This paper is an analytical investigation of a proposed vacuum barrier window that isolates the proton beam transport vacuum envelope from the Ultra Cold Neutron (UCN) experimental target system at atmospheric pressure. The window is subjected to static pressure and cyclic thermal stresses as the accelerated particle beam passes through it and deposits a small amount of energy in the window. The analysis investigates various beam rms sizes for two beam delivery time structures. The 0.1-mm thick, 52 mm diameter window is made of inconel alloy 718 and is welded to the beamline tube at its outer edge. For some combinations of delivery time structure and beam size, the window under differential pressure and proton beam heating experiences stress that is well above yield and possibly large enough to break the inconel foil. In order to analyze the induced temperature and stress, a finite element model has been developed. The model has been written parametrically to allow the beam characteristics, window material properties, dimensions and mesh densities to be easily adjusted. The heat load is applied to the model through the use of a 3-dimentional table containing the calculated volumetric heat rates. The heat load is based on a radial distribution for a circular Gaussian beam under both normal and extensional operation cases. In this analysis, a radial-centered, circular beam is assumed. The results of several analyses are presented in this paper.