Seabed pipelines undergo temperature cycles that create axial load which can be relieved through controlled lateral buckling. The prediction of lateral buckling in design requires accurate assessment of the lateral breakout resistance. This Technical Note describes upper bound plasticity analysis of a partially-embedded pipe on undrained soil. The purpose is to generate failure envelopes for vertical and horizontal loading to provide a theoretical basis for estimating breakout resistance. The following cases have been considered: smooth and rough pipes, with and without separation at the rear face of the pipe. The envelopes are similar to those developed previously for surface foundations, but capture additional effects that are due to the curved geometry of the pipe surface. The breakout resistance and the movement of the pipe at failure are strongly influenced by the separation condition. Pipe roughness and soil self-weight have a relatively minor effect on breakout resistance. Existing empirical expressions usually assume a linear variation in breakout resistance with embedment and vertical load. This theoretical analysis demonstrates that these relationships are non-linear. The resulting envelopes provide a more rigorous basis for predicting the breakout resistance of partially-embedded pipelines.