Reliability-based design optimization of composite stiffened panels in post-buckling regime

This paper focuses on Deterministic and Reliability Based Design Optimization (DO and RBDO) of composite stiffened panels considering post-buckling regime and progressive failure analysis. The ultimate load that a post-buckled panel can hold is to be maximised by changing the stacking sequence of both skin and stringers composite layups. The RBDO problem looks for a design that collapses beyond the shortening of failure obtained in the DO phase with a target reliability while considering uncertainty in the elastic properties of the composite material. The RBDO algorithm proposed is decoupled and hence separates the Reliability Analysis (RA) from the deterministic optimization. The main code to drive both the DO and RBDO approaches is written in MATLAB and employs Genetic Algorithms (GA) to solve the DO loops because discrete design variables and highly nonlinear response functions are expected. The code is linked with Abaqus to perform parallel explicit nonlinear finite element analyses in order to obtain the structural responses at each generation. The RA is solved through an inverse Most Probable failure Point (MPP) search algorithm that benefits from a Polynomial Chaos Expansion with Latin Hypercube Sampling (PCE-LHS) metamodel when the structural responses are required. The results led to small reductions in the maximum load that the panels can bear but otherwise assure that they will collapse beyond the shortening of failure imposed with a high reliability.

     

Design and analysis of stiffened composite panels including post-buckling and collapse

The European aircraft industry demands reduced development and operating costs, by 20% and 50% in the short and long term, respectively. Contributions to this aim are provided by the completed project POSICOSS (5th FP) and the running follow-up project COCOMAT (6th FP), both supported by the European Commission. As an important contribution to cost reduction a decrease in structural weight can be reached by exploiting considerable reserves in primary fibre composite fuselage structures through an accurate and reliable simulation of post-buckling up to collapse. The POSICOSS team developed fast procedures for the post-buckling analysis of stiffened fibre composite panels, created comprehensive experimental data bases and derived suitable design guidelines. COCOMAT builds up on the POSICOSS results and considers in addition the simulation of collapse by taking degradation into account. The results comprise an extended experimental data base, degradation models, and improved certification and design tools as well as extended design guidelines.One major task of POSICOSS and COCOMAT is the development of improved analysis tools that are validated by experiments performed within the framework of the projects. Because the new tools must comprise a wide range of various aspects a considerable number of different structures had to be tested. These structures were designed under different objectives (e.g. large post-buckling region). For the design process, the consortiums applied state-of-the-art simulation tools and brought in their own design experience. This paper deals with the design process as performed within both projects and with the applied analysis procedures. It is focused on the DLR experience in the design and analysis of stringer-stiffened CFRP panels gained within the scope of these two projects.     

Two-stage size-layout optimization of axially compressed stiffened panels

In this study, a two-stage optimization framework is proposed for cylindrical or flat stiffened panels under uniform or non-uniform axial compression, which are extensively used in the aerospace industry. In the first stage, traditional sizing optimization is performed. Based on the buckling or collapse-like deformed shape evaluated for the optimized design, the panel can be divided in sub-regions each of which shows characteristic deformations along axial and circumferential directions. Layout optimization is then performed using a stiffener spacing distribution function to represent the location of each stiffener. A layout coefficient is assigned to each sub-region and the overall layout of the panel is optimized. Three test problems are solved in order to demonstrate the validity of the proposed optimization framework: remarkably, the load-carrying capacity improves by 17.4 %, 66.2 % and 102.2 % with respect to the initial design.     

Optimum design and evaluation of stiffened panels with practical loading

Optimum design of a blade-stiffened panel of composite/honeycomb sandwich construction and a metal T-stiffened panel is considered using the buckling and strength constraint program VICONOPT. Both panels have practical loadings which produce a nonlinear out-of-plane bending moment, calculated using beam-column expressions. Large deflection finite element analysis of the optima shows that modifications to these expressions are necessary when the panels are shear loaded. The use of integrally machined stiffeners, as opposed to a conventional, built-up panel designed using PANDA2, is shown to permit 20% mass saving when the latter has no postbuckling strength and 3% saving when postbuckling strength is allowed for.     

