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I put a picture of my original frame. The internal diameter of the straight cylinder is 31mm.

In this frame was installed an original fork of diameter 1''.

I would like to buy this fork : https://www.bike-components.de/fr/Surly/Fourche-Rigide-Straggler-Disc-28-700c-p38261/

but the diameter is 1 1/8''.

I wonder, since there is some space between the frame and the 1'' fork, if I can installed this 1 1/8'' fork ?

I will need also to buy a new stem, but it's ok, my problem is more about the compatibility frame/fork.

Frame Cylinder

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Your frame seems to have a non-standard head tube diameter or you mis-measured it.

Here you can see a list of headset standards: https://www.parktool.com/blog/repair-help/headset-standards

There is no headset standard where head tube internal diameter would be 31mm.

For 1+1/8" forks and bicycles with external headset cups, the inner diameter of head tube should be 33.8-33.9 mm.

Only if you can find a headset that is compatible both with your frame and with the 1+1/8" fork, you can install it.

I would measure the head tube internal diameter again. The external pressed-in headset cups might be affecting your measurement.

Do note that 1" fork has 1" = 25.4mm outer diameter. The inner diameter is 22.2mm

A 1+1/8" fork has 28.6mm outer diameter and 25.4mm inner diameter.

So if your fork had 25.4mm inner diameter, it's a 1+1/8" fork.

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  • you are right, the outer diameter is 1 1/8'', perfect !
    – Smilia
    Commented Mar 14, 2021 at 12:32

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