Abstract

     In order to obtain stable and realistic truss structures in practical applications, it is essential to include nodal and local buckling stability in truss topology optimization. There have been several approaches to address the challenges including nodal stability or local buckling stability. However, these approaches often lead to ill-conditioned optimization problems, such as convergence problems due to the concavity of the problem or high computational costs. In this study, two novel but conceptually simple methodologies, the nominal perturbing force (NPF) approach, and the allowable stress iteration (ASI) approach, are proposed to address nodal instability and local buckling instability problems in truss topology optimization, respectively. Initially, in the NPF approach, an infinite number of disturbing forces that a node may suffer are incorporated into the truss topology optimization problem in the form of nominal perturbing force conditions, whose magnitude and direction are discussed to capture the worst case. In the ASI approach, the allowable stress for each compressive bar is redefined in each iteration to ensure that the Euler critical buckling constraint is active. In this way, the concave local buckling instability problem is linearized in each iteration and can be solved efficiently by a linear programming solver. Finally, based on the finite element limit analysis (FELA) method, a truss topology optimization formulation incorporating the NPF and ASI approaches is proposed to solve the nodal stability and local buckling stability problems simultaneously. The proposed formulation is demonstrated through several numerical examples showing significant effects of including nodal stability and local buckling stability in the optimized designs, while at the same time demonstrating the validity and potential of the proposed approaches.

Introduction

     The pin-jointed truss topology (or “layout”) optimization problem is concerned with finding an optimal arrangement of structural bars subject to prescribed constraints. A typical truss topology optimization problem might comprise a design domain containing an array of fixed nodal points connected by potential members, called the ground structure [4,11,33,37,42]. The optimization objective might typically then be to minimize the total volume, mass [21,37], or compliance[28] of structure, with constraints ensuring that each node is in static equilibrium and that bar stresses are within predefined limits [22,47,48]. The optimal cross-sectional areas for all the bars in the ground structure are then determined as part of the optimization process; typically, many of these will be zero, leaving the optimal topology comprising only of bars with non-zero cross-sectional areas[20].

     Mathematically, the ground structure provides a feasible domain for the truss structure optimization problem. Typically, the truss topology optimization based on the ground structure method is mainly divided into four steps [19]. Firstly, as shown in Fig. 1a, the design domain with appropriate supports and loads is determined. Secondly, as shown in Fig. 1b, the nodes are arranged, usually evenly, to discrete the design domain. Thirdly, as shown in Fig. 1c, these nodes are connected with potential bars to generate the ground structure. Finally, as shown in Fig. 1d, some optimization algorithms are used to remove the redundant bars to generate the optimal structure.

Conclusions

     The main contribution of this study is the development of a truss topology optimization formulation including nodal and local buckling stability constraints. A brief review of literature has shown that a convenient approach to state the basic plastic problem of truss topology optimization relies on the FELA method.

     The novel nominal perturbing force (NPF) approach has been proposed to address the nodal instability in the truss topology optimization problem. In the NPF approach, nominal perturbing forces consistent with the directions of the Cartesian axes are used to model the possible perturbations of the unstable nodes. Compared with the conventional nominal lateral force (NLF) approach, (a) the NPF approach has higher computational efficiency because it contains fewer optimization variables; (b) the effect of tension bars connected to the nodes is considered in determining the unstable nodes, which can lead to a better optimization result.

     In addition, the novel nominal allowable stress iteration (ASI) approach is proposed to address the local buckling instability in the truss topology optimization problem. In this way, the concave local buckling instability problem is linearized in each iteration and can be solved efficiently by a linear programming solver. Numerical examples have shown the practical applicability of the proposed method for the preliminary design of truss structures.