Extreme disasters such as earthquakes, fires, and collisions can cause the partial or widespread collapse of building structures. Existing research on the resistance of building structures to progressive collapse has mainly considered the contribution of the slab to the beam stiffness, but not the continuous tension of the rebar, which implies that the contribution of the slab to the whole structure is conservatively evaluated. In this study, we selected as the research object the composite joints of a circular concrete-filled steel tubular structure. Using ABAQUS software, we established a numerical model with two different boundary conditions for the steel end in the plate. We analyzed the joint failure mechanism during failure of the vertical middle column and observed the influence of the constraint conditions of reinforcement in the plate on the bearing capacity of the joints' girder and catenary mechanisms. The results show that there are four stages in the joint collapse process:the beam mechanism, transformation mechanism, catenary mechanism, and failure stages. By comparing the resistance curves under two different constraint conditions, we found that the bearing capacity of the beam mechanism is increased by about 7% when considering the tensile action of reinforcement in the plate, whereas the bearing capacity of the catenary stage is not obvious.