A problem about virtual work principle for continuous system

[athosanian]

dear all, the virtual work pinciple can be used to derive the equilibrium equations for the mechanical systems. however, when I want to apply it to a continuous system, I found it can not give out the simple equilibrium equations. there should be something wrong with my thinking. I expect some expert could give me some advice. thanks very much.

the problem detail is shown in the pictures:

 

[optophotophys]

The virtual work needed to compress (or elongate) the small element is the stress times the amount of the compression. So,
$$\begin{align} &TA(\delta u+\frac{d \delta u}{dx} dx - \delta u)=TA\frac{d \delta u}{dx} dx,\\ &\delta W=\int_0^L TA\frac{d \delta u}{dx} dx=TA\delta u|_{x=0}^{x=L}-A \int_0^L \delta u\frac{dT}{dx}dx =TA\delta u|^{x=L}-A \int_0^L \delta u\frac{dT}{dx}dx, \end{align}$$
where we use integration by parts.
As $\delta u$ is arbitrary, we have $\frac{dT}{dx}$ anywhere other than the open end$x=L$. Happily we know that $T=P$ at $x=L$. We get equation (1).