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 Title
 Revisiting BresseTimoshenko theory for beams.
 Creator
 Hache, Florian, Elishakoff, Isaac, Challamel, Noël, Graduate College
 Abstract/Description

In this study, a variational derivation of the simpler and more consistent version of BresseTimoshenko beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of the different energy contributions is established, including the shear effect and the...
Show moreIn this study, a variational derivation of the simpler and more consistent version of BresseTimoshenko beams equations, taking into account both shear deformation and rotary inertia in vibrating beams, is presented. Whereas Timoshenko gets his beam equations in terms of the equilibrium, the governing equations and the boundary conditions are here derived using the Hamilton’s principle. First, a list of the different energy contributions is established, including the shear effect and the rotary inertia. Second, the Hamilton’s principle is applied demanding the stationary of an appropriate functional, leading to two different equations of motion. The resolution of these equations provides the governing differential equation. It turns out that an additional term appears. The derived equations are intended for dynamic stability applications. Specifically, the parametric vibrations will be studied when the axial force varies periodically. This problem has important aerospace applications.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00005880
 Format
 Document (PDF)
 Title
 Dynamic response and stability of viscoelastic structures by interval mathematics.
 Creator
 Duan, Dehe., Florida Atlantic University, Elishakoff, Isaac
 Abstract/Description

It is demonstrated in this thesis that the interval mathematics is a powerful tool to deal with uncertain phenomena especially when the uncertainty in bounded. In this thesis, we apply interval mathematics to several engineering problems, apparently for the first time in the world literature. The following topics are included: (1) The application of interval mathematics in several applied mechanics problems. A brief review of basis concepts is given, and some problems are presented to...
Show moreIt is demonstrated in this thesis that the interval mathematics is a powerful tool to deal with uncertain phenomena especially when the uncertainty in bounded. In this thesis, we apply interval mathematics to several engineering problems, apparently for the first time in the world literature. The following topics are included: (1) The application of interval mathematics in several applied mechanics problems. A brief review of basis concepts is given, and some problems are presented to illustrate the application of interval mathematics. (2) The stability and dynamic response of viscoelastic plate are studied. The effect of viscoelastic parameters on critical velocity is elucidated. (3) The application of QiuChenElishakoff theorem in uncertain string and beam problems is investigated.
Show less  Date Issued
 1994
 PURL
 http://purl.flvc.org/fcla/dt/15090
 Subject Headings
 Interval analysis (Mathematics), Viscoelasticity, Fractional calculus
 Format
 Document (PDF)
 Title
 Deterministic, stochastic and convex analyses of one and twodimensional periodic structures.
 Creator
 Zhu, Liping., Florida Atlantic University, Lin, Y. K., Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The periodic structures considered in the dissertation are onedimensional periodic multispan beams, and twodimensional periodic grillages with elastic interior supports. The following specific topics are included: (1) Deterministic VibrationExact solutions are obtained for free vibrations of both multispan beams and grillages, by utilizing the wave propagation concept. The wave motions at the periodic supports/nodes are investigated and the dispersion equations are derived from which...
Show moreThe periodic structures considered in the dissertation are onedimensional periodic multispan beams, and twodimensional periodic grillages with elastic interior supports. The following specific topics are included: (1) Deterministic VibrationExact solutions are obtained for free vibrations of both multispan beams and grillages, by utilizing the wave propagation concept. The wave motions at the periodic supports/nodes are investigated and the dispersion equations are derived from which the natural frequencies of the periodic structures are determined. The emphasis is placed on the calculation of mode shapes of both types of periodic structures. The general expressions for mode shapes with various boundary conditions are obtained. These mode shapes are used to evaluate the exact dynamic response to a convected harmonic loading. (2) Stochastic VibrationA multispan beam under stochastic acoustic loading is considered. The exact analytical expressions for the spectral densities are derived for both displacement and bending moment by using the normal mode approach. Nonlinear vibration of a multispan beam with axial restraint and initial imperfection are also investigated. In the latter case, the external excitation is idealized as a Gaussian white nose. An expression for the joint probability density function in the generalized coordinates is obtained and used to evaluate the mean square response of a multispan beam system. (3) Convex Modeling of Uncertain Excitation FieldIt is assumed that the parameters of the stochastic excitation field are uncertain and belong to a multidimensional convex set. A new approach is developed to determine the multidimensional ellipsoidal convex set with a minimum volume. The most and least favorable responses of a multispan beam are then determined for such a convex set, corresponding to a stochastic acoustic field. The procedure is illustrated in several examples.
Show less  Date Issued
 1994
 PURL
 http://purl.flvc.org/fcla/dt/12366
 Subject Headings
 Grillages (Structural engineering), GirdersVibration, Wavemotion, Theory of, Vibration
 Format
 Document (PDF)
 Title
 Buckling of composite cylindrical shells with geometric, thickness and material imperfections.
 Creator
 Li, Yiwei., Florida Atlantic University, Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

