VTechWorks Repository :: Browsing by Author "Nikolaidis, Efstratios"

Browsing by Author "Nikolaidis, Efstratios"

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Analytical and experimental comparison of deterministic and probabilistic optimization
Ponslet, Eric (Virginia Tech, 1994)
The probabilistic approach to design optimization has received increased attention in the last two decades. It is widely recognized that such an approach should lead to designs that make better use of the resources than designs obtained with the classical deterministic approach by distributing safety onto the different components and/or failure modes of a system in an optimal manner. However, probabilistic models rely on a number of assumptions regarding the magnitude of the uncertainties, their distributions, correlations, etc. In addition, modelling errors and approximate reliability calculations (first order methods for example) introduce uncertainty in the predicted system reliability. Because of these inaccuracies, it is not clear if a design obtained from probabilistic optimization will really be more reliable than a design based on deterministic optimization. The objective of this work is to provide a partial answer to this question through laboratory experiments — such experimental validation is not currently available in the literature. A cantilevered truss structure is used as a test case. First, the uncertainties in stiffness and mass properties of the truss elements are evaluated from a large number of measurements. The transmitted scatter in the natural frequencies of the truss is computed and compared to experimental estimates obtained from measurements on 6 realizations of the structure. The experimental results are in reasonable agreement with the predictions, although the magnitude of the transmitted scatter is extremely small. The truss is then equipped with passive viscoelastic tuned dampers for vibration control. The controlled structure is optimized by selecting locations for the dampers and for tuning masses added to the truss. The objective is to satisfy upper limits on the acceleration at given points on the truss for a specified excitation. The properties of the dampers are the primary sources of uncertainties. Two optimal designs are obtained from deterministic and probabilistic optimizations; the deterministic approach maximizes safety margins while the probability of failure (i.e. exceeding the acceleration limit) is minimized in the probabilistic approach. The optimizations are performed with genetic algorithms. The predicted probability of failure of the optimum probabilistic design is less than half that of the deterministic optimum. Finally, optimal deterministic and probabilistic designs are compared in the laboratory. Because small differences in failure rates between two designs are not measurable with a reasonable number of tests, we use anti-optimization to identify a design problem that maximizes the contrast in probability of failure between the two approaches. The anti-optimization is also performed with a genetic algorithm. For the problem identified by the anti-optimization, the probability of failure of the optimum probabilistic design is 25 times smaller than that of the deterministic design. The rates of failure are then measured by testing 29 realizations of each optimum design. The results agree well with the predictions and confirm the larger reliability of the probabilistic design. However, the probabilistic optimum is shown to be very sensitive to modelling errors. This sensitivity can be reduced by including the modelling errors as additional uncertainties in the probabilistic formulation.
Applications of fuzzy logic to mechanical reliability analysis
Touzé, Patrick A. (Virginia Tech, 1993-06-28)
In this work, fuzzy sets are used to express data or model uncertainty in structural systems where random numbers used to be utilized.
Closed-form approximations and series representations for partially saturated ocean acoustic processes
Perakis, Anastassios N.; Nikolaidis, Efstratios; Katzouros, Emmanuel (Acoustical Society of America, 1988-04-01)
An approximate, closed-form expression for the value of the integral encountered in the calculation of the probability density function (PDF) of the envelope of a partially saturated ocean acoustic process is obtained. Furthermore, an expression of this PDF as a series of modified Bessel functions is presented. The results may also be directly applied to the evaluation of the PDF encountered in the structural reliability analysis of rotating machinery components. Numerical applications show that the closed-form expression is always within 1-2% of the exact result. The required computational effort is substantially lower than that required by direct numerical integration. Copyright 1988 Acoustical Society of America
Comparing Probabilistic and Fuzzy Set Approaches for Designing in the Presence of Uncertainty
Chen, Qinghong (Virginia Tech, 2000-08-08)
Probabilistic models and fuzzy set models describe different aspects of uncertainty. Probabilistic models primarily describe random variability in parameters. In engineering system safety, examples are variability in material properties, geometrical dimensions, or wind loads. In contrast, fuzzy set models of uncertainty primarily describe vagueness, such as vagueness in the definition of safety. When there is only limited information about variability, it is possible to use probabilistic models by making suitable assumptions on the statistics of the variability. However, it has been repeatedly shown that this can entail serious errors. Fuzzy set models, which require little data, appear to be well suited to use with designing for uncertainty, when little is known about the uncertainty. Several studies have compared fuzzy set and probabilistic methods in analysis of safety of systems under uncertainty. However, no study has compared the two approaches systematically as a function of the amount of available information. Such a comparison, in the context of design against failure, is the objective of this dissertation. First, the theoretical foundations of probability and possibility theories are compared. We show that a major difference between probability and possibility is in the axioms about the union of events. Because of this difference, probability and possibility calculi are fundamentally different and one cannot simulate possibility calculus using probabilistic models. We also show that possibility-based methods tend to be more conservative than probability-based methods in systems that fail only if many unfavorable events occur simultaneously. Based on these theoretical observations, two design problems are formulated to demonstrate the strength and weakness of probabilistic and fuzzy set methods. We consider the design of tuned damper system and the design and construction of domino stacks. These problems contain narrow failure zones in their uncertain variables and are tailored to demonstrate the pitfalls of probabilistic methods when little information is available for uncertain variables. Using these design problems we demonstrate that probabilistic methods are better than possibility-based methods if sufficient information is available. Just as importantly, we show possibility-based methods can be better if little information is available. Our conclusion is that when there is little information available about uncertainties, a hybrid method should be used to ensure a safe design.
