Chapter 9, Problem 55P

For a balanced double-lap joint cured at room temperature, Volkersen’s equation simplifies to

$\tau \left(x\right)=\frac{P\omega \cosh \left(\omega x\right)}{4b\sinh \left(\omega l/2\right)}=A_1\cosh \left(\omega x\right)$

1. (a)   Show that the average stress $\bar{\tau }$ is P/(2bl).
2. (b)   Show that the largest shear stress is Pω/[4b tanh(ωl/2)].
3. (c)   Define a stress-augmentation factor K such that

   $\tau \left(l/2\right)=k\bar{\tau }$

and it follows that

$K=\frac{P\omega }{4b\tanh \left(\omega l/2\right)}\frac{2bl}{P}=\frac{\omega l/2}{\tanh \left(\omega l/2\right)}=\frac{\omega l}{2}\frac{\exp \left(\omega l/2\right)+\exp \left(-\omega l/2\right)}{\exp \left(\omega l/2\right)-\exp \left(-\omega l/2\right)}$