# Virtual work (internal = external)

1. Oct 27, 2014

### disclaimer

1. The problem statement, all variables and given/known data
How can one show/prove that for a beam (hinged supports on both ends) subjected to bending due to a uniformly distributed load over its entire length, the virtual work of internal forces is equal to the virtual work of external forces? Given are the length of the beam (L), uniformly distributed load (P = constant), Young's modulus (E), and second moment of area (I).

2. Relevant equations (I guess)
$$\delta W_{in}={\int_{V}^{}}\delta \tilde{\varepsilon} ^T\tilde{\sigma} dV$$ and $$\delta W_{ex}={\int_{V}^{}}\delta \tilde{u} ^T\tilde{\textbf{f}} dV+{\int_{S}^{}}\delta \tilde{u} ^T\tilde{\textbf{t}} dS$$
3. The attempt at a solution
I think that in this particular case the first equation can be simplified to
$$W_{in}={\int_{V}^{}}(\sigma_x\tilde{\varepsilon}_x+\tau_{xy}\tilde{\gamma}_{xy})dV$$
Can the shear stress (τxy) be neglected here? If so, we would get
$$W_{in}={\int_{V}^{}}(\sigma_x\tilde{\varepsilon}_x)dV$$
I'm not sure what I should do with the other equation. Am I even approaching this correctly? If not, what are the right steps to follow? Any suggestions welcome. Thank you.

Last edited: Oct 27, 2014
2. Nov 1, 2014