Cyclostationary Random Vibration of a Ship Propeller and a Road Vehicle
Jha, Akhilesh K. (Virginia Tech, 2000-08-07)
A special class of nonstationary processes with periodically varying statistics, called cyclostationary (CS), is investigated. These processes are encountered in many engineering problems involving rotating machinery such as turbines, propellers, helicopter rotors, and diesel engines. We analyze a cyclostationary process model in order to show its advantages compared to a traditional stationary process model and present a methodology for calculating the statistics of the response of a linear system subjected to CS excitations. We demonstrate that a CS model estimates the statistics of the response of a linear dynamic system subjected to CS excitations more accurately by considering (1) a vehicle traveling on a rough road and (2) a propeller rotating in the wake of a ship in the presence of turbulence. In the case of the vehicle, the road consists of concrete plates of fixed length. We model the road excitation using a CS process and calculate the standard deviation (root mean square) of the vehicle response. In the case of the ship propeller, we calculate the hydrodynamic forces acting on the propeller using the vortex panel method and the vortex theory of propeller. Considering the randomness in the axial and the tangential components of velocity, we calculate the mean and the covariance of the forces. This analysis shows that the hydrodynamic forces acting on the propeller are CS processes. Then we perform finite element analysis of the propeller and calculate the mean and the standard deviation of the blade response. We do the parametric analysis to demonstrate the effects of some physical quantities such as the standard deviation, the correlation coefficient, the decorrelation time, and the scale of turbulence of the axial and the tangential components of the wake velocity on the standard deviation of the blade deflection. We found that the CS model yields the time-wise variation of the statistics of the excitation and the response (e.g., the root mean square) and their peaks correctly. This is important information for the calculation of probability of failure of the propeller. A traditional stationary model cannot provide this information.
Design of Automotive Joints Using Response Surface Polynomials and Neural Networks
Ling, Qi (Virginia Tech, 1998-09-08)
In the early design stages of a car body, a simplified model, which represents the constituent components of the car body by their performance characteristics, is used to optimize the overall car. The determined optimum performance characteristics of the components are used as performance targets to design these components. Since designers do not know the relation between the performance characteristics of the components and their dimensions and mass, this may lead to unreasonable performance targets for the components. Moreover, this process is inefficient because design engineers use empirical procedures to design the components that should meet these targets. To design the component more efficiently, design tools are needed to link the performance targets with the physical design variables of the components. General methodologies for developing two design tools for the design of car joints are presented. These tools can be viewed as translators since they translate the performance characteristics of the joint into its dimensions and vice-versa. The first tool, called translator A, quickly predicts the stiffness and the mass of a given joint. The second tool, called translator B, finds the dimensions and mass of the most efficient joint design that meets given stiffness requirements, packaging, manufacturing and styling constraints. Putting bulkheads in the joint structure is an efficient way to increase stiffness. This thesis investigates the effect of transverse bulkheads on the stiffness of an actual B-pillar to rocker joint. It also develops a translator A for the B-pillar to rocker joint with transverse bulkheads. The developed translator A can quickly predict the stiffness of the reinforced joint. Translator B uses optimization to find the most efficient, feasible joint design that meets given targets. Sequential Linear Programming (SLP) and the Modified Feasible Direction (MFD) method are used for optimization. Both Response Surface Polynomial (RSP) translator B and Neural Network (NN) translator B are developed and validated. Translator A is implemented in an MS-Excel program. Translator B is implemented in a MATHEMATICA program. The methodology for developing translator B is demonstrated on the B-pillar to rocker joint of an actual car. The convergence of the optimizer is checked by solving the optimization problem many times starting from different initial designs. The results from translator B are also checked against FEA results to ensure the feasibility of the optimum designs. By observing the optimum designs and by performing parametric studies for the effect of some important design variables on the joint mass we can establish guidelines for design of joints.
