Design of non-linear steel frames for stress and displacement constraints with semi-rigid connections via genetic optimization

A genetic-algorithm-based optimum design method is presented for non-linear steel frames with semi-rigid connections and column bases. The design algorithm obtains the minimum total cost, which comprises total member plus connection costs, by selecting suitable sections from a standard set of steel sections such as American Institute of Steel Construction (AISC) wide-flange (W) shapes. A genetic algorithm is employed as the optimization method, which utilizes reproduction, crossover and mutation operators. Displacement and stress constraints of AISC Allowable Stress Design (ASD) specification and size constraints for beams and columns are imposed on the frame. The algorithm requires a large number of non-linear analyses of frames. The analyses cover both the non-linear behavior of beam-to-column connection and P- effects of beam-column members. The Frye and Morris polynomial model is used for modeling semi-rigid connections. Two design examples with various types of connections are presented to demonstrate the application of the algorithm. The semi-rigid connection and column base modeling results in more economical solutions than rigid connection modeling, but increases the sway of frames.     

On the minimum-maximum bending moment and the least-weight design of semi-rigid beams

Semi-rigid steel construction has been adopted in modern design specifications of steel structures for many years. Unlike traditional analysis and design of steel structures in which connections are usually idealized as either pinned or fully rigid, the analysis of semi-rigid structures accounts for the rotational behaviour of connections. Therefore, more realistic predictions of the responses and strength of such structures can be obtained for the purposes of generating cost-effective designs. This paper first presents and proves the achievable minimum value of the maximum bending moment for a semi-rigid beam under an applied load due to the variation of connection stiffness. A second theorem concerns the relationship between the achievable least-weight design and the minimum-maximum moment for the semi-rigid beam through the modification of connection stiffness, and is shown to be useful in the design of semi-rigid structures. Received November 10, 1999     

Optimum weight design of steel space frames with semi-rigid connections using harmony search and genetic algorithms

In this paper, an optimization process using MATLAB-SAP2000 Open Application Programming Interface (OAPI) is presented for optimum design of space frames with semi-rigid connections. A specified list including W-profiles taken from American Institute of Steel Construction (AISC) is used in the selection of suitable sections. The stress constraints as indicated in load and resistance factor design of AISC, lateral displacement constraints being the top- and inter-storey drift and geometric constraints are considered in the optimization process. Genetic algorithm method based on biological principles and harmony search algorithm method based on the processes of musical harmony are used for optimum designs. Two different space frames are solved for the cases of rigid and semi-rigid connections, separately. A computer program is coded in MATLAB for the purpose interacting with SAP2000 OAPI. Results obtained from the analyses show that type of semi-rigid connections plays a crucial role in the optimization of steel space frames and increases the optimum weight.

     

Harmony search algorithm for minimum cost design of steel frames with semi-rigid connections and column bases

Harmony search-based algorithm is developed to determine the minimum cost design of steel frames with semi-rigid connections and column bases under displacement, strength and size constraints. Harmony search (HS) is recently developed metaheuristic search algorithm which is based on the analogy between the performance process of natural music and searching for solutions of optimum design problems. The geometric non-linearity of the frame members, the semi-rigid behaviour of the beam-to-column connections and column bases are taken into account in the design algorithm. The results obtained by semi-rigid connection and column base modelling are also compared to one developed by rigid connection modelling. The efficiency of HS algorithm, in comparison with genetic algorithms (GAs), is verified with three benchmark examples. The results indicate that HS could obtain lighter frames and less cost values than those developed using GAs.     

Optimum weight analysis of steel frames with semirigid connections

A fact that should be initially emphasized is that this research work does not mainly aim at development of optimization techniques which have already been well established for some time ago. Our aim here is only to make use of these techniques for another purpose namely; to obtain the optimum weight solution for steel frames using the semirigid connections concept. The purpose of this study was to determine the following: (1) The percent of rigidity of the semirigid connections that would give the optimum steel weight for the common types of steel frame structures. (2) The exact percent of steel saving when using the semirigid connections concept, compared to the classical approach. (3) Which is cheaper, the gable or the portal frame, considering the same condition of loading and geometrical configuration. Also the relative cheapness of the double bay or triple-bay multistory frames having the same number of storys and also same conditions of loading and height of columns and total span. (4) Whether the deflection is a governing parameter which might hinder the benefits of the use of semirigid connections. The problem here is to find the optimum weight of plane semirigid connected steel frames. The objective function is given by the weight of structure in terms of the geometrical properties of the elements and the density of steel. The design variables are the breadth of flange and the height of web for each element. For the portal, the gable, the triple bay three story, and the double bay three story frames the number of design variables are four, four, ten, and ten respectively. The design constraints represent strength and stiffness design code requirements. Side constraints are respected to assure nonviolating of the practical available dimensions. The model is solved using the nonlinear programming techniques. The unconstrained formulation using the interior penalty function technique is selected. Powell's algorithm is chosen for the generation of the search vector, and the quadratic interpolation technique is chosen for the determination of the step size. The conclusions are as follows: (1) The use of semirigid connections will provide a saving in weight equal to 28% in the portal frame for corresponding rigidity ratio (R.R.) of 0.9, and a saving of 19% in the gable frame for a corresponding R.R. of 0.733. (2) The use of semirigid connections will provide a saving in weight of 10.75% in the double bay three story frame for a corresponding R.R = 0.733, and 23% for the triple bay three story frame with a corresponding R.R = 0.75. (3) The variation in the percent of the rigidity of the column to girder connection will not cause a corresponding change in the amount of steel allocated to columns and girders. (4) The saving in weight will occur mainly due to girders rather than columns. (5) The vertical deflections were not regarded as a dominating factor as its value did not exceed the allowable limits.     

