The vibration of a uniform geometrically nonlinear shear deformable beam subjected to a transverse harmonic excitation is investigated by the method of numerical simulation. Rotatory and axial inertia are included in the model. The beam is simply supported with supports a fixed distance apart. The nonlinear partial differential equations of motion are discretized in space by the Rayleigh-Ritz method, resulting in a set of nonlinear ordinary differential equations in time. The ordinary differential equations are integrated numerically to produce a time history of the solution of the equations. Transverse displacement, axial displacement, and cross sectional rotation are approximated by series of the corresponding linear natural mode shapes of the beam. Solutions of the equations of motion are compared to corresponding solutions where shear deformation and rotatory inertia are neglected. The effect of slenderness on the difference between the shear deformable case and the non shear deformable case is investigated by considering two beam configurations. In the simulations considered, the difference between the shear deformable model and the non shear deformable model increases as excitation frequency is increased and the length to thickness ratio of the beam is decreased. / Master of Science

We consider the application of high-order spectral/hp finite element technology to the numerical solution of boundary-value problems arising in the fields of fluid and solid mechanics. For many problems in these areas, high-order finite element procedures offer many theoretical and practical computational advantages over the low-order finite element technologies that have come to dominate much of the academic research and commercial software of the last several decades. Most notably, we may avoid various forms of locking which, without suitable stabilization, often plague low-order least-squares finite element models of incompressible viscous fluids as well as weak-form Galerkin finite element models of elastic and inelastic structures. The research documented in this dissertation includes applications of spectral/hp finite element technology to an analysis of the roles played by the linearization and minimization operators in least-squares finite element models of nonlinear boundary value problems, a novel least-squares finite element model of the incompressible Navier-Stokes equations with improved local mass conservation, weak-form Galerkin finite element models of viscoelastic beams and a high-order seven parameter continuum shell element for the numerical simulation of the fully geometrically nonlinear mechanical response of isotropic, laminated composite and functionally graded elastic shell structures. In addition, we also present a simple and efficient sparse global finite element coefficient matrix assembly operator that may be readily parallelized for use on shared memory systems. We demonstrate, through the numerical simulation of carefully chosen benchmark problems, that the finite element formulations proposed in this study are efficient, reliable and insensitive to all forms of numerical locking and element geometric distortions.

This research endeavor develops Isogeometric approach for analysis of composite structures and take advantage of higher order continuity, smoothness and variation diminishing property of Nurbs basis for stress analysis of composite and sandwich beams and plates. This research also computes stress concentration factor in a composite plate with a hole. Isogeometric nonlinear/linear finite element code is developed for static and dynamic analysis of laminated composite plates. Nurbs linear, quadratic, higher-order and k-refined elements are constructed using various refinement procedures and validated with numerical testing. Nurbs post-processor for in-plane and interlaminar stress calculation in laminated composite and sandwich plates is developed. Nurbs post-processor is found to be superior than regular finite element and in good agreement with the literature. Nurbs Isgoemetric analysis is used for stress analysis of laminated composite plate with open-hole. Stress concentration factor is computed along the hole edge and good agreement is obtained with the literature. Nurbs Isogeometric finite element code for free-vibration and linear dynamics analysis of laminated composite plates also obtain good agreement with the literature. Main highlights of the research are newly developed 9 control point linear Nurbs element, k-refined and higher-order Nurbs elements in isogeometric framework. Nurbs elements remove shear-locking and hourglass problems in thin plates in context of first-order shear deformation theory without the additional step and compute better stresses than Lagrange finite element and higher order shear deformation theory for comparatively thick plates i.e. a/h = 4. Also, Nurbs Isogeometric analysis perform well for vibration and dynamic problems and for straight and curved edge problems. / Ph. D.

Meshless methods provide a highly continuous approximation field, convenient for thin structures like shells. Nevertheless, the lack of Kronecker Delta property makes the formulation of essential boundary conditions not straightforward, as the trial and test fields cannot be tailored to boundary values. Similar problem arise when different approximation regions must be joined, in a multi-region problem, such as kinks, folds or joints. This work presents three approaches to impose both kinematic conditions: the well known Lagrange Multiplier method, used since the beginning of the Element Free Galerkin method; a pure penalty approach; and the recently rediscovered alternative of Nitsche\'s Method. We use the EFG discretization technique for thick Reissner-Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local refinement on multi-region problems. / Métodos sem malha geram campos de aproximação com alta continuidade, convenientes para estruturas finas como cascas. No entanto, a ausência da propriedade de Delta de Kronecker dificulta a formulação de condições essenciais de contorno, já que os campos de aproximação e teste não podem ser moldados aos valores de contorno. Um problema similar aparece quando diferentes regiões de aproximação precisam ser juntadas em um problema multi-regiões como dobras, vincos ou junções. Este trabalho apresenta três métodos de imposição ambas condições cinemáticas: o já conhecido método dos multiplicadores de Lagrange, usado desde o começo do método de Galekin sem elementos (EFG); uma abordagem de penalidade pura; e o recentemente redescoberto método de Nitsche. Nós usamos a técnica de discretização com EFG para cascas espessas de Reissner-Mindlin e adaptamos a forma fraca de forma a separar graus de liberdade de deslocamento e rotação e obter coeficientes de estabilização diferentes e apropriados. Essa abordagem permite a modelagem de cascas discontínuas e o refinamento local em problemas multi-regiões.

Meshless methods provide a highly continuous approximation field, convenient for thin structures like shells. Nevertheless, the lack of Kronecker Delta property makes the formulation of essential boundary conditions not straightforward, as the trial and test fields cannot be tailored to boundary values. Similar problem arise when different approximation regions must be joined, in a multi-region problem, such as kinks, folds or joints. This work presents three approaches to impose both kinematic conditions: the well known Lagrange Multiplier method, used since the beginning of the Element Free Galerkin method; a pure penalty approach; and the recently rediscovered alternative of Nitsche\'s Method. We use the EFG discretization technique for thick Reissner-Mindlin shells and adapt the weak form as to separate displacement and rotational degrees of freedom and obtain suitable and separate stabilization parameters. This approach enables the modeling of discontinuous shells and local refinement on multi-region problems. / Métodos sem malha geram campos de aproximação com alta continuidade, convenientes para estruturas finas como cascas. No entanto, a ausência da propriedade de Delta de Kronecker dificulta a formulação de condições essenciais de contorno, já que os campos de aproximação e teste não podem ser moldados aos valores de contorno. Um problema similar aparece quando diferentes regiões de aproximação precisam ser juntadas em um problema multi-regiões como dobras, vincos ou junções. Este trabalho apresenta três métodos de imposição ambas condições cinemáticas: o já conhecido método dos multiplicadores de Lagrange, usado desde o começo do método de Galekin sem elementos (EFG); uma abordagem de penalidade pura; e o recentemente redescoberto método de Nitsche. Nós usamos a técnica de discretização com EFG para cascas espessas de Reissner-Mindlin e adaptamos a forma fraca de forma a separar graus de liberdade de deslocamento e rotação e obter coeficientes de estabilização diferentes e apropriados. Essa abordagem permite a modelagem de cascas discontínuas e o refinamento local em problemas multi-regiões.