Dale A. Hopkins

Ballistic impact of GLARE fiber-metal laminates

Abstract Analytical solutions to predict the ballistic limit and energy absorption of fully clamped GLARE panels subjected to ballistic impact by a blunt cylinder were derived. The analytical solutions were based on test results from NASA Glenn. The ballistic limit was found through an iterative process such that the initial kinetic energy of the projectile would equal the total energy dissipated by panel deformation, delamination/debonding and fracture. The transient deformation of the panel as shear waves propagate from the point of impact was obtained from an equivalent mass–spring system, whereby the inertia and stiffness depend on the shear wave speed and time. Predictions of the ballistic limit from the resulting non-linear differential equation were within 13% of the test data. The deformation energy due to bending and membrane accounted for most of the total energy absorbed (84–92%), with the thinner panels absorbing a higher percentage of deformation energy than the thicker panels. Energy dissipated in delamination represented 2–9% of the total absorbed energy, with the thinner panels absorbing a lower percentage of delamination energy than the thicker panels. About 7% of the total energy was attributed to tensile fracture energy of the glass/epoxy and aluminum.

Reliability based structural optimization : a simplified safety index approach

Abstract A probabilistic optimal design methodology for complex structures modelled with finite element methods is presented. The main emphasis is on developing probabilistic analysis tools suitable for optimization. An advanced second-moment method is employed to evaluate the failure probability of the performance function. The safety indices are interpolated using the information at the mean and most probable failure point. The minimum weight design with an improved safety index limit is achieved by using the extended interior penalty method of optimization. Numerical examples covering beam and plate structures are presented to illustrate the design approach. The results obtained by using the proposed approach are compared with those obtained by using the existing probabilistic optimization techniques.

COMPARATIVE EVALUATION OF DIFFERENT OPTIMIZATION ALGORITHMS FOR STRUCTURAL DESIGN APPLICATIONS

Non-linear programming algorithms play an important role in structural design optimization. Fortunately, several algorithms with computer codes are available. At NASA Lewis Research Centre, a project was initiated to assess the performance of eight different optimizers through the development of a computer code CometBoards. This paper summarizes the conclusions of that research. CometBoards was employed to solve sets of small, medium and large structural problems, using the eight different optimizers on a Cray-YMP8E/8128 computer. The reliability and efficiency of the optimizers were determined from the performance of these problems. For small problems, the performance of most of the optimizers could be considered adequate. For large problems, however, three optimizers (two sequential quadratic programming routines, DNCONG of IMSL and SQP of IDESIGN, along with Sequential Unconstrained Minimizations Technique SUMT) outperformed others. At optimum, most optimizers captured an identical number of active displacement and frequency constraints but the number of active stress constraints differed among the optimizers. This discrepancy can be attributed to singularity conditions in the optimization and the alleviation of this discrepancy can improve the efficiency of optimizers.

Modified Fully Utilized Design (MFUD) Method for Stress and Displacement Constraints

The traditional fully stressed method performs satisfactorily for stress-limited structural design. When this method is extended to include displacement limitations in addition to stress constraints, it is known as the Fully Utilized Design (FUD). Typically, the FUD produces an overdesign, which is the primary limitation of this otherwise elegant method. We have modified FUD in an attempt to alleviate the limitation. This new method, called the Modified Fully Utilized Design (MFUD) method, has been tested successfully on a number of problems that were subjected to multiple loads and had both stress and displacement constraints. The solutions obtained with MFUD compare favourably with the optimum results that can be generated by using non-linear mathematical programming techniques. The MFUD method appears to have alleviated the overdesign condition and offers the simplicity of a direct, fully stressed type of design method that is distinctly different from optimization and optimality criteria formulations. The MFUD method is being developed for practicing engineers who favour traditional design methods rather than methods based on advanced calculus and non-linear mathematical programming techniques. The Integrated Force Method (IFM) was found to be the appropriate analysis tool in the development of the MFUD method. In this paper, the MFUD method and its optimality are examined along with a number of illustrative examples.

Improved accuracy for finite element structural analysis via an integrated force method

Abstract Finite element structural analysis based on the original displacement (stiffness) method has been researched and developed for over three decades. Although today it dominates the scene in terms of routine engineering use, the stiffness method does suffer from certain deficiencies. Various alternate analysis methods, commonly referred to as the mixed and hybrid methods, have been promoted in an attempt to compensate for some of these limitations. In recent years two methods for finite element analyses of structures, within the framework of the original force method concept, have been introduced. These are termed the ‘integrated force method’ and the ‘dual integrated force method’. A comparative study was carried out to determine the accuracy of finite element analyses based on the stiffness method, a mixed method, and the new integrated force and dual integrated force methods. The numerical results were obtained with the following software: MSC/NASTRAN and ASKA for the stiffness method; an MHOST implementation for a mixed method; a GIFT for the integrated force methods. For the cases considered, the results indicate that, on an overall basis, the stiffness and mixed methods present some limitations. The stiffness method typically requires a large number of elements in the model to achieve acceptable accuracy. The MHOST mixed method tends to achieve a higher level of accuracy for coarse models than does the stiffness method as implemented by MSC/NASTRAN and ASKA. The two integrated force methods, which bestow simultaneous emphasis on stress equilibrium and strain compatibility, yield accurate solutions with fewer elements in a model. The full potential of these new integrated force methods remains largely unexploited, and they hold the promise of spawning new finite element structural analysis tools.

