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Thesis (M.S.)--University of New Orleans, 2004. / Title from electronic submission form. "A thesis ... in partial fulfillment of the requirements for the degree of Masters of Science in Hospitality and Tourism Management"--Thesis t.p. Vita. Includes bibliographical references.

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A geotechnical composite foundation system that has become increasingly popular over the years is a column-supported, geosynthetic-reinforced embankment. This system consists of strong columns or piles placed in soft clay, a bridging layer of sand or sand and gravel, and one or more layers of geosynthetic reinforcement. It is often used in soft ground situations where there is a need for faster construction and/or where there are adjacent structures that would be affected by settlement caused by the new embankment. The geosynthetic reinforcement is placed in the bridging layer to help transfer the load to the columns and decrease the total and differential settlements. Current methods of analysis for this material are extremely simplified, and do not thoroughly model the behavior of the system. Therefore, a more comprehensive analysis needs to be conducted that will better predict the true effect of the geosynthetic layer or layers. In this thesis, one geosynthetic layer was considered. Models were developed using two different computer programs: Mathematica and ABAQUS. In Mathematica, the Rayleigh-Ritz method was used to approximate the deflections and tensile forces in the membrane. This method considered the geosynthetic reinforcement as a plate and minimized the total energy of the system. In ABAQUS, a finite element modeling program, the membrane was analyzed as a shell, and results were compared with some results from Mathematica. A parametric study was completed in Mathematica to determine the effects of different parameters. The parameters varied involved the geogrid properties (Poisson's ratio, modulus of elasticity, and thickness), the vertical load, the soil stiffness above the piles, the soil stiffness between the piles, the size of the piles, and the distance between the piles. / Master of Science

Στα πλαίσια της παρούσας διατριβής προσεγγίστηκε το πρόβλημα της κρούσης των σύνθετων υλικών μέσω μιας συνολικής μεθοδολογίας. Ο όρος συνολική μπορεί να χαρακτηρίσει την παρούσα εργασία της διατριβής γιατί η αναλυτική και αριθμητική υλοποίηση του προβλήματος που παρουσιάσθηκε επιλύει πολλά παράπλευρα προβλήματα. Η επίλυση του προβλήματος της κρούσης χαμηλής ταχύτητας σε σύνθετα υλικά κινήθηκε σε τέσσερις φάσεις. Σε πρώτη φάση αναπτύχθηκε το κύριο εργαλείο με το οποίο εξελίσσεται η διαδικασία μοντελοποίησης του προβλήματος. Η κατάστρωση του μοντέλου επίλυσης φυσικών συχνοτήτων και η αριθμητική επίλυση αυτού δίνει εκτιμήσεις για την ακρίβεια της μεθόδου Ritz με τη χρήση των αποδεκτών συναρτήσεων p-Ritz. Η μέθοδος Ritz επιτρέπει την πλήρη μοντελοποίηση του πολύστρωτου λαμβάνοντας υπόψη και τους καμπτικούς όρους σύζευξης κάτι που μέχρι στιγμής δεν έχει παρουσιαστεί στη βιβλιογραφία. Κατά τη δεύτερη φάση και μετά την τεκμηρίωση της πρώτης φάσης και τον έλεγχο της ακρίβειας των υπολογισμών των φυσικών συχνοτήτων, διατυπώθηκε το δυναμικό πρόβλημα. Η ακρίβεια των υπολογισμών ήταν εξαιρετική και σε ικανοποιητική σύγκλιση με τα αποτελέσματα του κώδικα πεπερασμένων στοιχείων LSDYNA3D. Παράλληλα αποδείχτηκε ότι το δυναμικό μοντέλο, επιλύει το δυναμικό φαινόμενο για όλα τα σχήματα και τους συνδυασμούς συνοριακών συνθηκών. Σε εξέλιξη των ανωτέρω έγινε η διατύπωση του συζευγμένου προβλήματος της κρούσης και επιλύθηκε για διάφορα υλικά και συνοριακές συνθήκες που αποτελούν μοντέλα υποδείγματα στη διεθνή βιβλιογραφία. Η ικανότητα πρόβλεψης των μοντέλων χαρακτηρίζεται ιδιαίτερα ικανοποιητική ενώ παρουσιάστηκε ο τρόπος εισαγωγής μη γραμμικών νόμων επαφής, στο συζευγμένο σύστημα διαφορικών εξισώσεων. Με σκοπό τον πλήρη χαρακτηρισμό του φαινομένου αλλά και τη μείωση των παραμέτρων που χαρακτηρίζουν το πρόβλημα σε μια προσπάθεια χαρακτηρισμού των κύριων παραμέτρων που διέπουν την κρούση παρουσιάζεται ο τρόπος υπολογισμού των αδιάστατων ομάδων που περιγράφουν την κρούση. Στα πλαίσια αυτής της ανάλυσης υπολογίστηκε το Νομογράφημα Κρουστικής Απόκρισης. Το Νομογράφημα Κρουστικής Απόκρισης μπορεί να χρησιμοποιηθεί στην οργάνωση πειραμάτων χαρακτηρισμού της κρουστικής αντοχής πολύστρωτων διατάξεων. Για την επίλυση του κύριου προβλήματος προτάθηκε μια διαδικασία ιεραρχικής αντιμετώπισης