This dissertation deals with the determination of buckling loads of composite cylindrical shell structures which involve uncertainty either in geometry, namely thickness variation, or in material properties. Systematic research has been carried out, which evolves from the simple isotropic cases to anisotropic cases. Since the initial geometric imperfection has a dominant role in the reduction of those imperfectionsensitive structures such as cylindrical shells, the combined effect of...
Show moreThis dissertation deals with the determination of buckling loads of composite cylindrical shell structures which involve uncertainty either in geometry, namely thickness variation, or in material properties. Systematic research has been carried out, which evolves from the simple isotropic cases to anisotropic cases. Since the initial geometric imperfection has a dominant role in the reduction of those imperfectionsensitive structures such as cylindrical shells, the combined effect of thickness variation and initial imperfection is also investigated in depth. Both analytic and numerical methods are used to derive the solutions to the problems and asymptotic formulas relating the buckling load to the geometric (thickness variation and/or initial imperfection) parameter are established. It is shown that the axisymmetric thickness variation has the most detrimental effect on the buckling load when the modal number of thickness variation is twice as much as that of the classical buckling mode. For the composite shells with uncertainty in material properties, the convex modelling is employed to evaluate the variability of buckling load. Based on the experimental data for the elastic moduli of the composite laminates, the upper and lower bounds of the buckling load are derived, which are numerically substantiated by the results from nonlinear programming. These bounds will be useful in practice and can provide engineers with a better view of the real loadcarrying capacity of the composite structure. Finally, the elastic modulus is modeled as a function of coordinates to complete the study on variability of material property so that the result can be obtained to account for the situation where the elastic modulus is different from one place to another in the structure.
Show less  Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/12444
 Subject Headings
 Composite materials, Buckling (Mechanics), Shells (Engineering), Structural dynamics
 Format
 Document (PDF)
 Title
 Finite element methods for stochastic structures and conditional simulation.
 Creator
 Ren, Yongjian., Florida Atlantic University, Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

