Theodore F. Johnson

Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments

Designs of composite laminates are investigated for hydrogen tanks in cryogenic environments. Large residual strains, which can develop due to thermal mismatch between matrix and fibers, result in matrix cracking at cryogenic temperatures and increase hydrogen leakage through the tank wall. To reduce thermal mismatch, ply angles need to be close to each other, but this leads to a substantial weight increase under biaxial loading. First deterministic optimization is used to investigate possible weight reduction measures. Reducing axial loads on walls by auxiliary stiffening mechanisms led to significant weight reduction. Reliability-based optimizations were performed to identify the uncertainties in composite material properties with the largest influences on the optimum design. Then measures for reducing uncertainty in important parameters are examined. The results indicate that the most effective measure for reducing thickness is quality control.

Tradeoff of Uncertainty Reduction Mechanisms for Reducing Weight of Composite Laminates

Inspired by work on allocating risk between the different components of a system for a minimal cost, we explore the optimal allocation of uncertainty in a single component. The tradeoffs of uncertainty reduction measures on the weight of structures designed for reliability are explored. The uncertainties in the problem are broadly classified as error and variability. Probabilistic design is carried out to analyze the effect of reducing error and variability on the weight. As a demonstration problem, the design of composite laminates at cryogenic temperatures is chosen because the design is sensitive to uncertainties. For illustration, variability reduction takes the form of quality control, while error is reduced by including the effect of chemical shrinkage in the analysis. Tradeoff plots of uncertainty reduction measures, probability of failure and weight are generated that could allow choice of optimal uncertainty control measure combination to reach a target probability of failure with minimum cost. In addition, the paper also compares response surface approximations to direct approximation of a probability distribution for efficient estimation of reliability.

Challenges in comparing numerical solutions for optimum weights of stiffened shells

Optimizations of stiffened shells with different stiffener shapes performed to rank and identify the optimum designs during the preliminary design trade studies require a large number of analyses and hence rely on the useof efficient but approximate analysis methods. In the design of shells, the treatment of imperfections on buckling loads and stresses is of paramount importance. It is demonstrated how conservativeness of the approximate analyses used in buckling load calculation, the number of variables optimized (design freedom), and nonstructural constraints influence the weight of optimum designs. This demonstration is based on the results of a trade study performed to compare minimum weight designs of stiffened shells optimized under stress and buckling constraints for a reusable launch vehicle tank. PANDA2 was selected for the present study because it uses approximate analysis procedures that permit the many thousands of structural analyses needed for global optimization and it also has sophisticated machinery for generating imperfections and accounting for their effects. Optimum weights were influenced not only by material choice, number of optimization variables, and manufacturing constraints, but also by the analysis model conservativeness. Optimization of shells with effect of initial imperfections exhibited substantial weight differences between different stiffened-shell concepts, partly because of conservativeness in the analysis.

Being Conservative with a Limited Number of Test Results

combined loads. It is a common practice to repeat the element tests and then select the lowest test result as a conservative estimate of the mean failure stress. Thispractice is equivalent to reducing the average test failure stress by a knockdown factor (one that is quite variable). Instead, we propose using the average test result with an explicit knockdown factor obtained from statistical distribution of the test data. We show reductions in the variability of the estimated mean failure stress as well as the likelihood of unconservative estimate. In addition, when the initial distribution or confidence interval of the mean failure stresses is available, we can further decrease the chance of unconservative estimate using Bayesian updating. We demonstrate the gains associated with Bayesian updating when the upper and lower bounds of errors in the analytical predictions are available. Examples with uniform and lognormal distributions of failure stresses compare the lowest-result approach with the two alternatives with the explicit knockdown factor. Both approaches significantly reduce the likelihood of unconservative estimates of the mean failure stress. The average approach reduced this likelihood by about a half and the Bayesian approach by up toanorderofmagnitude(from12.5to1%).Wealsoexaminescenariosinwhichestimatesoferrorandvariabilityare substantially inaccurate. We show that, even then, the likelihood of unconservative estimates reduces significantly. Remarkably,anunderestimateofvariabilityalsoresultsinabouta2%higheraverageoftheestimatedmeanfailure stress. Thus, we are able to simultaneously use higher average failure stress (leading to lower weight) and reduce the likelihood of unconservative estimates.

Design of Shell Structures for Buckling Using Correction Response Surface Approximations

Several programs for the analysis and design of shells of revolution against buckling, including PANDA2, BOSOR4 and STAGS, are discussed in the paper. It is demonstrated that inexpensive (and possibly less accurate) solutions based on simplified models available in PANDA2 can be effectively combined with more expensive and more accurate solutions available from BOSOR4 and STAGS. This is achieved by using response surfaces to approximate the ratio of the buckling loads from the expensive and less expensive programs. The utility of the procedure is demonstrated with an optimization of a ring stiffened cylinder. In addition, the effect of modeling decisions, such as the choice of finite element to model rings, are discussed.