του προβλήματος. Για να καθοριστεί η κρουστική απόκριση του πολύστρωτου η διάταξη υποβάλλεται σε δυναμικό και στατικό χαρακτηρισμό με τη χρήση αριθμητικών μοντέλων που βασίζονται στη μέθοδο Ritz. Υλοποιήθηκε μια μέθοδος που δύναται να επιλύει διάφορες γεωμετρίες σε ποικιλία συνοριακών συνθηκών. Στη βιβλιογραφία μέχρι στιγμής έχουν παρουσιαστεί μοντέλα για παραλληλόγραμμες πλάκες, ειδικώς ορθότροπες και σε συνοριακές συνθήκες απλής έδρασης. Υλοποιήθηκαν προγράμματα σε Fortran που επιλύουν στατικά και δυναμικά προβλήματα σε καμπτικά φορτία και για ποικίλους συνδυασμούς συνοριακών συνθηκών. Υλοποιήθηκε το συζευγμένο πρόβλημα της κρούσης πολύστρωτης διάταξης από σφαίρα (διεισδυτή) σε χαμηλές ταχύτητες Έγινε σύγκριση ελαστοπλαστικών μη γραμμικών μοντέλων επαφής με γραμμικά μοντέλα επαφής. Με τη χρήση της θεωρίας αδιαστατοποίησης καθορίστηκαν οι αδιάστατες ομάδες που διέπουν το πρόβλημα της κρούσης και δημιουργήθηκε το Νομογράφημα Κρουστικής Απόκρισης. Αποδείχτηκε ότι το Νομογράφημα Κρουστικής Απόκρισης ισχύει για οιαδήποτε συμμετρική πολύστρωτη διάταξη σε όποια συνοριακή συνθήκη και αν είναι και οιοδήποτε σχήμα πλάκας. Αποδείχτηκε ότι το Νομογράφημα Κρουστικής Απόκρισης μπορεί να χρησιμοποιηθεί ως χάρτης αντοχής και ανοχής σε κρούση με τη χρήση της αδιάστατου Κατωφλιού Δύναμης για Διαστρωματική Αποκόλληση. / Low-velocity impact in composite laminates is a destructive loading condition since it leads to significant internal damage, no detectable by visual inspection. Numerous researchers have made significant efforts to model the impact response of composite plates and facilitate the analysis and design for impact resistance, using simple linear spring-mass models or a combination of continuum mechanics models and contact laws. Many investigators have used simple engineering structures beams, plates and shells to demonstrate the impact response of composite structures. The basic concept for modelling the impact of an object to a target was proposed by Timoshenko (1913). He introduced the procedure where the description of the interaction between the impactor and the structure (Euler beam) was implemented using the Hertzian contact law. This approach was further extended in isotropic plates and shells. Lee, 1940; Greszcuk, 1982; Lee et al., 1983 and Shivakumar et al, 1985, proposed simple models of the low-velocity impact problem. Although many important contributions exist in the bibliography for the characterization of the impact response of laminated plates, analytical solutions are considered to be very few. Most of the models proposed are for analyzing specially orthotropic plates subjected to a local dominated impact. In addition, these models do not take into account the shear coupling terms of the bending stiffness matrix, e.g. D16 and D26. Sun and Chattopadhyay (1975), Dobyns (1981), and Ramkumar and Chen (1983) employed the first order shear deformation theory developed by Whitney and Pagano (1970), and used it in conjunction with the Hertzian contact law or plasticity contact law for characterizing the impact of laminated composite plates. In their analysis they have studied the impact response of a simply supported orthotropic plate subjected to central impact using the lamination theory that includes transverse shear deformations. It is evident that the contact force history must be computed as part of the solution of the dynamic response problem solving the nonlinear integral equation. Christoforou and Swanson (1991) and Carvalho (1996), obtained an analytical solution of the impact problem using the Laplace transform technique. Qian and Swanson (1990) obtained analytical solutions by linearising the contact deformation law and compared this with a Rayleigh-Ritz approach with numerical integration in time. The three dimensional finite element method with an explicit formulation where also used in the dynamic analysis of laminated plates subjected to low velocity impact. Moreover, many researchers have proposed simple models for characterizing the impact phenomenon. Shivakumar et.al. (1985) developed a simple model to predict the impact force and the duration of the impact phenomenon on composite plates. The composite plates were modeled as three springs while their stiffness was calculated by the plate properties and the contact parameters. Other researchers instead of a classical Hertzian law used the statical indentation law presented by Yang and Sun (1982) and Tan and Sun (1985). The non Hertzian Contact impact was considered