To what extent does pressurisation increase the rigidity of an airframe?
Disclaimer: I am not a pilot, nor do I have anything with planes. I am a mechanical engineering student, so I do know quite a bit about the strength of materials.

What you should know is that whenever something is filled with air, that air is trying to achieve a state in which it has the lowest pressure, meaning it is trying to increase in volume. This is the reason balloons are roughly round, because a sphere has the largest volume for the lowest surface area. The same pretty much applies to planes. What generally happens when you bend a tube (and a fuselage is in essence a big tube) is that two sides are getting closer, making it easier to bend because the second moment of area decreases. This does however mean that the hull becomes less round, thereby decreasing the volume. The air inside will try to prevent this volume change,  which makes it harder to bend the tube, which means the stiffness increases. Note however that more stiffness does not necessarilly mean it is also stronger.