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.     

Buckling analysis of imperfect frames using a stochastic finite element method

A stochastic finite element method is developed for the buckling analysis of frames with random initial imperfections, uncertain sectional and material properties. The random geometrical imperfections of the frames are described by member initial crookednesses which are modeled as given initial displacement functions with amplitudes treated as random variables. The effects of the random initial geometric imperfections are formulated as a set of equivalent random nodal coordinates in the finite element discretization of the members. The mean-centered second-order perturbation technique is used to formulate the stochastic finite element method for the buckling analysis of the imperfect frames. Use of the present method is illustrated by several examples of buckling analysis of random frames. Results derived from the Monte Carlo method are also obtained for comparison.     

Optimum least-cost design of a truss roof system

An algorithm is presented encompassing the application of optimization methods to the least-cost elastic design of roof systems composed of rigid steel trusses, web joists and steel roof deck. The method is capable of designing rigid trusses that can be fabricated from various grades of steel and several types of standard sections. The selection of open web joist is presently limited to standard H-series, and decking material is standard 22 gage. The design is based upon AISC allowable values where combined stresses resulting from axial forces and secondary bending moments are considered. The effective column lengths are computed using the characteristic buckling equation for a member whose ends are elastically restrained against rotation. The procedure developed considers changes in the mechanical properties of the members, geometric variations in the truss configuration and changes in topology. Selected sets of members may be chosen to be identical, and chord members may be defined as continuous over several panels. Also investigated is the problem of finding the design containing the optimum number of trusses. A number of examples are presented which demonstrate the flexibility and generality of the design approach developed.     

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.     

Optimum design of steel frames with stability constraints

Optimum design algorithms based on the optimality criteria approach are proven to be efficient and general. They have the flexibility of accomodating variety of design constraints such as displacement, stress, stability and frequency in the design problem. The design methods developed recently, although considering one or more of these constraints, lack the necessity of referring to any relevant design code. The algorithm presented for the optimum design of street frames implements the displacement and combined stress limitations according to AISC. The recursive relationship for design variables in the case of dominant displacement constraints is obtained by the optimality criteria approach. The combined stress inequalities which include in-plane and lateral buckling of members are reduced into nonlinear equations of design variables. The solution of these equations gives the values of bounds for the variables in the case where the stress constraints are dominant in the design problem. The use of effective length in the combined stress constraints makes it possible to study the effect of the end rigidities on the final designs. The design procedure is simple and easy to program which makes it particularly suitable for microcomputers. A number of design examples are considered to demonstrate the practical applicability of the method. It is also shown that the design procedure can be employed in selecting the optimum topology of steel frames.     

Topology optimization for the seismic design of truss-like structures

A practical optimization method is applied to design nonlinear truss-like structures subjected to seismic excitation. To achieve minimum weight design, inefficient material is gradually shifted from strong parts to weak parts of a structure until a state of uniform deformation prevails. By considering different truss structures, effects of seismic excitation, target ductility and buckling of the compression members on optimum topology are investigated. It is shown that the proposed method could lead to 60% less structural weight compared to optimization methods based on elastic behavior and equivalent static loads, and is efficient at controlling performance parameters under a design earthquake.     

Optimum design of pitched roof steel frames with haunched rafters by genetic algorithm

The pitched roof steel frames appear to be the simplest structural form used in single storey buildings. However, its design necessitates consideration of many different structural criteria that are required in the design of complex structures. In this paper, a genetic algorithm is used to develop an optimum design method for pitched roof steel frames with haunches for the rafters in the eaves. The algorithm selects the optimum universal beam sections for columns and rafters from the available steel sections tables. Furthermore, it determines the optimum depth of the haunch at the eaves and the length of the haunch required for reaching the most cost-effective form. Formulation of the design problem is based on the elastic design method. The serviceability and the strength constraints are included in the design problem as defined in BS 5950. Furthermore, the overall buckling of columns and rafters in the torsional mode between effective torsional restraints to both flanges is also checked. A pitched roof frame is designed by the algorithm developed to demonstrate its practical application.     

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.

     

铝合金受压板件局部屈曲承载力设计方法

为考察现有铝合金设计规范预测非焊接受压板件承载力的准确性,采用壳单元建立单向受压四边简支板、方形截面管柱和十字形截面轴压柱有限元模型,使用ANSYS计算得到铝合金板件弹塑性屈曲临界应力和极限承载力。数值计算结果与理论分析结果对比发现：现有板件弹塑性屈曲应力理论能够给出铝合金板件屈曲临界应力的下限值,误差较小。中国规范非加劲板件有效厚度预测公式偏保守,因此给出修正的有效厚度预测公式。修正后的非加劲板件有效厚度公式可以提高板件利用率,增强经济性。     

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 design of frames with multiple constraints using an evolutionary method

This paper presents a simple evolutionary method for the optimum design of structures with stress, stiffness and stability constraints. The evolutionary structural optimization method is based on the concept of slowly removing the inefficient material and/or gradually shifting the material from the strongest part of the structure to the weakest part until the structure evolves towards the desired optimum. The iterative method presented here involves two steps. In the first step, the design variables are scaled uniformly to satisfy the most critical constraint. In the second step, a sensitivity number is calculated for each element depending on its influence on the strength, stiffness and buckling load of the structure. Based on the element sensitivity number, material is shifted from the strongest to the weakest part of the structure. These two steps are repeated in cycles until the desired optimum design is obtained. Illustrative examples are given to show the applicability of the method to the optimum design of frames and trusses with a large number of design variables.     

