Efficient optimum seismic design of reinforced concrete frames with nonlinear structural analysis procedures

Performance-based seismic design offers enhanced control of structural damage for different levels of earthquake hazard. Nevertheless, the number of studies dealing with the optimum performance-based seismic design of reinforced concrete frames is rather limited. This observation can be attributed to the need for nonlinear structural analysis procedures to calculate seismic demands. Nonlinear analysis of reinforced concrete frames is accompanied by high computational costs and requires a priori knowledge of steel reinforcement. To address this issue, previous studies on optimum performance-based seismic design of reinforced concrete frames use independent design variables to represent steel reinforcement in the optimization problem. This approach drives to a great number of design variables, which magnifies exponentially the search space undermining the ability of the optimization algorithms to reach the optimum solutions. This study presents a computationally efficient procedure tailored to the optimum performance-based seismic design of reinforced concrete frames. The novel feature of the proposed approach is that it employs a deformation-based, iterative procedure for the design of steel reinforcement of reinforced concrete frames to meet their performance objectives given the cross-sectional dimensions of the structural members. In this manner, only the cross-sectional dimensions of structural members need to be addressed by the optimization algorithms as independent design variables. The developed solution strategy is applied to the optimum seismic design of reinforced concrete frames using pushover and nonlinear response-history analysis and it is found that it outperforms previous solution approaches.     

Multi-database exploration of large design spaces in the framework of cascade evolutionary structural sizing optimization

In discrete sizing optimization of truss and frame structures the design variables take values from databases, which are usually populated with a relatively small number of cross-section types and sizes. The aim of this work is to allow the use of large-size databases in discrete structural sizing optimization problems, in order to enrich the set of design variable options and increase the potential of achieving high-quality optimal designs. For this purpose, the concept of coarse database is introduced, according to which smaller-size versions of an appropriately ordered large database can be constructed. This concept is combined with the idea of cascading, which allows a single optimization problem to be tackled with a number of autonomous optimization stages. Under this context, several coarse versions of the same full-size database are formed, in order to utilize a different database in each cascade stage executed with an evolutionary optimization algorithm. The first optimization stages of the resulting multi-database cascade procedure make use of the coarsest database versions available and serve the purpose of basic design space exploration. The last stages exploit finer databases (including the original full-size database) and aim in fine tuning the achieved optimal solution. Based on the reported numerical results, multi-database cascading proves to be an effective tool for the handling of large databases and corresponding extensive design spaces in the framework of discrete structural sizing optimization applications.     

Adaptive topology optimization

Topology optimization of continuum structures is often reduced to a material distribution problem. Up to now this optimization problem has been solved following a rigid scheme. A design space is parametrized by design patches, which are fixed during the optimization process and are identical to the finite element discretization. The structural layout is determined, whether or not there is material in the design patches. Since many design patches are necessary to describe approximately the structural layout, this procedure leads to a large number of optimization variables. Furthermore, due to a lack of clearness and smoothness, the results obtained can often only be used as a conceptual design idea.To overcome these shortcomings adaptive techniques, which decrease the number of optimization variables and generate smooth results, are introduced. First, the use of pure mesh refinement in topology optimization is discussed. Since this technique still leads to unsatisfactory results, a new method is proposed that adapts the effective design space of each design cycle to the present material distribution. Since the effective design space is approximated by cubic or Bézier splines, this procedure does not only decrease the number of design variables and lead to smooth results, but can be directly joined to conventional shape optimization. With examples for maximum stiffness problems of elastic structures the quality of the proposed techniques is demonstrated.     

