Design optimization and sensitivity analysis on buckling stability of piezoelectric truss structures

The design optimization of buckling behavior is studied for piezoelectric intelligent truss structures. First, on the basis of mechanical–electric coupling equation and considering electric load and mechanical loads together, the finite element model of piezoelectric trusses has been built up. Then, the computational formula has been derived for the design sensitivities of critical buckling load factor of the structure with respect to size and shape design variables. The electric voltage is taken as a new kind of design variable and the calculation method of critical load buckling factor with respect to the electric voltage variables is proposed. Particularly, the variations of the loads and pre-buckling inner forces with design variables have been accounted for. Finally, the sequential linear programming algorithm is employed to solve the optimization problem, and a new method of controlling structural buckling stability by optimizing the voltages of piezoelectric active bars is proposed. Numerical examples given in the paper have demonstrated the effectiveness of the methods presented.     

A technique for optimally designing fibre-reinforced laminated plates under in-plane loads for minimum weight with manufacturing uncertainties accounted for

A procedure to design symmetrically laminated plates under buckling loads for minimum mass with manufacturing uncertainty in the ply angle, which is the design variable, is described. A minimum buckling load capacity is the design constraint implemented. The effects of bending–twisting coupling are neglected in implementing the procedure, and the golden section method is used as the search technique, but the methodology is flexible enough to allow any appropriate problem formulation and search algorithm to be substituted. Three different tolerance scenarios are used for the purposes of illustrating the methodology, and plates with varying aspect ratios and loading ratios are optimally designed and compared.     

Elasto-plastic buckling of stiffener plates in beam-to-column flange connections

The problem of plastic buckling of steel plates is reviewed in relation to the load carrying capacity of stiffener plates in beam-to-column flange connections. Due to the non-uniformity of the stress distribution in these plates, the finite element method is used to compute the stresses in the elastic and plastic ranges. A bifurcation analysis is performed using both flow and deformation theory to evaluate the elasto-plastic buckling of the stiffener. A scaled inverse iterative version of the power method is employed to evaluate the bifurcation load. A parametric study is conducted on stiffeners and design curves are obtained showing the relationship between the critical stress and the slenderness ratio for different plate aspect ratios.     

Multi-objective optimization of functionally graded materials,thickness and aspect ratio in micro-beams embedded in an elastic medium

Optimal design of micron-scale beams as a general case is an important problem for development of micro-electromechanical devices. For various applications, the mechanical parameters such as mass, maximum deflection and stress, natural frequency and buckling load are considered in strategies of micro-manufacturing technologies. However, all parameters are not of equal importance in each operating condition but multi-objective optimization is able to select optimal states of micro-beams which have desirable performances in various micro-electromechanical devices. This paper provides optimal states of design variables including thickness, distribution parameter of functionally graded materials, and aspect ratio in simply supported FG micro-beams resting on the elastic foundation using analytical solutions. The elastic medium is assumed to be as a two-layered foundation including a shear layer and a linear normal layer. Also, the size effect on the mechanical parameters is considered using the modified strain gradient theory and non-dominated sorting genetic algorithm-II is employed to optimization procedure. The target functions are defined such that the maximum deflection, maximum stress and mass must be minimized while natural frequency and critical buckling load must be maximized. The optimum patterns of FG micro-beams are presented for exponential and power-law FGMs and the effect of theory type and elastic foundation discussed in details. Findings indicate that the elastic foundation coefficients and internal length scale parameters of materials have the significant influences on the distribution of design variables. It is seen that the optimum values of inhomogeneity parameter and aspect ratio for E-FG micro-beams predicted by the modified strain gradient theory are larger than those of the classical continuum theory. Also, the multi-objective optimization is able to improve the normalized values of mass, maximum deflection, buckling load and natural frequency of P-FG micro-beams.     

