The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations. / Ph. D.

O presente trabalho trata da análise numérica de cascas abatidas com o uso do método dos elementos de contorno (MEC). A formulação é desenvolvida a partir do acoplamento entre as equações integrais para flexão de placas delgadas e para estado plano de tensão. No esquema implementado, os termos sobre o contorno são avaliados a partir de processos analíticos e numéricos de integração. No caso das integrais de domínio, aplica-se um procedimento semi-analítico de cálculo sobre células discretas. A validação do modelo computacional desenvolvido é feita com base em resultados da literatura, obtidos com uso do método dos elementos finitos e dos elementos de contorno, além de soluções analíticas. / The present work deals with the numerical analysis of shallow shells using boundary element method (BEM). The formulation is developed by coupling integral equations of plate bending and plane stress elasticity. In the implemented scheme, the boundary terms are evaluated with analytical and numerical processes of integration. In the case of domain integrals, a semi-analytical calculation procedure is applied on discrete cells. The validation of developed computational model is made with results from other works, obtained by use of BEM or finite element method, besides analytical solutions.

O presente trabalho trata da análise numérica de cascas abatidas com o uso do método dos elementos de contorno (MEC). A formulação é desenvolvida a partir do acoplamento entre as equações integrais para flexão de placas delgadas e para estado plano de tensão. No esquema implementado, os termos sobre o contorno são avaliados a partir de processos analíticos e numéricos de integração. No caso das integrais de domínio, aplica-se um procedimento semi-analítico de cálculo sobre células discretas. A validação do modelo computacional desenvolvido é feita com base em resultados da literatura, obtidos com uso do método dos elementos finitos e dos elementos de contorno, além de soluções analíticas. / The present work deals with the numerical analysis of shallow shells using boundary element method (BEM). The formulation is developed by coupling integral equations of plate bending and plane stress elasticity. In the implemented scheme, the boundary terms are evaluated with analytical and numerical processes of integration. In the case of domain integrals, a semi-analytical calculation procedure is applied on discrete cells. The validation of developed computational model is made with results from other works, obtained by use of BEM or finite element method, besides analytical solutions.