Minimum Weight Truss Research Articles

A generative design-to-BIM workflow for minimum weight plane truss design

Structural design has a significant impact on the overall cost of truss structures. In order to reduce the cost of a structure it is important to support designers' decision making starting from the early design phase. In this study, an optimization workflow is proposed, developed and implemented using well known generative design and Building Information Modeling (BIM) tools to achieve a cost-optimal design of a truss structure in the early design phase. Generative design aims to develop products that are lighter, stronger, more efficient and tailored to the specific needs of end users. Generative design tools allow users to create efficient designs via optimizing factors such as cost, weight, energy efficiency and performance. The aim of this study was to develop an optimization workflow to find and model the minimum weight / minimum cost design alternative in the early design phase using generative design, structural analysis and BIM tools in an integrated manner. The goal for the optimization was determined as finding the minimum weight (thus minimum cost) structure among the generated design alternatives. The single span steel truss was selected as the structure to be optimized, and optimization scenarios were prepared and implemented to determine the structural components of the truss with minimum weight. The results demonstrated that through integrated use of structural analysis, generative design and BIM tools minimum-weight truss design can be realized easily and practically.

Minimum-Weight Truss Reinforcement of a Composite Beam to Increase the Free-Vibrations Fundamental Eigenfrequency

In this contribution, we design a minimum-weight truss reinforcement of a thin-walled composite beam, such that the fundamental free-vibrations eigenfrequency of the beam is increased to a specific value. The reinforcement structure is designed using techniques of topology optimization and produced using additive manufacturing, in order to achieve economical design and minimize manual interventions in the fabrication process. Finally, an experimental validation of theoretical outcomes is performed on an additively manufactured prototype.

Randomized line search techniques in combined GA for discrete sizing optimization of truss structures

This paper presents two randomized line search techniques, each combined with a genetic algorithm (GA), to improve the convergence and the accuracy ratio for discrete sizing optimization of truss structures. The first technique is a simple one-dimensional line search in which design variable axes are selected randomly as search directions. The second is a line search technique whose search direction is determined randomly by fitness function values and differences in the genotypes of individuals. To apply the above-mentioned line search techniques without difficulty, real coding is adopted for discrete problems. The line search techniques are applied to discrete optimization problems of minimum-weight truss structures subjected to stress and displacement constraints. The proposed techniques provide convergence to better solutions than a conventional GA.

A novel optimization approach for minimum cost design of trusses

This paper describes new optimization strategies that offer significant improvements in performance over existing methods for bridge-truss design. In this study, a real-world cost function that consists of costs on the weight of the truss and the number of products in the design is considered. We propose a new sizing approach that involves two algorithms applied in sequence – (1) a novel approach to generate a “good” initial solution and (2) a local search that attempts to generate the optimal solution by starting with the final solution from the previous algorithm. A clustering technique, which identifies members that are likely to have the same product type, is used with cost functions that consider a cost on the number of products. The proposed approach gives solutions that are much lower in cost compared to those generated in a comprehensive study of the same problem using genetic algorithms (GA). Also, the number of evaluations needed to arrive at the optimal solution is an order of magnitude lower than that needed in GAs. Since existing optimization techniques use cost functions like those of minimum-weight truss problems to illustrate their performance, the proposed approach is also applied to the same examples in order to compare its relative performance. The proposed approach is shown to generate solutions of not only better quality but also much more efficiently. To highlight the use of this sizing approach in a broader optimization framework, a simple geometry optimization algorithm that uses the sizing approach is presented. This algorithm is also shown to provide solutions better than the existing results in literature.

Global optimization of minimum weight truss topology problems with stress, displacement, and local buckling constraints using branch‐and‐bound

AbstractWe present a convergent continuous branch‐and‐bound algorithm for global optimization of minimum weight truss topology problems with displacement, stress, and local buckling constraints. Valid inequalities which strengthen the problem formulation are derived. The inequalities are generated by solving well‐defined convex optimization problems. Computational results are reported on a large collection of problems taken from the literature. Most of these problems are, for the first time, solved with a proof of global optimality. Copyright © 2004 John Wiley & Sons, Ltd.

