## 1 Introduction  
A spiral strand is a composite structure that consists of helical wires twisted around an initially straight core and grouped in concentric layers; it is a critical component of cable structure or wire rope. The helical structures are widely applied for long-distance force delivery or signal transmission in many engineering applications, such as suspension cables in cable-stayed bridges and electrical conductors as well as wire rope in mining hoists. These structures are mainly designed for bearing axial loads, but they inevitably experience bending deformations and/or transverse vibrations caused by the gravity, hanging, or wind disturbance in practice. In these cases, bending stiffness is an essential parameter for the estimation of lateral deformation near the terminations or transverse vibration with high-order nature frequencies. For instance, neglecting bending stiffness will lead to unacceptable errors in the estimation of high-order nature frequencies for large-diameter sag cables experiencing transverse vibration [1]. Moreover, cyclic bending is a vital factor in fitting fatigue damage which happens at or near the terminations where the strand curvature rapidly changes. Although the bending stiffness of cables is small, it has a large influence on the deformed shape near the constraints [2, 3]. In addition, flexural hysteresis damping properties play an important role in dynamic behaviors [4], and flexural self-damping depends on the nonlinear hysteresis bending stiffness [5]. However, the spiral strand exhibits extremely complicated bending behavior because of intractable interaction in the strand. In fact, the bending stiffness for spiral strands may vary with their tension, curvature and deformation history [6]. Therefore, accurately characterizing the effect of friction on the bending behaviors of spiral strands is essential for dynamic analysis, fatigue life prediction and self-damping estimation. Earlier studies on the bending behaviors of spiral strands mainly focus on experimental measurements, and a direct and effective experimental approach is to measure the deflection of spiral strands subjected to external loads under free bending. [PERSON] and [PERSON] [7] measured the flexural stiffness of multistrand electrical conductors under tension, and they found that the bending stiffness increased with tension. Similarly, [PERSON] and [PERSON] [8] pointed out that the bending stiffness of two electrical conductors approach minimum theoretical bending stiffness under most combinations of axial tension and lateral deformation. However, for very large axial tension, the bending stiffness is closer to the maximum possible bending stiffness. Apart from electrical conductors, sag deflections for several structural cables are also measured by [PERSON] et al. [9]. These tests showed elastic-plastic like force-displacement responses under bending. However, the experimental method costs huge human efforts and financial resources. Additionally, the effective bending stiffness is derived based on the assumption that the bending stiffness is constant along the strand centerline, but this is not consistent with the realistic one. To avoid the drawbacks of experimental methods, many researchers have tried to derive the bending deformation of spiral strands based on mechanical principles. A discrete model which consists of six nonlinear differential equations of equilibrium is derived based on [PERSON]'s thin rod theory [10]. [PERSON] [11] proposed a formula for the minimum bending stiffness by treating the spiral strand as an assembly of individual helical springs. [PERSON] and [PERSON] [12] put forward a change law of the secondary helix angle and established a new mechanical model for a multi-strand wire rope under a bending state. However, the inner friction was ignored. Later on, several mechanical models were proposed to evaluate the impact of friction on the bending behavior.