Minimum cost design of RC frames using the DCOC method Part I: Columns under uniaxial bending actions

The paper solves the minimum-cost design problem of RC plane frames. The cost to be minimized includes those of concrete, reinforcing steel and formwork, whereas the design constraints include limits on maximum deflection at a specified node, on bending and shear strengths of beams and on combined axial and bending strength of columns, in accordance with the limit state design (LSD) requirements. The algorithms developed in this work can handle columns under uniaxial bending actions. In the companion paper the numerical procedure is generalized to include columns subjected to biaxial bending. On the basis of discretized continuum-type optimality criteria (DCOC), the design problem is systematically formulated, followed by explicit mathematical derivation of optimality criteria upon which iterative procedures are developed for the solution of design problems when the design variables are the cross-sectional parameters and steel ratios. For practical reasons, the cross-sectional parameters are chosen to be either uniform per member or uniform for several members at a given floor level. The procedure is illustrated on several test examples. It is shown that the DCOC-based methods are particularly efficient for the design of large RC frames.     

Optimum design of geometrically nonlinear space trusses

A structural optimization algorithm is developed for geometrically nonlinear three-dimensional trusses subject to displacement, stress and cross-sectional area constraints. The method is obtained by coupling the nonlinear analysis technique with the optimality criteria approach. The nonlinear behaviour of the space truss which was required for the steps of optimality criteria method was obtained by using iterative linear analysis. In each iteration the geometric stiffness matrix is constructed for the deformed structure and compensating load vector is applied to the system in order to adjust the joint displacements. During nonlinear analysis, tension members are loaded up to yield stress and compression members are stressed until their critical limits. The overall loss of elastic stability is checked throughout the steps of algorithm. The member forces resulted at the end of nonlinear analysis are used to obtain the new values of design variables for the next cycle. Number of design examples are presented to demonstrate the application of the algorithm. It is shown that the consideration of nonlinear behaviour of the space trusses in their optimum design makes it possible to achieve further reduction in the overall weight. The other advantage of the algorithm is that it takes into account the realistic behaviour of the structure, without which an optimum design might lead to erroneous result. This is noticed in one of the design example where a tension member changed into a compression one at the end of nonlinear analysis.     

Optimum design of rigidly jointed frames

In this paper a method is presented for the optimum, minimum weight design of rigidly jointed frames. The problem is formulated using the matrix displacement method for which the design variables are not only the areas of members but also the displacements of joints. Both the stress and displacement requirements, as stated in B.S. 449, are taken into consideration. The problem turns out to be one of non-linear programming. This is linearised by using the first two terms of a Taylor expansion. To ensure a feasible solution move limits are employed so that the iteration procedure avoids the reduction of vital structural variables to zero. The design procedure developed do not require the structural analysis equations to be solved during the optimisation process. Examples are given to demonstrate the method.     

An efficient optimality criteria approach to the minimum weight design of elastic structures

This paper is concerned with the optimality criteria approach to the minimum weight design of elastic structures analyzed by finite elements. It is first shown that the classical methods apply the lagrangian multiplier technique to an explicit problem. This one results from high quality, first order approximations of the displacement constraints and cruder, zero order approximations of the stress constraints. A generalized optimality criterion is then proposed as the explicit Kuhn-Tucker conditions of a first order approximate problem. Hence a hybrid optimality criterion is developed by using both zero and first order approximations of the stress constraints, according to their criticality. Efficient solution algorithms of the explicit approximate problem are suggested. Its dual statement generalizes the classical lagrangian approaches. Its primal statement leads to a rigorous definition of the optimality criteria approach, which appears to be closely related to the linearization methods of mathematical programming. Finally some numerical applications clearly illustrate the efficiency of the generalized and hybrid optimality criteria.     

Optimal design of planar frames based on structural performance criteria

The aim of this study is to present a new approach, nearer to engineer's point of view, for optimizing the structures, in particular for the planar frames. As a means of improving the overall stability of the structure the objective function chosen to be maximized is the eigenvalues of the buckling modes. The optimization process involves two stages: a preliminary design when a prescribed value for the natural fundamental vibration period is the main constraint and a second stage in which the usual stress, displacement, and side constraints are taken into account. The algorithm used for optimization is based on a classical optimality criteria approach. The steps of the algorithm are illustrated by some examples which demonstrate the effectiveness of the proposed procedure in improving design.     

