Optimum spacing design of grillage systems using a genetic algorithm

In this study a genetic algorithm based method is developed for the optimum design of grillage systems. The algorithm not only selects the optimum sections for the grillage elements from a set of standard universal beam sections, but also finds the optimum spacing required for the grillage system. Deflection limitations and allowable stress constraints are considered in the formulation of the design problem. Due to the fact that grillage elements are thin walled sections, warping plays an important role in their design, particularly, when they are subjected to significant torsional loading. The algorithm developed has the flexibility of including or excluding the effect of warping in the design process. Grillage structures are designed for uniformly distributed loading. The optimum spacings are determined both considering and not taking into account the effect of warping in the design. The comparison of the results shows that inclusion of warping in the design process has a significant effect on the optimum spacing.     

Sectorial properties of straight thin-walled beams

The sectorial method of analysis for thin-walled beams subjected to torsional loading offers many advantages. In particular it enables warping restraint effects due to non-uniform torsion to be incorporated into a general beam theory covering all solid, thick-walled and thin-walled beams of non-deformable cross-section. The distribution of warping restraint stresses around the section is defined in a similar way as for bending by a system of sectorial co-ordinates and several additional geometrical terms. A wider understanding and acceptance of this useful method of analysis is only hindered by difficulties in calculating the various sectorial functions. Solutions are readily available for open sections and regular single cell closed sections. For more complex sections, particularly asymmetric or multicellular sections or those with tapering walls, pierced walls or bracing, these calculations are often tedious. A general computer program is described which analyses any cross-section with open or closed parts. Examples are given of its application to the analysis of several straight thin-walled beams.     

Optimum topological design of geometrically nonlinear single layer latticed domes using coupled genetic algorithm

Single layer latticed domes are lightweight and elegant structures that provide cost-effective solutions to cover the large areas without intermediate supports. The topological design of these structures present difficulty due to the fact that the number of joints and members as well as the height of the dome keeps on changing during the design process. This makes it necessary to automate the numbering of joints and members and the computation of the coordinates of joints in the dome. On the other hand the total number of joints and members in a dome is function of the total number of rings exist in the dome. Currently no study is available that covers the topological design of dome structures that give the optimum number of rings, the optimum height of crown and the tubular cross-sectional designations for the dome members under the given general external loading. The algorithm presented in this study carries out the optimum topological design of single layer lattice domes. The serviceability and strength requirements are considered in the design problem as specified in BS5950. The algorithm takes into account the nonlinear response of the dome due to effect of axial forces on the flexural stiffness of members. The optimum solution of the design problem is obtained using coupled genetic algorithm. Having the total number of rings and the height of crown as design variables provides the possibility of having a dome with different topology for each individual in the population. It is shown in the design example considered that the optimum number of joints, members and the optimum height of a geodesic dome under a given external loading can be determined without designer’s interference.     

Cross-section optimal design of composite laminated thin-walled beams

This paper aims to perform optimal design of cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. The finite element method is used in discretizing the theory. For design sensitivity calculations, the cross-section is modelled throughout design elements. Geometrically, these elements may coincide with the laminates that constitute the cross-section. The developed formulation is based on the concept of adjoint structure. After a warping function is calculated for the cross-section, an adjoint problem may be formulated for each of the properties and a corresponding adjoint warping is determined. It can be applied in a unified way to open, closed or hybrid cross-sections. Design optimization is performed by nonlinear programming techniques. Laminate thickness and lamina orientations are considered as design variables.     

A thin-walled box beam finite element for curved bridge analysis

Practical design of single and multispan curved bridges requires an analysis procedure which is easy and economical to use, and provides a physical insight into structural response under general loading conditions. In the work presented, the thin-walled beam theory has been directly combined with the finite element technique to provide a new thin-walled box beam element. The beam element includes three extra degrees-of-freedom over the normal six degrees-of-freedom beam formulation, to take into account the warping and distortional effects as well as shear. The beam may be curved in space and variable cross-sections may be included. The performance of the box beam element has been compared favourably against results obtained from full 3D shell element analysis, differential equation solutions and experimental results.     

Microgimbal torsion beam design using open, thin-walled crosssections

A thin-film micromolding process enabled the construction of microtorsional springs with unique cross-sectional designs by combining high-aspect-ratio beams with horizontal surface features. Cross sections such as T-bars, pi sections, and channels were utilized in creating torsional springs with low torsional stiffnesses and high in- and out-of-plane bending stiffnesses. Experimental modal analysis was used to determine torsional stiffnesses as low as 0.13 μN·m/deg with T-bar springs 45 μm tall, 50 μm wide, and 100 μm long. Springs of the same outer dimensions but with solid rectangular cross sections were calculated to have torsional stiffnesses of at least two orders of magnitude greater. Several microgimbals were constructed using the thin-film micromolding process with various torsional spring designs. Modal analysis was used to experimentally determine pitch and roll natural frequencies. Torsional stiffness models for open, thin-walled sections that included warping effects were developed and used to analytically predict the torsional natural frequencies of tested spring designs to within 20%     

Torsional vibrations of composite bars of variable cross-section by BEM

In this paper a boundary element method is developed for the nonuniform torsional vibration problem of doubly symmetric composite bars of arbitrary variable cross-section. The composite bar consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli and are firmly bonded together. The beam is subjected to an arbitrarily distributed dynamic twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. A distributed mass model system is employed which leads to the formulation of three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions. These problems are solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced torsional vibrations are considered and numerical examples are presented to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The discrepancy in the analysis of a thin-walled cross-section composite beam employing the BEM after calculating the torsion and warping constants adopting the thin tube theory demonstrates the importance of the proposed procedure even in thin-walled beams, since it approximates better the torsion and warping constants and takes also into account the warping of the walls of the cross-section.     

