Virtual work (internal = external)

[disclaimer]

Homework Statement

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).

Homework Equations

[/B](I guess)
$$\delta W_{in}={\int_{V}^{}}\delta \tilde{\varepsilon} ^T\tilde{\sigma} dV$$ and $$\delta W_{ex}={\int_{V}^{}}\delta \tilde{u} ^T\tilde{\textbf{f}} dV+{\int_{S}^{}}\delta \tilde{u} ^T\tilde{\textbf{t}} dS$$

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 (τ_xy) 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.

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?