Using http://www.roymech.co.uk/Useful_Tables/Mechanics/Rotating_cylinders.shtml for information about how to calculate maximum stress in cylindrical flywheels, and using numbers for granite, as representative of a fairly strong solid rock, I calculated a maximum stress about 9 billion Newtons per square meter for the asteroid in Eon (25 km internal radius, 100 km external radius), while granite's strength in tension is about only about 5 million Newtons per square meter.
Even a 100-meter external radius asteroid spun up to 1 gee would be operating with stresses about twice the expected strength of granite, a fact that surprised me when I first did the calculation.
After I wrote this slide, I did run the numbers for Ringworld; the strength of scrith must be greater than about 8 * 10^16 kg per square meter. For context, that's about a half million times stronger than graphene, which is the stronger substance I could find numbers for. These numbers are a little uncertain, because the cross-section of the Ring isn't clearly laid out in the book - the Ring is definitely about a million miles wide, but its thickness is never given (though it's clear that it's not much thicker than 50 feet). Niven does have his characters remark that scrith is incredibly strong (indeed, comparable to the strong nuclear force in strength).
Niven has noted in various places that Ringworld can be analyzed by treating it as a "suspension bridge with no endpoints," which was one of the inspirations for this talk (and its title).