aveilus bais hamikdash

Rashi in Succah 25A says that aveilus is considered a Tirdah De'rishus because a person is not obligated to feel sorrow but rather act in a sorrowful manner to show respect for the lost relative. We can adapt this concept possibly to the obligation of aveilus for the Bais Hamikdash during Tisha Ba'av or for the whole Av for that matter. Mishenichnas Av Mi'ma'atim Be'simcha can mean that we have to act in a less joyous way to show respect for the Bais Hamikdash but it might not be an actual obligation to be sorrowful just as the aveilus of  a relative doesn't require that we be sad. 

It can be proven from the gemara in ta'anis that states a kal vachomer regarding the isur ta'anug on all fast days vs the inyan of aveilus on tisha ba'av and the gemara says that kal vachomer from a regular avel which is worse than tisha ba'av so we see that there is seemingly no mitzva to be sad on tisha ba'av just to act as such.