for combining the overall deformation of the structure to the local deformation in the contact area and was used to predict the impact response of transversely isotropic beams and plates. In general, it is impossible to obtain the exact solution for an impact problem except by ignoring the local contact deformation or pre-assuming the contact force. The first step for understanding the problem is to predict the force applied by the projectile on the structure during impact. In order to characterize the contact force history, the model should account for the motion of the target, the effective stiffness of the projectile The present work deals with the development of a numerical scheme for the calculation of the dynamic response of any type of laminated composite plates subjected to any type (and/or combination) of boundary conditions under low-velocity impact. Using the p-version Ritz polynomials any shape that can be represented in Cartesian coordinates can be formulated. The governing second order differential equations are derived and allow furnishing solutions with a variety of boundary conditions along the edges of the plate shape represented in Cartesian coordinates. Statically determined non-linear contact laws, for loading and unloading conditions, are coupled with the partial differential equations governing the dynamic response of the composite plate. These nonlinear governing equations of the contact-impact problem are decoupled according to the second order terms by the method of principal transformation and then solved numerically. The dynamic response of fully clamped cross-ply and a variation of simply support and fixed boundary conditions for angle ply and cross ply composite laminates was investigated. Rectangular, circular and elliptical plates where analysed using the developed numetrical scheme. A total analysis method is used in order to check the efficiency of the results. Starting from the homogeneous formulation of the structure the eigenfrequencies and eigenshapes were calculated for all material and boundary conditions cases. Using the “transformation of the principal” method all cases were subjected in force and pressure pulses in order to check the efficiency of the dynamic solver developed. The efficiency of the present method was investigated comparing the respective results with well-known benchmark problems, experimental results and FEM analysis using MSC-NASTRAN and LSDYNA 3D codes. Following this exhaustive analysis, the coupled contact-impact problem was formulated and numerous contact schemes were applied to the problem. For the impact problem the type of the response is intimately related to the contact stiffness, impact energy, ply thickness and orientation, geometrical configuration and boundary conditions, as well as the orthotropic material properties and the longitudinal, transverse and interlaminar strength both in compression and in tension. Three types of dynamic responses were found to exist, local, global and transitional. All types are categorised and analysed. Using Buckingham Theorem the numerous parameters of the dynamic contact-impact problem were grouped to non-dimensional groups in order to provide a flexible analysis scheme for theoretical and experimental evaluation of the low velocity impact problem. A general normalised graph for characterising the impact response of structures was constructed. The numerous advantages for analysing impact phenomena using the normalised curve developed were presented. Moreover its use for predicting composite material damage area on low velocity impact conditions was presented.

The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for B-spline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions.

A displacement-based Rayleigh-Ritz finite element model is developed to simulate the static and dynamic behavior of stiffened plates. By con1paring natural frequency, time-history, and power density predictions with experimental results, it is shown that the model can be used to predict the vibratory behavior of wood floor systems constructed With either solid-sawn joists, I-Joists, or parallel-chard-trusses. Furthermore. using the model. it is shown that appropriate structural modifications can be used to improve the performance of wood floor systems by increasing natural frequency and reducing peak time-history velocity. Using the techniques described. a design example is included that indicates ho,v floor acceptability can be achieved. / Ph. D.