Design-Oriented Translators for Automotive Joints
Long, Luohui (Virginia Tech, 1998-09-13)
A hierarchical approach is typically followed in design of consumer products. First, a manufacturer sets performance targets for the whole system according to customer surveys and benchmarking of competitors' products. Then, designers cascade these targets to the subsystems or the components using a very simplified model of the overall system. Then, they try to design the components so that they meet these targets. It is important to have efficient tools that check if a set of performance targets for a component corresponds to a feasible design and determine the dimensions and mass of this design. This dissertation presents a methodology for developing two tools that link performance targets for a design to design variables that specify the geometry of the design. The first tool (called translator A) predicts the stiffness and mass of an automotive joint, whose geometry is specified, almost instantaneously. The second tool (called translator B) finds the most efficient, feasible design whose performance characteristics are close to given performance targets. The development of the two translators involves the following steps. First, an automotive joint is parameterized. A set of physical parameters are identified that can completely describe the geometry of the joint. These parameters should be easily understood by designers. Then, a parametric model is created using a CAD program, such as Pro/Engineer or I-Deas. The parametric model can account for different types of construction, and includes relations for styling, packaging, and manufacturing constraints. A database is created for each joint using the results from finite element analysis of hundreds or thousands of joint designs. The elements of the database serve as examples for developing Translator A. Response surface polynomials and neural networks are used to develop translator A. Stepwise regression is used in this study to rank the design variables in terms of importance and to obtain the best regression model. Translator B uses optimization to find the most efficient design. It analyzes a large number of designs efficiently using Translator A. The modified feasible direction method and sequential linear programming are used in developing translator B. The objective of translator B is to minimize the mass of the joint and the difference of the stiffness from a given target while satisfying styling, manufacturing and packaging constraints. The methodologies for Translators A and B are applied to the B-pillar to rocker and A-pillar to roof rail joints. Translator B is demonstrated by redesigning two joints of actual cars. Translator B is validated by checking the performance and mass of the optimum designs using finite element analysis. This study also compares neural networks and response surface polynomials. It shows that they are almost equally accurate when they are used in both analysis and design of joints.
Development and Applications of a Flat Triangular Element for Thin Laminated Shells
Mohan, P. (Virginia Tech, 1997-11-18)
Finite element analysis of laminated shells using a three-noded flat triangular shell element is presented. The flat shell element is obtained by combining the Discrete Kirchhoff Theory (DKT) plate bending element and a membrane element similar to the Allman element, but derived from the Linear Strain Triangular (LST) element. Though this combination has been employed in the literature for linear static analysis of laminated plates, the results presented are not adequate to ascertain that the element would perform well in the case of static and dynamic analysis of general shells. The element is first thoroughly tested for linear static analysis of laminated plates and shells and is extended for free vibration, thermal, and geometrically nonlinear analysis. The major drawback of the DKT plate bending element is that the transverse displacement is not explicitly defined within the interior of the element. Hence obtaining the consistent mass matrix or the derivatives of the transverse displacement that are required for forming the geometric stiffness matrix is not straight forward. This problem is alleviated by borrowing shape functions from other similar elements or using simple displacement fields. In the present research, free vibration analysis is performed both by using a lumped mass matrix and a so called consistent mass matrix, obtained by borrowing shape functions from an existing element, in order to compare the performance of the two methods. The geometrically nonlinear analysis is performed using an updated Lagrangian formulation employing Green strain and Second Piola-Kirchhoff (PK2) stress measures. A linear displacement field is used for the transverse displacement in order to compute the derivatives of the transverse displacement that are required to compute the geometric stiffness or the initial stress matrix. Several numerical examples are solved to demonstrate the accuracy of the formulation for both small and large rotation analysis of laminated plates and shells. The results are compared with those available in the existing literature and those obtained using the commercial finite element package ABAQUS and are found to be in good agreement. The element is employed for two main applications involving large flexible structures. The first application is the control of thermal deformations of a spherical mirror segment, which is a segment of a multi-segmented primary mirror used in a space telescope. The feasibility of controlling the surface distortions of the mirror segment due to arbitrary thermal fields, using discrete and distributed actuators, is studied. This kind of study was required for the design of a multi-segmented primary mirror of a next generation space telescope. The second application is the analysis of an inflatable structure, being considered by the US Army for housing vehicles and personnel. The tent structure is made up of membranes supported by arches stiffened by internal pressure. The updated Lagrangian formulation of the flat shell element has been developed primarily for the nonlinear analysis of the tent structure, since such a structure is expected to undergo large deformations and rotations under the action of environmental loads like the wind and snow loads. The wind load is modeled as a nonuniform pressure load and the snow load as lumped concentrated loads. Since the direction of the pressure load is assumed to be normal to the current configuration of the structure, it changes as the structure undergoes deformation. This is called the follower action. As a result, the pressure load is a function of the displacements and hence contributes to the tangent stiffness matrix in the case of geometrically nonlinear analysis. The thermal load also contributes to the system tangent stiffness matrix. In the case of the thermal load this contribution is similar to the initial stress matrix and hence no additional effort is required to compute this contribution. In the case of the pressure load, this contribution (called the pressure stiffness) is in general unsymmetric but can be systematically derived from the principle of virtual work. The follower effects of the pressure load have been included in the updated Lagrangian formulation of the flat shell element and have been validated using standard examples in the literature involving deformation-dependent pressure loads. The element can be used to obtain the nonlinear response of the tent structure under wind and snow loads.
