Firstly, a horizontal grip provides better stability when recieving blows and the shield is held out with the forarm at right angles to the ground so the strain does go on the bicep. There is no evidence to suggest that the pila were clipped onto the shield as far as I know.
The Pila were thrown and as such were held in the sword arm for the most part. I suspect the spare was driven into the ground while the first one was thrown. Once you entered melee I can't see any way to keep ahold of the pilum, it would get shatterd by the first downward cut the scutum recieved anyway.
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