A problem about virtual work principle

[athosanian]

hello, I want to apply the virtual work principle to a continuous mechanical system to derive equilibrium equation, naemly, a long thin bar with one end fixed in the wall and other end applied with pressure P. but I can not derive the correct equilibrium equations. I hope some expert could give some advice.
my thhinking is shown below:

 

[voko]

Obviously, if the two terms from (4) result in $T \delta u |_0^L = 0$, which gives you no condition on $T$ because $\delta u(0) = \delta u(L) = 0$, it is the neglected third term that you must take into account.