We use polymer self-consistent field theory to study the physics involved in the permeation of charged dendrimer molecules across anionic lipid bilayer membranes. We specifically examine the influence of dendrimer shape deformations, neutral grafts, and pH conditions on the interactions between dendrimers and membranes. Our results indicate that the ability of the dendrimer to undergo conformational rearrangements plays a crucial role in influencing the interactions between the dendrimer and the membrane. At neutral pH, we observe that dendrimers with grafted chains are repelled by the anionic bilayers. However, decreasing the solution pH to endosomal conditions results in attractive dendrimer-membrane interactions under some parametric conditions. We observe that dendrimer insertion into the membrane results in a decreased value in membrane tension at which rupture occurs and, furthermore, that the rupture tension decreases with the addition of grafts to the dendrimer. Our results suggest that dendrimers grafted with neutral polymers can serve as effective pH sensitive delivery vectors.