DarkSide-20k, currently under construction at the INFN Gran Sasso Laboratory, involves the deployment of a large-scale cryogenic system within a underground environment. The detector is hosted inside a cubic cryostat with a characteristic size of approximately 8 m per side, operating at cryogenic temperatures. Its integration requires the design of a highly customised steel support system together with an extended set of auxiliary facilities, including multi-level platforms, clean-room structures, lifting devices and precision handling assemblies. In this context, the installation phases represent critical transient conditions requiring dedicated structural assessment. This contribution focuses on the computational mechanics framework developed to design and verify both the primary structural system and the auxiliary infrastructures. A hierarchical suite of finite element models was implemented to capture thermo-mechanical coupling, contact and friction interfaces, and load-transfer mechanisms across hybrid assemblies. Particular attention is devoted to the interaction between the cryostat and temporary support systems during positioning, sliding and lowering operations, where local stress concentrations and strict deformation limits become dominant design drivers.