Pipelines tend to buckle laterally under thermal expansion. In existing analytical solutions by Kerr and Hobbs, it is assumed that the seabed resistance q0 to lateral pipe movements is constant in magnitude and opposite in direction to the total displacement. Here, it is opposite to the velocity instead, i.e., the seabed is taken to be frictional rather than nonlinear elastic with a V-shaped potential function. A three-lobe (“mode 3f”) analytical solution is provided for the frictional case, using the same approximate end-of-buckle condition v = v′ = v″ = 0 used by Hobbs in his “mode 3” solution for the nonlinear elastic case. For both modes 3 and 3f solutions, the shape of the buckle does not change as it grows with increasing thermal expansion, though the scaling factors in the axial and lateral directions are different, i.e., the solutions are self-similar. A single finite element solution for the frictional case with an initial imperfection imposed by a bumper can be scaled to cover all such cases. It shows that the shape of the buckle depends on the amplitude of the initial triggering imperfection and is close to the mode 3f solution for very small initial imperfections. The difference between modes 3 and 3f is significant in regard to buckle shape and the relative size of the buckle lobes, but small in regard to the maximum bending moment for a given amount of thermal expansion accommodated by the buckle.