## 2 Materials and Methods
### A Beam FE Modeling Strategy for a Simple Strand under Multi-Axial Loads
According to the penalty stiffness algorithm, the normal contact force of two adjacent beam elements holds:  
\[F_{n}=k^{*}\delta \tag{6}\]  
\(k^{*}\) represents the integrated interface contact stiffness. Its magnitude depends on the contact stiffness of two contacting beam elements, and there exists:  
\[k^{*}=\frac{k_{i}k_{j}}{k_{i}+k_{j}} \tag{7}\]  
where \(k_{i}\) and \(k_{j}\) is the contact stiffness of beam I and beam J, respectively. The contact stiffness of a beam element is related to its material properties and geometric parameters and it can be calculated by:  
\[k=\frac{f_{s}\pi KR^{2}}{L} \tag{8}\]  
Figure 2: (**a**) The schematic for the beam-to-beam contact; (**b**) penetration for the two contacting beam elements in a cross-section view. where \(K\) is the Bulk modulus and \(K=\frac{E}{3(1-2 eV)}\), \(f_{s}\) is the scale factor and the default value is 0.1, \(R\) is the radius of the beam element and \(L\) is the beam element length. A Coulomb friction model, which is analogous to an elastic-plastic spring, is adopted to evaluate the friction which resists the relative movement between two contacting beams. The incremental update algorithm is more convenient to describe the friction model since it is path dependent. Figure 3 shows that the friction varies with the relative displacement \(\Delta e\) perpendicular to the normal contact force between two contacting beams. Assuming the friction is \(f^{n}\) at the \(n\)th time step, and then at the (\(n+1\))  
Furthermore, in order to assess the accuracy of the present model to simulate the bending deformation of the spiral strand, several pure bending quasi-static simulations are conducted by applying a moment to the loading end, as shown in Figure 5a. Friction inside the strand is neglected here. The magnitudes of the moment are specified as 1 Nm, 5 Nm, 10 Nm and 20 Nm, respectively. As a comparison, the bending curvature of a spiral strand under pure bending is calculated by [PERSON]'s formula. [PERSON] treated a whole strand as an assembly of six individual helical wires and a straight core. For pure bending deformation, no interaction that exists inside the strand was assumed. Hence, the moment only induces the bending and torsional deformations for each wire with no axial extension. Based on these assumptions, the bending stiffness of the present spiral strand under pure bending deformation can be calculated by:  
\[EI_{min}=\frac{\pi E}{4}(R_{c}^{4}+\frac{2m\sin\beta}{2+\
u\cos^{2}\beta}R_{w} ^{4}) \tag{11}\]  
Figure 4: Convergence analysis for beam model: Tc–Ns for uniaxial tension at \(T=5\) kN. Figure 1:0 (a) Schematic of the pure bending deformation; (b) comparison of the bending moment–curvature curves obtained from [PERSON]’s theory and the presented beam FE model. Figure 5b shows the comparison of the results from [PERSON]'s theory and the present FE model. It is seen that the FE results agree well with those from [PERSON]'s theory. It demonstrates the presented beam FE method is suitable to calculate the bending stiffness of the frictionless spiral strand. Therefore, it is reasonable to infer that the present beam model is suitable for simulating the bending behavior of the spiral strand subjected to multi-axial loads. A complete simulation model of a simple strand under free bending is present in Figure 1:1 The lateral load \(P\) is exerted on the mid-span of the whole strand. The boundary condition stays identical to the aforementioned model. The loading sequence is given as follows: the axial force exerted on the central node at the loading end increases from zero to maximum within 0-0.1 s, and then the lateral force increases from zero to maximum within 0.1-1.1 s.