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.     

Analytical benchmarks for topological optimization IV: square-shaped line support

Michell’s problem of optimizing truss topology for stress or compliance constraints under a single load condition is solved analytically for plane trusses having a square-shaped line support. Geometrical characteristics of the Hencky nets giving the truss layout are expressed in terms of Lommel functions. Analytically derived truss volumes for the above problem are compared with those of trusses supported along circles of equivalent area. Some general implications of the results are also discussed.     

Discrete optimal design of elasto-plastic trusses using compliance and stability constraints

In this paper two discrete optimization methods are presented for the minimum volume design of elasto-plastic trusses with given geometry. The design is based on given sets of discrete cross-sectional sizes. Both methods enable the use of the plastic reserve of the truss; the plastic deformations, however, are controlled by compliance constraints on plastic deformations. In the second solution method, the shakedown of the truss is also taken into consideration. The stability of the bars is also controlled by using permissible stresses for compression. The methods are based on continuous optimal elasto-plastic design methods and on the discrete optimization method of elastic trusses using segmental approach. By the iterative application of these methods, solution procedures that use standard linear programming have been developed.Dedicated to Prof. F. Ziegler, for his 60-th birthday; extended version of a paper presented at the Second World congress of Structural and Multidisciplinary Optimization (WCSMO-2), in Zakopane, Poland, May 1997     

On optimal physically nonlinear trusses

Optimization problems for physically nonlinear hyperelastic trusses statically loaded by a single system of loads are investigated. Several possible formulations of the problems for a prescribed truss layout are considered. It is demonstrated that the problems lead to the design of equal absolute stress values in some rods and of active technology constraints in other rods. Some general properties of the trusses are examined. Optimal physically nonlinear hyperelastic truss layout problems under static loading are also considered. It is shown that the well-known results of Maxwell, Michell and Prager concerning linear-elastic structures are generalized for the considered problems. It is demonstrated that the results of the paper are extended to optimization problems for structures in stationary creep with an arbitrary concave stress-strain velocity relation.     

Force density method for simultaneous optimization of geometry and topology of trusses

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.     

Two approaches for truss topology optimization: a comparison for practical use

This paper discusses ground structure approaches for topology optimization of trusses. These topology optimization methods select an optimal subset of bars from the set of all possible bars defined on a discrete grid. The objectives used are based either on minimum compliance or on minimum volume. Advantages and disadvantages are discussed and it is shown that constraints exist where the formulations become equivalent. The incorporation of stability constraints (buckling) into topology design is important. The influence of buckling on the optimal layout is demonstrated by a bridge design example. A second example shows the applicability of truss topology optimization to a real engineering stiffened membrane problem.     

Automated design of truss and frame geometry

It is shown that the optimization of truss and frame structures where layout geometry variables are included as design variables is inherently poorly scaled and that a two-step procedure can be used to overcome this problem. Using a standard optimization package this procedure is used to find the optimal geometry and sizing of a two-dimensional and a three-dimensional frame structure.     

On using genetic algorithms for optimum damper placement in space trusses

Although similar in some ways to the design of aircraft and other lightweight structures, the optimal design of space structures has several unique challenges. A flexible optimization system allowing for multiple analysis techniques and including continuous and discrete design variables is desired. Although not as computationally efficient as traditional optimization techniques, genetic algorithms meet this requirement.The present investigation used a genetic algorithm to place passive viscous dampers in space trusses. The flexibility of the system was demonstrated through the use of fixed and free boundary conditions. The results showed that four dampers are generally sufficient to suppress bending motion in a seventy-two-bar fixed truss and a seventy-eight-bar free truss. The results were intuitive, demonstrating the suitability of the genetic algorithm to this class of problem.     

Some general results for optimal structures

This paper considers some approaches to structural optimization that meet real design requirements to a greater extent than has previously been achieved. In all of the problems considered, the structures are statically loaded, strains and displacements are small, and buckling effects are neglected.Topology optimization problems for multi-material (both linear and non-linear) elastic trusses, in particular, bi-material ones, are considered. Several general theorems are proven. Well-known classical results are generalized to minimal mass structures with restricted stresses or stiffness values. Some general theorems for trusses of given geometry are also proven. Moreover, it is proven that in the case of restricted stresses, the minimum mass non-linear-elastic truss of a given geometry is not heavier than the corresponding optimal linear-elastic one.     

