Formula??es num?ricas para an?lise de vigas em contato com bases el?sticas.

Programa de P?s Gradua??o em Engenharia Civil. Departamento de Engenharia Civil, Escola de Minas, Universidade Federal de Ouro Preto.Este trabalho tem como objetivo principal o desenvolvimento de duas metodologias capazes de resolver o problema de equil?brio de vigas com restri??es de contato. Essas restri??es de contato s?o impostas aqui por bases el?sticas modeladas com um par?metro de rigidez (modelo de Winkler ou molas discretas), e duas situa??es de contato s?o consideradas, a saber: bilateral e unilateral. No caso de contato unilateral, a funda??o el?stica reage somente ?s solicita??es de compress?o; j? na situa??o de contato bilateral, a base reage ?s solicita??es de tra??o e compress?o. Na primeira parte do trabalho, uma metodologia geral de solu??o baseada no emprego do m?todo de Rayleigh?Ritz ? proposta e usada em seguida para resolver tr?s problemas particulares de vigas com restri??es unilaterais de contato. Uma estrat?gia de solu??o iterativa, baseada no m?todo de Newton?Raphson, ? usada para resolver o sistema de equa??es n?o-lineares resultante da formula??o do problema. Na segunda parte da pesquisa, o m?todo dos elementos finitos ? usado para discretizar a viga e a funda??o el?stica, e o problema de contato ? tratado diretamente como um problema de minimiza??o, envolvendo somente as vari?veis originais do problema, sujeitas ?s restri??es de desigualdade e ? condi??o de complementaridade. Duas formula??es s?o ent?o desenvolvidas (primal e dual) onde as equa??es relevantes para a solu??o do problema de contato s?o escritas na forma de um problema de complementaridade linear (PCL) e resolvidas atrav?s do algoritmo de Lemke. As duas metodologias propostas s?o analisadas e testadas atrav?s de v?rios exemplos e as respostas obtidas atrav?s das implementa??es computacionais realizadas s?o comparadas com os resultados encontrados na literatura. Por fim, algumas conclus?es sobre as metodologias e as formula??es desenvolvidas e implementadas, e sobre as aproxima??es dos resultados s?o apresentadas no final do trabalho.Two numerical methodologies capable of solving equilibrium problems of beams with contact restraints are presented in this work. These contact constraints are imposed by one parameter elastic foundations model (Winkle?s model and discrete spring model) and two contact conditions are considered, i.e.: bilateral and unilateral. In the formulation, special attention is given to the case in which the elastic foundation reacts in compression only, characterizing the contact as unilateral. In the first part, a general Rayleigh?Ritz type methodology with moveable boundaries is proposed and used here to solve three particular problems of beams under unilateral contact constraint. An iterative solution strategy, based on the Newton? Raphson method, is used to solve the non-linear equation system obtained in the problem formulation. In the second part, the finite element method is used to model the beam and foundation, and the contact problem is dealt with directly as a minimization problem, involving only the original variables, subjected to inequality constraints and complementarity condition. The relevant equations are presented for two alternative linear complementarity problems (LCP) and solved by Lemke?s algorithm. In the first formulation, named primal, the LCP variables are the beam displacements and the elastic foundation reaction, while in the second formulation (dual), the LCP is derived in terms of the elastic foundation reaction only. The methodologies are illustrated and analyzed by many examples and the results are compared with existing numerical results found in the literature. Some conclusions about the precision of the results, implemented formulations and computational efficiency of these methodologies are presented at the end of the work