Behavior of tall buildings with mixed use of rigid and semi-rigid connections

Semi-rigid connections are rarely used in tall building frames. In this paper, the mixed use of rigid connections with semi-rigid connections for tall building frames as a way to reduce costs is investigated. Using the three-parameter power model for semi-rigid connections, a four-bay eight-story frame is numerically analyzed with four combinations of mixed use of rigid and semi-rigid connections. It is shown that normalized building drift Δ/H can be kept under 1/400 by properly selecting the combination of rigid and semi-rigid connections.     

A genetic algorithm for design of moment-resisting steel frames

This paper presents computational approaches that can be implemented in a decision support system for the design of moment-resisting steel frames. Trade-off studies are performed using genetic algorithms to evaluate the savings due to the inclusion of the cost of connections in the optimization model. Since the labor costs and material costs vary according to the geographical location and time of construction, the trade-off curves are computed for several values of the ratio between the cost of rigid connection and the cost of steel. A real-life 5-bay 5-story frame is used for illustration. Results indicate that the total cost of the frame is minimal when rigid connections are present only at certain locations. Finally, “Modeling to Generate Alternatives—MGA,” is proposed to generate good design alternatives as the solution from optimization may not be optimal with respect to the unmodeled objectives and constraints. It provides a set of alternatives that are near-optimal with respect to the modelled objectives and that are also farther from each other in the decision space. Results show that a final design could be chosen from the set of alternatives or obtained by tinkering one of the alternatives.     

Minimal cycle bases for analysis of frames with semi-rigid joints

For an efficient force method of frame analysis, special cycle bases should be generated for the formation of localized self-equilibrating systems, leading to sparse flexibility matrices. In this paper, an algorithm is presented using a fundamental cycle basis, where the selected cycles are improved via an algebraic exchange method. Optimal analysis is performed for frames with semi-rigid joints. In this method, flexibility matrices are generated which are highly sparse. An ordering algorithm is also used for profile reduction to acquire an efficient method for the solution of the corresponding equations. Thus a complete force method analysis of semi-rigid frames is formulated and a computer code is developed. Examples are analyzed with the present approach and the results are compared to those of a stiffness method program with semi-rigid joints.     

Equivalent post-buckling models for the flexural behaviour of steel connections

Analytical solutions for the evaluation of the behaviour of steel connections are presented which are able to reproduce their full non-linear behaviour. Because usual models for the analysis of steel connections consist of translational springs and rigid links whereby the springs exhibit a non-linear force–deformation response, usually taken as a bi-linear approximation, they require an incremental non-linear analysis. Using a substitute elastic post-buckling model where each bi-linear spring is replaced by two equivalent elastic springs in the context of a post-buckling stability analysis using an energy formulation, closed-form solutions are obtained for a connection loaded in bending. Application to a beam-to-column welded connection using the component (spring) characterisation of code regulations yields the same results in terms of moment resistance and initial stiffness, being additionally able to trace the full unstiffening response.     

The effects of rehabilitation objectives on near optimal trade-off relation between minimum weight and maximum drift of 2D steel X-braced frames considering soil-structure interaction using a cluster-based NSGA II

A cluster-based non-dominated sorting genetic algorithm (NSGA) II has been considered to investigate the effects of rehabilitation objectives on multi-objective design optimization of two-dimensional (2D) steel X-braced frames in the presence of soil-structure interaction. The substructure elasto-perfect plastic model has been adopted for modeling of the soil-structure interaction and the nonlinear pushover analysis is used to evaluate the performance level of the frames for a specified hazard level. Cross-sections of grouped elements of the frames are considered to be discontinuous design variables of the problem. Via implementing some of the constraints, which are independent of doing the time-consuming nonlinear analysis, input population of the optimization technique has been clustered. By using the nonlinear analysis technique in conjunction with the cluster-based NSGA II, near optimal trade-off relation between minimum weight and maximum story drifts of the frames are obtained. The allowable rotations, geometry, and resistance constraints of the structural elements are considered in the optimization design of the frames. The effects of the enhanced basic safety and limited selective rehabilitation objectives on optimum design of the frame are studied. The results show differences between the optimum results of the three mentioned rehabilitation objectives and effects of soil types.