Optimum structural design of spatial steel frames via biogeography-based optimization

Metaheuristic algorithms have provided an efficient tool for designers by which discrete optimum design of real-size steel space frames under design code requirements can be obtained. In this study, the optimum sizing design of steel space frames is formulated according to provisions of Load and Resistance Factor Design—American Institute of Steel Construction. The weight of the steel frame is taken as objective function. The design algorithm selects the appropriate W sections for members of the steel frame such that the frame weight is the minimum and design code limitations are satisfied. The biogeography-based optimization algorithm is utilized to find out the optimum solution of the discrete programming problem. This algorithm is one of the recent additions to metaheuristic techniques which are based on theory of island biogeography where each habitat is assumed to be potential solution for the design problem. The performance of the biogeography-based optimization algorithm is compared with other recent metaheuristic algorithms such as adaptive firefly algorithm, teaching and learning-based optimization, artificial bee colony optimization, dynamic harmony search algorithm, and ant colony algorithm. It is shown that biogeography-based optimization algorithm outperforms other metaheuristic techniques in the design examples considered.

     

Matrix-free aerostructural optimization of aircraft wings

In structural optimization subject to failure constraints, computing the gradients of a large number of functions with respect to a large number of design variables may not be computationally practical. Often, the number of constraints in these optimization problems is reduced using constraint aggregation at the expense of a higher mass of the optimal structural design. This work presents results of structural and coupled aerodynamic and structural design optimization of aircraft wings using a novel matrix-free augmented Lagrangian optimizer. By using a matrix-free optimizer, the computation of the full constraint Jacobian at each iteration is replaced by the computation of a small number of Jacobian-vector products. The low cost of the Jacobian-vector products allows optimization problems with thousands of failure constraints to be solved directly, mitigating the effects of constraint aggregation. The results indicate that the matrix-free optimizer reduces the computational work of solving the optimization problem by an order of magnitude compared to a traditional sequential quadratic programming optimizer. Furthermore, the use of a matrix-free optimizer makes the solution of large multidisciplinary design problems, in which gradient information must be obtained through iterative methods, computationally tractable.     

School-based optimization for performance-based optimum seismic design of steel frames

The performance-based optimum seismic design of steel frames is one of the most complicated and computationally demanding structural optimization problems. Metaheuristic optimization methods have been successfully used for solving engineering design problems over the last three decades. A very recently developed metaheuristic method called school-based optimization (SBO) will be utilized in the performance-based optimum seismic design of steel frames for the first time in this study. The SBO actually is an improved/enhanced version of teaching–learning-based optimization (TLBO), which mimics the teaching and learning process in a class where learners interact with the teacher and between themselves. Ad hoc strategies are adopted in order to minimize the computational cost of SBO results. The objective of the optimization problem is to minimize the weight of steel frames under interstory drift and strength constraints. Three steel frames previously designed by different metaheuristic methods including particle swarm optimization, improved quantum particle swarm optimization, firefly and modified firefly algorithms, teaching–learning-based optimization, and JAYA algorithm are used as benchmark optimization examples to verify the efficiency and robustness of the present SBO algorithm. Optimization results are compared with those of other state-of-the-art metaheuristic algorithms in terms of minimum structural weight, convergence speed, and several statistical parameters. Remarkably, in all test problems, SBO finds lighter designs with less computational effort than the TLBO and other methods available in metaheuristic optimization literature.

     

A hybrid multilevel approach for aeroelastic optimization of composite wing-box

The quest for finding optimum solutions to engineering problems has been existing for a long time. In the last decade several optimization techniques have been applied to the structural design of composite wing structures. Generally many of these proposed procedures have dealt with different disciplines such as aerodynamics, structures, or dynamics separately. However an aeronautical design process is multidisciplinary since it involves strong couplings and interactions among, for instance, aerodynamics, dynamics, flight mechanics and structures. The main problem in a multidisciplinary aircraft design is usually the development of an efficient method to integrate structures or structural properties, which can be considered both as “global” and “local” design variables. This paper describes an integrated aerodynamic / dynamic / structural optimization procedure for a composite wing-box design. The procedure combines an aeroelastic optimization of a composite wing based on a general purpose optimizer such as the Sequential Quadratic Programming (SQP) and a composite optimization using Genetic Algorithm (GA). Both the optimizations are implemented through a hybrid multilevel decomposition technique.     

