Elongational behavior of four narrow molar mass distribution polystyrene melts of masses 50 000, 100 000, 200 000, and 390 000, g/mol, respectively was investigated up to Hencky strains of 5. All melts show strain hardening behavior. For the two highest molar mass polystyrenes, strain hardening starts at elongation rates larger than the inverse reptation time, and the steady-state elongational viscosities decrease with increasing elongation rate according to a power law with a power-law index of approximately - 1 / 2 instead of - 1 as predicted by the original Doi-Edwards tube model. Marrucci and Ianniruberto [Macromolecules 37, 3934 (2004)] have introduced an interchain pressure term arising from lateral forces between the chain and the tube wall into the Doi-Edwards model to account for the latter effect. Based on the molecular stress function theory allowing for a strain-dependent tube diameter, we show that the transient and steady-state elongational viscosities of the nearly monodisperse polystyrene melts can be modeled quantitatively by assuming affine chain deformation balanced by the interchain pressure term of Marrucci and Ianniruberto. The interchain pressure is governed by a tube diameter relaxation time tau(a), which is found to be larger than the Rouse time tau(R) of the chain, and which is the only parameter of the model. For monodisperse polystyrene melts of sufficient low molar mass, tau(a), is larger than the reptation time, and a maximum in the steady-state elongational viscosity is predicted.