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(covers information from several alternate timelines)

A turboshaft, or more precisely turbolift shaft, was a passageway allowing the turbolift to move through key sections of the ship.

Kelvin-type starships had exposed turboshafts in engineering. (Star Trek)

In the 2270s, Rear Admiral James T. Kirk asked a yeoman where Turboshaft 8 was located on the refitted USS Enterprise. (Star Trek: The Motion Picture)

Kirk, Spock and McCoy were forced to climb the turboshaft of the USS Enterprise-A while attempting to escape Sybok, as the turbolifts were down. Despite the fact that Kirk and McCoy began climbing, Spock opted for using his jet boots to expedite the journey. (Star Trek V: The Final Frontier)

Aboard a Galaxy-class starship, the shortest route from the bridge to transporter room 1 on deck 6 included using a turbolift that traveled through turboshaft 4. (TNG: "Brothers") A failure in turboshaft 4 would prevent direct turbolift access from the bridge to main engineering. (TNG: "Remember Me")

Emergency bulkheads were installed in turboshafts and could be initiated to stop a turbolift from falling or to restrict access for unwanted passengers of a turbocar. Turbocars were also equipped with emergency clamps that would act as brakes and anchor the car inside the turboshaft in case of power loss. (TNG: "Power Play", "Disaster")

Jean-Luc Picard was trapped with three young children in an unstable turbolift when the USS Enterprise-D was hit by a series of quantum filaments. His group used a ladder mounted on the turboshaft wall to escape. (TNG: "Disaster")

In the Star Trek: The Original Series first season episode "The Naked Time", the turboshaft was called a 'tube'.
The inclusion of a turboshaft in "Disaster" was thought up by Ronald D. Moore. The idea was inspired by Moore reading about how NASA's Vehicle Assembly Building at the Kennedy Space Center – the enclosed facility where the Apollo and Saturn rockets were assembled, as was the Space Shuttle – was so enormous it had developed an atmosphere, including clouds and rain. Recalled Moore, "I put Picard in the turboshaft because I wanted to make it rain in there [....] I had this notion that in a really tall turboshaft–an enclosed space–that when all the power went out, it might start to rain. The danger would be with lightning lancing through the tube. Unfortunately," Moore continued with a sigh, "anything that has to do with water on a set is just an enormous pain in the ass. It's too cost prohibitive and takes forever. So we couldn't do it. But that's the only reason Picard's in the turboshaft." (Star Trek: The Next Generation 365, p. 225)