Design space optimization using design space adjustment and refinement

To deal with large-scale problems that often occur in industry, the authors propose design space optimization with design space adjustment and refinement. In topology optimization, a design space is specified by the number of design variables, and their layout or configuration. The proposed procedure has two efficient algorithms for adjusting and refining design space. First, the design space can be adjusted in terms of design space expansion and reduction. This capability is evolutionary because the design domain expands or reduces wherever necessary. Second, the design space can be refined uniformly or selectively wherever and whenever necessary, ensuring a target resolution with fewer elements, especially for selective refinement. Accordingly, the proposed procedure can handle large-scale problems by solving a sequence of smaller problems. Two examples show the efficiency of the proposed approach.     

Design of structure/control systems with transient response constraints exhibiting relative minima

Structural optimization problems involving dynamic behaviour constraints often exhibit nonconvex design spaces. The direct application of a global optimization algorithm requires a large number of function evaluations which in turn require a large number of dynamic structural analyses. This work presents a strategy aimed at finding the global optimum for problems with transient dynamic behaviour constraints based on approximation concepts. The method consists of generating and solving a sequence of approximate problems using a global optimizer. The approximations are explicit and capture most of the inherent nonconvexity of the exact functions. A simple example. problem is presented to illustrate the procedure set forth.     

A mixed GA-PSO-based approach for performance-based design optimization of 2D reinforced concrete special moment-resisting frames

A mixed genetic algorithm and particle swarm optimization in conjunction with nonlinear static and dynamic analyses as a smart and simple approach is introduced for performance-based design optimization of two-dimensional (2D) reinforced concrete special moment-resisting frames. The objective function of the problem is considered to be total cost of required steel and concrete in design of the frame. Dimensions and longitudinal reinforcement of the structural elements are considered to be design variables and serviceability, special moment-resisting and performance conditions of the frame are constraints of the problem. First, lower feasible bond of the design variables are obtained via analyzing the frame under service gravity loads. Then, the joint shear constraint has been considered to modify the obtained minimum design variables from the previous step. Based on these constraints, the initial population of the genetic algorithm (GA) is generated and by using the nonlinear static analysis, values of each population are calculated. Then, the particle swarm optimization (PSO) technique is employed to improve keeping percent of the badly fitted populations. This procedure is repeated until the optimum result that satisfies all constraints is obtained. Then, the nonlinear static analysis is replaced with the nonlinear dynamic analysis and optimization problem is solved again between obtained lower and upper bounds, which is considered to be optimum result of optimization solution with nonlinear static analysis. It has been found that by mixing the analyses and considering the hybrid GA-PSO method, the optimum result can be achieved with less computational efforts and lower usage of materials.     

A hierarchical design concept for shape optimization based on the interaction of CAGD and FEM

The numerical treatment of shape optimization problems requires sophisticated software tools such as Computer Aided Design (CAD), the Finite Element Method (FEM) and a suitable Mathematical Programming (MP) algorithm. Efficiency of the overall procedure is guaranteed if these tools interact optimally. The theoretical and numerical effort for sensitivity analysis reflect the complexity of this engineering problem.In this paper we outline a general modelling concept for shape optimization problems. Hierarchical design models within Computer Aided Geometrical Design (CAGD) and the interaction of geometry and FEM lead to an efficient overall optimization procedure. Our concept has been derived, implemented and tested for shell structures but it is seen to be generally applicable.After a short introduction containing the state of the art we give an overview of the numerical tools used and outline the interaction of CAGD and FEM within the overall optimization procedure.The paper is mainly devoted to the hierarchical design space based on a hierarchical geometrical modelling. The major part of computational effort is consumed by sensitivity analysis related to the number of design variables. Therefore, this number should be limited and only few powerful design variables corresponding to the special interests of the considered problem should be defined. This procedure may lead to a considerable limitation of the design space. Based on a hierarchy in the geometrical model different types of design variables are introduced: design variables with global, regional and local influence. The new method is based on successive activation of these types of design variables. This procedure leads to a considerable reduction of computational time for the sensitivity analysis without loss of geometrical flexibility.A new method of geometrical refinement and a successive adaptively driven expansion and reduction of the design space is described. It is based on the degree elevation or degree reduction of parametric curves and surfaces, respectively.A numerical example illustrates the new method and the efficiency of the overall optimization procedure.     

