are given in Table 1. Table 1 shows that the optimal solution is not unique; instead, a set of optima from which the designer may choose the most suitable solution is available. There is certainly some subjectivity in choosing the best solution by trading off one objective for another, but the fact that there are several optimal solutions renders the design process more flexible. The following remarks may be made: * As the minimum initial camber increases, the minimum slab cost decreases, as the two criteria are of conflicting nature. * Successive improvements of the minimum cost are: made by increasing the prestressing force P and decreasing the slab depth. The opposite trend occurs on improving the minimum initial camber. * In the single objective minimization of |f.sub.1~ and |f.sub.2~ in turn, the ULS flexural strength is the only active constraint. In the case of multiobjective optimization, the only active constraint is the initial camber because all solutions obtained are Pareto optima. * Since the initial camber in all seven cases is small, Table 1 suggests that the solution corresponding to the minimum cost solution 7 may be adopted as optimal. Example 2: Multiobjective Optimal Bridge Design The reinforced concrete slab on precast protensioned I-griders with the cross section, tendon layout and loading in Fig. 2 is to be designed for three objective functions: (1) Minimum number of girders; (2) minimum weight of prestressing steel; and (3) minimum initial camber. The constraints include all serviceability and ultimate limit state requirements of the Ontario Highway Bridge Design Code (OHBDC: Ontario 1983). Design variables TABULAR DATA OMITTED are the girder spacing S, slab overhang |Mathematical Expression Omitted~, prestressing force P, and tendon eccentricities at end and midspan |e.sub.e~ and |e.sub.c~, respectively. The bridge has three design lanes of 4 m width each. The midspan and support sections are considered in the design. The slab is 225 mm thick with |Mathematical Expression Omitted~ and is reinforced with the minimum OHBDC reinforcement ratio of 0.3% (both top and bottom steel with |f.sub.y~ = 400 MPa) in the longitudinal and transverse directions. The loads on the girders are the slab and girder own weights, the weight of a 90 mm thick asphalt pavement and the OHBDC truck load shown in Fig. 2. The bonded tendons are held down at third span points and have |f.sub.pu~ = 1,860 MPa and an effective prestress |f.sub.se~ = 1,116 MPa. The dynamic load allowance factor is 0.4 and the effective slab width is taken as the girder spacing S. For the concrete, in girder |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |E.sub.ci~ = 29,400 MPa and the allowable stresses are: tension at transfer |f.sub.tt~ = 1.3 MPa; compression at transfer |f.sub.tc~ = 18 MPa; tension at service |f.sub.st~ = 3 MPa; compression at service |f.sub.sc~ = 18 MPa and the allowable compression in slab is |f.sub.ss~ = 12 MPa. Prestress losses of 15% are assumed. The bridge analysis is carried out using the OHBDC simplified method described elsewhere (OHBDC: Ontario 1983; Lounis and Cohn 1992). The satisfaction of the first two objective functions (minimum number of girders with minimum prestressing) yields the minimum superstructure cost. The girder number n, spacing S and slab overhangs |Mathematical Expression Omitted~ are related to the bridge width W by the obvious relation |Mathematical Expression Omitted~. However, instead of solving the optimization problem with the minimization of the number of griders n as objective function (which will then result in a complex mixed integer programming problem), we solve the equivalent but simpler nonlinear programming problem of maximizing the girder spacing S and slab overhang |Mathematical Expression Omitted~. Thus the primary objective becomes Max |f.sub.1~(x) = Max S = Min (-S) the secondary objective is the minimization of the prestressing force Min |f.sub.2~(x) = Min P and the tertiary objective is the minimization of the initial camber |Mathematical Expression Omitted~ i.e. Min |f.sub.3~(x) = Min ||delta~.sub.i~ where ||delta~.sub.p~ and ||delta~.sub.g~ = camber and deflection due to prestressing and girder own weight, respectively, and |M.sub.g~ = midspan moment due to girder own weight. Hence, the multiobjective optimization problem may be formulated as: Minimize ||-S, P, |delta.sub.i~.sup.T~ (19a) Such that: ||sigma~.sub.tt~ |is less than or equal to~ |f.sub.tt~ (19b) ||sigma~.sub.tc~ |is less than or equal to~ |f.sub.tc~ (19c) ||sigma~.sub.st~ |is less than or equal to~ |f.sub.st~ (19d) ||sigma~.sub.sc~ |is less than or equal to~ |f.sub.sc~ (19e) ||sigma~.sub.ss~ |is less than or equal to~ |f.sub.ss~ (19f) |M.sub.u~ |is less than or equal to~ 0.85|M.sub.n~ (19g) |V.sub.u~ |is less than or equal to~ 0.75|V.sub.n~ (19h) |V.sub.uh~ |is less than or equal to~ 0.75|V.sub.nh~ (19i) |omega~ |is less than or equal to~ 0.30 (19j) |M.sub.n~ |is greater than or equal to~ 1.25|M.sub.cr~ (19k) |C.sub.bot~ |is greater than or equal to~ 100 mm (19l) |C.sub.top~ |is greater than or equal to~ 135 mm (19m) 2.0 |is less than or equal to~ S |is less than or equal to~ 15t (19n) |Mathematical Expression Omitted~ |is less than or equal to~ min{1.8 m, 0.6S} Eqs. (19b)-(19f~ represent the transfer and service stress constraints, where |sigma~ = applied maximum stress; and f = corresponding allowable value. Eqs. (19g)-(19i) represent the ultimate limit states constraints on flexural, shear, and interface shear strengths, respectively. Eqs. (19j) and (19k) are the limits on maximum and minimum reinforcements and on minimum bottom and top covers, respectively. Finally (19n) and (19o) are some side constraints on the girder spacing, slab thickness, and slab overhang (OHBDC: Ontario 1983; Lounis and Cohn 1992). We start by performing a single-objective optimization of the number of girders, i.e. maximizing the girder spacing and ignoring other objectives for the time being. We get |f.sub.1 max~ = |S.sub.max~ = 3.37 m, |Mathematical Expression Omitted~, P = 2,676 kN, |e.sub.e~ = 0; and |e.sub.c~ = 535 mm. These girder spacing and slab overhang yield the minimum number of girders n = 4. The active constraints are the tensile stress at transfer, (19b), and minimum bottom concrete cover, (19l). A better solution (minimum superstructure cost) is achieved by choosing the smallest girder spacing that allows four girders transversely. This is equivalent to maximizing the slab overhang, and yields |Mathematical Expression Omitted~, S = 2.90 m, and n = 4 girders. Hence the initial three-objective optimization problem reduces to a two-objective optimization problem by setting S and S' as preassigned parameters (equal to the above values). We then adopt the minimization of the prestressing force as the primary objective |f.sub.1~ and transform the camber minimization objective |f.sub.2~ into a constraint by the |epsilon~-constraint approach. Solving problem (19a)-(19o) with minimization of P alone as objective function yields the following optimal design vector: |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. The active constraints are the ULS flexural strength (19g) and the minimum bottom concrete cover (19l). Solving problem (19a)-(19o) with minimization of the initial camber alone as objective function yields the following optimal design vector: |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. The only active constraint is the ULS flexural strength. Using (4a) and (4b) we get the following values for the lower and upper bounds on ||epsilon~.sub.2~, respectively: |Mathematical Expression Omitted~ |Mathematical Expression Omitted~ As it can be seen from (20a) and (20b), there is no significant difference between the lower and upper bounds on the initial camber. This suggests that the initial camber may not be critical as long as the transfer stresses are within some limits. However, to illustrate the generation of Pareto optima for this example, we solve the optimization problem for three intermediate values of the ||epsilon~.sub.2~-constraint in the preceding interval and the results are summarized in Table 2. For this example, too, minimization of the prestressing force and initial camber are conflicting objectives, because any decrease in one objective leads to an increase in the other objective. From Table 2 we note that: * For all Pareto optima the ULS flexural strength and initial camber constraints are active. * As the allowable initial camber ||epsilon~.sub.2~ decreases, the prestressing force TABULAR DATA OMITTED increases (P |is greater than~ |P.sub.min~ = 2,268 kN) while the midspan eccentricity decreases (|e.sub.c~ |is less than~ |e.sub.c max~ = 535 mm). * To minimize the initial camber, the optimal values of P and |e.sub.c~ are obtained from the ULS flexural strength and initial camber constraint equations. * Any solution from Table 2 may be adopted as optimal. For this problem, since the initial camber is not critical, solution 5 may be chosen as optimal as it yields the lowest value of the prestressing force. Thus, the optimum solution for this bridge design problem is S = 2.90 m, |Mathematical Expression Omitted~, (n = 4 girders), P = 2,268 kN, |e.sub.e~ = 0 and |e.sub.c~ = 535 mm, which yields an initial camber of 13.4 mm. * For all Pareto optima the end eccentricity is zero. CONCLUSIONS The multiobjective optimization approach presented in this paper demonstrates the potential of the constraint approach coupled with nonlinear programming techniques to solve a variety of prestressed concrete design problems in a very efficient way. The major merits of the approach are: (1) Inclusion of all possible (even conflicting) objective functions for a given structural design problem; (2) satisfaction of all constraints as in any optimization approach (always feasible designs); and (3) nonunique optimal solutions (Pareto optima). In multiobjective optimization, the designer enjoys some flexibility in the