Dissertations / Theses on the topic 'Euler Beam Theory'

In this thesis, transverse vibration of a cracked beam on an elastic foundation and the effect of crack and foundation parameters on transverse vibration natural frequencies are studied. Analytical formulations are derived for a beam with rectangular cross section. The crack is an open type edge crack placed in the medium of the beam and it is uniform along the width of the beam. The cracked beam rests on an elastic foundation. The beam is modeled by two different beam theories, which are Euler-Bernoulli beam theory and Timoshenko beam theory. The effect of the crack is considered by representing the crack by rotational springs. The compliance of the spring that represents the crack is obtained by using fracture mechanics theories. Different foundation models are discussed
these models are Winkler Foundation, Pasternak Foundation, and generalized foundation. The equations of motion are derived by applying Newton'
s 2nd law on an infinitesimal beam element. Non-dimensional parameters are introduced into equations of motion. The beam is separated into pieces at the crack location. By applying the compatibility conditions at the crack location and boundary conditions, characteristic equation whose roots give the non-dimensional natural frequencies is obtained. Numerical solutions are done for a beam with square cross sectional area. The effects of crack ratio, crack location and foundation parameters on transverse vibration natural frequencies are presented. It is observed that existence of crack reduces the natural frequencies. Also the elastic foundation increases the stiffness of the system thus the natural frequencies. The natural frequencies are also affected by the location of the crack.

This study focuses on assessing the accuracy of the Euler-Bernoulli beam theory as computational bases to calculate strain and deflection of composite sandwich beam subjected to three-point and four-point bending. Two groups of composite sandwich beams tests results will be used for comparison purposes. Mechanical properties for the laminated skin are provided by researchers from University of Mississippi (Ellen Lackey et al., 2000). Mechanical properties for the balsa wood core are provided by Alcan Baltek Corporation. Appropriate material properties and test geometries are then used in the Euler-Bernoulli-based algorithm in order to generate analytical data for comparison to experimental data provided by researchers from University of New Orleans (UNO, 2005). The resulting single material cross section is then analyzed in the traditional manner using the Euler-Bernoulli beam theory. In general, the Euler-Bernoulli beam theory provides an appropriate analytical approach in predicting flexural behavior of composite sandwich beams.

Research is performed to assess the viability of applying the least squares model to one-dimensional heat transfer and Euler-Bernoulli Beam Theory problems. Least squares models were developed for both the full and mixed forms of the governing one-dimensional heat transfer equation along weak form Galerkin models. Both least squares and weak form Galerkin models were developed for the first order and second order versions of the Euler-Bernoulli beams. Several numerical examples were presented for the heat transfer and Euler- Bernoulli beam theory. The examples for heat transfer included: a differential equation having the same form as the governing equation, heat transfer in a fin, heat transfer in a bar and axisymmetric heat transfer in a long cylinder. These problems were solved using both least squares models, and the full form weak form Galerkin model. With all four examples the weak form Galerkin model and the full form least squares model produced accurate results for the primary variables. To obtain accurate results with the mixed form least squares model it is necessary to use at least a quadratic polynominal. The least squares models with the appropriate approximation functions yielde more accurate results for the secondary variables than the weak form Galerkin. The examples presented for the beam problem include: a cantilever beam with linearly varying distributed load along the beam and a point load at the end, a simply supported beam with a point load in the middle, and a beam fixed on both ends with a distributed load varying cubically. The first two examples were solved using the least squares model based on the second order equation and a weak form Galerkin model based on the full form of the equation. The third problem was solved with the least squares model based on the second order equation. Both the least squares model and the Galerkin model calculated accurate results for the primary variables, while the least squares model was more accurate on the secondary variables. In general, the least-squares finite element models yield more acurate results for gradients of the solution than the traditional weak form Galkerkin finite element models. Extension of the present assessment to multi-dimensional problems and nonlinear provelms is awaiting attention.

A computational framework for linear static aeroelastic analyses is presented. The overall aeroelasticity model is applicable to conceptual aircraft design studies and other low-fidelity aero-structural analyses. A partitioned approach is used, i. e. separate solvers for aerodynamics and structure analyses are coupled in a suitable way, together forming a model for aeroelastic simulations. Aerodynamics are modelled using the vortexlattice method (VLM), a simple computational fluid dynamics (CFD) model based on potential flow. The structure is represented by a three-dimensional (3D) Euler-Bernoulli beam model in a finite element method (FEM) formulation. A particular focus was put on the modularity and loose coupling of aforementioned models. The core of the aeroelastic framework was abstracted, such that it does not depend on any specific details of the underlying aerodynamics and structure modules. The final aeroelasticity model constitutes independent software tools for the VLM and the beam FEM, as well as a framework enabling the aeroelastic coupling. These different tools have been developed as part of this thesis work. A wind tunnel experiment with a simple wing model is presented as a validation test case. An aero-structural analysis of a fully elastic unmanned aerial vehicle (UAV) (OptiMale) is described and results are compared with an existing higherfidelity study.
Rapporten beskriver en beräkningsmodell för linjära, statisk aeroelastiska analyser. Modellen kan användas för konceptuella designstudier av flygplan. En partitionerad metod används, d v s separata lösare för aerodynamik- och strukturanalyser kopplas på ett lämpligt sätt, och bildar tillsammans en modell för aeroelastiska simulationer. Aerodynamik modelleras med hjälp av en så kallad vortex-lattice method (VLM), en enkel modell för beräkningsströmningsdynamik (CFD) som är baserad på friktionsfri strömning. Strukturen representeras av en tredimensionell (3D) Euler-Bernoulli-balkmodell implementerad med hjälp av en finita elementmetod (FEM). Ovannämnda modeller har utvecklats med fokus på modularitet och lös koppling. Kärnan i den aeroelastiska modellen har abstraherats så att den inte beror på specifika detaljer i de underliggande aerodynamik- och strukturmodulerna. Aeroelasticitetsmodellen i sin helhet består av separata mjukvaruprogram för VLM och balk-FEM, såväl som ett ramverk som möjliggör den aeroelastiska kopplingen. Dessa olika program har utvecklats som en del av examensarbetet. Ett vindtunnelförsök med en enkel vingmodell presenteras som ett valideringstest. Dessutom beskrivs en analys av ett elastiskt obemannad flygplan (OptiMale) och resultaten jämförs med en befintlig studie som har genomförts med modeller av högre trovärdighet.

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.