Development and Applications of Finite Elements in Time Domain
Park, Sungho (Virginia Tech, 1996-12-04)
A bilinear formulation is used for developing the time finite element method (TFM) to obtain transient responses of both linear, nonlinear, damped and undamped systems. Also the formulation, used in the h-, p- and hp-versions, is extended and found to be readily amenable to multi-degree-of-freedom systems. The resulting linear and nonlinear algebraic equations for the transient response are differentiated to obtain the sensitivity of the response with respect to various design parameters. The present developments were tested on a series of linear and nonlinear examples and were found to yield, when compared with other methods, excellent results for both the transient response and its sensitivity to system parameters. Mostly, the results were obtained using the Legendre polynomials as basis functions, though, in some cases other orthogonal polynomials namely, Hermite, Chebyshev, and integrated Legendre polynomials were also employed (but to no great advantage). A key advantage of TFM, and the one often overlooked in its past applications, is the ease in which the sensitivity of the transient response with respect to various design parameters can be obtained. Since a considerable effort is spent in determining the sensitivity of the response with respect to system parameters in many algorithms for parametric identification, an identification procedure based on the TFM is developed and tested for a number of nonlinear single-and two-degree-of-freedom system problems. An advantage of the TFM is the easy calculation of the sensitivity of the transient response with respect to various design parameters, a key requirement for gradient-based parameter identification schemes. The method is simple, since one obtains the sensitivity of the response to system parameters by differentiating the algebraic equations, not original differential equations. These sensitivities are used in Levenberg-Marquardt iterative direct method to identify parameters for nonlinear single- and two-degree-of-freedom systems. The measured response was simulated by integrating the example nonlinear systems using the given values of the system parameters. To study the influence of the measurement noise on parameter identification, random noise is added to the simulated response. The accuracy and the efficiency of the present method is compared to a previously available approach that employs a multistep method to integrate nonlinear differential equations. It is seen, for the same accuracy, the present approach requires fewer data points. Finally, the TFM for optimal control problems based on Hamiltonian weak formulation is proposed by adopting the p- and hp-versions as a finite element discretization process. The p-version can be used to improve the accuracy of the solution by adding more unknowns to each element without refining the mesh. The usage of hierarchical type of shape functions can lead to a significant saving in computational effort for a given accuracy. A set of Legendre polynomials are chosen as higher order shape functions and applied to two simple minimization problems for optimal control. The proposed formulation provides very accurate results for these problems.