Shape optimization of interior cutouts in composite panels

This paper presents validated results of the optimization of cutouts in laminated carbon-fibre composite panels by adapting a recently developed optimization procedure known as Evolutionary Structural Optimization (ESO). An initial small cutout was introduced into each finite element model and elements were removed from around this cutout based on a predefined rejection criterion. In the examples presented, the limiting ply within each plate element around the cutout was determined based on the Tsai-Hill failure index. Plates with values below the product of the average Tsai-Hill number and a rejection ratio (RR) were subsequently removed. This process was iterated until a steady state was reached and the RR was then incremented by an evolutionary rate (ER). The above steps were repeated until a cutout of a desired area was achieved.Nomenclature ₁ stress in fibre direction - ₂ stress in transverse direction - ₁₂ shear stress - X tensile/compressive strength in fibre direction - Y tensile/compressive strength in transverse direction - S shear strength - e plate element number lying on edge of cutout - p plate element number - P total number of plates - TH Tsai-Hill number - TH(l) limiting Tsai-Hill number of platee - TH(p) limiting Tsai-Hill number of platep - RR rejection ratio - [K] global stiffness matrix - {} displacement vector - {F} force vector - {} stress vector - l ply number - j number of plies per plate - FI failure index - T specified condition on which to terminate evolution (e.g. area of cutout) - _A stress at major vertex of ellipse - _B stress at minor vertex of ellipse - a ellipse major axis - b ellipse minor axis - i increment number - ER evolutionary rate     

Weight and cost oriented multi-objective optimisation of impact damage resistant stiffened composite panels

In this paper, a multi-objective optimisation procedure, based on the adoption of genetic algorithms, is presented. The optimal configuration with minimum weight and minimum cost of a damage resistant stiffened, composite panel with buckling constraints has been determined. The numerical procedure is based on an in-house optimisation code used in conjunction with the ANSYS FEM code. The presence of both continuous and high sensitivity discrete design variables, suggested for GA the adoption of a special bit-masking data structure able to increase the overall computational efficiency. Optimal configurations of the stiffened panel, are finally analysed and discussed focusing on the influence of the damage resistance constraint on the overall costs.     

A framework combining meshfree analysis and adaptive kriging for optimization of stiffened panels

Structural and Multidisciplinary Optimization - A framework is developed for structural optimization using an Element Free Galerkin (EFG) method for analyzing the structure, a kriging for surrogate...     

Optimization of composite stiffened panels subject to compression and lateral pressure using a bi-level approach

Composite stiffened panel optimization is typically a mixed discrete-continuous design problem constrained by buckling and material strength. Previous work applied a bi-level optimization strategy to the problem by decomposing the mixed problem to continuous and discrete levels to reduce the optimization search space and satisfy manufacturing constraints. A fast-running optimization package, VICONOPT, was used at the continuous optimization level where the buckling analysis was accurately and effectively performed. However, the discrete level was manually adjusted to satisfy laminate design rules. This paper develops the strategy to application on continuously long aircraft wing panels subjected to compression and lateral pressure loading. The beam-column approach used to account for lateral loading for analysis during optimization is reported. A genetic algorithm is newly developed and applied to the discrete level for automated selection of laminated designs. The results that are presented show at least 13% weight saving compared with an existing datum design.     

Multi-fidelity design of stiffened composite panel with a crack

Crack propagation is an important concern in the design of aircraft composite fuselage and wing panels. However, numerical simulation of crack propagation is computationally expensive. This work proposes combining high-fidelity analysis model with low-fidelity model to calculate the crack propagation constraint in the design optimization process. Correction response surfaces are employed to relate the high-fidelity models to the low-fidelity models. Four different forms of correction response surface methods are explored and their prediction capabilities are compared. The multi-fidelity approach is found to be more accurate than single-fidelity response surface method at the same computational cost.     

Shape optimization of stiffeners in stiffened composite plates with thermal residual stresses

The finite element analysis method is used to examine the influence of manufacturing-induced thermal residual stresses on the optimal shape of stiffeners in stiffened, symmetrically laminated plates. Three stiffener arrangements are studied via an optimization process in which the objective is to maximize the first natural frequency of the stiffened plate. The optimization problem is solved using the method of moving asymptotes (MMA). The numerical simulations indicate that thermal residual stresses can either cause a dispersion of stiffeners along the perimeter or a concentration around the centre. Further, the optimum fundamental frequency tends to increase with increasing temperature difference.     

Multilevel optimization of composite panels under complex load and boundary conditions

Due to their complexity and large numbers of design variables, aerospace structures, such as aircraft wings, are best optimized using a multi-level process. In addition to simplifying the optimization procedure, such an approach allows a combination of different methods to be used, increasing the efficiency of the analysis. This paper presents a technique based on the usage of exact finite strip software, VICONOPT, with the finite element analysis package, ABAQUS. The computer programme VICONOPT is computationally efficient but provides solutions for a restricted range of geometries and loading conditions. Finite element analysis allows accurate models of structures with complex geometries to be created but is computationally expensive. By combining the two, these limitations are minimised, whilst the strengths of each are exploited. The fundamental principles of this multi-level procedure are demonstrated by optimizing a series of curved composite panels under combined shear and in-plane bending subject to buckling constraints. Presented at the 7th World Congress on Computational Mechanics, LA, USA, July 2006.     

