Torsion catapults were used from the 4th century onwards by the Roman Empire, using the catapult designs known as Onagers and Mangonels. Originally designed as a lighter version of the Roman ballista, the Onager, and Mangonel quickly replaced them, being simpler than their more complex counterparts. The Roman Empire was technologically advanced for its time period, having access to roads for transportation, and having advanced technology such as lighthouses and sawmills. When constructing these siege weapons, the Romans would have likely used said roads as a method of transportation over land, though naval routes would have benefitted from the usage of lighthouses. Being made of wood, the materials used in said catapults would likely come from some of the more forested regions of the empire, such as the Balkans or Austria, and then transported either by a horse and cart, or boats, which would be wind-powered. The wood would then have likely been cut into usable lumber at a saw mill, either before they arrived at the location of the siege engine's construction or at the location itself. After gathering the required lumber to construct the Torsion Catapult, the Romans would then assemble the siege weapon, either by tying the timber together with rope (likely made from animal sinews) or by hammering in nails. This sinews would either come from a Roman farm or simply from the wildlife, while the nails, if used, would have been made from iron, and thus mined from the various iron deposits located in the empire, or possibly even imported from somewhere else. The arm would be fired by using the tension (hence the name) in a rope (made from animal sinews, as mentioned before), which would travel at intense speeds when released, flinging whatever projectile the siege engine was loaded with over a long distance.

Sources:

http://www.lordsandladies.org/mangonel.htm

http://www.medieval-castle-siege-weapons.com/onager-catapult.html

https://en.wikipedia.org/wiki/Roman_technology#Technologies_developed_or_invented_by_the_Romans