In this paper the effects of adding buckling constraints to truss sizing optimization for minimizing mass are investigated. Introduction of buckling testing increases the complexity of the optimization process as Euler buckling criteria changes with each iteration of the optimization process due to the changes in element cross section dimensions. The resulting models which consider this criteria are practically applicable. For the purposes of showing the effects of dynamic constraints for buckling, optimal parametric standard test models of 10 bar, 17 bar, and 25 bar trusses from the literature are tested for buckling and compared to the models with the added constraint. Models which do not consider buckling criteria have a considerable number of elements which do not meet buckling criteria. The masses of these models are substantially smaller than their counterparts which consider buckling.

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Effects of introducing dynamic constraints for buckling to truss sizing optimization problems

The main research goal is the development of a numerical methodology for solving parametric optimization problems of steel structures with orientation on software implementation in a computer-aided design system. The paper has proposed a new mathematical model for parametric optimization problems of steel structures. The design variable vector includes geometrical parameters of the structure (node coordinates), cross-sectional dimensions of the structural members, as well as initial pre-stressing forces introduced into the specified redundant members of the structure. The system of constraints covers load-carrying capacities constraints formulated for all design sections of structural members of the steel structure subjected to all ultimate load case combinations. The displacements constraints formulated for the specified nodes of the steel structure subjected to all serviceability load case combinations have been also included into the system of constraints. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used for solving the parametric optimization problem. A numerical algorithm for solving the formulated parametric optimization problems of steel structures has been developed in the paper. The comparison of the optimization results of truss structures presented by the paper confirms the validity of the optimum solutions obtained using the proposed numerical methodology.

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Parametric optimization of steel structures based on gradient projection method

The problems of optimal design of flexible rod members connected to a carrying body undergoing programmed motion with respect to the center of mass of the system, are examined. The problems are solved by solving directly optimal-control problems with phase constraints in the form of inequalities. The results of optimization of the shape of a flexible member, the distribution of mass and stiffness along a flexible member with an azimuthat contour of the axis, and determination of the optimal distribution of mass and stiffness along the member with the optimal contour of the axis are presented. It is concluded that inertially loaded flexible rods are extremely sensitive to the geometry and the distribution of material.

Optimal design of flexible rod structural members

The paper considers parametric optimization problems for the steel bar structures formulated as nonlinear programming ones with variable unknown cross-sectional sizes of the structural members, as well as initial prestressing forces introduced into the specified redundant members of the structure. The system of constraints covers load-bearing capacity constraints for all the design sections of the structural members subjected to all the design load combinations at ultimate limit state, as well as displacement constraints for the specified nodes of the bar system, subjected to all design load combinations at serviceability limit state. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the parametric optimization problem. A numerical technique to determine the optimal number of the redundant members to introduce the initial prestressing forces has been offered for high-order statically indeterminate bar structures. It reduces the dimension for the design variable vector of unknown initial prestressing forces for considered optimization problems.

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Optimal numbers of the redundant members for introducing initial pre-stressing forces into steel bar structures

The paper has proposed a mathematical model for parametric optimization problem of the steel lattice portal frame. The design variable vector includes geometrical parameters of the structure (node coordinates), as well as cross-sectional dimensions of the structural members. The system of constraints covers load-carrying capacities constraints formulated for all design sections of structural members of the steel structure subjected to all ultimate load case combinations. The displacements constraints formulated for the specified nodes of the steel structure subjected to all serviceability load case combinations have been also included into the system of constraints. Additional requirements in the form of constraints on lower and upper values of the design variables, constraints on permissible minimal thicknesses, constraints on permissible maximum diameter-to-thickness ratio for the structural members with circle hollow sections, as well as the conditions for designing gusset-less welded joints between structural members with circle hollow sections have been also considered in the scope of the mathematical model. The method of the objective function gradient projection onto the active constraints surface with simultaneous correction of the constraints violations has been used to solve the formulated parametric optimization problem. New optimal layouts of the steel lattice portal frame by the criterion of the minimum weight, as well as minimum costs on manufacturing and erection have been presented.

