Beam Theory

$E_{\text{bend}}=\frac{1}{2}{K}_{\text{eff}}{\int }_0^L{\left|\frac{\mathrm{d}\vec{t}}{\mathrm{d}s}\right|}^2\mathrm{d}s$

• Energy to bend a beam
• $K_{\text{eff}}=EI$
• $\vec{t}$ is the unit tangent
• $L$ is the length of the beam
• ${\xi }_p$ is the persistence length

$f{\xi }_p=\frac{z}{L}+\frac{1}{4{\left(1-\frac{z}{L}\right)}^2}-\frac{1}{4}$

• $f$ is the force applied to the beam divided by $k_{B}T$
• $z$ is the extension length of the object
• $L$ is the length of the beam