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.     

Fast procedure for Non-uniform optimum design of stiffened shells under buckling constraint

For tailoring the non-uniform axial compression, each sub-panel of stiffened shells should be designed separately to achieve a high load-carrying efficiency. Motivated by the challenge caused by numerous variables and high computational cost, a fast procedure for the minimum weight design of non-uniform stiffened shells under buckling constraint is proposed, which decomposes a hyper multi-dimensional problem into a hierarchical optimization with two levels. To facilitate the post-buckling optimization, an efficient equivalent analysis model of stiffened shells is developed based on the Numerical Implementation of Asymptotic Homogenization Method. In particular, the effects of non-uniform load, internal pressure and geometric imperfections are taken into account during the optimization. Finally, a typical fuel tank of launch vehicle is utilized to demonstrate the effectiveness of the proposed procedure, and detailed comparison with other optimization methodologies is made.

     

Reinforcement layout and sizing optimization of composite submarine sail structures

A topology optimization approach that makes use of nonlinear design variable-to-sizing relationship is presented. A finite element (FE) model is used to describe the loaded structure, but unlike the microstructure approach, the decision is whether an element in the continuum should have maximum or minimum cross-sectional dimension while its material density and moduli are held constant. This approach is applied to reinforcement layout optimization of a very large and geometrically complex Composite Advanced Sail (CAS) structure under an asymmetric wave slap loading condition. A high-complexity model in the form of multilayered shell and a low-complexity model in the form of stiffened shell are developed for the layout optimization of the CAS and solved for minimum strain energy. The effects of constraints such as buckling instability on optimal placement of internal stiffeners are also explored. Based on the results of the layout optimization, a new FE model of the CAS is developed and optimized for minimum weight. Depending upon the degree of variability in skin thickness, the results show a weight saving of up to 19% over the original model.     

Design optimization of non-uniform stiffened steel plate girders—a computer code

The paper presents a direct search design procedure that has been automated to obtain a minimum weight design of non-uniform stiffened steel plate girders. The direct search procedure does not require evaluation of design gradients and the search for minimum weight design is according to either the ASD or the LRFD design specifications of the American Institute of Steel Construction. An algorithm for the two design procedures is presented to facilitate a general utilization of the computer code. The comzputer code is tested by comparing results of minimum weight designs of a three-span stiffened steel plate girder to results obtained from a generalized reduced gradients algorithm. Results of the comparative design studies are summarized to (i) compare effectiveness of the procedure presented in this paper over a generalized gradient-based design procedure; (ii) illustrate the simplified input–output format of the computer code through a set of design examples; and (iii) compare designs obtained from the two design specifications implemented in the code.     

Minimum weight design of stiffened cylinders subjected to pure torsion

A minimum weight design procedure along with actual designs of two typical fuselage type of stiffened circular cylindrical shell geometries subjected to pure torsion is presented. By formulating the weight of the composite shell as the objective function, an optimization technique is adopted to minimize it against general instability. In the design, all other possible failure modes, i.e. panel instability, skin wrinkling, local instability of stringers, yielding of skin and stiffener materials as well as failure mode interactions have been avoided. Typical opened type, like rectangular, tee, I, etc. and closed type, hat stiffener with all possible combinations are studied. For each shell geometry the best suited combination of stiffener geometries-are shown. In the first trial without minimum gauge (WMG), a design is obtained utilizing rectangular stringers and rectangular rings. Then the procedure is extended for other stiffener geometries to obtain a minimum gauge (MG) design. The effect of relaxing the MG on the weight of the stiffened shell is shown graphically and in tabular form which will be of added advantage to the designer while deciding about the MG. Since the smeared technique is employed in the analysis of stiffened shells, an upper bound on the stiffener spacing is initially employed. Then this bound is relaxed and its effect on the minimum weight design is studied for those types of stiffeners which proved economically feasible with bounded spacings. The procedure adopted in this work can be used for any other shell and stiffener geometry.     

Optimization of geometrically imperfect stiffened cylindrical shells under axial compression

A procedure is outlined for optimizing stiffened, thin, circular, cylindrical shells under uniform axial compression against general instability, in the presence of initial geometric imperfection. The procedure consists of two parts (a) optimization on the basis of a linear buckling analysis and perfect geometry, and (b) parametric studies on a reasonable region in the design space surrounding the optimum point (as obtained from part (a)) to assess the effect of initial geometric imperfections. This procedure is demonstrated through two design examples, for which it is concluded, that the presence of initial geometric imperfections does not alter the optimum weight and the corresponding design variables appreciably.     

Two approaches for truss topology optimization: a comparison for practical use

This paper discusses ground structure approaches for topology optimization of trusses. These topology optimization methods select an optimal subset of bars from the set of all possible bars defined on a discrete grid. The objectives used are based either on minimum compliance or on minimum volume. Advantages and disadvantages are discussed and it is shown that constraints exist where the formulations become equivalent. The incorporation of stability constraints (buckling) into topology design is important. The influence of buckling on the optimal layout is demonstrated by a bridge design example. A second example shows the applicability of truss topology optimization to a real engineering stiffened membrane problem.     

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.     