This dissertation deals with the nonperturbative finite element methods for stochastic structures and conditional simulation techniques for random fields. Three different nonperturbative finite element schemes have been proposed to compute the first and second moments of displacement responses of stochastic structures. These three methods are based, respectively, on (i) the exact inverse of the global stiffness matrix for simple stochastic structures; (ii) the variational principles for...
Show moreThis dissertation deals with the nonperturbative finite element methods for stochastic structures and conditional simulation techniques for random fields. Three different nonperturbative finite element schemes have been proposed to compute the first and second moments of displacement responses of stochastic structures. These three methods are based, respectively, on (i) the exact inverse of the global stiffness matrix for simple stochastic structures; (ii) the variational principles for staticallydeterminate beams; and (iii)Â the elementlevel flexibility for general stochastic statically indeterminate structures. The nonperturbative finite element method for stochastic structures possesses several advantages over the conventional perturbationbased finite element method for stochastic structures, including (i) applicability to large values of the coefficient of variation of random parameters; (ii) convergence to exact solutions when the finite element mesh is refined; (iii) requirement of less statistical information than that demanded by the highorder perturbation methods. Conditional simulation of random fields has been an extremely important research field in most recent years due to its application in urban earthquake monitoring systems. This study generalizes the available simulation technique for onevariate Gaussian random fields, conditioned by realizations of the fields, to multivariate vector random field, conditioned by the realizations of the fields themselves as well as the realizations of the fields derivatives. Furthermore, a conditional simulation for nonGaussian random fields is also proposed in this study by combining the unconditional simulation technique of nonGaussian fields and the conditional simulation technique of Gaussian fields. Finally, the dissertation incorporates the simulation technique of random field into the nonperturbation finite element method for stochastic structures, to handle the cases where only onedimensional probability density function and the correlation function of the random parameters are available, the demanded twodimensional probability density function is unavailable. Simulation technique is applied to generate the samples of random fields which are used to estimate the correlation between flexibilities over elements. The estimated correlation of flexibility is then used in finite element analysis for stochastic structures. For each proposed approach, numerous examples and numerical results have been implemented.
Show less  Date Issued
 1998
 PURL
 http://purl.flvc.org/fcla/dt/12580
 Subject Headings
 Finite element method, Stochastic processes, Random fieldsMathematical models
 Format
 Document (PDF)
 Title
 Dynamic stability of fluidconveying pipes on uniform or nonuniform elastic foundations.
 Creator
 Vittori, Pablo J., Florida Atlantic University, Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The dynamic behavior of straight cantilever pipes conveying fluid is studied, establishing the conditions of stability for systems, which are only limited to move in a 2Dplane. Internal friction of pipe and the effect of the surrounding fluid are neglected. A universal stability curve showing boundary between the stable and unstable behaviors is constructed by finding solution to equation of motion by exact and highdimensional approximate methods. Based on the BoobnovGalerkin method, the...
Show moreThe dynamic behavior of straight cantilever pipes conveying fluid is studied, establishing the conditions of stability for systems, which are only limited to move in a 2Dplane. Internal friction of pipe and the effect of the surrounding fluid are neglected. A universal stability curve showing boundary between the stable and unstable behaviors is constructed by finding solution to equation of motion by exact and highdimensional approximate methods. Based on the BoobnovGalerkin method, the critical velocities for the fluid are obtained by using both the eigenfunctions of a cantilever beam (beam functions), as well as the utilization of Duncan's functions. Stability of cantilever pipes with uniform and nonuniform elastic foundations of two types are considered and discussed. Special emphasis is placed on the investigation of the paradoxical behavior previously reported in the literature.
Show less  Date Issued
 2004
 PURL
 http://purl.flvc.org/fcla/dt/13167
 Subject Headings
 Strains and stresses, Structural dynamics, Structural stability, Fluid dynamics, Vibration
 Format
 Document (PDF)
 Title
 Vibration tailoring of inhomogeneous beams and circular plates.
 Creator
 Pentaras, Demetris., Florida Atlantic University, Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

The vibrational behavior of inhomogeneous beams and circular plates is studied, utilizing the semiinverse method developed by I. Elishakoff and extensively discussed in his recent monograph (2005). The main thread of his methodology is that the knowledge of the mode shape is postulated. The candidate mode shapes can be adopted from relevant static, dynamic or buckling problems. In this study, the exact mode shapes are sought as polynomial functions, in the context of vibration tailoring, i.e...
Show moreThe vibrational behavior of inhomogeneous beams and circular plates is studied, utilizing the semiinverse method developed by I. Elishakoff and extensively discussed in his recent monograph (2005). The main thread of his methodology is that the knowledge of the mode shape is postulated. The candidate mode shapes can be adopted from relevant static, dynamic or buckling problems. In this study, the exact mode shapes are sought as polynomial functions, in the context of vibration tailoring, i.e. designing the structure that possesses the prespecified value. Apparently for the first time in the literature, several closedform solutions for vibration tailoring have been derived for vibrating inhomogeneous beams and circular plates. Twelve new closedform solutions for vibration tailoring have been derived for an inhomogeneous polar orthotropic plate that is either clamped or simply supported around its circumference. Also, the vibration tailoring of a polar orthotropic circular plate with translational spring is analyzed. There is considerable potential of utilizing the developed method for design of functionally graded materials.
Show less  Date Issued
 2006
 PURL
 http://purl.flvc.org/fcla/dt/13344
 Subject Headings
 Acoustical engineering, Plates (Engineering)VibrationMathematical models, Buckling (Mechanics), Structural analysis
 Format
 Document (PDF)
 Title
 Convex identification and nonlinear random vibration analysis of elastic and viscoelastic structures.
 Creator
 Fang, Jianjie, Florida Atlantic University, Elishakoff, Isaac, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