Optimal strengthening of elasto-plastic trusses with plastic deformation and stability constraints

In this paper a design method for the optimal strengthening of trusses is presented. Trusses with given configuration composed of linearly elastic-perfectly plastic material are considered and it is assumed that because of the increase or change of the loads their strength and/or stiffness is not satisfactory. The problem is to strengthen the truss by the application of additional elastoplastic bars and/or supports, such that the truss be strong enough to carry the loads, does not undertake excessive plastic deformations, and the cost of the strengthened truss be a minimum. For the solution of the highly nonlinear problem an iterative method is presented. Besides the constraints on overall plastic deformations, the stability effects are also taken into consideration during iteration. The application is illustrated by the solution of numerical examples.     

Improved genetic algorithm with two-level approximation using shape sensitivities for truss layout optimization

Truss layout optimization is a procedure for optimizing truss structures under the combined influence of size, shape and topology variables. This paper presents an Improved Genetic Algorithm with Two-Level Approximation (IGATA) that uses continuous shape variables and shape sensitivities to minimize the weight of trusses under static or dynamic constraints. A uniform optimization model including continuous size/shape variables and discrete topology variables is established. With the introduction of shape sensitivities, the first-level approximations of constraint functions are constructed with respect to shape/topology/size variables. This explicit problem is solved by implementation of a real-coded GA for continuous shape variables and binary-coded GA for 0/1 topology variables. Acceleration techniques are used to overcome the convergence difficulty of the mixed-coded GA. When calculating the fitness value of each member in the current generation, a second-level approximation method is embedded to optimize the continuous size variables effectively. The results of numerical examples show that the usage of continuous shape variables and shape sensitivities improves the algorithm performance significantly.     

Simultaneous size and geometry optimization of steel trusses under dynamic excitations

During the past decades, the main focus of the research in steel truss optimization has been tailored towards optimal design under static loading conditions and limited work has been devoted to investigating the optimum structural design considering dynamic excitations. This study addresses the simultaneous size and geometry optimization problem of steel truss structures subjected to dynamic excitations. Using the well-known big bang-big crunch algorithm, the minimum-weight design of steel trusses is conducted under both periodic and non-periodic excitations. In the case of periodic excitations, in order to examine the effect of the exciting period of the dynamic load on the final results, the design instances are optimized under different exciting periods and the obtained results are compared. It is observed that by increasing the excitation period of the considered sinusoidal loading as well as the finite rise time of the non-periodic step force, the optimization results approach the minimum design weight obtained under the static loading counterpart. However, in the case of the studied rectangular periodic excitation, the results obtained do not approach the optimum design associated with the static loading case even for higher values of the exciting period.     

A variational inequality approach to optimal plastic design of structures via the Prager-Rozvany theory

The theory of optimal plastic design of structures via optimality criteria (W. Prager approach) transforms the optimal design problem into a certain nonlinear elastic structural analysis problem with appropriate stress-strain laws, which are derived by the adopted specific cost function for the members of the structure and which generally have complete vertical branches. Moreover, the concept of structural universe (introduced by G.I.N. Rozvany) permits us to tackle complicated optimal layout problems.On the other hand, a significant effort in the field of nonsmooth mechanics has recently been devoted to the solution of structural analysis problems with complete material and boundary laws, e.g. stress-strain laws or reaction-displacement laws with vertical branches.In this paper, the problem of optimal plastic design and layout of structures following the approach of Prager-Rozvany is revised within the framework of recent progress in the area of nonsmooth structural analysis and it is treated by means of techniques primarily developed for the solution of inequality mechanics problems. The problem of the optimal layout of trusses is used here as a model problem. The introduction of general convex, continuous and piecewise linear specific cost functions for the structural members leads to the formulation of linear variational inequalities or equivalent piecewise linear, convex but nonsmooth optimization problems. An algorithm exploiting the particular structure of the minimization problem is then described for the numerical solution. Thus, practical structural optimization problems of large size can be treated. Finally, numerical examples illustrate the applicability and the advantages of the method.On leave from the Institute of Applied Mechanics, Department of Engineering Sciences, Technical University of Crete, GR-73100 Chania, Greece     

On optimal geometrically non-linear trusses

The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated. Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure.Extensions of some theorems (Maxwells theorem, Michells theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved.Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained.It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.     