Optimum shape design of cold-formed thin-walled steel sections

The algorithm presented in this study obtains the optimum cross-sectional dimensions of cold-formed thin-walled steel beams subjected to general loading. It has the flexibility of considering different cross-sectional shapes such as symmetrical or unsymmetrical channel, lipped channel or Z-sections. The algorithm treats the cross-sectional dimensions such as width, depth and wall thickness as design variables and considers the displacement as well as stress limitations. The presence of torsional moments causes warping of thin-walled sections. The effect of warping in the calculation of normal stresses is included using Vlasov theorems. These theorems require the computation of sectorial properties of cross-sections. A general numerical procedure is presented for obtaining these properties. The optimum design problem of thin-walled open sections subjected to combined loading turns out to be a highly nonlinear problem. It is shown that optimality criteria method can effectively be used to obtain its solution. A number of design examples are presented to demonstrate the application of the algorithm.     

Optimality criteria and mathematical programming in structural weight optimization

It is shown that the optimality criteria approach to the structural weight minimization results from a proper linearization of the displacement constraints but not of the stress constraints in terms of the reciprocal design variables. On the basis of this interpretation, two new ideas are suggested. First, a “mixed method” is proposed, that can be regarded either as a pure mathematical programming or as an optimality criterion approach. It allows for a convergence control of the optimization process. Secondly, a proper linearization of the stress constraints is introduced by considering the stress components as linear combinations of the generalized displacements. The numerical applications presented in the paper show that both modifications of the optimization scheme lead to a significant improvement in the convergence properties.     

A unified approach to structural weight minimization

It is shown that the two classical approaches to structural optimization have now reached a stage where they employ the same basic principles. Indeed, the well-known optimality criteria approach can be viewed as transforming the initial problem in a sequence of simple explicit problems in which the constraints are approximated from virtual work considerations. On the other hand, the mathematical programming approaches have progressively evoluated to a linearization method using the reciprocals of the design variables — this powerful method is proven here to be identical to a generalized optimality criteria approach. Finally, new efficient methods are proposed: (a) a hybrid optimality criterion based on first-order approximations of the most critical stress constraints and zeroth-order approximations of the others and (b) a mixed method which lies between a strict primal mathematical programming method and a pure optimality criteria (or linearization) approach. Simple numerical problems illustrate the concepts established in the paper.     

A new approach to optimal design of elastic structures

A new approach to the optimization of elastic trusses under stress constraints is discussed. The stress constraints are transformed into compliance constraints, which makes possible the derivation of a simple optimality condition. The condition provides a simple test to check whether a fully stressed design is optimal, for single as well as multiple loading conditions. It is readily extended to optimization problems, in which both stress and displacement constraints are imposed. An iterative routine, that is both simple and efficient, is derived from the optimality condition. Numerical examples of the application of the routine are presented.     

Improved optimum structural design by passive control

The problem of optimum structural design by passive control is stated in a nonlinear programming form. A solution procedure, based on a successive selection of design and control variables, is presented. Neglecting the implicit analysis equations, the solution becomes independent of the control variables and a lower bound (LB) on the optimum can easily be obtained. The control variables are then selected to satisfy all constraints. If this cannot be achieved, the LB constraints are modified and the control variables are chosen for the revised optimal design. These two steps are repeated until the final optimum is reached.Employing the proposed procedure on various structural systems subjected to static loads showed that the final optimum has been achieved after a very small number of iteration cycles. The numerical examples illustrate a solution with two types of control devices: a linear spring device and a limited displacement device. It has been found that the final optimum is often identical or close to the initial LB solution. Savings of 14 to 63 percent in weight, compared with conventional optima without control, have been demonstrated for some common structures.     

Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.     

Optimum configurational and dimensional design of truss structures

The feasibility of simultaneous optimization of member sizing and structural configuration of truss structures is demonstrated. The structural analysis is treated by the finite element displacement method and the optimization accomplished by the steepest descent method. Inequality constraints including limitations on both state variables (stress and displacement) and design variables (element cross sectional areas and nodal point placement) are included.The computational results show that in the presence of displacement constraints, the configuration of the optimum design sometimes differs considerably from the fully stressed design. The techniques can be extended to other structures such as beams, frames, plates, etc. and to include the possibility of Euler buckling.     