Optimality of a fully stressed design

For a truss a fully stressed state is reached when all its members are utilized to their full strength capacity. Historically, engineers considered such a design optimum. But recently this optimality has been questioned, especially since the weight of the structure is not explicitly used in fully stressed design calculations. This paper examines optimality of the fully stressed design (FSD) with analytical and graphical illustrations. Solutions for a set of examples obtained by using the FSD method and optimization methods numerically confirm the optimality of the FSD. The FSD, which can be obtained with a small amount of calculation, can be extended to displacement constraints and to nontruss-type structures.

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

The integrated force method has been developed in recent years for the analysis of structural mechanics problems. In the intgrated force method all independent forces are treated as unknown variables, which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. The development of a finite element library for the analysis of two-dimensional problems using the integrated force method is presented in this paper. Elements of triangular and quadrilateral shapes, capable of modeling arbitrary domain configurations are developed. The element equilibrium and flexibility matrices are derived by discretizing expressions for corresponding potential and complementary energies, respectively. Independent approximations of displacement and stress fields within finite elements are performed. Interpolation of the displacement field is done similarly as in the standard displacement method. The stress field is approximated using full polynomials of correct orders. A procedure for deriving the stress interpolation polynomials that utilizes the definitions of stress components in terms of Airys stress function is developed. Such derived stress fields identically satisfy equations of equilibrium, and the resulting element matrices are insensitive to the orientation of local coordinate systems. A method to calculate the number of rigid body modes is devised, and it is shown that the present elements do not possess spurious zero energy modes. A number of example problems are solved using the present library and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the integrated force method, compared to the displacement method, in stress calculations, is observed.

Neural Network and Regression Approximations in High-Speed Civil Aircraft Design Optimization

Nonlinear mathematical-programming-based design optimization can be an elegant method. However, the calculations required to generate the merit function, constraints, and their gradients, which are frequently required, can make the process computational intensive. The computational burden can be greatly reduced by using approximating analyzers derived from an original analyzer utilizing neural networks and linear regression methods. The experience gained from using both of these approximation methods in the design optimization of a high speed civil transport aircraft is the subject of this paper. The Langley Research Centers Flight Optimization System was selected for the aircraft analysis. This software was exercised to generate a set of training data with which a neural network and a regression method were trained, thereby producing the two approximating analyzers. The derived analyzers were coupled to the Lewis Research Centers CometBoards test bed to provide the optimization capability. With the combined software, both approximation methods were examined for use in aircraft design optimization, and both performed satisfactorily. The CPU time for solution of the problem, which had been measured in hours, was reduced to minutes with the neural network approximation and to seconds with the regression method. Instability encountered in the aircraft analysis software at certain design points was also eliminated. On the other hand, there were costs and difficulties associated with training the approximating analyzers. The CPU time required to generate the input-output pairs and to train the approximating analyzers was seven times that required for solution of the problem.

A CASCADE OPTIMIZATION STRATEGY FOR SOLUTION OF DIFFICULT DESIGN PROBLEMS

A research project to evaluate comparatively ten different non-linear optimization algorithms was completed recently. A conclusion was that no single optimizer could successfully solve all the 40 structural design problems in the test-bed, even though most optimizers successfully solved at least one-third of the problems. We realized that improvements to search directions and step lengths, available in the ten optimizers compared, were not likely to alleviate the convergence difficulties. For the solution of those difficult problems we have devised an alternate approach called, the cascade optimization strategy. The strategy utilizes several optimizers, one followed by another in a specified sequence, to solve a problem. A pseudo-random dumping scheme perturbs the design variables between the optimizers. The cascade strategy has been tested out successfully in the design of supersonic and subsonic aircraft configurations and air breathing engines for high-speed civil transport applications. These problems could not be successfully solved by an individual optimizer. The cascade optimization strategy, however, generated feasible optimum solutions for both aircraft and engine problems. This paper presents the cascade strategy, solution of aircraft and engine problems along with discussions and conclusions.

Three-dimensional structural analysis by the integrated force method

Abstract The “integrated force method”, which has been recently developed for analyzing structures, is extended in this paper for three-dimensional structural analysis. A general formulation to generate the stress interpolation matrix in terms of complete polynomials of the required order is developed first. The formulation is based on the definitions of stress tensor components in terms of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. After eliminating dependent coefficients, the expressions for stress components are obtained as complete polynomials, where coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy Naviers equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero energy modes. The formulation provides a rational way to calculate the exact number of independent forces for the required order of approximation with complete polynomials. The reduction in the number of independent forces and its influence on the accuracy of the response are also analyzed. The stress fields derived here are next used to develop a comprehensive finite element library for the three-dimensional structural analysis using the “integrated force method”. Elements of both tetrahedral and hexahedral shapes, capable of modeling arbitrary geometric configurations, are developed. A number of example problems with available analytical solutions are solved using the present developments and a good agreement of results with the analytical solutions is observed. The responses obtained using the “integrated force method” are also compared with those generated with the standard displacement method. A better overall performance of the “integrated force method” is observed.