This master thesis aims to investigate some Pathological issues of the FEM on static analysis of plane problems. Its objective is to analyze the results of the plane elements and find a way to improve them in simple and efficient way. Four types of plane elements are being discussed; Constant Strain Triangle CST, Linear Strain Triangle LST, Bilinear Quadratic element Q4 and Quadratic Quadrature element Q8. Each type has its advantages and defects. Some locking problems as shear locking and compressibility locking are discussed too. They are the main reasons of the wrong results of the plane elements. How the choice of the numerical integration can affect the analysis and overcome the shear locking problem. An illustrative example is analyzed to understand the locking problems. Computer program for each example has been written and included in the appendices. The program used for the analysis is MatLab.