Optimization formulations for the maximum nonlinear buckling load of composite structures

This paper focuses on criterion functions for gradient based optimization of the buckling load of laminated composite structures considering different types of buckling behaviour. A local criterion is developed, and is, together with a range of local and global criterion functions from literature, benchmarked on a number of numerical examples of laminated composite structures for the maximization of the buckling load considering fiber angle design variables. The optimization formulations are based on either linear or geometrically nonlinear analysis and formulated as mathematical programming problems solved using gradient based techniques. The developed local criterion is formulated such it captures nonlinear effects upon loading and proves useful for both analysis purposes and as a criterion for use in nonlinear buckling optimization.     

On compliance and buckling objective functions in topology optimization of snap-through problems

This paper deals with topology optimization of static geometrically nonlinear structures experiencing snap-through behaviour. Different compliance and buckling criterion functions are studied and applied for topology optimization of a point loaded curved beam problem with the aim of maximizing the snap-through buckling load. The response of the optimized structures obtained using the considered objective functions are evaluated and compared. Due to the intrinsic nonlinear nature of the problem, the load level at which the objective function is evaluated has a tremendous effect on the resulting optimized design. A well-known issue in buckling topology optimization is artificial buckling modes in low density regions. The typical remedy applied for linear buckling does not have a natural extension to nonlinear problems, and we propose an alternative approach. Some possible negative implications of using symmetry to reduce the model size are highlighted and it is demonstrated how an initial symmetric buckling response may change to an asymmetric buckling response during the optimization process. This problem may partly be avoided by not exploiting symmetry, however special requirements are needed of the analysis method and optimization formulation. We apply a nonlinear path tracing algorithm capable of detecting different types of stability points and an optimization formulation that handles possible mode switching. This is an extension into the topology optimization realm of a method developed, and used for, fiber angle optimization in laminated composite structures. We finally discuss and pinpoint some of the issues related to buckling topology optimization that remains unsolved and demands further research.     

Single-level composite wing optimization based on flexural lamination parameters

An optimization procedure based on flexural lamination parameters is used to integrate unstiffened composite panel design and wing structural design. The lamination parameters are constrained to a hexagonal domain when the amounts of 0°, ±45°, and 90° plies are given. The single-level optimization based on continuous flexural lamination parameters for the minimization of wing weight is compared with a two-level optimization using response surfaces of maximal buckling load for a simple wing box design example. Reasonable agreement between the two procedures indicates that the two-level approach leads to near-optimal designs.     

On buckling optimization under uncertain loading combinations

This paper proposes a technique to optimize structural components for buckling when the applied loads are partially unknown or unpredictable. As opposed to the traditional buckling optimization situation where the loading configuration is specified, the load ratios are assumed uncertain and are incorporated as variables in the optimization problem formulation. As a result, the optimal designs obtained are insensitive to load variations within an admissible convex set. Additionally, in order to generalize the results and therefore provide a systematic solution procedure, a theorem concerning the shape of the stability boundary of structures whose buckling loads are the solution of linear eigenproblems is stated and proven. Recevied February 2, 2000     

Reliability-based and deterministic design optimization of a FSAE brake pedal: a risk allocation analysis

While probabilistic designs can translate into significant weight savings through better risk allocation, deterministic design optimization remains widely used in industry. To promote the use of probabilistic designs among engineering students and practitioners, this work solves reliability based design optimization (RBDO) and deterministic design optimization (DDO) models of a FSAE brake pedal with multiple failure modes (stress and buckling) with their relative performance evaluated through a risk allocation analysis. The problems of interest were systematically solved through the following steps: i) topology optimization to specify the brake-pedal shape, ii) numerical 3D brake-pedal modeling under uncertainty for stress and buckling analysis, iii) mass (M), maximum von Mises stress (S_max) and buckling load factor (f_buck) surrogate modeling, iv) global sensitivity analysis and surrogate model selection, and v) surrogate-based RBDO and DDO with risk allocation analysis. Results show that when compared to DDO with alternative safety factors, for the same probability of system failure, the RBDO brake pedal designs were significantly lighter and more robust (less mass variability).     