This thesis is about using structural-dynamics based methods to address the existing challenges in the field of Structural Health Monitoring (SHM). Particularly, new structural-dynamics based methods are presented, to model areas of damage, to do damage diagnosis and to estimate and predict the sensitivity of structural vibration properties like natural frequencies to the presence of damage. Towards these objectives, a general analytical procedure, which yields nth-order expressions governing mode shapes and natural frequencies and for damaged elastic structures such as rods, beams, plates and shells of any shape is presented. Features of the procedure include the following: 1. Rather than modeling the damage as a fictitious elastic element or localized or global change in constitutive properties, it is modeled in a mathematically rigorous manner as a geometric discontinuity. 2. The inertia effect (kinetic energy), which, unlike the stiffness effect (strain energy), of the damage has been neglected by researchers, is included in it. 3. The framework is generic and is applicable to wide variety of engineering structures of different shapes with arbitrary boundary conditions which constitute self adjoint systems and also to a wide variety of damage profiles and even multiple areas of damage. To illustrate the ability of the procedure to effectively model the damage, it is applied to beams using Euler-Bernoulli and Timoshenko theories and to plates using Kirchhoff's theory, supported on different types of boundary conditions. Analytical results are compared with experiments using piezoelectric actuators and non-contact Laser-Doppler Vibrometer sensors. Next, the step of damage diagnosis is approached. Damage diagnosis is done using two methodologies. One, the modes and natural frequencies that are determined are used to formulate analytical expressions for a strain energy based damage index. Two, a new damage detection parameter are identified. Assuming the damaged structure to be a linear system, the response is expressed as the summation of the responses of the corresponding undamaged structure and the response (negative response) of the damage alone. If the second part of the response is isolated, it forms what can be regarded as the damage signature. The damage signature gives a clear indication of the damage. In this thesis, the existence of the damage signature is investigated when the damaged structure is excited at one of its natural frequencies and therefore it is called ``partial mode contribution". The second damage detection method is based on this new physical parameter as determined using the partial mode contribution. The physical reasoning is verified analytically, thereupon it is verified using finite element models and experiments. The limits of damage size that can be determined using the method are also investigated. There is no requirement of having a baseline data with this damage detection method. Since the partial mode contribution is a local parameter, it is thus very sensitive to the presence of damage. The parameter is also shown to be not affected by noise in the detection ambience.

A Study Of Shock Analysis Using The Finite Element Method Verified With Euler-Bernoulli Beam Theory; Mechanical Effects Due To Pulse Width Variation Of Shock Inputs; And Evaluation Of Shock Response Of A Mixed Flow Fan David Jonathan González Campos For many engineers that use finite element analysis or FEA, it is very important to know how to properly model and obtain accurate solutions for complicated loading conditions such as shock loading. Transient acceleration loads, such as shocks, are not as common as static loads. Analyzing these types of problems is less understood, which is the basis for this study. FEA solutions are verified using classical theory, as well as experimental results. The complex loading combination of shock and high speed rotation is also studied. Ansys and its graphic user interface, Workbench Version 14.5, are the programs used to solve these types of problems. Classical theory and Matlab codes, as well as experimental results, are used to verify finite element solutions for a simple structure, such as a cantilevered beam. The discrepancy of these FEA results is found to be 2.3%. The Full Method and the Mode Superposition Method in Ansys are found to be great solution tools for shock loading conditions, including complex acceleration and force conditions. The Full Method requires less pre-processing but solutions could take days, as opposed to hours, to complete in comparison with the Mode Superposition Method, depending on the 3D Model. The Mode Superposition Method requires more time and input by the user but solves relatively quickly. Furthermore, a new representation of critical pulse width of the shock inputs is presented. Experimental and finite element analyses of a complete mixed flow fan undergoing ballistic shock is also completed; deformation results due to shock loading, combined with rotation and aerodynamic loading, account for 32.3% of the total deformation seen from experimental testing. Solution methods incorporated in Ansys, and validation of FEA results using theory, have great potential implications as powerful tools for engineering students and practicing engineers.

The highly optimized performance of nature’s creations and biological assemblies has inspired the development of their engineered counter parts that can potentially outperform conventional systems. In particular, bat wings are populated with air flow hair receptors which feedback the information about airflow over their surfaces for enhanced stability and maneuverability during their flight. The hairs in the bat wing membrane play a role in the maneuverability tasks, especially during low-speed flight. The developments of artificial hair sensors (AHS) are inspired by biological hair cells in aerodynamic feedback control designs. Current mathematical models for hair receptors are limited by strict simplifying assumptions of creeping flow hair Reynolds number on AHS fluid-structure interaction (FSI), which may be violated for hair structures integrated on small-scaled Unmanned Aerial Vehicles (UAVs). This study motivates by an outstanding need to understand the dynamic response of hair receptors in flow regimes relevant to bat-scaled UAVs. The dynamic response of the hair receptor within the creeping flow environment is investigated at distinct freestream velocities to extend the applicability of AHS to a wider range of low Reynolds number platforms. Therefore, a threedimensional FSI model coupled with a finite element model using the computational fluid dynamics (CFD) is developed for a hair-structure and multiple hair-structures in the airflow. The Navier-Stokes equations including continuity equation are solved numerically for the CFD model. The grid independence of the FSI solution is studied from the simulations of the hairstructure mesh and flow mesh around the hair sensor. To describe the dynamic response of the hair receptors, the natural frequencies and mode shapes of the hair receptors, computed from the finite element model, are compared with the excitation frequencies in vacuum. This model is described with both the boundary layer effects and effects of inertial forces due to fluid-structure xiv interaction of the hair receptors. For supporting the FSI model, the dynamic response of the hair receptor is also validated considering the Euler-Bernoulli beam theory including the steady and unsteady airflow.