Elastoplastic truss design using a displacement based optimization

A displacement based optimization (DBO) method is applied to truss design problems with material nonlinearities, to explore feasibility and verify efficiency of the method as compared to traditional structural optimization. Minimum-weight truss sizing problems with various path-independent elastoplastic laws, including elastic perfectly plastic, linear strain hardening, and Ramberg–Osgood models, are investigated. This type of material nonlinearity allows us to naturally extend the linear elastic truss sizing in the DBO setting to nonlinear problems. To implement the methodology a computer program that uses the commercially available optimizer DOT by VR&D and IMSL Linear Programming solver by Visual Numerics is developed. Several test problems are successfully solved using the DBO approach and solutions are compared to those available in the literature, demonstrating significant reduction of computational time in comparison to the traditional structural optimization method. In particular, the DBO approach is found to be suitable for truss topology design since the method allows member areas to have cross-sectional areas equal to zero exactly.

Truss topology optimization including bar properties different for tension and compression

The problem of optimum truss topology design based on the ground structure approach is considered. It is known that any minimum weight truss design (computed subject to equilibrium of forces and stress constraints with the same yield stresses for tension and compression) is—up to a scaling—the same as a minimum compliance truss design (subject to static equilibrium and a weight constraint). This relation is generalized to the case when different properties of the bars for tension and for compression additionally are taken into account. This situation particularly covers the case when a structure is optimized which consists of rigid (heavy) elements for bars under compression, and of (light) elements which are hardly/not able to carry compression (e.g. ropes). Analogously to the case when tension and compression is handled equally, an equivalence is established and proved which relates minimum weight trusses to minimum compliance structures. It is shown how properties different for tension and compression pop up in a modified global stiffness matrix now depending on tension and compression. A numerical example is included which shows optimal truss designs for different scenarios, and which proves (once more) the big influence of bar properties (different for tension and for compression) on the optimal design.

Structural optimization based on preconditioned conjugate gradient analysis methods

AbstractAn efficient method for structural optimization is presented. Instead of classical direct decomposition methods, Preconditioned Conjugate Gradient (PCG) methods, in conjunction with two proposed starting‐vector generation schemes, are used to solve the systems of linear equations associated with the finite element analysis and behaviour sensitivity analysis problems. These inherently iterative analysis procedures are then used to carry out the analyses needed at the beginning of each stage in an approximation concepts approach to structural optimization. This technique has been implemented in a research program and used to solve a collection of minimum weight truss sizing design problems subject to static deflection and stress constraints. The effectiveness of the PCG methods of analysis in structural optimization is demonstrated. Comparison among different preconditioners is made. The effect of the proposed starting‐vector generation schemes is shown. The comparative merits of analytical sensitivity analysis and finite difference sensitivity analysis, when using PCG methods of analysis, are assessed. The parallel computation potential of PCG methods is discussed. Because of the iterative nature of PCG analysis methods and the prospects they offer for parallel computation, it is found that PCG analysis methods show promise in the context of structural optimization.

A BRANCH AND BOUND ALGORITHM FOR TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES

An algorithm for the selection of a minimum weight truss, out of a set of possible candidate trusses, is presented. The trusses are subject to stress and deflection constraints. Multiple loading conditions are considered. Starling with a ground structure, a sequence of subtrusses called candidate trusses are generated and analyzed. Several criteria are used to rapidly discard non-optimal configurations. Sequential Quadratic Programming is used to solve a non-linearly constrained problem which is part of the algorithm. A technique, frequently used to keep second derivative approximations positive definite, is found to give numerical instabilities. Possible modifications to improve stability are discussed. Finally, three examples are given to demonstrate the algorithm.