Variable stiffness composite material design by using support vector regression assisted efficient global optimization method

In this work, a surrogate assisted optimization method is utilized to optimize buckling loads of variable stiffness composites made by fiber steering. To improve the efficiency of optimization procedure, an expected improvement criterion is employed. Moreover, considering uncertainties of the fiber placement, a robust surrogate, least square support vector regression (LSSVR) considering empirical and structural risks is integrated with the expected improvement (EI) criterion and applied to two applications. The first case is the fiber path design of a variable stiffness plate under the compression load. The second one is the fiber path design of a variable stiffness cylinder under the bending load. According to results of the optimization, the buckling load of the variable stiffness plate has 52.63% improvement than the constant stiffness plate and 24.3% improvement than the quasi-isotropic plate. The buckling load of the variable stiffness cylinder has 40.22% improvement than the constant stiffness cylinder and 31.25% improvement than the quasi-isotropic cylinder. Furthermore, to verify the robustness of optimal design variables for the variable stiffness cylinder, the perturbed optimum design is presented and demonstrates that the results are reliable.     

Optimization of the buckling load for composite structures taking thermal effects into account

This paper deals with optimization of the buckling load for laminated composite structures. A new methodology has been developed where thermal residual stresses introduced in the manufacturing process are included in the buckling analysis. The thermal effects are also included in the calculation of the buckling load sensitivities, and it is therefore possible to “tailor” the thermal residual stresses in order to increase the buckling load. Rectangular plates and circular cylindrical shells subjected to axial compression are considered. The structures are optimized twice; the first time the thermal residual stresses are ignored in the optimization, and the second time the thermal residual stresses are included in the optimization. These two sets of optimizations give two important results. Firstly, it is possible to increase the buckling load for the structures significantly when the thermal residual stresses are taken into account. Secondly, structures which have been optimized ignoring the effects of thermal residual stresses, may have a buckling load which is much less than expected when the effects of the thermal residual stresses are included. Received November 7, 1999     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Post-buckling behaviour of moderately thick cylindrically orthotropic annular plates

A finite element formulation including the effects of shear deformation and cylindrically orthotropic material properties is described for studying the post-buckling behaviour of annular plates. Numerical results for the buckling load parameter and ratios of nonlinear load parameter to buckling load parameter for various values of orthotropic properties, thicknesses and radii ratios of the plates are presented.     

Generalized topology design of structures with a buckling load criterion

Material based models for topology optimization of linear elastic solids with a low volume constraint generate very slender structures composed mainly of bars and beam elements. For this type of structure the value of the buckling critical load becomes one of the most important design criteria and so its control is very important for meaningful practical designs. This paper tries to address this problem, presenting an approach to introduce the possibility of critical load control into the topology optimization model.Using the material based formulation for topology design of structures, the problem of optimal structural reinforcement for a critical load criterion is formulated. The stability problem is conveniently reduced to a linearized eigenvalue problem assuming only material effective properties and macroscopic instability modes. The respective optimality criteria are presented by introducing the Lagrangian associated with the optimization problem. Based on this Lagrangian a first-order method is used as a basis for the numerical update scheme. Two numerical examples to validate the developments are presented and analysed.     

Multi-fidelity optimization of laminated conical shells for buckling

Optimum laminate configuration for minimum weight of filament-wound laminated conical shells is investigated subject to a buckling load constraint. In the case of a composite laminated conical shell, due to the manufacturing process, the thickness and the ply orientation are functions of the shell coordinates, which ultimately results in coordinate dependence of the stiffness matrices (A,B,D). These effects influence both the buckling load and the weight of the structure and complicate the optimization problem considerably. High computational cost is involved in calculating the buckling load by means of a high-fidelity analysis, e.g. using the computer code STAGS-A. In order to simplify the optimization procedure, a low-fidelity model based on the assumption of constant material properties throughout the shell is adopted, and buckling loads are calculated by means of a low-fidelity analysis, e.g. using the computer code BOCS. This work proposes combining the high-fidelity analysis model (based on exact material properties) with the low-fidelity model (based on nominal material properties) by using correction response surfaces, which approximate the discrepancy between buckling loads determined from different fidelity analyses. The results indicate that the proposed multi-fidelity approaches using correction response surfaces can be used to improve the computational efficiency of structural optimization problems.     

