This paper studies the effect of staged cooling of compounded cylinders in avoiding cracking due to the presence of large interference stresses and low fracture toughness in the presence of cryogenic temperatures. This study is motivated by the assembly procedure of the fulcrum (a compounded trunnion-hub assembly) of bascule bridges, where the fulcrum is shrunk by immersion in liquid nitrogen so that it can then be fitted into the girder of the bridge. In a few cases, the fulcrum developed cracks during the immersion in liquid nitrogen. To study the effect of staged cooling to avoid such cracking, a finite difference model was developed of a long compounded cylinder with axisymmetric response with temperature-dependent properties. The study showed that the resistance to failure was increased by as much as 50 per cent when the compounded cylinder is cooled first in a refrigerated air chamber and followed by immersion in liquid nitrogen.