Within the scope of the High-Luminosity LHC project, the collaboration between CERN and U.S. LARP is developing new low-$\beta$ quadrupoles using the Nb3Sn superconducting technology for the upgrade of the LHC interaction regions. The magnet support structure of the first short model was designed, and two units were fabricated and tested at CERN and at LBNL. The structure provides the preload to the collar-coil subassembly by an arrangement of outer aluminum shells pretensioned with water-pressurized bladders. For the mechanical qualification of the structure and the assembly procedure, superconducting coils were replaced with solid aluminum 'dummy coils,' and the structure was preloaded at room temperature and then cooled-down to 77 K. The mechanical behavior of the magnet structure was monitored with the use of strain gauges installed on the aluminum shells, the dummy coils, and the axial preload system. This paper reports on the outcome of the assembly and the cooldown tests with dummy coils, which were performed at CERN and at LBNL, and presents the strain gauge measurements compared with the 3-D finite-element model predictions.