Automated optimization of transverse frame layouts for ships by elastic-plastic finite element analysis

This paper presents a model for the optimum design of ship transverse frames. An elastic-plastic finite element analysis algorithm for plane frames has been incorporated in the model to evaluate the ultimate strength of the overall frame, and different effects of design loads. Using these strengths and load effects, appropriate design constraints are then formulated to prevent different failure categories; the overall collapse, ultimate limit state failures and serviceability failures. Possible instabilities and effects of combined loads are accounted for in formulating these constraints. Scantlings of the frame structure have been modelled as free design variables. The weight function and different constraint functions are then derived relating design variables in such a way that once parameters for finite element analysis are input, the scheme automatically forms the objective function and all constraints, and then interacts with the simplex algorithm through sequential linearization to find the optimum solution. Thus the scheme is almost automatic. Different layouts of the frame structure have been designed by executing this scheme, which demonstrates the capability of the model and the possibility of weight savings by choosing the appropriate layout. Finally, it is suggested how this model would interact with the design of longitudinal materials to ensure the overall optimality in ship hull module design, to prevent grillage buckling and to validate underlying assumptions in analysis.     

Minimum cost design of reinforced concrete beams using continuum-type optimality criteria

This paper outlines a general procedure for obtaining, on the basis of continuum-type optimality criteria (COC), economic designs for reinforced concrete beams under various design constraints. The costs to be minimized include those of concrete, reinforcing steel and formwork. The constraints consist of limits on the maximum deflection, and on the bending and shear strengths. However, the formulation can easily cater for other types of constraints such as those on axial strength. Conditions of cost minimality are derived using calculus of variation on an augmented Lagrangian. An iterative procedure based on optimality criteria is applied to a test example involving a reinforced concrete propped cantilever beam whose cross-section varies continuously. Numerical examples are presented in which the design variables are both the width and the depth or the depth alone, and the optimal costs are compared. The solution of the test example with depth alone as the design variable is confirmed by an alternative approach using discretized continuum-type optimality criteria (DCOC).     

Minimum cost design of RC beams using DCOC Part I: Beams with freely-varying cross-sections

A procedure for the economic design of reinforced concrete beams under several design constraints is outlined on the basis of discretized continuum-type optimality criteria (DCOC). The costs to be minimized involve those of concrete, reinforcing steel and formwork. The design constraints include limits on the maximum deflection in a given span, on bending and shear strengths, in addition to upper and lower bounds on design variables. An explicit mathematical derivation of optimality criteria is given based on the well known Kuhn-Tucker necessary conditions, followed by an iterative procedure for designs when the design variables are the depth and the steel ratio, or the depth alone. The computer code developed in Part I can handle freely-varying design variables along the members of any multispan beam. In Part II the DCOC and computer code are developed for designs when the member cross-section is assumed to be uniform along its entire length. Several test examples have been solved to prove the accuracy and efficiency of the DCOC-based techniques.     

Short communication: Iterative design procedure in the presence of technological constraints

In this paper an iterative procedure for the design of frames in the elastic range is given. In this connection, the total weight resulting from the individual weight sum of structural elements or parts is assumed as a cost function. These weights have been assumed as design variables and their effects, in terms of stress or displacement, are roughly estimated by means of suitable computed coefficients. A linear programming problem is thus obtained, enabling an iterative process to be performed aimed at the achievement of a designed structure which is, at the same time, lighter and in accordance with the technological constraints and the limitations envisaged by code requirements.     

Generalized shape optimization using stress constraints under multiple load cases

Optimal shape design using numerical techniques is an increasingly useful engineering tool. Generalized or layout optimal design where the topology of the object is not fixed is one of the emerging applications. These problems are numerically difficult to solve due to the large number of design variables and equality/inequality constraints. Solutions have focused primarily on compliance based minimization under a fixed volume. A more usual engineering approach would be one of minimizing the volume under a stress or deflection constraint. This, however, can lead to problems as stress is a local quantity and volume minimization of multiple load cases under stress constraints may not result in the stiffest design for the remaining material. The approach adopted here is based on a differential rate equation governed by a local operator that defines the state of each element at each time step. This algorithm forms the optimality criteria for the problem. To satisfy the global stress constraints, a feedback derivative is used, analogous to a Lagrange multiplier. The original method for a single load case developed by these authors is extended to deal with multiple load cases. Additionally, a discussion of the global behaviour is included.     

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.     

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.     

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.