ON TOPOLOGY OPTIMIZATION OF TRUSSES

The selection of a minimum weight truss, out of a large set of candidate trusses, is treated. An initial configuration is generated by connecting all the n nodal points with n(n − 1)\2 truss members. A feasible displacement field, satisfying displacement and stress constraints, is obtained from a static analysis of this truss, followed by a uniform scaling of all truss dimensions. The finite element formulation for the initial configuration is then reformulated to yield a linear programming problem. The solution to this leads to a new configuration which is further optimized by solving a small nonlinear programming problem. With the method proposed, trusses subject to one loading condition and subject to stress and displacement constraints can be selected and optimized using a modest computational effort. Three examples are given to demonstrate the usefulness of the proposed method.

On the derivation of Michell's theorem

In the classical proofs (proofs along the lines of the original proof) of Michell's theorem on the limits of economy of material in frame structures [i], or in using Maxwell's lemma [2] to find the minimum weight truss for a restricted class of loadings [3], it has been assumed that the minimum weight structure will be fully stressed. The assumption is explicitly stated in the version of the proof given by Cox [3], but is only implied in the original proof [I]. The validity of this assumption is obvious for statically determinate trusses. For statically indeterminate trusses, the sizing of one member can affect the forces in other members and therefore can affect the minimum weights of other members. For this reason, if it is not known in advance that the minimum weight truss will be statically determinate, it is not obvious that the minimum weight truss will be fully stressed. However, even for statically determinate Michell or Maxwell structures, previous classical proofs are not valid. The statical determinancy of that structure, which results from a derivation that assumes the minimum weight structure will be fully stressed, cannot be used to justify the assumption which produced that result. It is felt that a proof of Michell's theorem or use of Maxwell's lemma that makes an unsubstantiated fully stressed assumption is incomplete and therefore incorrect. What follows are the modifications in the use of Maxwell's lemma and of the proof of ~ichell's theorem; following the format in Reference [3], that does not make a fully stressed assumption, but instead proves the minimum weight structure will be fully stressed. The final results in both cases are unchanged but correct proofs are provided.

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.

Minimum Weight Design of Elastic Redundant Trusses under Multiple Static Loading Conditions

A method for automated selection of a minimum weight truss, from a subset of configurations obtained by omitting various member combinations from a primary configuration, is presented. A feasible direction algorithm is used to find an upper bound solution for any selected configuration. A dual simplex algorithm is used to rapidly generate lower bound solutions for many subset configurations considering only equilibrium conditions and stress limits. The lower bound solutions guide the selection of subset configurations for which upper bound solutions of reduced weight are sought. Examples of planar and space trusses illustrate the efficacy of the bounding technique presented. I. Mathematical Formulation C ONSIDER an elastic redundant truss with given geometric configuration and support conditions. The truss is to sustain several given alternative systems of static loads applied at the joints. Only q of these load systems which are supposed to govern the design will be considered. For simplicity, the truss is conventionally idealized; that is, the members of the truss are assumed to be joined together by frictionless connections and only axial forces, tension or compression, occur in the members under the loading. The design problem addressed herein is the selection of a minimum weight optimum truss configuration, as well as its member sizing, from a subset of truss configurations obtained by omitting various members from the given primary configuration. The design is subject to the following constraints: the crosssectional area Aj of member j, when it remains in the truss, is restricted to a given continuous range, the upper limit A? of which is finite and the lower limit A! may be, and usually is, greater than zero; the stress ajk, defined as the internal force/^ of the member j under the kih loading condition divided by Aj, has given lower limit — ffjk and upper limit ajk, and these stress limits are assumed independent of the cross-sectional area of the member; and the displacement component of a joint under loading condition k also has specified lower — uikl and upper uik limits (each joint, except for fixed supports, has at least one degree of freedom, the corresponding displacement components are numbered in a fixed order and identified as ut). The problem can thus be formulated as with