where F and M are the internal force and moment, respectively, and p is the pressure (contact force per unit length) on the strand exerted by the neighbor strands. Taking into account the geometry of the strands in the multi-ply we can write for the pressure:

where p_j+1_,j and p_j-1,_j are the pressures exerted on strand (j) by strands (j+1) and (j-1), respectively. After using the topological constraint, one finds the expression for the total pressure on a strand as follows

where B, ρ and n are the bending stiffness of a strand, the ratio of the cylinder radius and strand radius and the number of strands, respectively. The end force F0 and end moment M0 are both applied axially to the overall yarn.

3. Pressure on the Strands of a Multi-ply Yarn

Some single yarns of different counts were modelled to investigate the inter-strand and total pressures at the balanced point of multi-ply yarns comprising 2, 4 and 6 strands. Figure 2 describes the inter-strand pressure and radial total pressure on strands of the 2, 4 and 6-ply yarns plotted against the spinning twist of the original single yarns at the balance point for strand count of 40 Tex, after removing external forces. These results identify some interesting behavior for the multi-ply yarns. While the differences in the total pressure are negligible at low spinning twist (for example, less than 300tpm for 40 Tex), a clear difference is observed for the inter-strand pressure. Generally, the effect of the number of strands on these two kinds of pressure shows two contrasting effects. As the number of strands increases, the total pressure decreases but the interstrand pressure increases. Furthermore, the total pressure tends to reach a stable value with increasing spinning twist. This stable total pressure is particularly evident for the 6-ply yarns from 40 Tex single yarns. The effect of inter-strand pressures may have very practical consequences in relation to fabric performance. The inter-strand pressures may influence the lateral compression of strands and influence the bulk in the strand and multi-ply yarn as well as the movement of fibers in the yarn structure. The latter effect has potential to affect the shedding of fibers and the formation of pills in textile products.

In order to further evaluate the influence of structural properties of a multiply yarn on the inter-strand and total pressures, three multi-ply yarns of the same resultant count (80 Tex) prepared from 2x40tex, 4x20tex and 6x13.3tex were considered. Figure 3 depicts the total pressures and interstrand pressure with respect to spinning twist for the three multi-ply yarns. The results in Figure 3 confirm the influence of the number of strands on the total pressure and inter-strand pressure on strands. Due to the limitation of this extended abstract further results will be presented at the conference.