Experiments on interactive buckling in optimized stiffened panels

Five aluminium blade stiffened panels of three different geometries were tested in compression in a displacement controlled loading apparatus. Panel designs were achieved using VICONOPT, a fast-running optimization package based on linear eigenvalue buckling theory, and embrace two different design philosophies. The panels were loaded beyond initial buckling to collapse, and the effects of initial overall imperfections were monitored. In all cases the final failure showed evidence of significant interaction between buckling modes. The tests draw particular attention to a violently unstable and unpredictable form of failure, involving a combination of overall and stiffener buckling, which can occur even when initial buckling in the skin has the effect of pushing the panel in the opposite sense. Received November 14, 2000     

Robust seismic design optimization of steel structures

Stochastic performance measures can be taken into account, in structural optimization, using two distinct formulations: robust design optimization (RDO) and reliability-based design optimization (RBDO). According to a RDO formulation, it is desired to obtain solutions insensitive to the uncontrollable parameter variation. In the present study, the solution of a structural robust design problem formulated as a two-objective optimization problem is addressed, where cross-sectional dimensions, material properties and earthquake loading are considered as random variables. Additionally, a two-objective deterministic-based optimization (DBO) problem is also considered. In particular, the DBO and RDO formulations are employed for assessing the Greek national seismic design code for steel structural buildings with respect to the behavioral factor considered. The limit-state-dependent cost is used as a measure of assessment. The stochastic finite element problem is solved using the Monte Carlo Simulation method, while a modified NSGA-II algorithm is employed for solving the two-objective optimization problem.     

Robust design of discrete-time systems via optimization

An optimization approach for the design of robust discrete-time control systems is presented. Using pole clustering, the design specifications for the system time response are recast in the form of inequality constraints governing the free design parameters. Quantitative robustness and disturbance rejection measures are discussed using Lyapunov matrix equations. A performance index combining the robustness and disturbance rejection measures is minimized subject to the parameter inequality constraints to obtain the desired design. A numerical example is given to illustrate the usefulness of the design approach.     

A multilevel genetic algorithm for optimization of geometrically nonlinear stiffened composite structures

A multilevel genetic algorithm aiming the global optimization of beam reinforced composite structures with nonlinear geometric behaviour is proposed. A unified approach based on load-displacement control for buckling and first ply failure analysis is adopted. The Newton-Raphson iterative scheme and the arc-length method are used for tracing the equilibrium path and later for updating the critical values. The proposed genetic algorithm performs several sequences of two optimization levels resulting from the decomposition of the original optimization problem. Independent genetic searches are implemented for each level where different fitness functions and sub-populations are considered. The genetic operators selection and crossover supported by an elitist strategy are used while the diversity of the sub-populations is guaranteed based on implicit mutation. A genetic material exchange between levels is performed using clones and so the offspring of matured sub-populations is guaranteed. To improve the efficiency of the multilevel genetic optimization a niche of population is induced after the first stage at both levels.     

Robust design modeling and optimization with unbalanced data

The usual assumption behind robust design is that the number of replicates at each design point during an experimental stage is equal. In practice, however, it is often the case that this assumption is not met due to physical limitations and/or cost constraints. In this situation, using the usual method of ordinary least squares (OLS) to obtain fitted response functions for the mean and variance of the quality characteristic of interest may not be an effective tool. In this paper, we first show simulation results, indicating that an alternative method, called the method of weighted least squares (WLS), outperforms the OLS method in terms of mean squared error. We then lay out the WLS-based robust design modeling and optimization. A case study is presented for numerical purposes.     

Compressive buckling analysis and design of stiffened flat plates with multilayered composite reinforcement

This paper describes an analysis and its application in design for compressive buckling of flat stiffened plates considered as an assemblage of linked orthotropic flat plate and beam elements. Plates can be multilayered, with possible coupling between bending and stretching. Structural lips and beads are idealized as beams. The plate and the beam elements are matched along their common junctions for displacement continuity and force equilibrium in an exact manner. Buckling loads are found as the lowest of all possible general and local failure modes. The mode shape is used to determine whether buckling is a local or general instability and is particularly useful to the designer in identifying the weak elements for redesign purposes. Typical design curves are presented for the initial buckling of a hat stiffened plate locally reinforced with boron fiber composite.