Limit Loads for Pipelines and Cylinders Partially Embedded in Frictional Materials

Accurately assessing the forces and reactions on cylindrical objects partially embedded in soils is fundamental in pipeline engineering as well as the analysis and design of off-road vehicles. With a view towards understanding the limiting forces acting over the contact area between the cylinder and the soil, this paper compares two different theoretical approaches for predicting ultimate loads under combined vertical and horizontal loading. The first is an approximate method based on idealizing the contact interface as a flat shallow strip footing, and the second is based on finite element limit analysis (FELA). Focus is on purely frictional material and plane strain conditions, assuming a rigid cylinder and perfectly plastic material obeying the Mohr-Coulomb yield condition with associated plastic flow. The approximate method based on an analogy to an equivalent shallow foundation (with no overburden) predicts results that are reasonably close to those from FELA but only when small embedment is considered. Moreover, the discrepancy between the two approaches for the case of free cylinder rotation is attributed to the existence of a contact moment over the soil-cylinder contact interface. Practical implications for pipeline engineering and off-road vehicle engineering are discussed. on the limit loads for partially embedded cylinders in frictional material (e.g., sand) are comparatively scarce. However, this area is of equal importance for engineering practice. Examples of practical importance include pipelines sitting on sand/silt and subjected to slow (undrained) loading, and the performance of vehicles which are designed to operate on the beach. Gao et al. (2015) proposed a slip-line solution to predict the vertical bearing capacity of pipelines partially embedded in drained soils. Based on experimentally obtained V-H failure envelopes, Zhang et al. (2012) proposed a bounding surface model to compute the forcedisplacement relation of a shallowly embedded pipeline in calcareous sand subjected to general loading conditions. In engineering practice, the assessment of the limit load for a partially embedded cylinder or pipeline is sometimes simplified by evaluating the bearing capacity of an equivalent shallow strip foundation (Small, 1971; Brown, 2001; O’Rourke and Liu 2012). Hambleton and Stanier (2017) applied the same strategy to predict the forces acting on the wheels of off-road vehicles, by again considering an analogous strip foundation at the interface between wheel and soil. In this way, the forces developing on the wheel are related to soil properties through existing bearing capacity theories. This approach based on the shallow foundation analogy is appealing for its simplicity compared with the aforementioned analytical or numerical approaches, but more importantly this simplification enables direct application of a body of knowledge regarding foundation bearing capacity for the purposes of predicting the limit loads on cylindrical objects. O’Rourke and Liu (2002) showed that the equivalent shallow foundation model can provide reasonably accurate prediction of vertical bearing capacity of pipelines as compared with more sophisticated methods. Kouretzis et al. (2015) concluded that this is valid only for deeply embedded pipes and loose-to-medium dense sand. Additional investigation is needed to assess whether this simplification is satisfactory when used to predict the limit loads for cylinders subjected to general loading (i.e., combined horizontal and vertical loading). This is the key question that this work attempts to answer. For this purpose, finite element limit analysis (FELA) was utilized to compute the limit loads for both a partially embedded cylinder and the corresponding equivalent shallow foundation, and comparisons are presented in the form of V-H failure envelopes. The analyses completed for this work were restricted to plane strain, a rigid cylinder and perfectly plastic cohesionless soil obeying the MohrCoulomb yield condition with associated plastic flow. Deviating from previous studies in which only translational movement was considered, this work also studies the impact of the cylinder’s rotation on the limit load, which is particularly vital for applications pertaining to off-road vehicles. 2. SHALLOW FOUNDATION ANALOGY Figure 1 illustrates the shallow foundation analogy used to assess the limit load for a partially embedded cylinder. As a hypothesis, the evaluation of the ultimate force acting on the cylinder, denoted by F and depicted in Fig. 1(a), is equivalent to assessing the ultimate force acting on a strip foundation (no overburden) located at the interface between soil and cylinder. In additional to the vertical and horizontal components of loading, V and H, a moment M may also act at the interface, as shown in Fig.1(b). The width of the strip foundation is related to both the diameter D and embedment depth z of the cylinder: