Using genetic algorithm for the optimization of seismic behavior of steel planar frames with semi-rigid connections

Beam-column connections have a significant role in the results of the analysis and the design of steel frames. In this paper, a genetic algorithm has been used for the non-linear analysis and design of steel frames. For minimizing the weight of frames, while satisfying the applied constraints and restraints such as the limits of normal and combined stresses, criteria such as target displacement(s) and the number and locations of plastic hinges were used. To analyze and design the frame elements, I and box-shaped standard sections were used for beams and columns, respectively. Finally, some clues for finding optimizing semi-rigid connection stiffness values for beam-to-column connections have been obtained. The degrees of these rigidities are obtained by a genetic algorithm during the procedure of optimization in order to reach a frame with the minimum weight. SAP2000 structural analysis program was used to perform modal analysis and linear and non-linear static solutions as well as the design of the elements. A MATLAB program was written for the process of optimization. The procedure of optimization was based on a weight minimization carried out for 9 steel frames. Thus, the optimum connection stiffness could be obtained for minimizing the weight of the structure. The results show that the non-linear analysis gives less weight for short period frames with semi-rigid connections compared to those of linear ones. However, by increasing the periods of frames, much less weights are obtained in the case of non-linear analysis with semi-rigid connections.     

A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation

A modified finite element formulation to study the post-buckling behaviour of elastic circular plates is presented in this paper. A discussion on the derivation of nonlinear stiffness matrix for post-buckling analysis is included and the present results are compared with continuum solutions.     

FEM-based analysis of the local-plate/distortional mode interaction in cold-formed steel lipped channel columns

This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by local-plate/distortional buckling mode interaction. The results presented and discussed were obtained through analyses performed using the finite element code Abaqus and discretising the columns by means of fine 4-node shell element meshes. The columns analysed (i) are simply supported (end sections locally/globally pinned and free-to-warp), (ii) have cross-section dimensions that ensure equal local-plate and distortional critical buckling stresses, thus maximising the local-plate/distortional mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections that exhibit different shapes but share the same combined amplitude. The numerical post-buckling results reported consist of (i) elastic and elastic–plastic non-linear equilibrium paths, (ii) curves and figures describing how the column deformed configuration (expressed as a linear combination of its local-plate and distortional components) evolves along the elastic post-buckling equilibrium paths and (iii) figures providing a clear visualisation of the (iii₁) evolution of the elastic–plastic column deformed configurations, (iii₂) the growth of the plastic strains and (iii₃) failure mechanisms exhibited by a fairly large portion of the elastic–plastic columns that were analysed in this work.     

Non-linear GBT formulation for open-section thin-walled members with arbitrary support conditions

This paper presents the development and illustrates the application of a beam finite element based on Generalised Beam Theory (GBT) and intended to analyse the elastic post-buckling behaviour of thin-walled steel members exhibiting arbitrary open cross-section and with non-standard support conditions (e.g., in-span bracing systems). After briefly reviewing the main concepts and procedures required to obtain the GBT system of non-linear equilibrium equations, the paper describes the steps involved in the numerical implementation (incremental–iterative strategy) of a non-linear beam finite element that incorporates the influence of the non-standard support conditions. Finally, the application and capabilities of the proposed GBT-based beam finite element are illustrated by means of the presentation and discussion of numerical results concerning the post-buckling behaviour of lipped I-section columns and lipped channel columns and beams with and without localised displacement restraints. For validation purposes, most GBT-based results are compared with values yielded by shell finite element analyses carried out in the commercial code Ansys.     

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.     

