## 2 Materials and Methods  
### A Beam FE Modeling Strategy for a Simple Strand under Multi-Axial Loads  
A simple strand, also namely 1 + 6 steel strand, consists of one straight core and six outer layered wires wound around the core, as shown in Figure 1. The geometry of the strand is characterized as follows: the radius of the core is R\({}_{c}\), the radius of the outer layered wire is R\({}_{w}\), and the layer angle is \(\alpha\). Here, the layer angle is defined as the included angle between the centerline of the outside wire and that of the core wire. The centerline of the outer wire is a first-order helical curve characterized by helical radius \(R_{h}\) and pitch length \(p\). The pitch length is defined as the distance between two nodes situated on the helical wire whose corresponding angular phases differ by \(2\pi\) along the longitudinal axis of the strand. The parametrical equation of the helical curve is expressed as:  
\[R=\left[R_{h}\cos\theta,R_{h}\sin\theta,\frac{p}{2\pi}\theta\right] \tag{1}\]  
According to geometrical relations, there exists:  
\[R_{h}=R_{c}+R_{w} \tag{2}\]  
and the pitch length:  
\[p=2\pi R_{h}\tan\beta \tag{3}\]  
where \(\beta\) represents the helical angle of the helical curve and \(\beta=\frac{\pi}{2}-\alpha\). All necessary parameters involving geometric dimensions and material properties of the simple strand are listed in Table 1. Note, that only the elasticity phase of steel is considered here due to the assumption of the small strain of the spiral strand under free bending. Additionally, both radial and circumferential contact exists in the present strand. A 2-node Hughes-Liu beam element [42] implemented in Ls-dyna is utilized to generate a FE model for investigating bending behaviors of the simple strand. For each node of the beam element, there are 6 degrees of freedom in total involving 3 translational and 3 rotational degrees of freedom This beam element has high computational efficiency and good robustness. In addition, it has no locking modes because the integral algorithm \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Meaning** & **Value** \\ \hline \(R_{c}\) & Radius of core wire & 1.52 mm \\ \(R_{w}\) & Radius of outer wire & 1.50 mm \\ \(p\) & Pitch length & 141.58 mm \\ \(\alpha\) & Layer angle & 0.1331 rad \\ \(m\) & Number of outer wires & 6 \\ \(E\) & Youngs modulus & 197,950 MPa \\ \(\
u\) & Poisson ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the simple strand. Figure 1: Geometry of a typical simple strand: cross-section view (**left**); side view (**right**). of the beam element is performed with one-point integration along the axis and multiple points in the cross section. Therefore, it can be utilized to deal with the finite strains that occur in many practical applications. To capture the interaction of the adjacent wires in the strand in the process of deformation, a typical penalty stiffness algorithm is applied to evaluate the normal contact force on the interface. As shown in Figure 2, once the beam element I and the beam element J contact with a so-called penetration \(\delta\), the penalty stiffness spring will generate a penalty force to eliminate the penetration. Assuming the radius of two beam elements are \(r_{1}\) and \(r_{2}\), respectively, and the distance between their cross-sectional centers is \(l\), the contact penetration \(\delta\) can be calculated by:  
\[\delta=r_{1}+r_{2}-l \tag{4}\]  
The distance of two adjacent beams is obtained through vector operation  
\[l=\parallel\frac{\mathbf{A}\cdot(\mathbf{B}\times\mathbf{C})}{\mathbf{B}\times\mathbf{C}}\parallel \tag{5}\]  
Here, \(\mathbf{A}\) denotes the vector pointing from node I\({}_{1}\) to J\({}_{1}\), \(\mathbf{B}\) denotes the vector pointing from node I\({}_{1}\) to I\({}_{2}\), and \(\mathbf{C}\) denotes the vector pointing from node J\({}_{1}\) to J\({}_{2}\). I\({}_{1}\) and I\({}_{2}\) are two nodes of beam I, and J\({}_{1}\) and J\({}_{1}\) are two nodes of beam J.