L’emploi de matériaux composites à matrice organique (CMO) pour la réalisation de structures aéronautiques « tièdes », peut exposer ces matériaux à l’action d’environnements agressifs, qui peuvent entrainer des phénomènes de vieillissement et de dégradation sévères associés à la diffusion d’espèces au sein du réseau macromoléculaire des matrices polymères. La complexité de la microstructure des CMO utilisés pour ces applications peut complexifier la compréhension de phénomènes de dégradation. Le vieillissement humide des CMO préoccupe particulièrement les industriels du secteur aéronautique ; la diffusion de l’eau dans la matrice polymère du composite peut entrainer des phénomènes de gonflement hygroscopique, des modifications des propriétés mécaniques. Des méthodes expérimentales existent pour la caractérisation de ces phénomènes et pour l’identification des paramètres associés : ces méthodes préconisent souvent l’emploi d’éprouvettes saturées en humidité,ce qui nécessite de longs temps de conditionnement et un nombre élevé d’échantillons. Cette thèse a pour but d’établir des protocoles d’identification rapide des propriétés diffuso-mécaniques de matériaux polymères et CMO pour applications aéronautiques. La démarche mise en place dans cette thèse s’organise autour de quatre chapitres. Le premier chapitre présente une étude bibliographique sur les outils de modélisation des couplages diffuso-mécaniques et sur les méthodes de caractérisation des propriétés diffuso-mécaniques. L’étude bibliographique permet de préciser le cadre de travail, qui prévoit l’emploi d’un modèle diffuso-mécanique faiblement couplé où la diffusion d’eau suit la loi de Fick et le comportement mécanique est hygroélastique linéaire, dépendant de la concentration en eau. Le second chapitre présente la mise en place et le développement d’une méthode d’identification rapide des propriétés de diffusion anisotrope, adaptée à des CMO à architecture complexe.La méthode s’appuie sur des mesures de prise de masse d’échantillons de CMO, l’anisotropie de diffusion de ces matériaux est obtenue par rotation des axes principaux d’orthotropie. La méthode proposée représente une extension de la « méthode de la pente » introduite par Shen et Springer pour l’identification des propriétés de diffusion de matériaux orthotropes, et basée sur l’exploitation des courbes gravimétriques aux temps courts. A travers cette méthode, les coefficients principaux et les axes principaux d’orthotropie peuvent être identifiés. Une discussion sur les conditions d’équivalence de la diffusion 3D à la diffusion 1D en fonction de l’épaisseur de l’échantillon est également présentée à la fin de ce chapitre. Le troisième chapitre explore à travers une étude numérique la possibilité d’identifier rapidement les propriétés mécaniques affectées par la concentration en eau de matériaux polymères à travers des essais mécaniques sur plaques minces avec des gradients de concentration en eau. Des essais de traction et de flexion sont considérés. Pour des matériaux isotropes, dans un cadre hygroélastique, il est montré que cette méthode permet l’identification du module d’Young et du coefficient de Poisson dépendant de la concentration en eau avec un gain remarquable de temps d’essai par rapport à des essais sur des échantillons saturés en humidité. Enfin, le dernier chapitre propose à travers une étude numérique une méthode d’identification rapide des propriétés diffuso-mécaniques de matériaux isotropes basée sur l’emploi de plaques sollicitées par un champ asymétrique de concentration en eau. L’identification est ainsi effectuée à partir du suivi des déflexions engendrées par les champs de concentration. Le coefficient de dilatation hygroscopique et le module d’Young dépendant de la concentration en eau peuvent être identifiés durant l’essai, à l’état transitoire du conditionnement, avec un gain remarquable du temps d’essai par rapport à des essais sur échantillons saturés en humidité. / The use of organic matrix composite materials (OMC) for the realization of "warm" aeronautical structures, may expose these materials to aggressive environments: wet or gaseous environments,high temperatures, which may promote severe aging and degradation phenomena related to species diffusion within the macromolecular network of the polymer matrices. The complexity of the OMC microstructure used for these applications can complicate the understanding of degradation phenomena : for example, species diffusion can be isotropic, orthotropic or anisotropic, depending on the texture of the fibrous reinforcement. Humid aging of OMC is of particular concern for the aeronautical industry ; the diffusion of water in the polymer matrix of the composite may promote hygroscopic swelling, changes in mechanical properties (stiffness, strength). Experimental methods exist for the characterization of these phenomena and for the identification of the associated parameters : these methods often recommend the use of moisture saturated specimens, which require long conditioning times, sometimes often a relevant number of samples (1 sample for each saturated state), high costs. The aim of this work is to establish protocols for fast identification of the diffusomechanical properties of polymers and polymer based OMC materials for aeronautical applications. The approach implemented in this thesis is organized in four chapters. The first chapter presents a bibliographic study on coupled diffuso-mechanics modeling tools and on methods of characterization/identification of diffuso-mechanical properties, more particularly for OMC for aeronautical applications. The bibliographic study allows specifying the framework of the present research, which foresees the employment of a weakly coupled diffuso-mechanical model, where water diffusion follows the Fick’s law and the mechanical behavior is linear hygroelastic, depending on water content. The second chap-ter presents the setting up and the development of a method for fast identification of anisotropic diffusion