Post-buckling behaviour and strength of cold-formed steel lipped channel columns experiencing distortional/global interaction

This paper reports the results of a numerical investigation concerning the elastic and elastic–plastic post-buckling behaviour of cold-formed steel lipped channel columns affected by distortional/global (flexural–torsional) buckling mode interaction. The results presented and discussed were obtained by means of analyses performed using the finite element code Abaqus and adopting column discretisations into fine 4-node isoparametric shell element meshes. The columns analysed (i) are simply supported (locally/globally pinned end sections that may warp freely), (ii) have cross-section dimensions and lengths that ensure equal distortional and global (flexural–torsional) critical buckling loads, thus maximising the distortional/global mode interaction effects, and (iii) contain critical-mode initial geometrical imperfections exhibiting different configurations, all corresponding to linear combinations of the two “competing” critical buckling modes. After briefly addressing the lipped channel column “pure” distortional and global post-buckling behaviours, one presents and discusses in great detail a fair number of numerical results concerning the post-buckling behaviour and strength of similar columns experiencing strong distortional/global mode interaction effects. These results consist of (i) elastic (mostly) and elastic–plastic non-linear equilibrium paths, (ii) curves or figures providing the evolution of the deformed configurations of several columns (expressed as linear combination of their distortional and global components) and, for the elastic–plastic columns, (iii) figures enabling a clear visualisation of (iii₁) the location and growth of the plastic strains and (iii₂) the characteristics of the failure mechanisms more often detected in the course of this research work.     

Analysis and collapse behaviour of inelastic frame structures

The pre- and post-collapse behaviour of inelastic frame structures is investigated under nonproportional quasi-static loading. Idealized model frames with properties defined by simple parameters are developed to represent structures with varying distributions of stiffness and strength. A bi-linear model for non-linear member response is incorporated into a linearized, second-order incremental procedure developed for the digital computer analysis of the structural response. The features of static collapse are examined by means of a parametric scheme to show the importance of parameters related to the structure and the loading condition.     

The large deformation,post-buckling and catastrophic stability of slender cantilever columns

Several solutions, analytical and numerical, exist for the cases of nonlinear bending of beams and post-buckling of columns. However, the solution for the general case of an arbitrarily inclined load does not seem to exist. In this paper, the governing equation of equilibrium for a cantilever of variable cross section under an arbitrarily inclined load is formulated, by using the axial and transverse displacement quantities. The solution is obtained by means of a simple numerical iterative procedure. The results for the post-buckling and non-linear bending of the structure under concentrated tip load, uniformily distributed load and linearly varying load are presented, and compared with the available results, wherever possible. Further, a type of catastrophic behaviour in post-buckling region is observed and studied using the same numerical scheme. The method is general and can be used for any load variation and for any tapered beam-column.     

Post-buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation

The post-buckling behaviour of cylindrically orthotropic circular plates is investigated through a finite element formulation, with the plates resting on an elastic foundation and their edges are elastically restrained against rotation. Results are presented in the form of linear buckling load parameters and empirical formulae for radial load ratios for various values of spring stiffness, foundation stiffness and orthotropy parameter.     

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads.

To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.     

A connection rule for alpha-carbon coarse-grained elastic network models using chemical bond information

A sparser but more efficient connection rule (called a bond-cutoff method) for a simplified alpha-carbon coarse-grained elastic network model is presented. One of conventional connection rules for elastic network models is the distance-cutoff method, where virtual springs connect an alpha-carbon with all neighbor alpha-carbons within predefined distance-cutoff value. However, though the maximum interaction distance between alpha-carbons is reported as 7 angstroms, this cutoff value can make the elastic network unstable in many cases of protein structures. Thus, a larger cutoff value (>11 angstroms) is often used to establish a stable elastic network model in previous researches. To overcome this problem, a connection rule for backbone model is proposed, which satisfies the minimum condition to stabilize an elastic network. Based on the backbone connections, each type of chemical interactions is considered and added to the elastic network model: disulfide bonds, hydrogen bonds, and salt-bridges. In addition, the van der Waals forces between alpha-carbons are modeled by using the distance-cutoff method. With the proposed connection rule, one can make an elastic network model with less than 7 angstroms distance cutoff, which can reveal protein flexibility more sharply. Moreover, the normal modes from the new elastic network model can reflect conformational changes of a given protein better than ones by the distance-cutoff method. This method can save the computational cost when calculating normal modes of a given protein structure, because it can reduce the total number of connections. As a validation, six example proteins are tested. Computational times and the overlap values between the conformational change and infinitesimal motion calculated by normal mode analysis are presented. Those animations are also available at UMass Morph Server (http://biomechanics.ecs.umass.edu/umms.html).     