A ground-structure-based representation with an element-removal algorithm for truss topology optimization

In this study, a new ground-structure-based representation for truss topology optimization is proposed. The proposed representation employs an algorithm that removes unwanted elements from trusses to obtain the final trusses. These unwanted elements include kinematically unstable elements and useless zero-force elements. Since the element-removal algorithm is used in the translation of representation codes into corresponding trusses, this results in more representation codes in the search space that are mapped into kinematically stable and efficient trusses. Since more representation codes in the search space represent stable and efficient trusses, the strategy increases meaningful competition among representation codes. This remapping strategy alleviates the problem of having large search spaces using ground structures, and encourages faster convergences. To test the effectiveness of the proposed representation, it is used with a simple multi-population particle swarm optimization algorithm to solve several truss topology optimization problems. It is found that the proposed representation can significantly improve the performance of the optimization process.     

A revised discrete Lagrangian-based search algorithm for the optimal design of skeletal structures using available sections

Issues relating to the application of the discrete Lagrangian method (DLM) to the discrete sizing optimal design of skeletal structures are addressed. The resultant structure, whether truss or rigid frame, is subjected to stress and displacement constraints under multiple load cases. The members’ sections are selected from an available set of profiles. A table that contains sectional properties for all the available profiles is used directly in structural optimization. Each profile in the table is assigned by a unique profile number, which is used as the integer design variable for each of the structural members. It is proposed that we use a revised DLM search algorithm with static weighting to design trusses and rigid frames for minimum weight. Five examples are used to demonstrate the feasibility of the method. It is shown that, for monotonic as well as nonmonotonic constraint functions, the DLM is effective and robust for the discrete sizing design of skeletal structures.     

Interval analysis based robust truss optimization with continuous and discrete variables using mix-coded genetic algorithm

The problem of optimizing truss structures in the presence of uncertain parameters considering both continuous and discrete design variables is studied. An interval analysis based robust optimization method combined with the improved genetic algorithm is proposed for solving the problem. Uncertain parameters are assumed to be bounded in specified intervals. The natural interval extensions are employed to obtain explicitly a conservative approximation of the upper and lower bounds of the structural response, and hereby the bounds of the objective function and the constraint function. This way the uncertainty design may be performed in a very efficient manner in comparison with the probabilistic analysis based method. A mix-coded genetic algorithm (GA), where the discrete variables are coded with binary numbers while the continuous variables are coded with real numbers, is developed to deal with simultaneously the continuous and discrete design variables of the optimization model. An improved differences control strategy is proposed to avoid the GA getting stuck in local optima. Several numerical examples concerning the optimization of plane and space truss structures with continuous, discrete or mixed design variables are presented to validate the method developed in the present paper. Monte Carlo simulation shows that the interval analysis based optimization method gives much more robust designs in comparison with the deterministic optimization method.     

Discrete sizing/layout/topology optimization of truss structures with an advanced Jaya algorithm

Discrete optimization of truss structures is a hard computing problem with many local minima. Metaheuristic algorithms are naturally suited for discrete optimization problems as they do not require gradient information. A recently developed method called Jaya algorithm (JA) has proven itself very efficient in continuous engineering problems. Remarkably, JA has a very simple formulation and does not utilize algorithm-specific parameters. This study presents a novel JA formulation for discrete optimization of truss structures under stress and displacement constraints. The new algorithm, denoted as discrete advanced JA (DAJA), implements efficient search mechanisms for generating new trial designs including discrete sizing, layout and topology optimization variables. Besides the JA’s basic concept of moving towards the best design of the population and moving away from the worst design, DAJA tries to form a set of descent directions in the neighborhood of each candidate designs thus generating high quality trial designs that are very likely to improve current population. Results collected in seven benchmark problems clearly demonstrate the superiority of DAJA over other state-of-the-art metaheuristic algorithms and multi-stage continuous–discrete optimization formulations.     

Applications of linear programming in structural layout and optimization

Various computer methods have been developed for the optimal design of indeterminate structures, but it is not possible to guarantee that the result of any method will be a global optimum, rather than merely a local optimum. By temporarily neglecting the conditions of elastic compatibility and formulating a mathematical optimization problem based on the equilibrium conditions and the stress constraints, it is possible to obtain an approximate design which avoids merely local optima. In the cases examined, this design is close to the exact global optimum obtained by enforcing the compatibility conditions and is therefore a good starting point for an optimizing procedure. Examples include a graphical solution of a simple grillage known to have multiple local optima, and a sequence of planar trusses under alternate loading conditions. Linear programming is used to find the minimum weight truss designs satisfying equilibrium; this method eliminates extraneous members and leads to better indeterminate truss configurations than does a stress-ratio type algorithm.