Steel truss bridges with welded box-section members and bowknot integral joints, Part II: Minimum weight optimization

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Abstract

The second part of the paper presents the minimum weight optimization based on the provisions of current design codes for both conventional and bowknot trusses. The optimization philosophies are illustrated first through the analytic derivation of minimum weight optimization of a single member. These results indicate that the member weight reduction increases as the primary stress to secondary stress ratio, or the end moment reduction, is increased. The minimum weight optimization of the truss then proceeds on the basis of linear finite element analysis of the same truss that was discussed in Part I, by the use of first-order optimization method. In the optimization formulation, the cost rise due to steel strength enhancement of shrunken segments is taken into account in the nominal weight of whole truss, and a series of requirements related to truss vertical stiffness, member strength, member stability, and truss stability are set as constraint functions. The optimization results show that the optimal weights of bowknot trusses are less than those of conventional trusses on the premise that the steel strength of shrunken segments has been moderately improved.

Highlights

► We carried out weight optimization of Warren truss with bowknot integral joints. ► We studied multiple cases of design variables related to section thickness. ► Structural constraints concerning strength, stiffness, and stability were employed. ► Optimal weights of bowknot trusses could be less than those of conventional ones. ► Section-shrinking parameters have significant influences on optimized weights.

Introduction

The results of the structural analyses presented in Part I [1] have shown two major benefits for steel trusses with welded box-section members and bowknot integral joints. One is a reduction in secondary moments at member ends, and the other is a reduction in sectional stresses of uniform segments. Meanwhile, disadvantages – including increases in sectional stresses of shrunken segments, degradation in the truss's vertical stiffness, deteriorations in the elastic stabilities of compression members and the truss itself, etc. – have also been observed for bowknot trusses. For a heavy class truss composed of stocky members whose slenderness ratios are relatively small, the enhancements in the stability safety degrees of members under combined axial compression and bending moment, which are caused by the moment reductions, are especially favorable to the cross-section optimization of members. At the same time, the reduction in the vertical stiffness and elastic stability of the truss is usually not critical to the design because the degraded indices of these properties are higher than the minimum allowable values specified in design codes. Therefore, it is the authors' conjecture that a lighter weight of a bowknot truss in which the steel strength of shrunken segments has been moderately improved to avoid the overstressing of shrunken sections, when compared to the conventional one, can be obtained from structural optimization within the constraints of design specifications.

The goal of truss optimization is to create topology, geometry and cross-sectional dimensions of members such that the weight or cost of the truss is minimized. Generally, continuous variables, discrete variables, or combined continuous and discrete variables such as nodal coordinate, member number, sectional area, section thickness, material type, and a set of specified constraints related to stress (strength), displacement (stiffness), stability, local buckling and frequency, are partially or totally considered in the optimization process. Research over the past several decades has proven that while classical gradient-based mathematical programming methods are well-suited for most continuous problems, they make it difficult to search for the optimal solutions of discrete variables [2]. Consequently, notable work on discrete optimization of trusses has been undertaken using deterministic methods such as mixed integer programming [3], branch and bound techniques [4], dual formulation [5], penalty approach [6], segmented approach with linear programming [7]. Besides, stochastic or random algorithms have also been developed and introduced to the truss optimization involving discrete variables. These were genetic algorithm [8], simulated annealing method [9], particle swarm algorithm [10], ant colony optimization [11], and hybridized approaches [12]. Complex optimization problems concerning truss structures are more efficiently and accurately solved using these novel search algorithms or techniques.

In this paper, size optimization considering only continuous variables is formulated to achieve the minimum weights of simply supported Warren trusses with welded box-section members and bowknot integral joints. The analytic derivation of minimum weight optimization is first executed for a single shrunken member in order to reveal the optimization design strategy for the entire truss. The finite element analysis (FEA) based minimum weight optimizations of trusses are then performed using the ANSYS software in which the first-order method is employed by the authors to obtain the optimal solutions. In the optimization formulation, the section thicknesses of welded box-section members and the section-shrinking parameters, including the height-to-height and the length-to-height ratios, are defined as design variables; the nominal weight of the truss is taken as a single objective function; and requirements related to truss vertical stiffness, member strength, member stability, and truss stability are set as constraint functions. The influence of section-shrinking parameters and of the steel strength of shrunken segments on minimum nominal weight, as well as on other mechanical behaviors of the optimized truss, is analyzed. Finally, the benefits in reducing structure weight as obtained from the adoption of bowknot integral joints are outlined by comparing the optimization results of bowknot trusses with conventional trusses.

Section snippets

FE model and optimization method

During the truss optimization process, a linear analysis of the truss structure should first be performed to provide necessary data, including internal force, sectional stress, vertical deflection, and the elastic stability coefficient, and should be subsequently repeated. Hence, the same Warren trusses whose dimensions, materials, supporting and load conditions, element type and mesh-controlling parameters are identical to those of linear finite element models, which have been described in

Optimization derivation

The minimum weight optimization of a single shrunken member is derived to illustrate the weight optimization strategy for the whole truss. During the derivation, the very small differences in axial forces of uniform and shrunken members are ignored, and the probable overstressing of shrunken sections is assumed to be eliminated by the adoption of higher strength steel.

Fig. 1 shows the details of two welded box-section members, a uniform member and a shrunken member, with a length of L0.

Given data

Four types of parameters used in minimum weight optimization of a Warren truss are given as follows:

  • (1)

    Geometric dimensions of the truss, including truss span (L), truss depth (H), and section width (b) and height (h) of truss members;

  • (2)

    Material properties of steel, including Young's modulus (E), the Poisson ratio (ν), and the maximum allowable stress (F1,d and F2,d);

  • (3)

    Applied concentrated dead load (Pd) and live load (Pl);

  • (4)

    Maximum allowable vertical deflection of truss (f0).

Design variables

Five parameters, including

Conclusions

In this paper, minimum weight optimization based on stress equivalence of uniform sections and minimum weight optimization based on linear finite element analysis have been developed, respectively, for shrunken members and bowknot trusses. The influences of the section-shrinking parameters and the steel strength of shrunken segments on truss weight are further investigated, and the following main insights are produced from the optimization results:

  • (1)

    The minimum weight of a shrunken member is

Acknowledgments

The present research was undertaken with support from the Morning-Star Young Scholars Award Program (grant no. TS0330101005) and the Innovative Practice Program (grant no. IPP1003), which are both funded by Shanghai Jiao Tong University.

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