     

Elastic-plastic instability of flexibly connected non-orthogonal frames

A nonlinear elastic-plastic instability analysis of plane non-orthogonal frames, with or without elastic rotational restraints at the supports and semi-rigid connections is presented. The nonlinear flexible connections are represented by polynomial models. Both stability and strength are incorporated in the analysis. The effect of slenderness ratio, elastic rotational restraint at the supports, rise to span ratio and semi-rigid connections are fully assessed. In most cases considered, failure occurs through plastic collapse. It is found that in some cases limit point instability can occur before plastic collapse. From the geometries analyzed, it is concluded that for design application the assumption of linear (instead of nonlinear, polynomial) connection behavior is not adequate for gabled frames.     

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.

     

Optimal design of steel frames subject to gravity and seismic codes' prescribed lateral forces

Allowable stress design of two-dimensional braced and unbraced steel frames based on AISC specifications subject to gravity and seismic lateral forces is formulated as a structural optimization problem. The nonlinear constrained minimization algorithm employed is the feasible directions method. The objective function is the weight of the structure, and behaviour constraints include combined bending and axial stress, shear stress, buckling, slenderness, and drift. Cross-sectional areas are used as design variables. The anylsis is performed using stiffness formulation of the finite element analysis method. Equivalent static force and response spectrum analysis methods of seismic codes are considered. Based on the suggested methodology, the computer program OPTEQ has been developed. Examples are presented to illustrate the capability of the optimal design approach in comparative study of various types of frames subjected to gravity loads and seismic forces according to a typical code.     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

Lightweight flatbed trailer design by using topology and thickness optimization

A new design for a lightweight flatbed trailer with high bending stiffness and torsional frequency is presented. The design procedure consists of two main steps: topology optimization and thickness optimization. During topology optimization, a creative frame layout different from existing ladder-type frames can be obtained by searching the best layout out of all possible layouts of a simplified design domain model. After approximating the result of topology optimization as a thin-walled structure, the approximated thicknesses of the plates are optimized to minimize the mass of a trailer. The bending stiffness and torsional frequency obtained by topology optimization are set as design constraints for thickness optimization. Due to the closed cross-section, the optimized trailer can efficiently increase the stiffness-to-mass ratio to a large extent. Discrete thicknesses are employed as design variables for thickness optimization so that the thicknesses of the plates of a trailer can be included in those of commercially available high-strength steel products. The final model has a 29% reduction in total mass, a 21% decrease in mean compliance with a uniform bending load, and a 169% increase in torsional frequency.     

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.     

Optimization of composite discs

This paper deals with the problem of maximizing or minimizing the integral stiffness of a linearly elastic fiber reinforced composite disc of given domain and with given boundary conditions and in-plane loading along the edge. The local concentrations and directions of one or two fields of fibers are used as design variables, and the amounts of available fiber and matrix materials are defined via a prescribed bound on the total cost or weight of the disc.Solutions to this non-linear optimization problem are obtained by means of an iterative, two-level procedure based on an optimality criterion approach and a method of mathematical programming using analytical sensitivity analysis. The analysis of the composite structure is performed with the aid of the finite element program MODULEF. The disc is assumed to be in plane stress and a simple law relating fiber concentration, orientation, materials data and the elastic moduliA _kl is used. A number of illustrative examples are presented at the end of the paper.     

A micro-computer program for the elastic-plastic analysis and optimum design of plane frames

A computerized version of an approximate second-order, elastic-plastic analysis of multi-storey sway frames is presented. The computer program which applies the method to steel frames on a storey-by-storey basis is described. The program is used to demonstrate by way of an example the applicability of the method for the analysis and design of large, multi-storey frames. It is argued that the technique can greatly reduce analysis time as well as simplify the optimization of members in the design of such frames.     

A strategy for automated modeling of frame structures

Structural analysis involves three main phases, namely, modeling, analysis, and results processing. Computer software exists in order to automate the analysis and results processing phases. However, modeling is still done largely by hand, especially for frame structures.This paper considers automated modeling of frame structures in the context of a computer integrated system for structural engineering design. In this system a structure is defined in terms of its components and connections, and it is necessary to create an analysis model in terms of nodes, elements, boundary conditions, and so on. The paper describes the logic of the modeling process, describes the output from the process, and shows how this output can be used as input data for a structural analysis program. The scope is limited to determination of the structure stiffness properties and assembly of the stiffness matrix.     

Topology optimization of frame structures—joint penalty and material selection

This paper deals with joint penalization and material selection in frame topology optimization. The models used in this study are frame structures with flexible joints. The problem considered is to find the frame design which fulfills a stiffness requirement at the lowest structural weight. To support topological change of joints, each joint is modelled as a set of subelements. A set of design variables are applied to each beam and joint subelement. Two kinds of design variables are used. One of these variables is an area-type design variable used to control the global element size and support a topology change. The other variables are length ratio variables controlling the cross section of beams and internal stiffness properties of the joints. This paper presents two extensions to classical frame topology optimization. Firstly, penalization of structural joints is presented. This introduces the possibility of finding a topology with less complexity in terms of the number of beam connections. Secondly, a material interpolation scheme is introduced to support mixed material design.