Le travail présenté dans cette thèse est consacré à la modélisation de la dynamique de locomotion des "soft robots", i.e. les systèmes multi-corps mobiles compliants. Ces compliances peuvent être localisées et considérées comme des liaisons passives du système,ou bien introduites par des flexibilités distribuées le long des corps. La dynamique de ces systèmes est modélisée en adoptant une approche Lagrangienne basée sur les outils mathématiques développés par l'école américaine de mécanique géométrique. Du point de vue algorithmique, le calcul de ces modèles dynamiques s'appuie sur un algorithme récursif et efficace de type Newton-Euler, ici étendu aux robots locomoteurs munis d'organes compliants. Poursuivant des objectifs de commande et de simulation rapide pour la robotique, l'algorithme proposé est capable de résoudre la dynamique externe directe ainsi que la dynamique inverse des couples internes. Afin de mettre en pratique l'ensemble de ces outils de modélisation, nous avons pris le vol battant des insectes comme exemple illustratif. Les équations non-linéaires qui régissent les déformations passives de l'aile sont établies en appliquant deux méthodes différentes. La première consiste à séparer le mouvement de l'aile en une composante rigide dite de "repère flottant" et une composante de déformation. Cette dernière est paramétrée dans le repère flottant par la méthode des modes supposés ici appliquée à l'aile vue comme une poutre d'Euler-Bernoulli soumise à la flexion et à la torsion. Quant à la seconde approche, les mouvements de l'aile n'y sont pas séparés mais directement paramétrés par les transformations finies rigides et absolues d'une poutre Cosserat. Cette approche est dite Galiléenne ou "géométriquement exacte" en raison du fait qu'elle ne requiert aucune approximation en dehors des inévitables discrétisations spatiale et temporelle imposées parla résolution numérique de la dynamique du vol. Dans les deux cas,les forces aérodynamiques sont prises en compte via un modèle analytique simplifié de type Dickinson. Les modèles et algorithmes résultants sont appliqués à la conception d'un simulateur du vol, ainsi qu'à la conception d'un prototype d'aile, dans le contexte du projet coopératif (ANR) EVA.

A control metric, the weighted sum of spatial gradients (WSSG), has been developed for use in active structural acoustic control (ASAC). Previous development of WSSG [1] showed that it was an effective control metric on simply supported plates, while being simpler to measure than other control metrics, such as volume velocity. The purpose of the current work is to demonstrate that the previous research can be generalized to plates with a wider variety of boundary conditions and on less ideal plates. Two classes of plates have been considered: clamped flat plates, and ribbed plates. On clamped flat plates an analytical model has been developed for use in WSSG that assumes the mode shapes are the product of clamped-clamped beam mode shapes. The boundary condition specific weights for use in WSSG have been derived from this formulation and provide a relatively uniform measurement field, as in the case of the simply supported plate. Using this control metric, control of radiated sound power has been simulated. The results show that WSSG provides comparable control to volume velocity on the clamped plate. Results also show, through random placement of the sensors on the plate, that similar control can be achieved regardless of sensor location. This demonstrates that WSSG is an effective control metric on a variety of boundary conditions. Ribbed plates were considered because of their wide use in aircraft and ships. In this case, a finite-element model of the plate has been used to obtain the displacement field on the plate under a variety of boundary conditions. Due to the discretized model involved, a numerical, as opposed to analytical, formulation for WSSG has been developed. Simulations using this model show that ASAC can be performed effectively on ribbed plates. In particular WSSG was found to perform comparable to or better than volume velocity on all boundary conditions examined. The sensor insensitivity property was found to hold within each section (divided by the ribs) of the plate, a slightly modified form of the flat plate insensitivity property where the plates have been shown to be relatively insensitive to sensor location over the entire surface of the plate. Improved control at natural frequencies can be achieved by applying a second control force. This confirms that ASAC is a viable option for the control of radiated sound power on non-ideal physical systems similar to ribbed plates.

The conventional finite element models (FEM) of problems in structural mechanics are based on the principles of virtual work and the total potential energy. In these models, the secondary variables, such as the bending moment and shear force, are post-computed and do not yield good accuracy. In addition, in the case of the Timoshenko beam theory, the element with lower-order equal interpolation of the variables suffers from shear locking. In both Euler-Bernoulli and Timoshenko beam theories, the elements based on weak form Galerkin formulation also suffer from membrane locking when applied to geometrically nonlinear problems. In order to alleviate these types of locking, often reduced integration techniques are employed. However, this technique has other disadvantages, such as hour-glass modes or spurious rigid body modes. Hence, it is desirable to develop alternative finite element models that overcome the locking problems. Least-squares finite element models are considered to be better alternatives to the weak form Galerkin finite element models and, therefore, are in this study for investigation. The basic idea behind the least-squares finite element model is to compute the residuals due to the approximation of the variables of each equation being modeled, construct integral statement of the sum of the squares of the residuals (called least-squares functional), and minimize the integral with respect to the unknown parameters (i.e., nodal values) of the approximations. The least-squares formulation helps to retain the generalized displacements and forces (or stress resultants) as independent variables, and also allows the use of equal order interpolation functions for all variables. In this thesis comparison is made between the solution accuracy of finite element models of the Euler-Bernoulli and Timoshenko beam theories based on two different least-square models with the conventional weak form Galerkin finite element models. The developed models were applied to beam problems with different boundary conditions. The solutions obtained by the least-squares finite element models found to be very accurate for generalized displacements and forces when compared with the exact solutions, and they are more accurate in predicting the forces when compared to the conventional finite element models.

Adhesively bonded joints have been investigated for several decades. In most analytical studies, the Bernoulli-Euler beam theory is employed to describe the behaviour of adherends. In the current work, three analytical models are developed for adhesively bonded joints using the Timoshenko beam theory for elastic material and a Bernoulli-Euler beam model for viscoelastic materials. One model is for the peeling test of an adhesively bonded joint, which is described using a Timoshenko beam on an elastic foundation. The adherend is considered as a Timoshenko beam, while the adhesive is taken to be a linearly elastic foundation. Three cases are considered: (1) only the normal stress is acting (mode I); (2) only the transverse shear stress is present (mode II); and (3) the normal and shear stresses co-exist (mode III) in the adhesive. The governing equations are derived in terms of the displacement and rotational angle of the adherend in each case. Analytical solutions are obtained for the displacements, rotational angle, and stresses. Numerical results are presented to show the trends of the displacements and rotational angle changing with geometrical and loading conditions. In the second model, the peeling test of an adhesively bonded joint is represented using a viscoelastic Bernoulli-Euler beam on an elastic foundation. The adherend is considered as a viscoelastic Bernoulli-Euler beam, while the adhesive is taken to be a linearly elastic foundation. Two cases under different stress history are considered: (1) only the normal stress is acting (mode I); and (2) only the transverse shear stress is present (mode II). The governing equations are derived in terms of the displacements. Analytical solutions are obtained for the displacements. The numerical results show that the deflection increases as time and temperature increase. The third model is developed using a symmetric composite adhesively bonded joint. The constitutive and kinematic relations of the adherends are derived based on the Timoshenko beam theory, and the governing equations are obtained for the normal and shear stresses in the adhesive layer. The numerical results are presented to reveal the normal and shear stresses in the adhesive.

