 Bending rigidity of an unbonded fibre bundle

The bending rigidity of an unbonded fibre bundle relates total bending moment to curvature in a stated bending model. If slender material fibres slide longitudinally without appreciable friction, each bends about its own neutral axis. With the same imposed small curvature $\kappa$ and no additional axial tension, strand $i$ contributes $M_i=E_i{I}_i\kappa$, givingHere $I_i$ is that strand's second moment of area about its own bending axis. The underlying linear elasticity relation is the beam moment-curvature law.^[ref] Adding offset-area terms as though all strands shared one neutral axis would impose a different mechanical constraint.

An original two-strand comparison makes the difference quantitative. Take two identical circular strands of radius $r$, with centres vertically separated by $2r$, bending in the plane containing their centres. Each has $A=\pi r^2$ and $I=\pi r^4/4$. In the freely sliding model,For a separate idealization, bond the strands perfectly along their contact so they share a common neutral axis halfway between the centres. Neglect the bond's own area and compliance. Integrating squared distance from that common axis givesFor stipulated $E=2\mathrm{GPa}$ and $r=1\mathrm{mm}$, these are approximately $3142$ and $15708\mathrm{N}\mathrm{m}{\mathrm{m}}^2$. At $\kappa =0.001\mathrm{m}{\mathrm{m}}^{-1}$, the corresponding moments are $3.142$ and $15.708\mathrm{N}\mathrm{mm}$. Both specimens contain the same strand material; their ability to exchange longitudinal force differs.

A physical comparison would impose end rotations, permit the relative end travel required for inter-fibre sliding, and measure bending moment and curvature. Clamping every strand to identical end displacements, adding axial tension or allowing friction can change the response. The factor five above belongs to this two-strand geometry and ideal bond, not to every bundle. The equal-material subdivision model in material fibre makes a different comparison with a solid circular rod.