## 1 Introduction
Unlike strain demand, evaluation of the strain capacity is essentially based on the inherent mechanical resistance of the pipe segments and has relatively less dependency on external factors, hence analytical evaluation and deterministic prediction of strain capacity is relatively more straightforward, as parameterization of influencing factors for incorporation in the SBD is relatively less complicated. Extensive studies have been conducted by a number of researchers to investigate the limit load instability and deformational capacity of pipe segments, and cylindrical shells in general, subjected to various loading conditions. The earliest recorded investigations on the buckling behavior of pipe segments were analytical studies, known as the classical shell buckling theories, performed by [PERSON] [21], [PERSON] [22], and [PERSON] [23] at the inception of the 20 th century, which focused on deriving constitutive theoretical formulations for predicting the elastic critical buckling stress of perfect isotropic cylindrical shells subjected to pure axial compression. Subsequent experimental and analytical studies [24; 25; 26; 27; 28] pointed out some shortcomings of the classical shell buckling theories, and prescribed the extension of the classical equations to include nonlinear large-deflection considerations, as well as adequate characterization of material, geometric, and boundary characteristics. An analytical study by [PERSON] [29] on the circumferential flattening mode of distortion (referred to as _\"ovalization\"_or _\"Brazier instability\"_) observed in an infinitely long, circular tube subjected to pure bending led to the derivation of an expression for the limit moment. [PERSON]'s theory is based on the relationship between the strain energy per unit tube length and the change in axial curvature, such that the limit moment is directly related to the ovalization of the circular tube. Subsequent experimental studies [30; 31; 32; 33], however, indicate that bifurcation instability (buckling) in the form of longitudinal wrinkling or bulging may precede the limit moment in pipes subjected to bending due to increase in axial stress on the compression side of the pipe. [PERSON] and [PERSON] [34] explained that bifurcation instability is more likely to precede _Brazier_ instability in \"thinner\" tubes, and vice versa in \"thicker\" tubes. They further speculated that the transition between one preceding instability phenomenon and the other occurs in the range of \(D/t\) (diameter/thickness) ratios between 35 and 45. [PERSON] and [PERSON] [35] performed statistical evaluation of results obtained from (1) experimental tests of circular tubes under pure bending [36], (2) empirically-derived analytical formulations for cylinders under combined axial compression and bending [37], and (3) semi-empirical derivation of the critical buckling stress for pressurized circular tubes subjected to uniform bending [38]. They observed that the critical buckling stress of cylinders under pure bending is generally between 20 to 60% higher, depending on the diameter-to-thickness ratio (\(D/t\) ratio), than that of cylinders under pure axial compression. Using a special-purpose non-linear finite element technique, [PERSON] and [PERSON] [39] studied the structural stability of long uniformly pressurized thin elastic tubular shells subjected to in-plane bending. They observed that external overpressure reduces the buckling moment and corresponding curvature but increases the buckling wavelength, whereas internal overpressure tends to increase the buckling moment but reduce the cross-sectional ovalization and the buckling wavelength. Following a systematic series of experiments and by [PERSON] et al. [40] involving stainless steel tubes bent to failure at fixed values of internal pressure, it was shown that cross-sectional ovalization and circumferential extension occur simultaneously, due to bending and internal pressure, respectively. Ref.