The beam with axial load is one of the fundamental models in the field of physics and engineering. The applications of this model span from a guitar string to the rotating blade of a helicopter. It has an important role in vibration theory and structural mechanics. Also, it establishes a relationship between the mathematical continuum concept and continuum mechanics as it is the most basic example of a continuum physical model. For the long and slender beams, the Euler-Bernoulli beam theory is used. In this thesis, Euler-Bernoulli beam with axial load is studied. For the wide application in industry, an accurate and simple closed form solution for the general problem is highly desirable. Here, general problem implies a beam with variable stiffness, variable mass, and variable axial load. The governing equation of motion is a partial differential equation of order four in the spatial variable and of order two in the temporal variable. However, there is a deficiency in the research on exact solutions of the above mentioned model. The modern numerical methods such as finite element methods can give an approximate solution but it requires huge computer time and sometimes extensive coding. In this thesis, we look at some basic methods such as the Galerkin method and Fourier transform method. We also apply Lie symmetry method to extract the closed form solution. There are three parts in the thesis. In Part I, the Galerkin method is written; in Part II, the Fourier transform method is discussed and finally, in Part III, the Lie method is given

Sandwich constructions are the most commonly used structures in aircraft and navy industries, traditionally. These structures are made up of the face sheets and the core, where the face sheets will be taking the load and is connected to other structural members, while the soft core material, will be used to absorb energy during impact like situation. Thus, sandwich constructions are mainly employed in light weight structures where the high energy absorption capability is required. Generally the face sheets will be thin, made up of either metallic or composite material with high stiffness and strength, while the core is light in weight, made up of soft material. Cores generally play very crucial role in achieving the desired properties of sandwich structures, either through geometric arrangement or material properties or both. Foams are in extensive use nowadays as core material due to the ease in manufacturing and their low cost. They are extensively used in automotive and industrial field applications as the desired foam density can be fabricated by adjusting the mixing, curing and heat sink processes. Modeling of sandwich beams play a crucial role in their design with suitable finite elements for face sheets and core, to ensure the compatibility between degrees of freedom at the interfaces. Unless the mathematical model simulates the physics of the model in terms of kinematics, boundary and loading conditions, results predicted will not be accurate. Accurate models helps in obtaining an efficient design of sandwich beams. In Structural Health Monitoring studies, the responses under the impact loading will be captured by carrying out the wave propagation analysis. The loads applied will be for a shorter duration (in the orders of micro seconds), where higher frequency modes will be excited. Wavelengths at such high frequencies are very small and hence, in such cases, very fine mesh generally is employed matching the wavelength requirement of the propagating wave. Traditional Finite element softwares takes enormous time and computational e ort to provide the solution. Various possible models and modeling aspects using the existing Finite element tools for wave propagation analysis are studied in the present work. There exists a huge demand for an accurate, efficient and rapidly convergent finite elements for the analysis of sandwich beams. E orts are made in the present work to address these issues and provide a solution to the sandwich user community. Super convergent and Spectral Finite sandwich Beam Elements with metallic or composite face sheets and soft core are developed. As a philosophy, the sandwich beam finite element is constructed with the combination of two beams representing the face sheets (top and bottom) at their neutral axis. The core effects are captured at the interface boundaries in terms of shear stress and normal transverse stress. In the case of wave propagation analysis, the equations are coupled in time domain and spatial domain and solving them directly is a difficult task. In Spectral Finite Element Method(SFEM), the displacement functions are derived by solving the transformed governing equations in the frequency domain. By transforming them and forces from time domain to frequency domain, the coupled partial differential equations will become coupled ordinary differential equations. These equations in frequency domain, can be solved exactly as they are normally ordinary differential equation with constant coefficients with frequency entering as a parameter. These solutions will be used as interpolating functions for spectral element formulation and in this respect it differs from conventional FE method wherein mostly polynomials are used as interpolating functions. In addition, SFEM solutions are expressed in terms of forward and backward moving waves for all the degrees of freedom involved in the formulations and hence, SFEM provides faster and efficient solutions for wave propagation analysis. In the present work, strong form of the governing differential equations are derived for a given system using Hamilton's principle. Super Convergent elements are developed by solving the static part of the governing differential equations exactly and hence the stiffness matrix derived is exact for point static loads. For wave propagation analysis, as the mass is not exactly represented, these elements are required in the optimal numbers for getting good results. The number of these elements required are generally much lesser than the number of elements required using traditional finite elements since the stiffness distribution is exact. Spectral elements are developed by solving the governing equations exactly in the frequency domain and hence the dynamic stiffness matrix derived is exact for the dynamic loads. Hence, one element between any two joints is enough to solve the whole system under impact loads for simple structures. Developing FE for sandwich beams is quiet challenging. Due to small thickness, the face sheets can be modeled using 1D idealization, while modeling of large core requires 2-D idealization. Hence, most finite or spectral elements requires stitching of these two idealizations into 1-D idealization, which can be accomplished in a variety of ways, some of which are highlighted in this thesis. Variety of finite and spectral finite elements are developed considering Euler and Timoshenko beam theories for modeling the sandwich beams. Simple element models are built with rigid core in both the theories. Models are also developed considering the flexible core with the variation of transverse displacements across depth of the core. This has direct influence on shear stress variation and also transverse normal stress in the core. Simple to higher order models are developed considering different variations in shear stress and transverse normal stress across depth of the core. Development of super convergent finite Euler Bernoulli beam elements Eul4d (4 dof element), Eul10d (10 dof element) are explained along with their results in Chapter 2. Development of different super convergent finite Timoshenko beam elements namely Tim4d (4 dof), Tim7d (7 dof), Tim10d (10 dof) are explained in Chapter 3. Validation of Euler Bernoulli and Timoshenko elements developed in the present work is carried out with test cases available in the open literature for displacements and free vibration frequencies are presented in Chapter 2 and Chapter 3. The results indicates that all developed elements are performing exceedingly well for static loads and free vibration. Super convergence performance for the elements developed is demonstrated with related examples. Spectral elements based on Timoshenko theory STim7d, STim6d, STim6dF are developed and the wave propagation characteristics studies are presented in Chapter 4. Euler spectral elements are derived from Timoshenko spectral elements by enforcing in finite shear rigidity, designated as SEul7d, SEul6d, SEul6dF and are presented. E orts were made in this present work to model the horizontal cracks in top or bottom face sheets using the spectral elements and the methodology is presented in Chapter 4. Wave propagation analysis using general purpose software N AST RAN and the super convergent as well as spectral elements developed in this work, are discussed in detail in Chapter 5. Modeling aspects of sandwich beam in N AST RAN using various combination of elements available and the performance of four possible models simulated were studied. Validation of all four models in N AST RAN, Super convergent Euler, Timoshenko and Spectral Timoshenko finite elements was carried out by simulating a homogenous I beam by comparing the longitudinal and transverse responses. Studies were carried out to find out the response predictions of a sandwich beam with soft core and all the predictions were compared and discussed. The responses in case of cracks in top or bottom face sheets under the longitudinal and transverse loading were studied in this chapter. In Chapter 6, Parametric studies were carried out for bringing out the sensitiveness of the important specific parameters in overall behaviour and performance of a sandwich beam, using Super convergent and Spectral elements developed. This chapter clearly brings out the various aspects of design of sandwich beam such as material selection of core, geometrical configuration of overall beam and core. Effects of shear modulus, mass density on wave propagation characteristics, effects of thick or thin cores with reference to the face sheets and dynamic effects of core are highlighted. Wave propagation characteristics studies includes the study of wave numbers, group speeds, cut off frequencies for a given configuration and identification of frequency zone of operations. The recommendations for improvement in design of sandwich beams based on the parametric studies are made at the end of chapter. The entire thesis, written in seven Chapters, presents a unified treatment of sandwich beam analysis that will be very useful for designers working in the area.

