## 3. Suitable Unit for Stress and Modulus  
The most suitable measurement unit for describing the stress and modulus of strands and ropes when applying CLT is one of the main topics of this study. As previously mentioned, the elastic modulus of ropes with a matrix have been predicted using CLT in several studies [8; 9; 10; 11; 12; 13]. These studies use the unit of MPa to express stress and modulus.  
It is well known that all synthetic fiber ropes are produced from many tiny fibers which are constructed into yarns or strands before they are twisted or braided into rope. As a result, the presence of voids (space between strands) is inevitable for all sizes of ropes. To account for voids when studying the mechanical properties of the rope, i.e., stress or modulus (MPa), the packing factor (_PF_), defined as the cross-sectional area of strands over the cross-sectional area of rope, is generally introduced (Equation (1)) [14; 15]:  
\[PF=n\frac{A_{s}}{A_{r}}=n\frac{d_{s}^{2}}{D_{r}^{2}} \tag{1}\]  
where \(n\) is number of strands in the rope, and \(A_{s},d_{s},\ A_{r}\) and \(D_{r}\) are the cross-section and diameter of the strand and rope, respectively.  
The _PF_ is generally used in analytical models to represent the reduction in stiffness of the rope due to the existence of voids. For example, [PERSON] et al. [15] accounted for _PF_ in their model by multiplying it with the stiffness of a virtual rope without voids for a multilayered fibrous structure. Thus, the stress-strain relationship of the rope can be described by Equation (2):  
\[\sigma=PF*E*\varepsilon \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline Maximum load (N) & 20\%MBL \\ Minimum load (N) & 0 \\ Crosshead speed (mm/min) & 1 \\ Cycle number & 5 \\ \hline \end{tabular}
\end{table}
Table 2: Experimental loading conditions for the strand and rope.  
Figure 2: Experimental setup for rope (**a**) and strand (**b**).  
where  
\(\sigma\) is stress in conventional unit (MPa)  
\(\varepsilon\) is strain determined by ratio of rope displacement to initial length of the rope  
\(E\) is an estimated elasticity modulus without considering voids in the rope in the unit of (MPa).  
As demonstrated by Equation (2), when the \(PF\) increases, the stress increases. Figure 3 shows the relationship between the stress and strain of the rope with different PFs (PF\({}_{1}\) < PF\({}_{2}\) < PF\({}_{3}\)), assuming \(n*A_{s}\) in Equation (1) is constant and \(A_{s}\) is varied due to changes in voids. Longitudinal modulus is changed by \(PF\) as shown in Figure 3. On the other hand, when the stress in Figure 3 is replaced with load, the lines in the figure converge. This demonstrates that stiffness of the rope is independent from \(PF\) and that \(PF\) is introduced only for discussing stress and modulus with unit of MPa.  
As the \(PF\) of rope without a matrix is easily changed during service, and is difficult to measure, these results show the limitations of using unit of MPa when discussing the modulus. Based on this, in the current study, the cross-sectional area of the rope is replaced by linear density, expressed in tex (1 tex = 1 g/km), when predicting the modulus using CLT. Conventional engineering stress and elastic modulus (expressed in MPa) are thus replaced by specific stress and modulus, respectively, in unit of N/tex. As tex is independent from \(PF\), specific stress and modulus (N/tex) are also independent from \(PF\). Because tex is based on the weight and length of the rope only. In the above discussion, only the change in voids in the rope is considered. \(PF\) is also changed by deformation of the strand, although the effect of the deformation is discussed in the next section, based on experimental data.