Stability and load-carrying capacity of three-dimensional long-span steel arch bridges

The behavior of several models of three-dimensional long-span steel arch bridges is investigated for evaluating the effects of various design parameters on both the strength and stability of these special structures. The major concerns in the design of a long-span steel arch bridge, from the structural safety point of view, are the yield and buckling failures. Different design parameters may affect the failure load for either type of failure in various ways. This study investigates how changes in certain design parameters would affect the behavior of steel arch bridges, which could lead to an optimum design of this type of bridge structures. The effects of the plate girder stiffness and arch bracing stiffness as well as the rise-to-span ratio and inclination of the arches towards each other are examined in this study. Both critical buckling load and the load-carrying capacity of each design alternative are investigated using the finite element method. All design alternatives are based on the latest AASHTO code for highway bridge design. It is concluded from this study that the inclined arch bridge using the maximum practical rise-to-span ratio (which is about 0.25) is the most favorable design. In addition, the increase in the stiffness of the plate girder does not reduce the bending moments in the arch ribs. However, providing a lateral bracing system with sufficient stiffness greatly reduces the out-of-plane bending moments and increases the load-carrying capacity and the critical buckling load of a long-span arch bridge.     

Comparison of global and local response surface techniques in reliability-based optimization of composite structures

This paper discusses the development and application of two alternative strategies, in the form of global and sequential local response surface (RS) techniques, for the solution of reliability-based optimization (RBO) problems. The problem of a thin-walled composite circular cylinder under axial buckling instability is used as a demonstrative example. In this case, the global technique uses a single second-order RS model to estimate the axial buckling load over the entire feasible design space (FDS), whereas the local technique uses multiple first-order RS models, with each applied to a small subregion of the FDS. Alternative methods for the calculation of unknown coefficients in each RS model are explored prior to the solution of the optimization problem. The example RBO problem is formulated as a function of 23 uncorrelated random variables that include material properties, the thickness and orientation angle of each ply, the diameter and length of the cylinder, as well as the applied load. The mean values of the 8 ply thicknesses are treated as independent design variables. While the coefficients of variation of all random variables are held fixed, the standard deviations of the ply thicknesses can vary during the optimization process as a result of changes in the design variables. The structural reliability analysis is based on the first-order reliability method with the reliability index treated as the design constraint. In addition to the probabilistic sensitivity analysis of the reliability index, the results of the RBO problem are presented for different combinations of cylinder length and diameter and laminate ply patterns. The two strategies are found to produce similar results in terms of accuracy, with the sequential local RS technique having a considerably better computational efficiency.     

Reliability-based optimum design of a symmetric laminated plate subject to buckling

This study is concerned with the buckling reliability maximization of a symmetric laminated composite plate with respect to the mean ply orientation angle. The reliability is evaluated by modelling the buckling failure as a series system consisting of potential eigenmodes. The mode reliability is obtained by the first-order reliability theory (FORM), where material constants and orientation angles of individual layers, as well as the applied loads are treated as random variables. In order to keep track of the intended buckling mode during the reliability analysis, the mode tracking method is utilized. Then, the failure probability of the series system is approximated by Ditlevsen's upper bound. The reliability maximization problem is formulated as a nested problem with two levels of optimization. Through numerical calculations, the reliability-based design is demonstrated to be important for the structural safety in comparison with the deterministic buckling load maximization design.