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.     

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

The DCOC-based numerical algorithms developed in Part I are generalized to the minimum cost design of RC frames in which columns are subjected to biaxial bending. All columns in each floor are assumed to have the same cross-section. Likewise, all beams are assumed to have the same section over their entire length. To account for the variation of forces along the height of the columns or the length of the beams, the steel ratio is allowed to vary freely. Real problems are solved to demonstrate the efficiency and practicability of the algorithm developed.     

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.     

Minimum cost design of RC beams with segmentation using continuum-type optimality criteria

Economic designs of reinforced concrete beams with segmentation involving costs of concrete, reinforcing steel and formwork are considered. For practical reasons, the design variables are prescribed by means of an unknown constant and given shape function, whereas the constraints in addition to lower and upper bounds on design variables include maximum deflection in the span, flexural and shear strengths. Conditions of minimality are rigorously derived using calculus of variation on an augmented Lagrangian. An iterative procedure describing the computational aspects of the design is presented using a segmented propped cantilever beam as an example. Numerical examples are presented when the design variables involved are the depth alone or both the depth and steel ratio. A practical application of a simply supported beam is also considered and results compared with those obtained using nonlinear programming techniques (Karihaloo 1993; Kanagasundaram and Karihaloo 1990). An examination of the rate of convergence shows that the present technique is very promising.     

Minimum cost design of multispan partially prestressed concrete T-beams using DCOC

A previous study on the minimum cost design of multispan partially prestressed concrete (PPC) beams using DCOC is extended to multispan T-beams. The cost of construction which includes the costs of concrete, prestressing steel, nonprestressing steel and formwork is minimized. The design constraints include limits on the maximum deflection, flexural and shear strengths, in addition to ductility requirements, and upper and lower bounds on design variables as stipulated by the Australian Code AS 3600. Based on Kuhn-Tucker necessary conditions, the optimality criteria are explicitly derived in terms of the design variables-effective depth, eccentricity of prestressing steel and non-prestressing steel ratio. The prestressing profile is prescribed by parabolic functions. The self-weight of the structure is included in the equilibrium equation of the real system, as is the secondary effect resulting from the prestressing force. An iterative procedure for updating the design variables is outlined. Two numerical examples of multispan PPC beams with T crosssection are solved to show the applicability and efficiency of the DCOC-based technique.     

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 material cost design of five-layer sandwich beams

Welded aluminium box beams have very low vibration damping capacity. Regarding damping it is better to use a three-layer beam constructed from two rectangular hollow sections and a rubber layer glued between them. These three-layer beams have good damping capacity, but the dynamic deflection is large due to shear deformation of the rubber layer. To decrease this deflection two fiber-reinforced plastic layers are used. The minimum material cost design is worked out for such five-layer sandwich beams using the Rosenbrock Hillclimb mathematical programming method. Constraints on stress and local buckling are considered. A comparison is made between the optimized versions of a simple welded aluminium beam, three-layer and five-layer beams. It is shown that the deflection of the five-layer beams is smaller than that of the three-layer ones, but the cost of the five-layer beams is greater.     

DCOC: An optimality criteria method for large systems Part II: Algorithm

The theoretical basis for a new optimality criteria method (DCOC) was presented in Part I of this study (Zhou and Rozvany 1992). It was shown in that paper that a considerable gain in efficiency can be achieved by the proposed method, since the Lagrangian multipliers associated with stress constraints can be, in general, evaluated explicitly and hence the size of the dualtype problem is greatly reduced. The superior efficiency of the proposed method, for problems involving a large number of active stress constraints, was also demonstrated through simple numerical examples. In Part II, the computational algorithm of DCOC is presented in detail and several standard test examples are used for verifying the correctness and efficiency of the proposed method.     

Optimum design of plates and shells using the DCOC method

The DCOC method developed recently by Zhou and Rozvany (1992, 1993) has been shown to highly improve the efficiency of optimality criteria methods. In the present paper, the application of this method to plates and shells is discussed. High efficiency of DCOC is guaranteed by the fact that the computational expense of DCOC is only influenced by the number of active displacement constraints, which is usually very small for the considered problem.     

Optimum cost design of welded box beams with longitudinal stiffeners using advanced backtrack method

The use of longitudinal stiffeners in box girders loaded in bending results in savings in weight and cost. To study these savings the optimized box beams without and with stiffeners are compared to each other. The minimum cross-sectional area design can be solved analytically. A cost function is defined containing material and fabrication (welding) costs. This function is nonlinear in the structural dimensions to be optimized, therefore an advanced backtrack method is worked out and applied. An illustrative numerical example shows the savings. Received June 30, 1999     

Minimum cost design of reinforced concrete structures

The aim of this paper is to demonstrate how the traditional design process can be imitated mathematically to arrive at designs of reinforced concrete structures which conform to the requirement of the new Australian Standard AS3600-1988 and are the least expensive to construct. The total cost includes the costs of concrete, reinforcing steel and formwork. The minimum cost problem is formulated as a non-linear programming problem whose solution is attempted by two techniques. Several numerical examples of multi-span beams and of columns are given using a computer package developed in standard FORTRAN 77. The sensitivity of the minimum cost designs to variations in the relative cost of formwork is also studied.     