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.     

Topological optimization of continuum structures with design-dependent surface loading – Part I: new computational approach for 2D problems

This paper describes a new computational approach for optimum topology design of 2D continuum structures subjected to design-dependent loading. Both the locations and directions of the loads may change as the structural topology changes. A robust algorithm based on a modified isoline technique is presented that generates the appropriate loading surface which remains on the boundary of potential structural domains during the topology evolution. Issues in connection with tracing the variable loading surface are discussed and treated in the paper. Our study indicates that the influence of the variation of element material density is confined within a small neighbourhood of the element. With this fact in mind, the cost of the calculation of the sensitivities of loads may be reduced remarkably. Minimum compliance is considered as the design problem. There are several models available for such designs. In the present paper, a simple formulation with weighted unit cost constraints based on the expression of potential energy is employed. Compared to the traditional models (i.e., the SIMP model), it provides an alternative way to implement the topology design of continuum structures. Some 2D examples are tested to show the differences between the designs obtained for fixed, design-independent loading, and for variable, design-dependent loading. The general and special features of the optimization with design-dependent loads are shown in the paper, and the validity of the algorithm is verified. An algorithm dealing with 3D design problems is described in Part II, which is developed from the 2D algorithm in the present Part I of the paper.     

Multilevel optimal design of thin-walled continuous beams

This paper presents a multilevel approach to the optimal design of continuous thin-walled beams subjected to multiple loading conditions. The structure is assumed to be an assembly of beam elements each of which is completely specified by four independent design variables. The objective is to minimize the volume of the beam subject to behavioral constraints as well as side constraints. The minimization process is carried out in a double scheme where the local minimization of the element design variables is embedded in a global or system minimization. At the system level the design space is spanned over the cross-sectional moments of inertia of the beam elements. The volume associated with a distribution of moments of inertia is obtained by scanning the local design space in a quest for optimal cross-sectional designs. A study of the local design space has enabled the size of the problem at the local level to be reduced to a unidirectional search. The theory is illustrated with examples from the literature, in particular to emphasize the importance of including side constraints on the variables controlling the design of the cross-sections.     

Optimum topology and shape design of prestressed concrete bridge girders using a genetic algorithm

The purpose of this study is to optimize the topology and shape of prestressed concrete bridge girders. An optimum design approach that uses a genetic algorithm (GA) for this purpose is presented. The cost of girders is the optimum design criterion. The design variables are the cross-sectional dimensions of the prefabricated prestressed beams, the cross-sectional area of the prestressing steel and the number of beams in the bridge cross-section. Stress, displacement and geometrical constraints are considered in the optimum design. AASHTO Standard Specifications for Highway Bridges are taken into account when calculating the loads and designing the prestressed beams. A computer program is coded in Visual Basic for this optimization. Many design examples from various applications have been optimized using this program. Several of these examples are presented to demonstrate the efficiency of the algorithm coded in the study.     

Dynamic characterization of thin-walled laminated channel beams with warping

The dynamic characteristics of a round cornered C-channel beam made of laminated composite material are studied. The sharp corners are rounded for manufacturing considerations. Thin-walled beam theory is used to formulate the coupled vibration of a rounded C-channel beam with fixed-free end conditions. It is shown that cross-sectional properties used for isotropic case are applicable for laminated composites and the material properties needed for the formulation can be obtained using the law of average. A comparative study is conducted to show the advantage of using composites. The effect of radius of corner and warping on the natural frequency is investigated. The functional of the equations of motion is formed using the variational method. The Ritz method has been used to formulate an eigenvalue problem and its frequency equation. The method proposed is systematic. The computerized procedure can be used as a fast design tool in the design of composite channel beam structures.     

Computer method for analysis of multistory structures

A computer method for three dimensional analysis of multistory structures is presented. There are no restrictions referring to types of stiffening elements, their layout and changes in floor plane and variations in cross-sectional properties with height of the structure. The number of stiffening elements which may be included in analysis is practically unlimited. The loads may be vertical and lateral at arbitrary locations. Elastic warping of thin-walled members is included in analysis. The macro-flow-chart of the computer program and results of a numerical example is given.     