Efficient Methods for Structural Analysis of Built-Up Wings
Liu, Youhua (Virginia Tech, 2000-04-28)
The aerospace industry is increasingly coming to the conclusion that physics-based high-fidelity models need to be used as early as possible in the design of its products. At the preliminary design stage of wing structures, though highly desirable for its high accuracy, a detailed finite element analysis(FEA) is often not feasible due to the prohibitive preparation time for the FE model data and high computation cost caused by large degrees of freedom. In view of this situation, often equivalent beam models are used for the purpose of obtaining global solutions. However, for wings with low aspect ratio, the use of equivalent beam models is questionable, and using an equivalent plate model would be more promising. An efficient method, Equivalent Plate Analysis or simply EPA, using an equivalent plate model, is developed in the present work for studying the static and free-vibration problems of built-up wing structures composed of skins, spars, and ribs. The model includes the transverse shear effects by treating the built-up wing as a plate following the Reissner-Mindlin theory (FSDT). The Ritz method is used with the Legendre polynomials being employed as the trial functions. Formulations are such that there is no limitation on the wing thickness distribution. This method is evaluated, by comparing the results with those obtained using MSC/NASTRAN, for a set of examples including both static and dynamic problems. The Equivalent Plate Analysis (EPA) as explained above is also used as a basis for generating other efficient methods for the early design stage of wing structures, such that they can be incorporated with optimization tools into the process of searching for an optimal design. In the search for an optimal design, it is essential to assess the structural responses quickly at any design space point. For such purpose, the FEA or even the above EPA, which establishes the stiffness and mass matrices by integrating contributions spar by spar, rib by rib, are not efficient enough. One approach is to use the Artificial Neural Network (ANN), or simply called Neural Network (NN) as a means of simulating the structural responses of wings. Upon an investigation of applications of NN in structural engineering, methods of using NN for the present purpose are explored in two directions, i.e. the direct application and the indirect application. The direct method uses FEA or EPA generated results directly as the output. In the indirect method, the wing inner-structure is combined with the skins to form an "equivalent" material. The constitutive matrix, which relates the stress vector to the strain vector, and the density of the equivalent material are obtained by enforcing mass and stiffness matrix equities with regard to the EPA in a least-square sense. Neural networks for these material properties are trained in terms of the design variables of the wing structure. It is shown that this EPA with indirect application of Neural Networks, or simply called an Equivalent Skin Analysis (ESA) of the wing structure, is more efficient than the EPA and still fairly good results can be obtained. Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used in structural design practices for searching the optimal solutions near a baseline design. In the present work, the modal response of general trapezoidal wing structures is approximated using shape sensitivities up to the second order, and the use of second order sensitivities proved to be yielding much better results than the case where only first order sensitivities are used. Also different approaches of computing the derivatives are investigated. In a design space with a lot of design points, when sensitivities at each design point are obtained, it is shown that the global variation in the design space can be readily given based on these sensitivities.
Elastic and inelastic analysis of panel collapse by stiffener buckling
Ma, Ming (Virginia Tech, 1994-08-05)
A method is developed for analyzing the flexural-torsional and lateral-torsional buckling ("tripping") behavior of flanged stiffeners subjected to axial force, end moment, lateral pressure and any combination of these. The effects of cross-sectional distortion, postbuckling behavior of the plate (incorporated by considering the plate effective width), initial imperfections and plasticity are included. The method uses the Rayleigh-Ritz approach. Based on an assumed strain distribution, a displacement field is obtained for the tripping model, and the total potential energy functional is then derived. The strain distribution assumptions coincide with van der Neut's assumption. However, unlike the somewhat obscure differential equation approach given by van der Neut, this study provides a simple, clear, energy approach. Also the resulting method is applicable in the inelastic range, which is not possible with van der Neut's approach. Both the rigid web case and the flexible web case are studied. The effect of plate rotational restraint in the elastic range is accounted for. The method requires only four degrees of freedom and therefore the solution process is rapid. In order to verify the method in the elastic range, a number of sample stiffened panels are analyzed using the ABAQUS foote element program; the results are in quite good agreement. An inelastic tripping model is then developed based on the established elastic model, using deformation theory. Results obtained using the inelastic tripping method are shown to be in good agreement with experimental results, and to be more accurate than other methods.
Experimental comparison of probabilistic methods and fuzzy sets for designing under uncertainty
Maglaras, George K. (Virginia Tech, 1995-11-05)
Recently, probabilistic methods have been used extensively to model uncertainty in many design optimization problems. An alternative approach for modeling uncertainties is fuzzy sets. Fuzzy sets usually require much less information than probabilistic methods and they rely on expert opinion. In principle, probability theory should work better in problems involving only random uncertainties, if sufficient information is available to model these uncertainties accurately. However, because such information is rarely available, probabilistic models rely on a number of assumptions regarding the magnitude of the uncertainties and their distributions and correlations. Moreover, modeling errors can introduce uncertainty in the predicted reliability of the system. Because of these assumptions and inaccuracies it is not clear if a design obtained from probabilistic optimization will actually be more reliable than a design obtained using fuzzy set optimization. Therefore, it is important to compare probabilistic methods and fuzzy sets and determine the conditions under which each method provides more reliable designs. This research work aims to be a first step in that direction. The first objective is to understand how each approach maximizes reliability. The second objective is to experimentally compare designs obtained using each method. A cantilevered truss structure is used as a test case. The truss is equipped with passive viscoelastic tuned dampers for vibration control. The structure is optimized by selecting locations for tuning masses added to the truss. The design requirement is that the acceleration at given points on the truss for a specified excitation be less than some upper limit. The properties of the dampers are the primary sources of uncertainty. They are described by their probability density functions in the probabilistic analysis. In the fuzzy set analysis, they are represented as fuzzy numbers. Two pairs of alternate optimal designs are obtained from the probabilistic and the fuzzy set optimizations, respectively. The optimizations are performed using genetic algorithms. The probabilistic optimization minimizes the system probability of failure. Fuzzy set optimization minimizes the system possibility of failure. Problem parameters (e.g., upper limits on the acceleration) are selected in a way that the probabilities of failure of the alternate designs differ significantly, so that the difference can be measured with a relatively small number of experiments in the lab. The main difference in the way each method maximizes safety is the following. Probabilistic optimization tries to reduce more the probabilities of failure of the modes that are easier to control. On the other hand, fuzzy set optimization tries to equalize the possibilities of failure of all failure modes. These optimum probabilistic and fuzzy set designs are then compared in the laboratory. Twenty-nine realizations of each optimum design are tested and the failure rates are measured. The results confirm that, for the selected problems, probabilistic methods can provide designs that are significantly more reliable than designs obtained using fuzzy set methods.
