The observation of quadruped animals has suggested that the body flexibility is important for the bounding gait. In particular, the body flexibility is assumed to be crucial for determining the ground reaction force (GRF). In this paper, we investigated the dynamic effect of the body flexibility on the GRF in quadruped bounding with simple physical models by focusing on the vertical motion. More specifically, we first showed that the body flexibility reduces the GRF using a simple two legged model in sagittal plane. Next, to clarify the GRF reduction mechanism, we further simplified the model, which has only one leg. This allowed us to analytically obtain periodic solutions. Based on the solutions, we investigated the mechanism that the body flexibility reduces the GRF during quadrupedal bounding.