Sandwich constructions are the most commonly used structures in aircraft and navy industries, traditionally. These structures are made up of the face sheets and the core, where the face sheets will be taking the load and is connected to other structural members, while the soft core material, will be used to absorb energy during impact like situation. Thus, sandwich constructions are mainly employed in light weight structures where the high energy absorption capability is required. Generally the face sheets will be thin, made up of either metallic or composite material with high stiffness and strength, while the core is light in weight, made up of soft material. Cores generally play very crucial role in achieving the desired properties of sandwich structures, either through geometric arrangement or material properties or both. Foams are in extensive use nowadays as core material due to the ease in manufacturing and their low cost. They are extensively used in automotive and industrial field applications as the desired foam density can be fabricated by adjusting the mixing, curing and heat sink processes. Modeling of sandwich beams play a crucial role in their design with suitable finite elements for face sheets and core, to ensure the compatibility between degrees of freedom at the interfaces. Unless the mathematical model simulates the physics of the model in terms of kinematics, boundary and loading conditions, results predicted will not be accurate. Accurate models helps in obtaining an efficient design of sandwich beams. In Structural Health Monitoring studies, the responses under the impact loading will be captured by carrying out the wave propagation analysis. The loads applied will be for a shorter duration (in the orders of micro seconds), where higher frequency modes will be excited. Wavelengths at such high frequencies are very small and hence, in such cases, very fine mesh generally is employed matching the wavelength requirement of the propagating wave. Traditional Finite element softwares takes enormous time and computational e ort to provide the solution. Various possible models and modeling aspects using the existing Finite element tools for wave propagation analysis are studied in the present work. There exists a huge demand for an accurate, efficient and rapidly convergent finite elements for the analysis of sandwich beams. E orts are made in the present work to address these issues and provide a solution to the sandwich user community. Super convergent and Spectral Finite sandwich Beam Elements with metallic or composite face sheets and soft core are developed. As a philosophy, the sandwich beam finite element is constructed with the combination of two beams representing the face sheets (top and bottom) at their neutral axis. The core effects are captured at the interface boundaries in terms of shear stress and normal transverse stress. In the case of wave propagation analysis, the equations are coupled in time domain and spatial domain and solving them directly is a difficult task. In Spectral Finite Element Method(SFEM), the displacement functions are derived by solving the transformed governing equations in the frequency domain. By transforming them and forces from time domain to frequency domain, the coupled partial differential equations will become coupled ordinary differential equations. These equations in frequency domain, can be solved exactly as they are normally ordinary differential equation with constant coefficients with frequency entering as a parameter. These solutions will be used as interpolating functions for spectral element formulation and in this respect it differs from conventional FE method wherein mostly polynomials are used as interpolating functions. In addition, SFEM solutions are expressed in terms of forward and backward moving waves for all the degrees of freedom involved in the formulations and hence, SFEM provides faster and efficient solutions for wave propagation analysis. In the present work, strong form of the governing differential equations are derived for a given system using Hamilton's principle. Super Convergent elements are developed by solving the static part of the governing differential equations exactly and hence the stiffness matrix derived is exact for point static loads. For wave propagation analysis, as the mass is not exactly represented, these elements are required in the optimal numbers for getting good results. The number of these elements required are generally much lesser than the number of elements required using traditional finite elements since the stiffness distribution is exact. Spectral elements are developed by solving the governing equations exactly in the frequency domain and hence the dynamic stiffness matrix derived is exact for the dynamic loads. Hence, one element between any two joints is enough to solve the whole system under impact loads for simple structures. Developing FE for sandwich beams is quiet challenging. Due to small thickness, the face sheets can be modeled using 1D idealization, while modeling of large core requires 2-D idealization. Hence, most finite or spectral elements requires stitching of these two idealizations into 1-D idealization, which can be accomplished in a variety of ways, some of which are highlighted in this thesis. Variety of finite and spectral finite elements are developed considering Euler and Timoshenko beam theories for modeling the sandwich beams. Simple element models are built with rigid core in both the theories. Models are also developed considering the flexible core with the variation of transverse displacements across depth of the core. This has direct influence on shear stress variation and also transverse normal stress in the core. Simple to higher order models are developed considering different variations in shear stress and transverse normal stress across depth of the core. Development of super convergent finite Euler Bernoulli beam elements Eul4d (4 dof element), Eul10d (10 dof element) are explained along with their results in Chapter 2. Development of different super convergent finite Timoshenko beam elements namely Tim4d (4 dof), Tim7d (7 dof), Tim10d (10 dof) are explained in Chapter 3. Validation of Euler Bernoulli and Timoshenko elements developed in the present work is carried out with test cases available in the open literature for displacements and free vibration frequencies are presented in Chapter 2 and Chapter 3. The results indicates that all developed elements are performing exceedingly well for static loads and free vibration. Super convergence performance for the elements developed is demonstrated with related examples. Spectral elements based on Timoshenko theory STim7d, STim6d, STim6dF are developed and the wave propagation characteristics studies are presented in Chapter 4. Euler spectral elements are derived from Timoshenko spectral elements by enforcing in finite shear rigidity, designated as SEul7d, SEul6d, SEul6dF and are presented. E orts were made in this present work to model the horizontal cracks in top or bottom face sheets using the spectral elements and the methodology is presented in Chapter 4. Wave propagation analysis using general purpose software N AST RAN and the super convergent as well as spectral elements developed in this work, are discussed in detail in Chapter 5. Modeling aspects of sandwich beam in N AST RAN using various combination of elements available and the performance of four possible models simulated were studied. Validation of all four models in N AST RAN, Super convergent Euler, Timoshenko and Spectral Timoshenko finite elements was carried out by simulating a homogenous I beam by comparing the longitudinal and transverse responses. Studies were carried out to find out the response predictions of a sandwich beam with soft core and all the predictions were compared and discussed. The responses in case of cracks in top or bottom face sheets under the longitudinal and transverse loading were studied in this chapter. In Chapter 6, Parametric studies were carried out for bringing out the sensitiveness of the important specific parameters in overall behaviour and performance of a sandwich beam, using Super convergent and Spectral elements developed. This chapter clearly brings out the various aspects of design of sandwich beam such as material selection of core, geometrical configuration of overall beam and core. Effects of shear modulus, mass density on wave propagation characteristics, effects of thick or thin cores with reference to the face sheets and dynamic effects of core are highlighted. Wave propagation characteristics studies includes the study of wave numbers, group speeds, cut off frequencies for a given configuration and identification of frequency zone of operations. The recommendations for improvement in design of sandwich beams based on the parametric studies are made at the end of chapter. The entire thesis, written in seven Chapters, presents a unified treatment of sandwich beam analysis that will be very useful for designers working in the area.

