This approximation involves considering the lateral soil movements as being uniform along the pipeline, which means that the effect of longitudinal bending deflections along the pipe is neglected. The current study seeks to develop a simplified 3D analysis method, but also includes more conventional 2D calculations for comparison purposes (to assess the relative ability of these two different approaches to capture the peak pulling forces and pressure distribution around and along the pipe). In order to facilitate the comparisons between these two types of analysis, pulling load (P) is presented in this article in units of load/unit length (the units resulting from 2D calculations), unless otherwise stated, despite the fact that load is not generally uniform in laterally loaded pipes subjected to bending (the 3D results, therefore, represent the average magnitude of load per unit length along the whole of the pipe being modelled).

Figure 6. Yu,m/D versus embedment ratio (Yu,m = Yu,e = Yu for rigid pipes). When a section of steel pipeline is subjected to forces associated with lateral soil movements, design calculations are needed to estimate the bending stresses that result and the impact of those stresses on pipe performance. Given the difficulty of undertaking 3D analysis of the soil-pipe interaction, a computationally efficient approach is to simulate the behaviour as a plane strain problem to estimate the lateral forces and bending moments.

Figure 8 a-c show the contact pressure distribution around the pipe circumference for the 2D analysis compared with the pressure distribution at the same load level obtained from the 3D analysis at the pipe mid-span. The three figures covering analyses for H/D = 3, 5 and 7 suggest that analyses using 3D models provide comparable calculations to the 2D models if the contact pressure is averaged around the pipe mid-span circumference. At burial depth of H/D = 3, the pressure is fairly uniformly distributed along the pipe when pulling loads are still at relatively low values (0.94 kN m−1) as can be seen in figure 8 d. However, the nonlinearity of pressure distribution increases for pulling loads between 4.7 and 6.12 kN m−1.

Figure 8. Lateral contact pressure distribution against pipe: (a-c) along the circumferential direction, (d) along the spring line direction. — the measured load versus deflection for the experiment on the steel test pipe 24 ; Because the present problem involves large deformations and the flow of soil around the pipe, the use of master-slave interaction is necessary.

The effect of soil modulus that varies with depth based on the Janbu stress function (denoted ‘Janbu initial’ subsequently) on the flexural pipe response is illustrated in figure 17 using load-displacement curves and bending moment distributions for soil zones where soil modulus is uniform with depth according to equation (3.4) (Es = 803 kPa used for H/D = 3), and the analyses reported earlier based on soil with the ‘Janbu initial’ modulus. The current study presents a 3D numerical modelling approach that is relatively simple to implement for 3D pipe-soil interaction problems.