properties, suitable for OMC with complex architecture, such as for instance, 2D or 3Dwoven OMC. The method relies on mass-gain measures of OMC samples, the diffusion anisotropy ofthese materials is obtained by rotating the axes of orthotropy. The proposed method represents an extension of the "slope method" introduced by Shen and Springer in the 1970s for the identification of the diffusion properties of orthotropic materials (such as laminated composites), and is based onthe exploitation of gravimetric curves at short times. Through this method, the principal coefficients and the principal axes of orthotropy can be identified. A discussion about the transition from 3Dto 1D diffusion as a function of the sample geometry is also presented at the end of this chapter. The third chapter explores through a numerical study the possibility of identifying in a fast way the mechanical properties affected by moisture of polymeric materials by the use of mechanical tests on thin plates with water concentration gradients. Traction and bending tests are taken into account.For isotropic materials, in a hygroelastic setting, it is showed that this method allows identifyng the water concentration dependent Young’s modulus and the Poisson’s ratio with a remarkable time gain compared to tests on moisture saturated samples. Finally, the last chapter proposes through a numerical study a method for fast identification of the diffuso-mechanical properties of isotropic materials based on the use of plates loaded by an asymmetric water concentration field. The identification is thus carried out from the monitoring deflections generated by the concentration fields. The moisturedependent hygroscopic expansion coefficient and Young’s modulus can be identified during the test,by exploiting the transient state of conditioning, with a remarkable time gain compared with moisture saturated samples.

Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov decompositions; matrix relations which, under certain conditions, are associated with these spaces. We investigate the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. We derive standard and harmonic Ritz extraction strategies for approximating the eigenpairs of a matrix and for approximating the action of a matrix function onto a vector. While these topics have been considered previously, our approach does not require the last pole to be infinite, which makes the extraction procedure computationally more efficient. Typically, the computationally most expensive component of the rational Arnoldi algorithm for computing a rational Krylov basis is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become poorly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this non orthogonal basis. As a consequence we obtain a more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using our high performance C++ implementation. We develop an iterative algorithm for solving nonlinear rational least squares problems. The difficulty is in finding the poles of a rational function. For this purpose, at each iteration a rational Krylov decomposition is constructed and a modified linear problem is solved in order to relocate the poles to new ones. Our numerical results indicate that the algorithm, called RKFIT, is well suited for model order reduction of linear time invariant dynamical systems and for optimisation problems related to exponential integration. Furthermore, we derive a strategy for the degree reduction of the approximant obtained by RKFIT. The rational function obtained by RKFIT is represented with the aid of a scalar rational Krylov decomposition and an additional coefficient vector. A function represented in this form is called an RKFUN. We develop efficient methods for the evaluation, pole and root finding, and for performing basic arithmetic operations with RKFUNs. Lastly, we discuss RKToolbox, a rational Krylov toolbox for MATLAB, which implements all our algorithms and is freely available from http://rktoolbox.org. RKToolbox also features an extensive guide and a growing number of examples. In particular, most of our numerical experiments are easily reproducible by downloading the toolbox and running the corresponding example files in MATLAB.

Proposed designs for Very Large Floating Structures motivate us to understand water-wave propagation through arrays of hundreds, or possibly thousands, of floating structures. The water-wave problems we study are each formulated under the usual conditions of linear wave theory. We study the frequency-domain problem of water-wave propagation through a periodically arranged array of structures, which are solved using a variety of methods. In the first instance we solve the problem for a periodically arranged infinite array using the method of matched asymptotic expansions for both shallow and deep water; the structures are assumed to be small relative to the wavelength and the array periodicity, and may be fixed or float freely. We then solve the same infinite array problem using a numerical approach, namely the Rayleigh-Ritz method, for fixed cylinders in water of finite depth and deep water. No limiting assumptions on the size of the structures relative to other length scales need to be made using this method. Whilst we aren t afforded the luxury of explicit approximations to the solutions, we are able to compute diagrams that can be used to aid an investigation into negative refraction. Finally we solve the water-wave problem for a so-called strip array (that is, an array that extends to infinity in one horizontal direction, but is finite in the other), which allows us to consider the transmission and reflection properties of a water-wave incident on the structures. The problem is solved using the method of multiple scales, under the assumption that the evolution of waves in a horizontal direction occurs on a slower scale than the other time scales that are present, and the method of matched asymptotic expansions using the same assumptions as for the infinite array case.