This paper presents the results of ASME Boiler and Pressure Vessel Code (BPVC) Section VIII Division 2 analyses on a containment pressure vessel system designed to host small-scale shock physics experiments containing hazardous materials. These shock physics experiments are imaged using proton radiography at Los Alamos Neutron Science Center (LANSCE). The pressure vessel system examined is separated into two subsystems, the Outer Pressure Containment Vessel (OPCV) and the Beam Pipe and Auxiliary Hardware (BPAH). The OPCV mates with the BPAH to transport the proton beam and form the outer containment pressure boundary. An Inner Pressure Confinement Vessel (IPCV) is mounted inside the containment pressure boundary. The IPCV can be used to host a variety of physics experiments, which can include high explosive and hazardous materials. The IPCV is an impulsively loaded pressure vessel, while the OPCV and beam pipes are statically loaded. The OPCV and BPAH are designed to satisfy ASME BPVC Section VIII Division 2 Alternative Rules, Part 5 Design by Analysis Requirements for all load conditions expected during service. The BPAH is a series of stainless-steel tubes with low-profile flange assemblies, bellows, and aluminum windows to allow for proton imaging. The low-profile flanges are required for clearance of magnet bores located periodically along the length of the BPAH. The bellows are included along the beamline for ease of installation. The OPCV is a pressure vessel designed to interface with the BPAH, IPCV, and proton imaging alignment system. The OPCV and BPAH are both designed to support full vacuum loads. Major considerations for the BPAH are thermal elongation, collapse of bellows, and stresses due to clamp supports. Lifting loads and static equivalent loads transmitted from the IPCV are additional considerations for the OPCV. Under ASME BPVC Section VIII Division 2 Part 5.2.4, Elastic-Plastic Stress Analysis Method, Paragraph 5.2.4.3(a), Global Criteria, the requirements for analysis and acceptance are defined for the load cases described. A finite element analysis (FEA) model, performed in ANSYS 2020 R1, evaluates the design of the OPCV and BPAH under the aforementioned ASME guidance and criteria.