Weight optimization of stiffened cylinders under axial compression

A methodology is developed, by which one may design a stiffened cylinder of specified material, radius, and length, such that it can safely carry a given uniform axial compression with minimum weight. The solution procedure is divided into two stages, Phase I and Phase II. In Phase I, an unconstrained minimization is performed against one of the active constraints (in this paper-general instability) and data are generated in a sufficiently large region of the design space by employing efficient mathematical search techniques, in Phase II, these data are employed to arrive at the minimum weight configuration that satisfies all other constraints. Two design examples are presented which demonstrate the methodology.     

A practical method for the minimum weight design of stiffened plates under uniform lateral pressure

A method for the plastic analysis and minimum weight design of grillages and orthogonally stiffened plates subjected to lateral uniform pressure, and under a varying degree of rotational restraint at the supports, is presented. A computer aided design procedure for orthogonally stiffened rectangular plates based on this method is also described. This includes two optimization stages: overall optimization, in which the plastic moments for the beam in each set are found, and stiffener optimization, in which the optimum stiffener geometry is obtained. This design methodology has proved to be very efficient and easy to apply, and it is particularly valuable in the case of ship structures, where the maximum total number of panel stiffeners is in general not large.     

Safety of optimally designed structures like cylinders and plates

In the minimum weight designs, the influence of initial imperfection and suddenly applied loads undermine the presumed safety of cylinderical shells and, on the other hand, the contribution of post-buckled strength enhances the safety of plates. The safety margin for both the structures is reduced due to the interaction of buckling and frequency constraints. These well known phenomena are examined with relation to the structural safety for the benefit of practical designers with the help of two design examples; one a stiffened cylinder and the other a waffle plate.     

Optimization of hybrid steel plate girders

An efficient procedure is presented for minimum weight design of unstiffened and stiffened hybrid steel plate girders subjected to arbitrary loading using the General Geometric Programming technique. The nonlinear optimization problem is formulated on the basis of the current American Institute of Steel Construction specification. Four examples are presented to show the application of the algorithm for practical optimization of hybrid steel plate girders.     

Minimum weight design of the stiffened cylindrical shell under pure bending

An attempt has been made in this work to obtain a minimum weight design software for the airplane fuselage type of stiffened cylindrical shell under pure bending load. This design problem has been formulated as a non-linear constrained minimization problem. The two numerical methods used for the solution of this problem are: (1) penalty function technique and (2) method of complex box.

The general computer programs based on the above methods are prepared. The minimum weight design is carried out by considering the shell which is stiffened in longitudinal and circumferential directions and treating their dimensions as well as their spacings and skin thickness as design variables. The numerical results have been obtained for a simplified case under the specified assumptions. The programs can be further extended for the practical problems such as design of a minimum weight fuselage.     

Optimal design of underwater stiffened shells

The optimal design parameters of stiffened shells are determined using a rational multicriteria optimization approach. The adopted approach aims at simultaneously minimizing the shell vibration, associated sound radiation, weight of the stiffening rings as well as the cost of the stiffened shell. A finite element model is developed to determine the vibration and noise radiation from cylindrical shells into the surrounding fluid domain. The production cost as well as the life cycle and maintenance costs of the stiffened shells are computed using the Parametric Review of Information for Costing and Evaluation (PRICE) model. A Pareto/min-max multicriteria optimization approach is then utilized to select the optimal dimensions and spacing of the stiffeners. Numerical examples are presented to compare the vibration and noise radiation characteristics of optimally designed stiffened shells with the corresponding characteristics of plain un-stiffened shells. The obtained results emphasis the importance of the adopted multicriteria optimization approach in the design of quiet, low weight and low cost underwater shells which are suitable for various critical applications. Received September 14, 2000 Communicated by J. Sobieski     

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.     

Some aspects of truss topology optimization

The present paper studies some aspects of formulations of truss topology optimization problems. The ground structure approach-based formulations of three types of truss topology optimization problems, namely the problems of minimum weight design for a given compliance, of minimum weight design with stress constraints and of minimum weight design with stress constraints and local buckling constraints are examined. The common difficulties with the formulations of the three problems are discussed. Since the continuity of the constraint or/and objective function is an important factor for the determination of the mathematical structure of optimization problems, the issue of the continuity of stress, displacement and compliance functions in terms of the cross-sectional areas at zero area is studied. It is shown that the bar stress function has discontinuity at zero crosssectional area, and the structural displacement and compliance are continuous functions of the cross-sectional area. Based on the discontinuity of the stress function we point out the features of the feasible domain and global optimum for optimization problems with stress and/or local buckling constraints, and conclude that they are mathematical programming with discontinuous constraint functions and that they are essentially discrete optimization problems. The difference between topology optimization with global constraints such as structural compliance and that with local constraints on stress or/and local buckling is notable and has important consequences for the solution approach.     

Numerical collapse analysis of compression members

A brief survey of weld-induced stresses and deflections in steel-plated structures is given. A finite element approach for analyzing stiffened panels is presented. The stiffener formulation accounts for torsional buckling. Yielding is taken into account and, based on the updated Lagrangian formulation, the effect of large deflections is allowed for. By using the so-called volume approach the development of plastic zones can be followed. The non-linear equations are solved using a combined step-iterative procedure.The examples presented comprise an unstiffened rectangular plate under biaxial compression, and two stiffened panels under uniaxial compression. The effect of initial deflections and stresses is considered.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

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.