Mechanical design analysis of MQXFB, the 7.2-m-long low-$\beta$ quadrupole for the High-Luminosity LHC upgrade

As part of the High-Luminosity Large Hadron Collider (LHC) Project, a set of Nb$_{3}$Sn quadrupoles are being developed, aiming to enhance the performance of the inner triplets. The new magnets, identified as MQXFA and MQXFB, will share the same cross section with two different lengths, 4.2 and 7.2 m, respectively. During the magnet development, three short models were tested, along with a number of mechanical models, demonstrating the capability of the magnet cross section to achieve the specified performances. The same performances are now required for the full-length magnets. To ensure this, the authors studied the impact of the magnet length on the capability of the structure to provide an adequate support to the coils. Finite element and simplified analytical models were used to evaluate the impact of the magnet length on the stresses in the magnet ends and coil elongation during powering. The models were calibrated using the results from the short model tests, and used to provide an indication on the required prestress and its foreseen impact on the magnet performance