Why Tsunade-sama did not have wood element even though she was direct descendant of Senju Hashirama-sama? Most of the time in Naruto I have seen that the descendants of a particular clan will inherit the elements from their clan. 

For example, Uchiha clans mostly have fire element in them.

But why didn't Tsunade-sama inherit the wood element from her grandfather?
Hashirama's Wood element is, as far as the series goes, completely unique to him. It was not inherited to his descendants, and even his own brother didn't have it.

There were only a few exceptions:


Yamato, who is the sole survivor from an experiment to inject Hashirama's DNA into babies.
Danzo, who injected himself with Hashirama's cells to be able to use Izanagi, but was too unstable to actually use it.
Madara, who injected himself with Hashirama's cells after acquiring them from their battle.
Zetsu, who has been born of the Shinju's power.
Obito (while wearing Zetsu).


Madara is the only other person capable of using the actual Wood element large scale techniques like Jukai Kotan and Kajukai Korin.