Effect of Face-Sheet Anisotropy on Buckling and Postbuckling of Sandwich Plates

Astudy oftheeffectofanisotropyoffacesheetsonbuckling andpostbuckling ofsandwiche atpanelssubjected to both mechanical and thermal loads is presented. Themechanical loadsconsist of uniaxial compressive/tensile edge loads, while the temperature is assumed to exhibit an antisymmetric variation through the panel thickness. The study iscarried outin thecontextofa geometrically nonlinear theory of sandwichstructuresthat also incorporates the effect of unavoidable stress-free initial geometric imperfections. A detailed analysis of the ine uence of a large number of parameters associated with the panel geometry, material properties of the core, e ber orientation and stacking sequence in the face sheet, and character of tangential boundary conditions (i.e., movable or immovable ) on buckling strength and postbuckling response is accomplished, and pertinent conclusions are outlined.

Buckling and Nonlinear Response of Sandwich Curved Panels to Combined Mechanical Loads-Implications of Face-Sheet Elastic Tailoring

Results concerning the linear and nonlinear static response of curved sandwich panels to combined mechanical loads are addressed, and in this context, the implications of elastically tailored face-sheets upon the enhancement of their load carrying capacity are emphasized. The analysis is based on a geometrically nonlinear theory of shallow sandwich shells restricted to the case of small strains and moderately small rotations. In addition to the implications of ply-angle, ply-sequence and ply-thickness of composite face-sheets, those related with the panel curvature and aspect ratio on their buckling and nonlinear static response are analyzed, and pertinent conclusions are outlined.

THERMAL-STRUCTURAL OPTIMIZATION OF INTEGRATED CRYOGENIC PROPELLANT TANK CONCEPTS FOR A REUSABLE LAUNCH VEHICLE 1

A next generation reusable launch vehicle (RLV) will require thermally efficient and light-weight cryogenic propellant tank structures. Since these tanks will be weight-critical, analytical tools must be developed to aid in sizing the thickness of insulation layers and structural geometry for optimal performance. Finite element method (FEM) models of the tank and insulation layers were created to analyze the thermal performance of the cryogenic insulation layer and thermal protection system (TPS) of the tanks. The thermal conditions of ground-hold and re-entry/soak- through for a typical RLV mission were used in the thermal sizing study. A general-purpose nonlinear FEM analysis code, capable of using temperature and pressure dependent material properties, was used as the thermal analysis code. Mechanical loads from ground handling and proof-pressure testing were used to size the structural geometry of an aluminum cryogenic tank wall. Nonlinear deterministic optimization and reliability optimization techniques were the analytical tools used to size the geometry of the isogrid stiffeners and thickness of the skin. The results from the sizing study indicate that a commercial FEM code can be used for thermal analyses to size the insulation thicknesses where the temperature and pressure were varied. The results from the structural sizing study show that using combined deterministic and reliability optimization techniques can obtain alternate and lighter designs than the designs obtained from deterministic optimization methods alone.

THERMOMECHANICAL LOAD-CARRYING CAPACITY OF SANDWICH FLAT PANELS

The study of the nonlinear response of sandwich flat panels exposed to thermomechanical loading systems is the topic of this article. The sandwich structure considered in this article consists of a thick core layer bonded by the face layers, which are assumed to be symmetrically located with respect to the midplane of the overall structure. The loads involved in this analysis consist of biaxial compressive edge loads, a lateral pressure, and a nonuniform temperature field. The effects of the unavoidable initial geometric imperfections and the character of tangential boundary conditions are incorporated, and their implications upon the structural response are explored. In short, the results of this study are intended to provide pertinent information on the thermomechanical load-carrying capacity of flat sandwich structures.

Options for Using Test Data to Update Failure Stress

In structural design, failure stresses are obtained from coupon tests and then used to predict failure under combined loads in structural elements. These tests are used to update the failure envelope of structural elements. In determining the failure stress of a single structural element, it is a common practice to replicate element tests and then use the lowest test result as a measure of conservativeness. This practice is equivalent to adding a knockdo wn factor to the stress allowable -one that is quite variable. Instead we propose using the average test result with an explicit knockdown factor and show that the variability is reduced and with it the likelihood of unconservative estimate of the failure s tress. In addition, since failure theories are getting ever more accurate, it may not be reasonable to toss out analytical predictions and rely entirely on test results, which are subject to experimental error and variability. Instead, if the designer has some measure of confidence in the analytical predictions, Bayesian updating may further narrow down variability and decrease the chance of unconservative estimates of failure stresses. We demonstrate the gains associated with Bayesian updating for the case where the designer has only upper and lower bounds on the error in the analytical predictions. Examples with uniform and lognormal distributions of test results are used to compare the worst -test approach to the two alternatives with explicit knockdown fa ctors. Both approaches yield large reductions in the likelihood of unconservative estimates of the failure stresses. The average approach reduced this likelihood by about a factor of two while the Bayesian approach by up to an order of magnitude (from 12.5 % to 1%). We also examine scenarios where estimates of error and variability are substantially off. We show that even then there are still substantial reductions in likelihood of unconservative estimates of failure stresses. Remarkably, the underestimate o f variability also results in about 2% higher average of estimated failure stresses. Thus, we are able to simultaneously use higher average stress allowables (leading to lower weight) and reduce the likelihood of unconservative estimates!