In most cases, piping systems will experience the combination action of internal pressure and temperature. The limit analysis can be used to estimate the carrying capacity of a piping system. This piping system was analysed in the paper with one dimensional elastic method. The key part of the piping system was also analyzed with the 3-D limit analysis method to estimate the carrying capacity of the whole piping system. A direct iterative algorithm was used to determine the kinematic limit load of the 3-D structure. The effects of four kinds of typical slots on the limit loads of pipe elbows were investigated in detail. The failure modes corresponding to the different slot configurations were studied.