As to long-span cable-stayed bridge with floating or half floating girder-tower style, researches reveal that proper elastic connects(EC) between tower (pier) and girder placed in longitudinal direction of the bridge can reduce the displacements between tower (pier) and girder and cause reasonable distributions of earthquake load in longitudinal direction. Firstly, the dynamic responses under complex loading in time domain of single degree of freedom(SDOF) attached with EC are studied to investigate the influence of EC on the dynamic response of SDOF. Secondly, a long-span cable-stayed bridge with half floating girder-tower style is taken as an example to investigate the influence of failure of EC on the seismic response, under two earthquake motions. It is found that the failure of EC has important effects on the dynamic response of long-span bridges and significant dynamic responses exist. If the failure of EC happens during the ground motion, the internal force of girder is much smaller when the failure of EC is considered than that of failure of EC is not considered, however the displacements of girder and top of the tower in longitudinal direction may increase significantly. ECs should be designed with reasonable failure limits to protect the main structures of the bridge during the earthquake.