An efficient method for reliability-based design optimization of nonlinear inelastic steel space frames

This paper proposes an effective numerical procedure for reliability-based design optimization (RBDO) of nonlinear inelastic steel frames by integrating a harmony search technique (HS) for optimization and a robust method for failure probability analysis. The practical advanced analysis using the beam-column approach is used for capturing the nonlinear inelastic behaviors of frames, while a detail implement of HS for discrete optimization of steel frames is introduced. The failure probability of structures is evaluated by using the combination of the improved Latin Hypercube (IHS) and a new effective importance sampling (EIS). The efficiency and accuracy of the proposed procedure are demonstrated through three mathematical examples and five steel frames. The results obtained in this paper prove that the proposed procedure is computationally efficient and can be applied in practical design. Furthermore, it is shown that the use of nonlinear inelastic analysis in the optimization of steel frames yields more realistic results.     

Hybridizing electromagnetism-like mechanism algorithm with migration strategy for layout and size optimization of truss structures with frequency constraints

The optimal design of a truss structure with dynamic frequency constraints is a highly nonlinear optimization problem with several local optimums in its search space. In this type of structural optimization problems, the optimization methods should have a high capability to escape from the traps of the local optimums in the search space. This paper presents hybrid electromagnetism-like mechanism algorithm and migration strategy (EM–MS) for layout and size optimization of truss structures with multiple frequency constraints. The electromagnetism-like mechanism (EM) algorithm simulates the attraction and repulsion mechanism between the charged particles in the field of the electromagnetism to find optimal solutions, in which each particle is a solution candidate for the optimization problem. In the proposed EM–MS algorithm, two mechanisms are utilized to update the position of particles: modified EM algorithm and a new migration strategy. The modified EM algorithm is proposed to effectively guide the particles toward the region of the global optimum in the search space, and a new migration strategy is used to provide efficient exploitation between the particles. In order to test the performance of the proposed algorithm, this study utilizes five benchmark truss design examples with frequency constraints. The numerical results show that the EM–MS algorithm is an alternative and competitive optimizer for size and layout optimization of truss structures with frequency constraints.     

Structural synthesis of frame reinforced,submersible, circular,cylindrical hulls

This paper describes a mathematical programming procedure for the automated optimal structural synthesis of frame stiffened, cylindrical shells. For a specified set of design parameters such as external pressure, shell radius and length and material properties, the method generates those values of the design variables that produce a minimum weight design. The skin, frame web and frame flange thicknesses and the flange width are treated as continuous variables. Frame spacing is considered a discrete variable. Constraint equations control local and general shell and frame instability and yield. Limits may be placed on the variable values, and certain geometric or space constraints can be applied. The mixed (continuous and discrete nonlinear programming problem is solved by a combination of a discrete ‘Golden Search’ for the optimal number of frames and the ‘Direct Search Design Algorithm’ which provides the optimum values of the continuous variables.     

