A simplified model was developed to calculate the maximum tensile and compressive strains due to a uniform movement of a block of soil in a direction parallel to the pipe axis using a closed-form solution of the governing differential equations. The model employs the theory of plasticity for modelling the pipe material based on normality plastic flow rule, the von Mises yield criterion, and isotropic strain hardening. While the pipe was assumed to have a bilinear, stress-strain curve with strain hardening, the pipe-soil friction was assumed to have an elastic-perfectly plastic force-deformation response. The model accounts for the initial thermal axial strains in the pipe and biaxial state of stress in the pipe due to internal pressure. The model is capable of accommodating pipe bends at the ends of the sliding zone. The relationship between the ground displacement and pipe axial force at each interface of stable and sliding zones was obtained from closed-form solutions of governing differential equations, assuming both the stable and sliding zones are infinitely long. To prevent the overestimation of the axial strains in the pipe, a limiting scenario was considered where the soil was assumed to have yielded over the entire sliding zone. Equilibrium and compatibility equations were used to calculate the pipe axial forces and strains at the two interfaces. The simplified model for longitudinal ground movement was validated against finite element solutions. The validation example presented involves a 20-inch straight pipeline subjected to longitudinal ground movement over slide lengths of 50, 100 and 200 metres, as well as a semi-infinite sliding zone case.
Closed-form solutions of some partial differential equations via quasi-solutions I
