## 1 Introduction
Under the assumption that no slippage exists in the strand, [PERSON] and [PERSON] [13] predicted the stresses and contact forces in a single-layered strand subjected to axial, bending and torsional loading. [PERSON] et al. [14] established a semi-analytical model for a spiral strand subjected to bending load, in which interwire contact deformation and the effects of Poisson's ratio were considered. However, the assumption adopted might be questionable, and as indicated by [PERSON] [15] the constituent wires of the spiral strand exhibit a strong tendency to rotate with respect to one another in a practical situation of bending deformation of spiral strands. Except for the discrete model, another different semi-continuous model, in which each layer of a strand is modeled as a combination of concentric orthotropic cylinders, is presented to deal with the tensile, torsional and bending stiffness of multi-layered strands [16; 17]. To more accurately characterize the bending behavior of spiral strands, the wire slippage criterion should be introduced to link the friction on the contact interface and the internal axial tension of the constituent wires. The relationship between the moment and the associated curvature was determined in [18] and the bending deformation process was divided into three stages. [PERSON] [19] determined the wire's axial force of multi-layered strand based on the Bernoulli-Euler beam assumption, and the diminishing of the bending stiffness caused by the interlayer slippage is involved. Later on, an instructive slippage criterion governing the relationship between the axial force and friction was proposed by [PERSON] [20]. This slippage criterion is widely accepted as it is consistent with the observation in the fatigue experiment carried out by [PERSON] and [PERSON] [21] that the first failed wire is invariably located near the bending neutral axis rather than the wires in the extreme-fiber position. In 2005, [PERSON] et al. [22] extended [PERSON]'s theoretical model into multilayered strands by considering the transmission of hoop force induced by axial tension from the outer-layer to the inner-layer. Recently, [PERSON] and [PERSON] [23; 24] presented a refined mechanical description for metallic strands and a hysteretic moment\(-\)curvature relationship was accounted for. Similarly, [PERSON] et al. [25] developed a new analytical formulation to predict the bending stiffness for multi-layered spiral strands by considering wire-wire line contact in the same layer and contact deformation. [PERSON] et al. [26] presented a nonlinear bending stiffness for multi-layered strands by considering the wire slippage delayed by bending load. With the rapid development of computation capability, more and more research focuses on finite element (FE) analysis of the bending behavior of spiral strands. Compared with the analytical method, the FE method is of the capability to capture the realistic inter-wire slippage and interaction inside the spiral strand without any assumptions. [PERSON] and [PERSON] [27] adopted the FE method to study the interwire motions of a simple spiral strand subjected to axial loadings and bending deformations. Later on, with the aid of the commercial FE software, several researchers investigated the mechanical behavior and stress distribution of various types of spiral strands. According to the basic elements implemented in different FE models, they can be categorized into four types, involving a full three dimensional (3D) solid model [28; 29; 30; 31; 32; 33; 34], a solid-beam mixed model [35], a beam-spring mixed model [36] and a beam model [37; 38; 39; 40; 41]. In the full 3D FE model, each constituent wire of the spiral strand is discretized with hexahedral solid elements and nonlinear properties, such as frictional contact or elastoplastic constitutive material model can be introduced to the model. [PERSON] et al. [28] examined the bending behavior and interface stress distributions for a simple strand under longitudinal and lateral loadings. [PERSON] et al.