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.     

Non-linear finite element dynamic analysis of two-dimensional concrete structures

This paper presents a numerical procedure for predicting the non-linear dynamic response of plane and axisymmetric reinforced concrete structures. Isoparametric elements with special embedded axial members are used to discretize concrete and steel in space. A summary of a rate and history dependent constitutive model for progressive failure analysis of concrete is given in which the compression behaviour is modelled as a strain rate sensitive elasto-viscoplastic material and in tension as strain rate dependent linear elastic strain softening material. The different rales governing the pre-failure and post-failure behaviour in compression and tension are developed in which the strain rate dependency is included. Steel is modelled as a strain rate dependent uniaxial elasto-viscoplastic material. Explicit central difference scheme in conjunction with an energy balance check is employed for time integration of equations of motion. A computer program for linear and non-linear dynamic analysis of concrete structures is described. Finally, some numerical applications are presented.     

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     

Optimal design of conical springs

Mechanical systems often use springs to store energy though their axial length must sometimes be significantly reduced. This leads to the use of conical springs as they are able to telescope. Designers of mechanical systems can call on a large number of tools to assist them though most of these are merely validation tools requiring concomitant trial and error strategies. Optimization strategies can be applied to provide synthesis assistance tools for which the designer simply specifies his requirement. Thus the tool directly indicates the spring best suited to standards and requirements. Recent advances in the study of constant pitch conical springs have provided analytical expressions of their behavior even in the non-linear phase. Considering this, we have used optimization strategies to provide a synthesis tool for conical spring design. An example of application is presented. The tool introduced here is thus a synthesis assistance tool that can be of considerable interest for designers who require a conical spring in their design.     

Numerical modelling of the cyclic behaviour of the basic components of steel end plate connections

This paper aims at developing a new methodology based on finite element (FE) techniques with a combination of several other methods to extend the component-based design philosophy of EC3 to the cyclic behaviour of end plate connection. In the first part, the experimental procedure for studying the behaviour of an isolated T-stub as the main component of an end plate connection under monotonic and cyclic loading is presented. The second part comprises the four steps necessary to build the analytical model. Firstly the monotonic and secondly the cyclic FE models of these specimens are discussed and validated. Then, a mathematical energy balance model is developed to reproduce the cyclic response. This model is the basis of a mechanical model which predicts the behaviour of a T-stub which constitutes the main component of connections. With these data, a spring model is used to predict the moment rotation curve for an isolated end plate connection.     

System dynamics simulation of Hopfield neural networks

Hopfield networks are a class of neural network models where non-linear graded response neurons organized into networks with effectively symmetric synaptic connections are able to implement interesting algorithms, thereby introducing the concept of information storage in the stable states of dynamical systems. In addition to opening up the possibility of using system dynamics as a vehicle to gain potentially useful insights into the behaviour of such networks, especially in the field or nonelectrical engineering, we study the dynamics of the state-space trajectory as well as time domain evolution of sensitivities of the states with respect to circuit parameters.     

Non-linear analysis of shells of revolution under arbitrary loads

A new finite element formulation is presented for the non-linear analysis of elastic doubly curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. A coupled harmonics approach is employed, in which coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. Key issues in the formulation, such as non-linear coupling and growth of harmonic modes, are carefully and systematically explained. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, post-buckling load–deflection paths can be traced efficiently and accurately. The formulation also employs a non-linear shell theory more complete than existing classical theories. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general.     

Vibration response of geometrically nonlinear elastic beams subjected to pulse and impact loading

The governing equations for the geometrically non-linear deformation of elastic beams subjected to dynamic bending loads are developed and solved for various initial conditions. Of primary interest is the response to pulse loading and simulated impact. Both transient and several cycle solutions are generated for the free vibration response to pulse loading. The results obtained are compared to a first mode analysis approximation.A new model is developed to simulate impact loading by the distribution of additional mass to the elastic system and subjecting it to a velocity pulse. The governing equations are solved using second order finite differences in space and time. The solutions obtained are in reasonable agreement with experimental results previously obtained [1].