This dissertation deals with the identification of boundary conditions of elastic structures, and nonlinear random vibration analysis of elastic and viscoelastic structures through a new energybased equivalent linearization technique. In the part of convex identification, convex models are utilized to represent the degree of uncertainty in the boundary condition modification. This means that the identification is actually the identification of the convex model to which the actual boundary...
Show moreThis dissertation deals with the identification of boundary conditions of elastic structures, and nonlinear random vibration analysis of elastic and viscoelastic structures through a new energybased equivalent linearization technique. In the part of convex identification, convex models are utilized to represent the degree of uncertainty in the boundary condition modification. This means that the identification is actually the identification of the convex model to which the actual boundary stiffness profile belongs. Two examples are presented to illustrate the application of the method. For the beam example the finite element analysis is performed to evaluate the frequencies of a beam with any specific boundary conditions. For the plate example, the Bolotin's dynamic edge effect method, generalized by Elishakoff, is employed to determine the approximate natural frequencies and normal modes of elastically supported isotropic, uniform rectangular plates. In the part of nonlinear random analysis, first a systematic finite element analysis procedure, based on the element's energy formulation, through conventional stochastic linearization technique, is proposed. The procedure is applicable to a wide range of nonlinear random vibration problem as long as element's energy formulations are presented. Secondly, the new energybased stochastic linearization method in finite element analysis setting is developed to improve the conventional stochastic linearization technique. The entire formulation is produced in detail for the first time. The theory is applied to beam problem subjected to spacewise and timewise white noise excitations. Finally, the new energybased stochastic linearization technique is applied to treat nonlinear vibration problems of viscoelastic beams.
Show less  Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/12467
 Subject Headings
 Elasticity, Viscoelasticity, Structural dynamicsMathematical models, VibrationMathematical models
 Format
 Document (PDF)
 Title
 Vibration of nonlocal carbon nanotubes and graphene nanoplates.
 Creator
 Hache, Florian, Elishakoff, Isaac, Challamel, Noël, Florida Atlantic University, College of Engineering and Computer Science, Department of Ocean and Mechanical Engineering
 Abstract/Description

This thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional BresseTimoshenko models for thick beams and UflyandMindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first...
Show moreThis thesis deals with the analytical study of vibration of carbon nanotubes and graphene plates. First, a brief overview of the traditional BresseTimoshenko models for thick beams and UflyandMindlin models for thick plates will be conducted. It has been shown in the literature that the conventionally utilized mechanical models overcorrect the shear effect and that of rotary inertia. To improve the situation, two alternative versions of theories of beams and plates are proposed. The first one is derived through the use of equilibrium equations and leads to a truncated governing differential equation in displacement. It is shown, by considering a power series expansion of the displacement, that this is asymptotically consistent at the second order. The second theory is based on slope inertia and results in the truncated equation with an additional sixth order derivative term. Then, these theories will be extended in order to take into account some scale effects such as interatomic interactions that cannot be neglected for nanomaterials. Thus, different approaches will be considered: phenomenological, asymptotic and continualized. The basic principle of continualized models is to build continuous equations starting from discrete equations and by using Taylor series expansions or Padé approximants. For each of the different models derived in this study, the natural frequencies will be determined, analytically when the closedform solution is available, numerically when the solution is given through a characteristic equation. The objective of this work is to compare the models and to establish the eventual superiority of a model on others.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013019
 Subject Headings
 Carbon nanotubes, Graphene, Vibration
 Format
 Document (PDF)