Minimum cost design of laterally loaded welded rectangular cellular plates

Material and fabrication costs are included in the cost function. The fabrication cost is calculated by three formulae relating to the preparation, welding and additional costs. The design constraints are related to bending stresses, the local buckling of ribs due to bending and shear and to the limitation of the plate thicknesses. The local buckling of the compressed face plate elements is considered by an effective width calculation. In the numerical examples, the variables are the plate dimensions and the numbers of ribs in two directions. The optimization is carried out for steel Fe 360 and Fe 510 and for various values of the fabrication cost factor. The computations are performed by using the backtrack discrete combinatorial method, Rosenbrock's Hillclimb method and the FSQP method developed by Zhou and Tits (1992), and the results are compared with each other.Partly presented at the international conference Structural Optimization '93, Rio de Janeiro, August 2–6, 1993.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections

This work relates to the analysis of triply coupled vibrations of thin-walled beams having nonsymmetrical open cross-sections. The governing differential equations for coupled bending and torsional vibrations are derived and solved exactly. A recent study on the same subject is criticized and discussed in theoretical and numerical aspects.     

Minimum cost design of a cellular plate loaded by uniaxial compression

Cellular plates are constructed from two base plates and an orthogonal grid of stiffeners welded between them. Halved rolled I-section stiffeners are used for fabrication aspects. The torsional stiffness of cells makes the plate very stiff. In the case of uniaxial compression the buckling constraint is formulated on the basis of the classic critical stress derived from the Huber’s equation for orthotropic plates. The cost function contains the cost of material, assembly and welding and is formulated according to the fabrication sequence. The unknown variables are the base plate thicknesses, height of stiffeners and numbers of stiffeners in both directions. The cellular plate is lighter and cheaper than the plate stiffened on one side. The Particle Swarm Optimization and the IOSO techniques are used to find the optimum. PSO contains crazy bird and dynamic inertia reduction criteria, IOSO is based on a response surface technology.     

Minimum cost design of a welded stiffened square plate loaded by biaxial compression

The optimized plate structure consists of a simply supported square base plate stiffened with an orthogonal grid of flat stiffeners welded to the base plate by fillet welds. The uniformly distributed compressive load acts biaxially in the plane determined by the centre of gravity of T-sections, which consist of a part of the base plate and of a stiffener. In the optimization process the number of stiffeners as well as the thicknesses of the base plate and flat stiffeners, which minimize the cost function and fulfil the design constraints, as sought. The cost function includes the cost of material, assembly, welding and painting. Constraints relate to the global buckling, local buckling of base plate parts and stiffeners as well as to the deflection due to shrinkage of welds. To illustrate the effectiveness of the mathematical methods, the problem is solved by the Rosenbrocks hill-climb algorithm as well as by entropy-based unconstrained minimization.     

Artificial Bee Colony (ABC) algorithm in the design optimization of RC continuous beams

The objective of this study is to obtain the optimum design for reinforced concrete continuous beams in terms of cross section dimensions and reinforcement details using a fine tuned Artificial Bee Colony (ABC) Algorithm while still satisfying the constraints of the ACI Code (2008). The ABC algorithm used in this paper has been slightly modified to include a Variable Changing Percentage (VCP) that further improves its performance when dealing with members consisted of multiple variables. The objective function is the total cost of the continuous beam which includes the cost of concrete, formwork and reinforcing steel bars. The design variables used are beam width, beam height, number and diameter of: bottom continuous reinforcing bars, bottom cutoff reinforcing bars, top continuous reinforcing bars and top cutoff reinforcing bars as well as the diameter of stirrups. Four RC beams of varying complexity are presented and optimized. The first three beams are used to fine tune the control parameters of the ABC algorithm, whereas the fourth beam was previously optimized by Arafa et al. (J Artif Intell 76–88, 2011) and is presented here to prove the superiority of this relatively new optimization algorithm.     

Minimum weight design of structures with random parameters

A general formulation of the minimum-weight optimization problem is presented in the paper. Formulation of the problem is based on the concept of the expected value. Solution of the corresponding mathematical programming problem has been obtained by means of indirect method. To evaluate the magnitude of the influence of random character of the structure parameters two numerical examples concerning plane truss and free vibrating plane strain structure are presented.