Using genetic algorithm for the optimization of seismic behavior of steel planar frames with semi-rigid connections

Beam-column connections have a significant role in the results of the analysis and the design of steel frames. In this paper, a genetic algorithm has been used for the non-linear analysis and design of steel frames. For minimizing the weight of frames, while satisfying the applied constraints and restraints such as the limits of normal and combined stresses, criteria such as target displacement(s) and the number and locations of plastic hinges were used. To analyze and design the frame elements, I and box-shaped standard sections were used for beams and columns, respectively. Finally, some clues for finding optimizing semi-rigid connection stiffness values for beam-to-column connections have been obtained. The degrees of these rigidities are obtained by a genetic algorithm during the procedure of optimization in order to reach a frame with the minimum weight. SAP2000 structural analysis program was used to perform modal analysis and linear and non-linear static solutions as well as the design of the elements. A MATLAB program was written for the process of optimization. The procedure of optimization was based on a weight minimization carried out for 9 steel frames. Thus, the optimum connection stiffness could be obtained for minimizing the weight of the structure. The results show that the non-linear analysis gives less weight for short period frames with semi-rigid connections compared to those of linear ones. However, by increasing the periods of frames, much less weights are obtained in the case of non-linear analysis with semi-rigid connections.     

Optimization of nonlinear trusses using a displacement-based approach

A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust. Received December 4, 2000     

Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.     

Composite stacking sequence optimization for aeroelastically tailored forward-swept wings

A method for stacking sequence optimization and aeroelastic tailoring of forward-swept composite wings is presented. It exploits bend-twist coupling to mitigate aeroelastic divergence. The method proposed here is intended for estimating potential weight savings during the preliminary aircraft design stages. A structural beam model of the composite wingbox is derived from anisotropic shell theory and the governing aeroelastic equations are presented for a spanwise discretized forward swept wing. Optimization of the system to reduce wing mass is undertaken for sweep angles of ?35° to 0° and Mach numbers from 0.7 to 0.9. A subset of lamination parameters (LPs) and the number of laminate plies in each pre-defined direction (restricted to {0°,±45°, 90°}) serve as design variables. A bi-level hybrid optimization approach is employed, making use of a genetic algorithm (GA) and a subsequent gradient-based optimizer. Constraints are implemented to match lift requirements and prevent aeroelastic divergence, excessive deformations, airfoil stalling and structural failure. A permutation GA is then used to match specific composite ply stacking sequences to the optimum design variables with a limited number of manufacturing constraints considered for demonstration purposes. The optimization results in positive bend-twist coupling and a reduced structural mass. Results are compared to an uncoupled reference wing with quasi-isotropic layups and with panel thickness alone the design variables. For a typical geometry and a forward sweep of ?25° at Mach 0.7, a wingbox mass reduction of 13 % was achieved.     

Bat inspired algorithm for discrete size optimization of steel frames

Bat inspired (BI) algorithm is a recently developed metaheuristic optimization technique inspired by echolocation behavior of bats. In this study, the BI algorithm is examined in the context of discrete size optimization of steel frames designed for minimum weight. In the optimum design problem frame members are selected from available set of steel sections for producing practically acceptable designs subject to strength and displacement provisions of American Institute of Steel Construction-Allowable Stress Design (AISC-ASD) specification. The performance of the technique is quantified using three real-size large steel frames under actual load and design considerations. The results obtained provide a sufficient evidence for successful performance of the BI algorithm in comparison to other metaheuristics employed in structural optimization.     

Optimum cost design of reinforced concrete slabs using neural dynamics model

For structural optimization algorithms to find widespread usage among practicing engineering they must be formulated as cost optimization and applied to realistic structures subjected to the actual constraints of commonly used design codes such as the ACI code. In this article, a general formulation is presented for cost optimization of single- and multiple-span RC slabs with various end conditions (simply supported, one end continuous, both ends continuous, and cantilever) subjected to all the constraints of the ACI code. The problem is formulated as a mixed integer-discrete variable optimization problem with three design variables: thickness of slab, steel bar diameter, and bar spacing. The solution is obtained in two stages. In the first stage, the neural dynamics model of Adeli and Park is used to obtain an optimum solution assuming continuous variables. Next, the problem is formulated as a mixed integer-discrete optimization problem and solved using a perturbation technique in order to find practical values for the design variables. Practicality, robustness, and excellent convergence properties of the algorithm are demonstrated by application to four examples.