Robust seismic design optimization of steel structures

Stochastic performance measures can be taken into account, in structural optimization, using two distinct formulations: robust design optimization (RDO) and reliability-based design optimization (RBDO). According to a RDO formulation, it is desired to obtain solutions insensitive to the uncontrollable parameter variation. In the present study, the solution of a structural robust design problem formulated as a two-objective optimization problem is addressed, where cross-sectional dimensions, material properties and earthquake loading are considered as random variables. Additionally, a two-objective deterministic-based optimization (DBO) problem is also considered. In particular, the DBO and RDO formulations are employed for assessing the Greek national seismic design code for steel structural buildings with respect to the behavioral factor considered. The limit-state-dependent cost is used as a measure of assessment. The stochastic finite element problem is solved using the Monte Carlo Simulation method, while a modified NSGA-II algorithm is employed for solving the two-objective optimization problem.     

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.

     

Vibrations of grids by the finite element method

The finite element method is applied to the free vibration analysis of grids with arbitrary configuration. Grid bars are of solid or thin-walled doubly symmetric cross-section. Stiffness and consistent mass matrices for flexural behavior include the effects of shear deformation and rotary inertia in bending. The torsional behavior of solid sections is approximated by a linear displacement field, and of thin-walled sections, by a cubic. Rotary inertia in torsion is included in both cases and warping inertia, in the latter.

The computer program performs the free vibration analysis starting from the element stiffness and consistent mass matrices. A numerical solution of a thin-walled beam and a parametric solution of an orthogonal and a skew grid with solid and thin-walled bars are presented.     

Automatic mesh generation and modification techniques for mixed quadrilateral and hexahedral element meshes of non-manifold models

A typical geometric model usually consists of both solid sections and thin-walled sections. Through using a suitable dimensional reduction algorithm, the model can be reduced to a non-manifold model consisting of solid portions and two-dimensional portions which represent the mid-surfaces of the thin-walled sections. It is desirable to mesh the solid entities using three-dimensional elements and the surface entities using two-dimensional elements. This paper proposes a robust scheme to automatically generate such a mesh of mixed two-dimensional and three-dimensional elements. It also ensures that the mesh is conforming at the interface of the non-manifold geometries. Different classes of problems are identified and their corresponding solutions are presented.     

Optimization of reinforced concrete frames subjected to historical time-history loadings using DMPSO algorithm

In this paper, the seismic design of reinforced concrete (RC) frames subjected to time-history loadings was formulated as an optimization problem. Because finding the optimum design is relatively difficult and time-consuming for structural dynamics problems, an innovative algorithm combining multi-criterion decision-making (DM) and Particle Swarm Optimization (PSO), called DMPSO, was presented for accelerating convergence toward the optimum solution. The effectiveness of the proposed algorithm was illustrated in some benchmark reinforced concrete optimization problems. The main goal was to minimize the cost or weight of structures subjected to time-history loadings while satisfying all design requirements imposed by building design codes. The results confirmed the ability of the proposed algorithm to find the optimal solutions for structural optimization problems subjected to time-history loadings.     

Optimum geometry design of nonlinear braced domes using genetic algorithm

Domes are lightweight and cost effective structures that are used to cover large areas. They are mainly comprised of a complex network of triangles made out of slender members. The behaviour of flexible dome is nonlinear under the external loads which makes it necessary to consider the geometrical non-linearity in their analysis to obtain realistic response of these structures. Furthermore, instability check during the nonlinear analysis is of prime importance. In this paper, an algorithm is presented for the optimum geometry design of nonlinear braced domes. The height of crown is taken as design variable in addition to the cross-sectional properties of members. A procedure is developed that calculates the joint coordinates automatically for a given height of the crown. The optimum design algorithm takes into account the nonlinear response of the dome due to the effect of axial forces on the flexural stiffnesses of members. It considers serviceability requirements as well as combined strength limitations set by BS 5950. The solution of the design problem is obtained by genetic algorithm. The elastic instability analysis is then carried out for each individual in the initial population until the ultimate load factor is reached. During this analysis, checks on the overall stability of the dome is conducted. If the loss of stability takes place, this individual is taken out of the population and replaced by a new randomly generated individual. This replacement policy is repeated until an individual is found which does not have instability problem. Once the initial population is established where all the individuals are free of instability problem, the regular genetic operations are applied to generate a new population. Number of design examples are included to demonstrate the application of the algorithm.     

Numerical method for analysing open thin-walled structures under interaction of bending and torsion

A computer program is developed to analyse concrete beams of open thin-walled sections, at different stages of loading from zero load to failure. The program is divided into two parts; the first part deals with the beam from zero load to cracking. Of course, loading is combined bending, shear and torsion (warping torsion and St Venant's torsion). In this part of the program, Vlassov's theory has been used. The cracking load is defined as that load which causes principal tensile stresses equal to the tensile strength of concrete. The second part of the program deals with all post-cracking stages of loading from cracking point to failure. An iteration procedure is used until full convergence occurs at a particular cross-section. The geometrical properties are calculated; these include the contribution of steel in the cross-section and that of concrete in the compressive zones. The mathematical model is given. The computer results are compared with earlier experimental results, and the two sets of results show reasonable agreement. The program is written in FORTRAN.