selection of the preferred solution from the set of Pareto optima. The inherent subjectivity involved in this selection may be minimized by specifying a rational hierarchy of the objective functions (primary, secondary, tertiary,... etc...). The result is a design in which a sound engineering compromise between conflicting objectives may be achieved. The advantage of the |epsilon~-constraint approach over other approaches for transforming the multiobjective optimization problem into a single objective problem lies in the rational determination of the bounds ||epsilon~.sub.i~ to be imposed on the secondary objectives, which are then transformed into constraints. The limits ||epsilon~.sub.i~ are determined by performing several preliminary single objective optimizations for each objective function in turn. The multiobjective optimization approach enables the solution of optimization problems for which adequate allowable limits on some structural responses (e.g. initial camber) are not known by treating the ill-defined constraints as objective functions. Finally, this approach enables some insight into the sensitivity of various objectives functions to the design variables of a structural design problem. ACKNOWLEDGMENTS The financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada under Grant A-4789, which made possible the research reported in this paper, is gratefully acknowledged. APPENDIX I. REFERENCES Brook, A., Kendruck, D., and Meeraus, A. (1988). GAMS-general algebraic modelling system, a user's guide, Scientific Press, Redwood City. Calif. 'Building code requirements for reinforced concrete.' (1989). ACI 318-89, American Concrete Institute (ACI), Detroit, Mich. Carmichael, D. G. (1980). 'Computation of Pareto optima in structural design.' Int. 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NOTATION The following symbols are used in this paper: A = concrete section area; A, C = constant matrices; B, D = constant vectors; B, S, N = components of matrix C associated with basic, superbasic, and nonbasic variables, respectively; |C.sub.bot~, |C.sub.top~ = bottom and top concrete covers, respectively; |c.sub.c~, |c.sub.p~ = unit costs of concrete and prestressing steel per unit volume and weight, respectively; |E.sub.ci~ = elastic modulus of concrete at time of transfer; e = tendon eccentricity at slab midspan; |e.sub.e~, |e.sub.c~ = tendon eccentricities at bridge girder end and midspan, respectively; |f.sub.i~ = ith objective function; |g.sub.k~ = kth inequality constraint; |Mathematical Expression Omitted~ = first-order Taylor's series expansion of |g.sub.k~; H = Hessian matrix; h = slab depth; |h.sub.l~ = lth equality constraint; |I.sub.g~ = moment of inertia of gross concrete section; J = Jacobian matrix; L = Lagrangian function, slab (bridge) length; |M.sub.u~, |M.sub.n~ = ultimate-load moment and nominal resisting moment of section, respectively; |M.sub.g~, |M.sub.SD~ = own-weight moment and superimposed dead-load moment, respectively; |M.sub.L~ = live-load moment; |n.sub.e~ = number of equality constraints; |n.sub.i~ = number of inequality constraints; |n.sub.n~ = number of nonlinear constraints; P = prestressing force; S = girder spacing; |Mathematical Expression Omitted~ = slab overhang; t = slab thickness; |V.sub.u~, |V.sub.n~ = ultimate load shear and nominal resisting shear of section, respectively; |V.sub.uh~, |V.sub.nh~ = ultimate load horizontal shear and nominal resisting horizontal shear of section, respectively; W = bridge width; |W.sub.p~ = weight of prestressing steel per unit slab area; |W.sub.g~ = slab own weight; |w.sub.p~ = equivalent load due to prestressing; x = vector of design variables; |x.sub.B~, |x.sub.S~, |x.sub.N~ = basic, superbasic, and nonbasic variables; |x.sup.l~ = lower bound on design variable vector; |x.sup.u~ = upper bound on design variable vector; |x.sup.*~ = Pareto optimum; ||delta~.sub.i~ = initial camber; ||delta~.sub.p~, ||delta~.sub.g~ = camber and deflection due to prestressing and own weight, respectively; |delta~x = search direction vector; ||epsilon~.sub.i~ = limit imposed on secondary objective |f.sub.i~ transformed into constraint; |lambda~, |mu~ = Lagrange multipliers; |rho~ = positive penalty parameter; ||sigma~.sub.tt~, |f.sub.tt~ = effective and allowable tensile stresses at transfer, respectively; ||sigma~.sub.tc~, |f.sub.tc~ = effective and allowable compressive stresses at transfer, respectively; ||sigma~.sub.st~, |f.sub.st~ = effective and allowable tensile stresses at service, respectively; ||sigma~.sub.sc~, |f.sub.sc~ = effective and allowable compressive stresses at service, respectively; ||sigma~.sub.ss~, |f.sub.ss~ = effective and allowable compressive stresses at service in slab, respectively; |omega~ = feasible set to which x belongs; and |omega~ = net reinforcement index. Z. Lounis, Res. Asst., Dept. of Civ. Engrg., Univ. of Waterloo, Ontario, Canada. M. Z. Cohn, Prof., Dept. of Civ. Engrg., Univ. of Waterloo, Waterloo, Ontario, Canada.