Fatigue design of oil tankers: a design approach
Franklin, Paul (Virginia Tech, 1993-12-15)
The oil tankers that operate on the Trans-Alaska Pipeline Service (TAPS) route have exhibited a large number of structural fatigue cracks. These cracks can be attributed to the increase in use of high strength steel in tanker construction and to the harsh operating environment in the Gulf of Alaska. In response to the TAPS fatigue problem, this project examines the topic of preliminary design for fatigue resistance. The TAPS tankers have previously been the target of several studies on the subject of fatigue cracking. Most of these studies have concentrated on reducing the costs and risks involved with operating the current tanker fleet. Preliminary design, however, is oriented at reducing the fatigue risk in future tanker designs. To that end, the design method outlined within concentrates on the level of analysis that is appropriate for preliminary design. The design method consists of four steps: the specification of a wave environment, generation of a hydrodynamic model and subsequent wave loads, evaluation of cyclic stresses and an assessment of fatigue damage. A series of example calculations that is typical of preliminary design has been performed for one of the TAPS tanker classes. These calculations employed Buckley's climatic wave spectra, a 3-dimensional panel based hydrodynamics package by Lin and a Miner's rule fatigue assessment based on the S-N curves of the British Welding Institute. The example calculations yield two important results. First, relatively inexpensive methods can yie1d important and accurate fatigue results; for a side shell longitudinal at the water line the example calculations predict a fatigue life of approximately 3 operating years. This corresponds quite well to the published inspection data and obviously represents insufficient fatigue life. Second, local panel pressures can have a significant contribution to, and even dominate, total fatigue damage in the side shell. This contrasts with conventional fatigue studies of ship hulls which focus on global loads; i.e., hull girder bending.
Finite element formulation of a thin-walled beam with improved response to warping restraint
Ghose, Dhrubajyoti (Virginia Tech, 1991-03-15)
Linear elastic theory of torsion and flexure of thin-walled beams as developed by Vlasov and Timoshenko respectively are well known and commonly used in everyday engineering practice. However there are noticeable differences between calculations and experimental results. The difference is partly due to one of the basic assumption of classical theory, namely that the secondary shear strains due to warping are negligible. In the present work a new three noded, with CO continuity, isoparametric beam finite element is developed based on a torsion theory by Benscoter. In the classical theory warping is assumed to be proportional to the rate of twist, whereas in Benscoter's theory it is assumed to be proportional to an independent quantity called the "warping function". The exact form of this function can be evaluated from equilibrium equations. This assumption of Benscoter's allows the formulation of a Co element based on the assumed displacement method. The other advantage of Benscoter's theory is that it takes into account the effects of secondary shear strains. These effects are quite significant for closed sections. The element is validated for several cases of a cantilevered beam of rectangular cross section and in every case the results are in good agreement with the exact solution. It is also shown that the element gives a very good representation of curved beams, for which there is torsional-flexural coupling. A number of cases of a curved I-beam under various loading and boundary conditions are analysed, and in every case the results agree closely with the analytical solution. In order to represent the torsional response the element uses seven degrees of freedom per node. This seventh degree of freedom is the "warping function" mentioned earlier. To make the element compatible with standard finite-element programs which have six degrees of freedom per node, static condensation is used.
Flexural-Torsional Coupled Vibration of Rotating Beams Using Orthogonal Polynomials
Kim, Yong Y. (Virginia Tech, 2000-05-01)
Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. The present work starts from a review of the development and analysis of four basic types of beam theories: the Euler-Bernoulli, Rayleigh, Shear and Timoshenko and goes over to a study of flexural-torsional coupled vibration analysis using basic beam theories. In obtaining natural frequencies, orthogonal polynomials used in the Rayleigh-Ritz method are studied as an efficient way of getting results. The study is also performed for both non-rotating and rotating beams. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the eigenfunctions of a pinned-free uniform beam, and the special trigonometric functions used in conjunction with Hermite cubics. Studied cases are non-rotating and rotating Timoshenko beams, bending-torsion coupled beam with free-free boundary conditions, a cantilever beam, and a rotating cantilever beam. The obtained natural frequencies and mode shapes are compared to those available in various references and results for coupled flexural-torsional vibrations are compared to both previously available references and with those obtained using NASTRAN finite element package.