Simulation of inelastic cyclic buckling behavior of steel box sections

In this study, a nonlinear structural model is developed to simulate the cyclic axial force-deformation behavior of steel braces including their buckling behavior using the commercially available nonlinear finite element based software ADINA. The nonlinear cyclic axial force-deformation simulation is done for braces with box sections. However, the structural model and simulation techniques described in this study may be applicable to braces with various section types using other commercially available structural analysis software capable of handling material and geometric nonlinearity. The developed nonlinear brace model is verified using available test results from the literature. It is found that the accuracy of the shapes of the analytical hysteresis loops and the energy dissipated compared to the experimental ones is satisfactory for analysis and design purposes in practice. The developed nonlinear brace structural model is then used to study the effect of various ground motion and structural parameters on the seismic response of single story, single bay concentrically braced frames with chevron braces.     

An approximate resizing system for the preliminary design of reinforced concrete frames

Techniques for the preliminary design of a multistorey-multibay, moment-resisting reinforced concrete frames are investigated. Two-level optimization patterns are constructed in this paper. The objective function at the system level is to minimize the total volume of reinforcing steel. The relationship between the area of longitudinal reinforcement and the fully plastic moments of cross-sections will be approximated by a quadratic expression. Once the optimum plastic moments result at the system level, and the member sizes and reinforcement at critical sections within the span of each member will be selected at component level to complete the automatic resizing system. Two examples of reinforced concrete frames are presented to illustrate the features of the proposed method.     

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.     

Optimal seismic design of 3D steel moment frames: different ductility types

Three different types of lateral resisting steel moment frames consisting of ordinary moment frame (OMF), intermediate moment frame (IMF) and special moment frame (SMF) are available for design of 3D frames in literature. In this paper, optimum seismic design of 3D steel moment frames with different types of lateral resisting systems are performed according to the AISC-LRFD design criteria. A comparison is made considering the results of the above mentioned frames of different ductility types. These frames are analyzed by Response Spectrum Analysis (RSA), and optimizations are performed using nine different well-established metaheuristic algorithms. Performances of these algorithms are then compared for introducing the most suitable metaheuristic algorithms for optimal design of the 3D frames.     

A Newmark-based method for the stability of columns

Based on the well-known Newmark method for the solution of stability problems, a new method for solving strut overall buckling problems, based on the Picard iteration technique for the solution of differential equations, is developed. The method is easily programmed, and has the advantages of simplicity and speed of convergence. Five case studies of elastic struts are examined, and the results are shown to be very accurate with only a small number of iterations. The method may easily be extended to treat material and geometric nonlinearity including nonprismatic members and linear and nonlinear spring restraints that would be encountered in frames.     

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.     

Improved system buckling analysis of effective lengths of girder and tower members in steel cable-stayed bridges

This paper proposes an iterative system buckling analysis to determine reasonable effective lengths of girder and tower members in steel cable-stayed bridges. The modifications include the addition of a fictitious axial force to the geometric stiffness matrix with iterative eigenvalue computations. After verifying the proposed method by analyzing steel frames, we apply it to cable-stayed bridges with different center spans and girder depths. The effective lengths of members in these example bridges by the proposed method are compared with those found using conventional buckling analysis. The effects of the proposed method on slenderness ratios and compressive strengths are also discussed.     

Optimum design of a reinforced plastic bridge girder

This paper describes the results of an analytical and experimental investigation concerned with the optimum design of a fiberglass-reinforced plastic flexural number. In the initial phase of the study, seven members were designed, fabricated, load-tested and subsequently analyzed. Design modifications were made in each subsequent specimen based on results from the experimental and analytical study. The initial configurations were selected on the basis of intuitive arguments and empirical optimum design concepts. In order to achieve a more efficient and economical structural system, an iterative algorithm for optimum geometric configuration was developed and used in conjunction with an optimality criterion for optimum member sizing. This procedure was applied to the design of five separate flexural members and final design data for these members are presented and discussed.     

Constrained optimization of structures with displacement constraints under various loading conditions

Optimization problems often involve constraints and restrictions which must be considered in order to obtain an optimum result and the resultant solution should not deviate from any of the imposed constraints. These constraints and restrictions are imposed either on the design variables or on the algebraic relations between them. Constraints of allowable stress, minimum size and buckling of members in the absence of allowable displacement constraint are the most important factors in optimization of the cross-sectional area of structural elements. When the allowable displacement constraint is included in the problem as a determinant parameter, since the specifications of most of elements affect the displacement rate, the way of imposing and considering this constraint requires special care. In this research the way of simultaneous imposition of multi displacement constraints for optimum design of truss structures in several load cases is described. In this method various constraints for different load cases are divided into active and passive constraints. The mathematical formulation is based on the classical method of Lagrange Multipliers. Overall, this simple method can be employed along with other constraints such as buckling, allowable stress and minimum size of members for imposing the displacement constraint in various load cases.