The present paper deals with the analysis of cylindrical shells by the application of the Weighed Residual Method, more specifically the Least Squared Method, to obtain approximate solutions of shells structural problems, especially the cylindrical reservoirs subjected to hydrostatic shipment in the linear behavior regime. The ways of obtaining the approximate solutions refer to the adoption of linear, polynomial approximate bases, as well as the possibility of enriching the approximation by adding functions with similar characteristics to the accurate solution itself. Such procedures can be useful in the analysis of structures to significantly prevent the rise of the computational effort, by means of an approximate base that corresponds to the characteristics required by the analytical solution of the problem.

The distribution of temperature, the deformations and the cutting forces in the cutting zone during chip formation are very important aspects of the process. Therefore, they have great influence in results of machining. Tool wear, the precision of the machining operations and the finishing superface are some of the results that affect the economy of the process. This study has the objective of an evaluation offered potentialities for simulating chip formation using Finite Element Method (FEM). Four models with sufficiently different principles are used to exploring the most recent innovations. In the models, WC tools coated TiN was used to machine AISI 4340 steel. The simulations provide a study and detailed examination of temperature distribution, deformations, cutting forces, flow of material and an enormous amount of information that could be useful for the analysis of new processes and optimization of existing processes. Additionally, they open new horizon in the study of the chip formation. The simulations also demonstrate the complexity of the chip formation process, which creates many difficulties for its analysis using based analytical equations in constants. Some comparisons are established with experimental results found in temperature measurements.

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