Free vibration and nonlinear transient analysis of imperfect laminated structures
Byun, Chansup (Virginia Tech, 1991-11-08)
The free vibration and nonlinear transient analysis of imperfect laminated structures with emphasis on computational methods for accurate and efficient analysis are studied. The evaluation of interlaminar stresses is also studied by approximating global displacements of laminated plates. Free vibration response of imperfect laminated structures is studied in the presence of geometric and stress imperfections. The stress imperfections are the initial stresses as caused by preloads. Using a 48 degrees of freedom thin shell element, the effect of complex, arbitrary in-plane and out-of-plane loads on the transverse vibrations of thin arbitrarily laminated plates, cylindrical panels, and hyperbolic shells without and with geometric imperfections is analyzed. The, effects of geometric parameters (aspect ratio and panel curvature) and material properties (varying the number of layers but keeping the same laminate thickness) of imperfect plates are examined. The nonlinear transient response of imperfect structures is next obtained using the direct time integration schemes as applied to the full set of equations and also using reduction methods. Two time integration schemes, the Newmark method and the Wilson () method, are first tested on a series of linear and nonlinear examples without and with geometric imperfections. Reduction methods using the normal modes and Ritz vectors as the base vectors are employed to reduce the size of the nonlinear problem and thus save computational resources. The resulting reduced (but still coupled) set of equations is integrated in a step-by-step fashion using the aforementioned time integration schemes along with an iterative scheme for dynamic equilibrium. Also, the nonlinear dynamic response of imperfect plates subjected to impact loads is studied. The evaluation of the loads (due to a projectile) depends on a contact law which relates contact forces with indentation. The well-known Hertzian law and its previously proposed modification are incorporated. The transient response of an example problem is obtained using both full and reduced equations of motion. Finally, for accurate determination of interlaminar shear and normal stresses of laminated structures, a postprocessor for displacement-based finite element solutions of laminated plates under transverse loads is developed. The postprocessor can be used for the finite element solutions that have been obtained using either the classical laminated plate theory or the first order shear deformation theory. The equilibrium equations of elasticity are integrated directly. These equations include the influence of the products of in-plane stresses for geometrically nonlinear problems. To obtain accurately the derivatives of in-plane stresses the finite element nodal displacement data is first interpolated using polynomials with global support (Le., the interpolating polynomials are defined over the whole domain). Two types of polynomials, Chebyshev and a class of orthogonal polynomials that can be generated for a given location of known data points are used.
Fuzzy logic and utility theory for multiobjective optimization of automotive joints
Guyot, Nicolas E. (Virginia Tech, 1996-02-05)
In the early design stage of automotive joints, fuzziness is omnipresent because designers reason in non quantitative terms and deal with imprecise data. Consequently, they need a design methodology that accounts for vagueness. Fuzzy sets and utility theory are appropriate tools because they link the vagueness in a problem formulation and the precise nature of mathematical models. Fuzzy multiobjective optimizations are performed on an automotive joint to maximize the overall designer's satisfaction. Several methods that account for all the attributes and the fuzziness in the goals are used. Three multiobjective fuzzy approaches, namely, the conservative, the aggressive and the moderate methods are investigated. Utility theory is also considered to optimize the joint. One of the performance attributes of the joint, the stiffness, is evaluated rapidly using approximate tools (neural networks and response surface polynomials) to overcome the high computational cost of PEA, which is traditionally used to calculate the stiffness. This research compares fuzzy set methods and utility theory in design of automotive components. These methods are applied on two examples where the same B-pillar to rocker joint of an actual car is optimized. Fuzzy set based methods and utility theory appear to be suitable for optimizing automotive joints because they allow for trading conflicting objectives. Fuzzy set based methods avoid trading objectives to the point of having a level of satisfaction equal to zero. When using the fuzzy set based methods investigated in this research, the trade-offs among the attributes are not explicitly defined by the user. Utility theory requires the user to quantify precisely the trade-offs among the attributes. When using utility theory, the overall satisfaction of a design can be non zero even if one or more attributes has a level of satisfaction equal to zero. The approximate tools enable us to perform the optimization efficiently by reducing considerably the computational cost.