碩士
國立成功大學
土木工程學系
103
A Reissner’s mixed variational theorem (RMVT)-based nonlocal Euler-Bernoulli beam theory (EBT) is developed for the bending, free vibration and buckling analyses of a single-layered nanobeam (SLNB) (or a single-walled carbon nanotube, SWCNT) embedded in an elastic medium and with combinations of simply-supported and clamped edges. The interaction effect between the SLNB/SWCNT and its surrounding elastic medium is simulated using either a Winkler or a Pasternak foundation model. The SLNB/SWCNT’s equations of motion and the associated possible boundary conditions are derived by using Hamilton’s principle combined with Eringen’s nonlocal constitutive relations. A meshless collocation method is applied to obtain the deflection and stress-resultant components induced in a loaded SLNB/SWCNT, frequency parameters of an unloaded SLNB/SWCNT, and critical load parameters of an axially-loaded one, in which a differential reproducing kernel interpolation method is used to construct the shape functions of each field variable.

Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using different beam theories. In most cases, the governing differential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions. The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six different boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three different axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded. We assume simple polynomial mode shapes which satisfy the different boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiffness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The effect of different parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases. The derived analytical solutions can also serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the derived stiffness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.

Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using different beam theories. In most cases, the governing differential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions. The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six different boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three different axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded. We assume simple polynomial mode shapes which satisfy the different boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiffness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The effect of different parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases. The derived analytical solutions can also serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the derived stiffness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.

Advances in the design and manufacturing technologies have greatly enhanced the utility of fiber reinforced composite materials in aircraft, helicopter and space- craft structural components. The special characteristics of composites such as high strength and stiffness, light-weight corrosion resistance make them suitable sub- stitute for metals/metallic alloys. However, composites are very sensitive to the anomalies induced during their fabrication and service life. Also, they are suscepti- ble to the impact and high frequency loading conditions because the epoxy matrix is at-least an order of magnitude weaker than the embedded reinforced carbon fibers. On the other hand, the carbon based matrix posses high electrical conductivity which is often undesirable. Subsequently, the metal matrix produces high brittleness. Var- ious forms of damage in composite laminates can be identified as indentation, fiber breakage, matrix cracking, fiber-matrix debonding and interply disbonding (delam- ination). Among all the damage modes mentioned above, delamination has been found to be serious for all cases of loading. They are caused by excessive interlaminar shear and normal stresses. The interlaminar stresses that arise in the case of composite materials due to the mismatch in the elastic constants across the plies. Delamination in composites reduce it’s tensile and compressive strengths by consid- erable margins. Hence the knowledge of these stresses is the most important aspect to be looked into. Basic theories like the Euler-Bernoulli’s theory and Timoshenko beam theory are based on many assumptions which poses limitation to determine these stresses accurately. Hence the determination of these interlaminar stresses accurately requires higher order theories to be considered. Most of the conventional methods of determination of the stresses are through the solutions, involving the trigonometric series, which are available only to small and simple problems. The most common method of solution is by Finite Element (FE) Method. There are only few elements existing in the literature and very few in the commercially available finite element software to determine the interlaminar stresses accurately in the composite laminates. Accuracy of finite element solution depends on the choice of functions to be used as interpolating polynomials for the field variable. In-appropriate choice will manifest in the form of delayed convergence. This delayed convergence and accuracy in predicting these stresses necessiates a formulation of elements with a completely new concept. The delayed convergence is sometimes attributed to the shear locking phenomena, which exist in most finite element formulation based on shear deformation theories. The present work aims in developing finite elements based on higher order theories, that alleviates the slow convergence and achieves the solutions at a faster rate without compromising on the accuracy. The accuracy primarily depends on the theory used to model the problem. Thus the basic theories (such as Elementary Beam theory and Timoshenko Beam theory) does not suffice the condition to accuratley determine the interlaminar stresses through the thickness, which is the primary cause for delamination in composites. Two different elements developed on the principle of super-convergence has been presented in this work. These elements are subjected to several numerical experiments and their performance is assessed by comparing the solutions with those available in literature. Spacecraft and aircraft structures are light in weight and are also lightly damped because of low internal damping of the material of construction. This increased exibility may allow large amplitude vibration, which might cause structural instability. In addition, they are susceptible to impact loads of very short duration, which excites many structural modes. Hence, structural dynamics and wave propagation study becomes a necessity. The wave based techniques have found appreciation in many real world problems such as in Structural Health Monitoring (SHM). Wave propagation problems are characterized by high frequency loads, that sets up stress waves to propagate through the medium. At high frequency, the wave lengths are small and from the finite element point of view, the element sizes should be of the same order as the wave lengths to prevent free edges of the element to act as a free boundary and start reflecting the stress waves. Also longer element size makes the mass distribution approximate. Hence for wave propagation problems, very large finite element mesh is an absolute necessity. However, the finite element problems size can be drastically reduced if we characterize the stiffness of the structure accurately. This can accelerate the convergence of the dynamic solution significantly. This can be acheived by the super-convergent formulation. Numerical results are presented to illustrate the efficiency of the new approach in both the cases of dynamic studies viz., the free vibration study and the wave propagation study. The thesis is organised into five chapters. A brief organization of the thesis is presented below, Chapter-1 gives the introduction on composite material and its constitutive law. The details of shear locking phenomena and the interlaminar stress distribution across the thickness is brought out and the present methods to avoid shear locking has been presented. Chapter-2 presents the different displacement based higher order shear deformation theories existing in the literature their advantages and limitations. Chapter-3 presents the formulation of a super-convergent finite element formulation, where the effect of lateral contraction is neglected. For this element static and free vibration studies are performed and the results are validated with the solution available in the open literature. Chapter-4 presents yet another super-convergent finite element formulation, wherein the higher order effects due to lateral contraction is included in the model. In addition to static and free vibration studies, wave propagation problems are solved to demonstrate its effectiveness. In all numerical examples, the super-convergent property is emphasized. Chapter-5 gives a brief summary of the total research work performed and presents further scope of research based on the current research.

