= Bending rigidity of an unbonded fibre bundle
{title2=$B_{\rm bundle}$}
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 $M_i=E_iI_i\kappa$, giving
$$
M=B_{\rm bundle}\kappa,\qquad B_{\rm bundle}=\sum_iE_iI_i.
$$
Here $I_i$ is that strand's <second moment of area> about its own bending axis. The underlying <linear elasticity> relation is the beam moment-curvature law.<Stresses in Beams by David Roylance>[Normal Stresses, pp.2–4, equations 3–6]{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,
$$
B_{\rm slide}=2EI=\frac{E\pi r^4}{2}.
$$
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 gives
$$
B_{\rm bonded}=2E(I+Ar^2)=5B_{\rm slide}.
$$
For stipulated $E=2\,\mathrm{GPa}$ and $r=1\,\mathrm{mm}$, these are approximately $3142$ and $15708\,\mathrm{N\,mm^2}$. At $\kappa=0.001\,\mathrm{mm^{-1}}$, the corresponding moments are $3.142$ and $15.708\,\mathrm{N\,mm}$. 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.
Back to article page
