Optimization of the initial post-buckling response of trusses and frames by an asymptotic approach

TL;DR

The paper addresses the challenge of incorporating post-buckling behavior into topology and sizing optimization of truss and frame structures. It develops a framework based on asymptotic post-buckling theory and the ANM to approximate the initial post-buckling path, and it introduces constraints on the two lowest post-buckling coefficients $\alpha$ and $\beta$ to control imperfection sensitivity. Through numerical examples on Euler beams, Rooda-Koiter frames, and large assemblies, it demonstrates that enforcing $|\alpha|$ and a minimum $\beta$ leads to more stable designs and better resilience to imperfections, while providing efficient nonlinear response approximations for optimization. The work highlights computational advantages of ANM (roughly 10–15x speed-up) and discusses current limitations, practical trade-offs, and prospects for extending the approach to continuum topology optimization.

Abstract

Asymptotic post-buckling theory is applied to sizing and topology optimization of trusses and frames, exploring its potential and current computational difficulties. We show that a designs' post-buckling response can be controlled by including the lowest two asymptotic coefficients, representing the initial post-buckling slope and curvature, in the optimization formulation. This also reduces the imperfection sensitivity of the optimized design. The asymptotic expansion can further be used to approximate the structural nonlinear response, and then to optimize for a given measure of the nonlinear mechanical performance such as, for example, end-compliance or complementary work. Examples of linear and nonlinear compliance minimization of trusses and frames show the effective use of the asymptotic method for including post-buckling constraints in structural optimization.

Figures (8)

• Figure 1: Types of bifurcations according to the signs of the post-buckling slope $\alpha$, and curvature $\beta$. (a) Asymmetric bifurcation ($\alpha < 0$), for the Rooda-Koiter frame rooda-chilver_70a_frameBucklingPerturbationolesen-byskov_82a_asymptoticPostbucklingStresses. (b, c) Symmetric bifurcations ($\alpha = 0$) for the Euler beam (b) and von Mises truss (c) book:bazant-cedolin2010. For the last two cases, the bifurcations are respectively stable ($\beta > 0$) and unstable ($\beta < 0$), independent of the sign of the initial imperfection $\Delta$. Multiple dashed red and blue curves correspond to increasing magnitudes of the imperfection $\Delta$
• Figure 2: Equilibrium paths computed by the arc-length method (black circles), compared to the first-order (blue dashed curve) and second-order (red continuous curve) approximations given by the ANM method. The red dashed lines in (a, b) correspond to the second-order "locked" asymptotic solution, following from inappropriate structural modeling and discretization. The plot in (c) also shows the nonlinear deformation at the buckling point, for the stable and unstable Rooda-Koiter configuration, respectively
• Figure 3: Initial design (IG) for the compressed column (a), Rooda frame (b) and deep clamped beam (c) configurations, examined in \ref{['sSec:MotivationExample']}-\ref{['sSec:ClampedDeepBeam']}. Here and in the following plots, bars subjected to tensile axial stress are colored in blue, whereas those in red are compressed
• Figure 4: Column designs obtained by linear compliance minimization, with volume and buckling constraints, considering truss (a) and frame (c) modeling. The volume fraction is $\bar{v}_{f} = 0.25$ and $\lambda_{\rm ref} = \ell\lambda_{1(0)}$, where $\lambda_{1(0)}$ is the BLF of the initial design (see \ref{['tab:Table1']}). (b) and (d) show the nonlinear response of the structures, computed by the arc-length algorithm up to reaching $v_{\rm Tip} \leq 0.075$. The horizontal dash-dotted lines mark the BLFs ($\lambda_{1}$) computed by the LBA, whereas the red dots mark the maximum loads reached by the nonlinear response ($\lambda_{c}$)
• Figure 5: (a) Column designs obtained when introducing the constraints on the post-buckling slope and curvature in the optimization problem \ref{['eq:optProblemYesStabilityConstraints']}. The objective function is linear compliance, the volume fraction $\bar{v}_{f} = 0.25$, and we set $g_{\alpha} = |\alpha|\leq 10^{-10}$, $g_{\beta} = \beta \geq 1.25\beta_{0}$. The buckling constraint is not active for the final designs. (b) shows the nonlinear post-evaluation of the designs' response