Advances in the design and manufacturing technologies have greatly enhanced the utility of fiber reinforced composite materials in aircraft, helicopter and space- craft structural components. The special characteristics of composites such as high strength and stiffness, light-weight corrosion resistance make them suitable sub- stitute for metals/metallic alloys. However, composites are very sensitive to the anomalies induced during their fabrication and service life. Also, they are suscepti- ble to the impact and high frequency loading conditions because the epoxy matrix is at-least an order of magnitude weaker than the embedded reinforced carbon fibers. On the other hand, the carbon based matrix posses high electrical conductivity which is often undesirable. Subsequently, the metal matrix produces high brittleness. Var- ious forms of damage in composite laminates can be identified as indentation, fiber breakage, matrix cracking, fiber-matrix debonding and interply disbonding (delam- ination). Among all the damage modes mentioned above, delamination has been found to be serious for all cases of loading. They are caused by excessive interlaminar shear and normal stresses. The interlaminar stresses that arise in the case of composite materials due to the mismatch in the elastic constants across the plies. Delamination in composites reduce it’s tensile and compressive strengths by consid- erable margins. Hence the knowledge of these stresses is the most important aspect to be looked into. Basic theories like the Euler-Bernoulli’s theory and Timoshenko beam theory are based on many assumptions which poses limitation to determine these stresses accurately. Hence the determination of these interlaminar stresses accurately requires higher order theories to be considered. Most of the conventional methods of determination of the stresses are through the solutions, involving the trigonometric series, which are available only to small and simple problems. The most common method of solution is by Finite Element (FE) Method. There are only few elements existing in the literature and very few in the commercially available finite element software to determine the interlaminar stresses accurately in the composite laminates. Accuracy of finite element solution depends on the choice of functions to be used as interpolating polynomials for the field variable. In-appropriate choice will manifest in the form of delayed convergence. This delayed convergence and accuracy in predicting these stresses necessiates a formulation of elements with a completely new concept. The delayed convergence is sometimes attributed to the shear locking phenomena, which exist in most finite element formulation based on shear deformation theories. The present work aims in developing finite elements based on higher order theories, that alleviates the slow convergence and achieves the solutions at a faster rate without compromising on the accuracy. The accuracy primarily depends on the theory used to model the problem. Thus the basic theories (such as Elementary Beam theory and Timoshenko Beam theory) does not suffice the condition to accuratley determine the interlaminar stresses through the thickness, which is the primary cause for delamination in composites. Two different elements developed on the principle of super-convergence has been presented in this work. These elements are subjected to several numerical experiments and their performance is assessed by comparing the solutions with those available in literature. Spacecraft and aircraft structures are light in weight and are also lightly damped because of low internal damping of the material of construction. This increased exibility may allow large amplitude vibration, which might cause structural instability. In addition, they are susceptible to impact loads of very short duration, which excites many structural modes. Hence, structural dynamics and wave propagation study becomes a necessity. The wave based techniques have found appreciation in many real world problems such as in Structural Health Monitoring (SHM). Wave propagation problems are characterized by high frequency loads, that sets up stress waves to propagate through the medium. At high frequency, the wave lengths are small and from the finite element point of view, the element sizes should be of the same order as the wave lengths to prevent free edges of the element to act as a free boundary and start reflecting the stress waves. Also longer element size makes the mass distribution approximate. Hence for wave propagation problems, very large finite element mesh is an absolute necessity. However, the finite element problems size can be drastically reduced if we characterize the stiffness of the structure accurately. This can accelerate the convergence of the dynamic solution significantly. This can be acheived by the super-convergent formulation. Numerical results are presented to illustrate the efficiency of the new approach in both the cases of dynamic studies viz., the free vibration study and the wave propagation study. The thesis is organised into five chapters. A brief organization of the thesis is presented below, Chapter-1 gives the introduction on composite material and its constitutive law. The details of shear locking phenomena and the interlaminar stress distribution across the thickness is brought out and the present methods to avoid shear locking has been presented. Chapter-2 presents the different displacement based higher order shear deformation theories existing in the literature their advantages and limitations. Chapter-3 presents the formulation of a super-convergent finite element formulation, where the effect of lateral contraction is neglected. For this element static and free vibration studies are performed and the results are validated with the solution available in the open literature. Chapter-4 presents yet another super-convergent finite element formulation, wherein the higher order effects due to lateral contraction is included in the model. In addition to static and free vibration studies, wave propagation problems are solved to demonstrate its effectiveness. In all numerical examples, the super-convergent property is emphasized. Chapter-5 gives a brief summary of the total research work performed and presents further scope of research based on the current research.

