On the critical axial forces of upheaval buckling for imperfect submarine pipelines

For pipelines with vertical imperfection, upheaval buckling may occur if the axial compressive force reaches the critical axial force of upheaval buckling. The critical axial force is sensitive to the pipeline imperfection and previous researchers have suggested that there is no universal analytical solutions for the critical axial force of upheaval buckling for imperfect pipelines. However with theory of dimensional analysis, it was proved that there should be a general form of the approximation formulas of the critical axial force, although the coefficients in the formulas are different for different imperfection shapes. And most recently, Zeng et al. proposed approximation formulas of the critical axial force accounting for the Out-of-Straightness (OOS) of the imperfection, while they haven't considered the influence of the imperfection size. In this paper, effect of the imperfection size on the critical axial force was proved significant even when the OOS and shape of the imperfection are determined. To account for this size effect, a parameter named the dimensionless imperfection length is proposed based on theory of dimensional analysis. This parameter combined the effects of the imperfection length, the vertical distributed force and the pipeline bending stiffness. A formula of the critical axial force, covering the newly proposed parameter and the OOS of the imperfection, is derived, and coefficients in the formula are determined with numerical results from the Vector Form Intrinsic Finite Element (VFIFE) simulations. Notably, the coefficients in the formulas are not constants but assumed to change with the OOS and the dimensionless imperfection length to account for the geometric nonlinearity of the initially curved pipeline. The proposed formulas are proved more accurate than previous ones and applicable for pipes with different cross-sectional properties and different buried conditions. They are also suggested within the error range of ±5% in the dimensionless scope of the OOS from 0.001 to 0.01 and the dimensionless imperfection length from 0.89 to 4.95.