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 the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures

Several geometrically nonlinear beam models are evaluated with respect to their utility in the analysis of buckling and post-buckling behavior of three-dimensional beam structures. The first two models are based on the so-called geometrically exact beam theory capable of representing finite rotations and finite displacements. The principal difference between these models concerns only the chosen parameterization of finite rotations, with the orthogonal matrix used in the first and the rotation vector used in the second one. The third beam model based on the second-order approximation of finite rotations is also discussed along with its application to constructing a consistent formulation of the linear eigenvalue problem for computing an estimate of the critical load. Exact linearized forms, which are crucial for facilitating the buckling load computation and assuring a robust performance of a Newton-method-based continuation strategy, are presented for all three beam models. An elaborate set of numerical simulations of buckling and post-buckling analysis of beam structures is given in order to illustrate the performance of each of the presented models. Finally, some conclusions are drawn.     

Including global stability in truss layout optimization for the conceptual design of large-scale applications

Including stability in truss topology optimization is critical to avoid unstable optimized designs in practical applications. While prior research addresses this challenge by implementing local buckling and linear prebuckling, numerical difficulties remain due to the global stability singularity phenomenon. Therefore, the goal of this paper is to develop an optimization formulation for truss topology optimization including global stability without numerical singularities, within the framework of the preliminary design of large-scale structures. This task is performed by considering an appropriate simultaneous analysis and design formulation, in which the use of a disaggregated form for the equilibrium equations alleviates the singularities inherent to global stability. By implementing a local buckling criterion for hollow truss elements, the resulting formulation is well-suited for the preliminary design of large-scale trusses in civil engineering applications. Three applications illustrate the efficiency of the proposed approach, including a benchmark truss structure and the preliminary design of a footbridge and a dome. The results demonstrate that including local buckling and global stability can considerably affect the optimized design, while offering a systematic means of avoiding unstable solutions. It is also shown that the proposed approach is in a good agreement with linear prebuckling assumptions.     

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.     

Sensitivity of structural response in context of linear and non-linear buckling analysis with solid shell finite elements

The paper is concerned with the sensitivity analysis of structural responses in context of linear and non-linear stability phenomena like buckling and snapping. The structural analysis covering these stability phenomena is summarised. Design sensitivity information for a solid shell finite element is derived. The mixed formulation is based on the Hu-Washizu variational functional. Geometrical non-linearities are taken into account with linear elastic material behaviour. Sensitivities are derived analytically for responses of linear and non-linear buckling analysis with discrete finite element matrices. Numerical examples demonstrate the shape optimisation maximising the smallest eigenvalue of the linear buckling analysis and the directly computed critical load scales at bifurcation and limit points of non-linear buckling analysis, respectively. Analytically derived gradients are verified using the finite difference approach.     

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.     

Prestressed pin-jointed structures—Flexibility analysis and prestress design

Two classes of prestressable structures exist. Class I are geometrically rigid and statically indeterminate structures. Class II are statically and kinematically indeterminate structures with infinitesimal mechanisms. In structures of class I, prestress by means of imposed lack of fit can substantially enhance the design, particularly when failure is governed by compression member buckling. Structures of class II depend on prestress for their geometric integrity.

A unified algorithm for the analysis and for the design of prestress of pin-jointed structures of both classes, consisting of any combination of bars and cables, is presented. The algorithm is based on the flexibility method of structural analysis, and assumes small displacements. It is capable of identifying states of prestress and mechanisms and optimizing the prestress in compression controlled structures.     

Sensitivity analysis and optimization of thin laminated structures with a nonsmooth eigenvalue based criterion

This paper presents a discrete model for the design sensitivity analysis of thin laminated angle-ply composite structures using a plate shell element based on a Kirchhoff discrete theory for the bending effects. To overcome the nondifferentiability of multiple eigenvalues, which may occur during a structural optimization involving free vibrations or buckling design situations, a nonsmooth eigenvalue based criterion is implemented. Angle-ply design variables and vectorial distances from the laminated midle surface to the upper surface of each layer are considered as design variables. The design sensitivities and the directional derivatives are evaluated analytically. The efficiency and accuracy of the model developed is discussed with two illustrative cases which show the need to compute sensitivities of multiple eigenvalues as directional derivatives for laminated composite structures.     

Buckling design optimization of complex built-up structures with shape and size variables

The design optimization of buckling behaviour is studied for complex built-up structures composed of various kinds of elements and implemented within JIFEX95, a general-purpose software for finite element analysis and design optimization. The direct and adjoint methods of sensitivity analysis for critical buckling loads are presented with detailed computational procedures. Particularly, the variations of prebuckling stresses and external loads have been accounted for. The design model and solution methods presented in this paper are available for both shape and size optimization, and buckling optimization can also be combined with static, frequency and dynamic response optimization. The numerical examples show the applications of the buckling optimization method and the effectiveness of the methods and the program of this paper. Received February 23, 1999     

Optimal discrete sizing of truss structures subject to buckling constraints

The selective dynamic rounding (SDR) algorithm previously developed by the authors, and based on a dual step rounding approach, is used for the optimal sizing design of truss structures subject to linear buckling constraints. The algorithm begins with a continuous optimum followed by a progressive freezing of individual variables while solving the remaining continuous problems. The allowable member stresses are predicted by the linear regression of the tabular section properties, while the exact allowable compressive stresses are back-substituted for those variables fixed on discrete values in each intermediate mixed-discrete nonlinear problem. It is shown that a continuous design based on the regression analysis of section effectiveness vs. area is effective as a starting point for the dual step discrete optimization phase. A range of examples is used to illustrate that with conservative regression, discrete designs can be achieved which are significantly lighter than those in which the variables have been rounded up.     

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.     

Buckling of webs with openings

The buckling of the webs of steel beams is investigated analytically and experimentally. The influence of circular and rectangular holes on the buckling strength of beams of the dimensions normally used in building structures is considered. The buckling analysis is based on an energy method, and relies on finite element analysis for the in-plane stress distributions in the perforated webs. For the proportions of the beams considered, buckling takes place after the initiation of plasticity in the web and this is considered in the analysis. Comparison of analytical results with results of tests on five beams shows excellent agreement. It is shown that for these beams, the range of slenderness for which elasto-plastic buckling occurs is quite limited, and that below a web slenderness of 50, or in some cases more, holes of the shape and size considered will not lead to buckling prior to the development of full plastic strength.     

Structural optimization with member dimensional imperfections

The paper deals with the effect of dimensional imperfections of truss members on the minimum weight design of a structure. It is assumed that for each element its imperfections cannot exceed given a priori maximum values, called tolerances. The incorporation of the considered imperfections into the design is achieved by diminishing the limit values of state functions by the product of assumed imperfections and appropriate sensitivities. Therefore, the given method allows the introduction of tolerances into the design in a relatively simple way. In the submitted paper both members’ cross-section and length imperfections are discussed. The paper is illustrated with several design examples, considering cases with multiple loading conditions and buckling analysis. The achieved optimum solutions for designs with admissible tolerances show significant differences in structural weight and material distribution compared to ideal structures (i.e. having nominal dimensions). The calculations for designs with buckling analysis also reveal changes in material distribution compared to the designs without buckling constraints.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

Probabilistic buckling analysis of fiber metal laminates under shear loading condition

The design allowables for compressive and shear buckling of fiber metal laminate (FML) panels need to be developed for the certification of these materials. In this paper, the shear buckling behavior of the two FML panels, ARALL3-3/2 and GLARE3-2/1, is investigated using a probabilistic analysis method in order to predict the distributions of the buckling load. The scatter associated with the material properties and the randomness in the loading conditions are accounted for in the probabilistic analysis. The shear buckling load of the panel is taken as the response function of the material properties and the load parameter. The response surface methodology (RSM) is employed for the development of the response function based on limited finite element data. Three RSM design methods: full factorial design, CCD design and saturated D-optimum design, are applied to construct the response function following which the shear buckling load distributions of the two FML panels are predicted. The predicted results are verified using independently generated finite element data.     

Optimal design of frames with substructuring

This paper presents a formulation for optimal design of large scale, two and three dimensional framed structures. Von Mises equivalent stress constraints and displacement constraints are imposed at all points in the structure. Member size constraints and constraints based on Schilling's approach for member buckling are also imposed. Three example problems of varying degrees of difficulty are solved, using a gradient projection algorithm with state space design sensitivity analysis and substructuring. Results of these examples are analyzed and conclusions are presented.     

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.     

DMTO – a method for Discrete Material and Thickness Optimization of laminated composite structures

This paper presents a gradient based topology optimization method for Discrete Material and Thickness Optimization of laminated composite structures, labelled the DMTO method. The capabilities of the proposed method are demonstrated on mass minimization, subject to constraints on the structural criteria; buckling load factors, eigenfrequencies, and limited displacements. Furthermore, common design guidelines or rules, referred to as manufacturing constraints, are included explicitly in the optimization problem as series of linear inequalities. The material selection and thickness variation are optimized simultaneously through interpolation functions with penalization. Numerical results for several parameterizations of a finite element model of a generic main spar from a wind turbine blade are presented. The different parameterizations represent different levels of complexity with respect to manufacturability. The results will thus give insight into the relation between potential weight saving and design complexity. The results show that the DMTO method is capable of solving the problems robustly with only few intermediate valued design variables.     

Aircraft morphing wing design by using partial topology optimization

A morphing wing concept has been investigated over the last decade because it can effectively enhance aircraft aerodynamic performance over a wider range of flight conditions through structural flexibility. The internal structural layouts and component sizes of a morphing aircraft wing have an impact on aircraft performance i.e. aeroelastic characteristics, mechanical behaviors, and mass. In this paper, a novel design approach is proposed for synthesizing the internal structural layout of a morphing wing. The new internal structures are achieved by using two new design strategies. The first design strategy applies design variables for simultaneous partial topology and sizing optimization while the second design strategy includes nodal positions as design variables. Both strategies are based on a ground structure approach. A multiobjective optimization problem is assigned to optimize the percentage of change in lift effectiveness, buckling factor, and mass of a structure subject to design constraints including divergence and flutter speeds, buckling factors, and stresses. The design problem is solved by using multiobjective population-based incremental learning (MOPBIL). The Pareto optimum results of both strategies lead to different unconventional wing structures which are superior to their conventional counterparts. From the results, the design strategy that uses simultaneous partial topology, sizing, and shape optimization is superior to the others based on a hypervolume indicator. The aeroelastic parameters of the obtained morphing wing subject to external actuating torques are analyzed and it is shown that it is practicable to apply the unconventional wing structures for an aircraft.     

Thermal buckling optimisation of composite plates using firefly algorithm

Abstract

Composite plates play a very important role in engineering applications, especially in aerospace industry. Thermal buckling of such components is of great importance and must be known to achieve an appropriate design. This paper deals with stacking sequence optimisation of laminated composite plates for maximising the critical buckling temperature using a powerful meta-heuristic algorithm called firefly algorithm (FA) which is based on the flashing behaviour of fireflies. The main objective of present work was to show the ability of FA in optimisation of composite structures. The performance of FA is compared with the results reported in the previous published works using other algorithms which shows the efficiency of FA in stacking sequence optimisation of laminated composite structures.