This thesis work provides an insight into the design of Force-amplifying Compliant Mechanisms (FaCMs) that are integrated with micromachined resonant accelerometers to increase their sensitivity. An FaCM, by mechanically amplifying the inertial force, enhances the shift in the resonance frequency of the beams used for sensing the acceleration whose effect causes an axial force on the beams. An extensive study on different configurations of resonators namely, single beam resonator, single-ended tuning fork (SETF), and double-ended tuning fork (DETF), is carried out to gain insights about their resonant behavior. The influence of the boundary conditions on the sensor’s sensitivity emerged from the study. We found that not only the force-amplification factor but also the multi-axial stiffness of the FaCM and proof-mass influence the resonance frequency of the resonator as well as the bandwidth of the modified sensor for certain configurations but not all. Thus, four lumped parameters were identified to quantify the effectiveness of an FaCM. These parameters determine the boundary condition of the sensing beams and also the forces and the moment transmitted to them. Also presented in this work is a computationally efficient model, called the Lumped Parameter Model (LPM) for evaluation of the sensitivity. An analytical expression for the frequency-shift of the sensing resonator beams is obtained by considering the FaCM stiffness parameters as well as the lumped stiffness of the suspension of the inertial mass. Various FaCMs are evaluated and compared to understand how the four lumped parameters influence the sensor’s sensitivity. The FaCMs are synthesized using topology optimization to maximize the net amplification factor with the volume constraint. One of the FaCMs outperforms the lever by a factor of six. Microfabrication of resonant accelerometer coupled with FaCM and comb-drive actuator is carried out using a silicon-on-insulator process. Finally, the selection map technique, a compliant mechanism redesign methodology is used for enhancing the amplification of FaCMs. This technique provides scope for further design improvement in FaCMs for given sensor specifications.

This thesis work provides an insight into the design of Force-amplifying Compliant Mechanisms (FaCMs) that are integrated with micromachined resonant accelerometers to increase their sensitivity. An FaCM, by mechanically amplifying the inertial force, enhances the shift in the resonance frequency of the beams used for sensing the acceleration whose effect causes an axial force on the beams. An extensive study on different configurations of resonators namely, single beam resonator, single-ended tuning fork (SETF), and double-ended tuning fork (DETF), is carried out to gain insights about their resonant behavior. The influence of the boundary conditions on the sensor’s sensitivity emerged from the study. We found that not only the force-amplification factor but also the multi-axial stiffness of the FaCM and proof-mass influence the resonance frequency of the resonator as well as the bandwidth of the modified sensor for certain configurations but not all. Thus, four lumped parameters were identified to quantify the effectiveness of an FaCM. These parameters determine the boundary condition of the sensing beams and also the forces and the moment transmitted to them. Also presented in this work is a computationally efficient model, called the Lumped Parameter Model (LPM) for evaluation of the sensitivity. An analytical expression for the frequency-shift of the sensing resonator beams is obtained by considering the FaCM stiffness parameters as well as the lumped stiffness of the suspension of the inertial mass. Various FaCMs are evaluated and compared to understand how the four lumped parameters influence the sensor’s sensitivity. The FaCMs are synthesized using topology optimization to maximize the net amplification factor with the volume constraint. One of the FaCMs outperforms the lever by a factor of six. Microfabrication of resonant accelerometer coupled with FaCM and comb-drive actuator is carried out using a silicon-on-insulator process. Finally, the selection map technique, a compliant mechanism redesign methodology is used for enhancing the amplification of FaCMs. This technique provides scope for further design improvement in FaCMs for given sensor specifications.

The aim of this work is to develop and validate a continuum model for the simulation of the thermomechanical response of a shape memory polymer (SMP). Rather than integral type viscoelastic model, the approach here is based on the idea of two inter-penetrating networks, one which is permanent and the other which is transient together with rate equations for the time evolution of the transient network. We find that the activation stress for network breakage and formation of the material controls the gross features of the response of the model, and exhibits a "thermal Bauschinger effect". The model developed here is similar to a thermoviscoelastic model, and is developed with an eye towards ease of numerical solutions to boundary value problems. The primary hypothesis of this model is that the hysteresis of temperature dependent activation-stress plays a lead role in controlling its main response features. Validation of this hypothesis is carried out for the uniaxial response from the experimental data available in the literature for two different SMP samples: shape memory polyurethane and Veriflex, to show the control of the evolution of the temperature sensitive activation stress on the response. We extend the validated 1D model to a three dimensional small strain continuum SMP model and carry out a systematic parameter optimization method for the identification of the activation stress coefficients, with different weights given to different features of the response to match the parameters with experimental data. A comprehensive parametric study is carried out, that varies each of the model material and loading parameters, and observes their effect on design-relevant response characteristics of the model undergoing a thermomechanical cycle. We develop "response charts" for the response characteristics: shape fixity, shape recovery and maximum stress rise during cooling, to give the designer an idea of how the simultaneous variation of two of the most influential material parameters changes a specific response parameter. To exemplify the efficacy of the model in practical applications, a thermoviscoelastic extension of a beam theory model will be developed. This SMP beam theory will account for activation stress governed inelastic response of a SMP beam. An example of a three point bend test is simulated using the beam theory model. The numerical solution is implemented by using an operator split technique that utilizes an elastic predictor and dissipative corrector. This algorithm is validated by using a three-point bending experiment for three different material cases: elastic, plastic and thermoplastic response. Time step convergence and mesh density convergence studies are carried out for the thermoviscoelastic FEM model. We implement and study this model for a SMP beam undergoing three-point bending strain recovery, stress recovery and cyclic thermomechanical loading. Finally we develop a thermodynamically consistent finite continuum model to simulate the thermomechanical response of SMPs. The SMP is modeled as an isotropic viscoplastic material where thermal changes govern the evolution of the activation stress of the material. The response of the SMP in a thermomechanical cycle is modeled as a combination of a rubbery and a glassy element in series. Using these assumptions, we propose a specific form for the Helmholtz potential and the rate of dissipation. We use the technique of upper triangular decomposition for developing the constitutive equations of the finite strain SMP model. The resulting model is implemented in an ODE solver in MATLAB, and solved for a simple shear problem. We study the response of the SMP model for shear deformation as well as cyclic shear deformation at different initial temperatures. Finally, we implement the thermomechanical cycle under shear deformations and study the behavior of the model.

Nonlinear finite element models of functionally graded beams with power-law variation of material, accounting for the von-Karman geometric nonlinearity and temperature dependent material properties as well as microstructure dependent length scale have been developed using the Euler-Bernoulli as well as the first-order and third- order beam theories. To capture the size effect, a modified couple stress theory with one length scale parameter is used. Such theories play crucial role in predicting accurate deflections of micro- and nano-beam structures. A general third order beam theory for microstructure dependent beam has been developed for functionally graded beams for the first time using a modified couple stress theory with the von Karman nonlinear strain. Finite element models of the three beam theories have been developed. The thermo-mechanical coupling as well as the bending-stretching coupling play significant role in the deflection response. Numerical results are presented to show the effect of nonlinearity, power-law index, microstructural length scale, and boundary conditions on the bending response of beams under thermo-mechanical loads. In general, the effect of microstructural parameter is to stiffen the beam, while shear deformation has the effect of modeling more realistically as a flexible beam.