Application of improved gradient projection method to parametric optimization of steel lattice portal frame

In this paper parametric optimization of plane two-hinged steel truss with hollow structural members has been considered. Cross sectional sizes of circular hollow brace and chord members and node coordinates of steel truss have been considered as design variables. The system of constraints included stresses and buckling checks for each structural member as well as displacements checks for defined structural nodes. A set of technological requirements have been also integrated to the system of constraints. The minimum structural weight and minimum construction budget was considered as the purpose function. An update gradient method has been proposed in order to solve the parametric optimization task. Suggested optimization technique has been implemented with elaborated software OPTCAD. Numerical example presented by paper has been demonstrated the effectiveness of the proposed optimization methodology.

Parametric Optimization of Steel Truss with Hollow Structural Members Based on Update Gradient Method

The object of this study is a structural member made from a C-shaped cold-formed profile, investigated to search for optimal cross-sectional dimensions. The parametric optimization task is stated as the problem to find the optimal cross-sectional dimensions of the structural member under axial compression conditions, taking into account its post-buckling behavior (local buckling of the web and flanges, as well as a distortional buckling of the cross-section) and structural requirements. In this case, the material consumption and mechanical characteristics of steel, as well as the design lengths of the structural member, were considered constant and predefined. The considered criterion of optimality was the maximization of the load-carrying capacity of the structural member for the overall buckling under the axial compression. The stated optimization problem is solved using the method of exhaustive search while applying the developed software. Additionally, for fixed steel consumption, compromise solutions were searched that do not depend on the thickness of the profile and the design lengths of the structural member. The resulting cold-formed C-shaped profiles with optimal cross-sectional dimensions are characterized by a higher load-carrying capacity for the overall buckling under axial compression (to 24.45 % and 22.19 %) at the same steel consumption compared to the profiles offered by the manufacturer. Analysis of the reported results made it possible to devise recommendations for optimal ratios of dimensions and geometric characteristics of the structural members made from C-shaped profiles operating under axial compression. The ratios could be used both at the stage of selection of cross-sections of structural members from cold-formed profiles, and in the development of effective assortments of cold-formed profiles

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Optimization of cross-section dimensions of structural members made of cold-formed profiles using compromise search

A parametric optimization problem of cross-sectional sizes for cold-formed steel lipped channel structural members subjected to axial compression has been considered by the paper. An optimization problem is formulated as to define optimum cross-sectional sizes of cold-formed structural member taking into account post-buckling behavior (web and flange local and distortional buckling) of the member as well as structural requirements when the profile perimeter (strip width), profile thickness, design lengths of the structural member as well as material properties are constant and specified in advance. Maximization of the load-carrying capacity of the cold-formed structural member has been assumed as purpose function. The formulated parametric optimization problem has been solved by exhaustive search method using the software written in Python. As optimization results the cold-formed steel lipped channels with optimum cross-sectional dimensions have been obtained depending on the profile thickness and design lengths of the structural member. In order to obtain optimum solutions for cross-sectional dimensions of the CFS lipped channel structural members which are independent from the design flexural lengths and profile thickness, searching for a compromise solution has been performed by exhaustive search method. The obtained cold-formed steel lipped channel structural members with optimum cross-sectional sizes have higher design buckling resistance under the axial compression at the same material consumption (stripe width) comparing with the cold-formed steel lipped channels proposed by the manufacturer. Web local buckling phenomenon has been occurred in all obtained CFS lipped channel cross-sections with optimum sizes.

Searching for a compromise solution in cross-section size optimization problems of cold-formed steel structural members

Abstract A parametric optimization problem of cross-sectional sizes for cold-formed steel lipped channel structural members subjected to axial compression has been considered by the paper. An optimization problem is formulated to define optimum cross-sectional sizes of cold-formed structural member taking into account post-buckling behavior and structural requirements when stripe width, profile thickness and profile type are constant and specified in advance. Maximization of the load-carrying capacity of the cold-formed structural member has been assumed as purpose function. As optimization results cold-formed steel lipped channels with optimum dimensions have been obtained. Steel lipped channels structural members with the optimum cross-sectional dimensions have higher load-carrying capacities at the same material consumption (stripe width) comparing with the ones proposed by the manufacturer when the material consumption.

Ivan Peleshko

Papers

Parametric Optimization of Steel Truss with Hollow Structural Members Based on Update Gradient Method

Parametric optimization of steel structures based on gradient projection method

Optimization of cross-section dimensions of structural members made of cold-formed profiles using compromise search

Searching for a compromise solution in cross-section size optimization problems of cold-formed steel structural members

Cross-Section Size Optimization of Cold-Formed Steel Lipped Channel Structural Members Subjected to Axial Compression