Hydrodynamic analysis and computer simulation applied to ship interaction during maneuvering in shallow channels
Kizakkevariath, Sankaranarayanan (Virginia Polytechnic Institute and State University, 1989)
A generalized hydrodynamic interaction force model is combined with a ship maneuvering simulator to provide a free-running, closed loop ship simulation capable of trajectory predictions of ships operating in close proximity in a shallow, asymmetric canal. The interaction force model is based on the generalized Lagally's theorem, properly accounting for the orientations and dynamic motions of the ships. Also included are the lift forces and the cross-flow drag forces, which are found to be important for bank suction phenomena. A simplified method is implemented for box shapes, applicable for barge-tows operating in rivers. Results of the calculations are found to be generally in good agreement with experimental and other theoretical results. This work would have utility in studying maneuvers involving ships and barges in close proximity and can be used in training pilots who operate in canals, harbors and rivers, and also in studying the effects of various steering control systems in the early design stages.
Hypersonic test facilities: requirements analysis and preliminary design
Drauch, Gregory Andrew (Virginia Tech, 1990-04-15)
There has come about, in recent years, a renewed interest in aerospace vehicles operating in the hypersonic regime. With this interest has come a need to not only reestablish the hypersonic test capability that was available in the 1960s but to enhance this capability to meet the demanding needs of today's proposed vehicles. This will require more capable hypersonic wind tunnels with larger test sections, longer run times, and test gases more closely resembling the fluid to be encountered by the vehicle being developed. This document will review the current hypersonic testing capability, examine the operating characteristics of several hypersonic vehicles to develop a set of hypersonic testing requirements, and develop a preliminary design of a required hypersonic facility that addresses the demonstrated requirements. An order of magnitude cost estimate is also presented.
Impact response of composite structures to rigid and flexible projectiles
Stoumbos, Tom James G. (Virginia Tech, 1995)
Numerical schemes are developed to study the impact response of composite structures to rigid (spherical masses) and flexible (uniform and nonuniform bars) projectiles. In the first phase of this study the impact response of imperfect ]aminated cylindrical panels to rigid projectiles is investigated. A 48 degree-of-freedom (DOF) shell finite element based on the classical laminated plate theory, which is capable of modeling geometric imperfections is used to model the shell. Linear and geometrically nonlinear transient responses are obtained using reduction methods based on the use of (i) natural modes and (ii) the Ritz vectors (also called Lanczos’ vectors) as the basis functions. The results obtained from these schemes are compared with those obtained using direct integration schemes, the Newmark-β and the Wilson-σ methods. The effect of number of reduced basis on the response is also studied. The impact loads are obtained using a modified Hertzian contact law by Tan and Sun. Effects of geometric imperfections and shell radius of curvature on the response are also studied. The present results are compared with those obtained experimentally at the NASA Langley Research Center. With one notable exception, a good agreement between the theoretical predictions and experimental results is observed. In the second phase, numerical schemes are developed to incorporate the effect of the projectile flexibility on the impact response of structures. A step by step approach, in which the impact responses of increasingly complex structures, namely, the axial bars, beams, and shear deformable plates subjected to flexible projectiles (uniform and nonuniform bars) are obtained, is used. The target axial bar is-:modeled using two degree-of-freedom axial bar elements. For the projectile, two different finite element models using, an axial bar element and a six-degree-of-freedom axisymmetric solid element with a triangular cross-section, are employed. The axisymmetric element (from the general purpose code MSC/NASTRAN) is used for those cases in which the target axial bar area is smaller than the projectile area and a two dimensional modelling of the projectile is needed. The impact response is obtained using an explicit algorithm based on the central difference scheme. In the algorithm developed, the target is assumed to be at rest and the projectile is assumed to be moving at a constant velocity, the impact velocity. At time t=0, the projectile hits the bar. At each time step, and as long as the two bars, are in contact, we assume that the two impacting bodies have the same velocity. For each time step, an iterative procedure is incorporated to predict the force that will enforce the velocity condition described previously. The results obtained from this approach are compared with other analytical and experimental results available in the literature for the impact response of a Hopkinson's bar. A good agreement is achieved. The algorithm developed here is next applied to study the impact response of beams and generally laminated, skew trapezoidal plates subjected to low velocity impact of a non-uniform linearly elastic composite projectile. The beam is modelled using two different approaches: a four degree-of-freedom beam element and an eight degree-of-freedom plane stress element. For the case of laminated plates, a Ritz method based approach developed by Kapania and Lovejoy is used. The present approach can be easily extended to study the nonlinear impact response of geometrically imperfect plates and shells.

Browsing by Author "Nikolaidis, Efstratios"

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