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6.BF.4. RAIL BUCKLING.
A railway rail of length L and ends fixed expands to length L + ΔL. Assuming the rail makes two hypotenuses, the middle rises by a height, H, satisfying H² = {(L+ΔL)/2}²   (L/2)², hence H  (LΔL/2).

However, one might assume the rail buckled into an arc of a circle of radius r. If we let the angle of the arc be 2θ, then we have to solve rθ = (L + ΔL)/2; r sin θ = L/2. Taking sin θ  θ - θ³/6, we get r²  (L + ΔL)³/ 24 ΔL. We have H = r (1 - cos θ)  rθ²/2 and combining this with earlier equations leads to H  {3(L+ΔL)ΔL/8} which is about 3 / 2 = .866... as big as the estimate in the linear case.

The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 149, prob. 12. L = 1 mile, ΔL = 1 ft or 2 ft -- text is not clear. "Answer: More than 54 ft." However, in the linear case, ΔL = 1 ft gives H = 51.38 ft and ΔL = 2 ft gives H = 72.67 ft, while the exact answers in the circular case are 44.50 ft and 62.95 ft.

Sullivan. Unusual. 1943. Prob. 15: Workin' on the railroad. L = 1 mile, ΔL = 2 ft. Answer: about 73 ft.

Robert Ripley. Mammoth Believe It or Not. Stanley Paul, London, 1956. If a railroad rail a mile long is raised 200 feet in the centre, how much closer would it bring the two ends? I.e. L = 1 mile, H = 200 ft. Answer is: "less than 6 inches". I am unable to figure out what Ripley intended.

Jonathan Always. More Puzzles to Puzzle You. Tandem, London, 1967. Gives the same question as Ripley with answer "approximately 15 feet". The exact answer is 15.1733.. feet or 15 feet 2.08 inches.

David Singmaster, submitter. Gleaning: Diverging lines. MG 69 (No. 448) (Jun 1985) 126. Quotes from Ripley and Always.

David Singmaster. Off the rails. The Weekend Telegraph (18 Feb 1989) xxiii & (25 Feb 1989) xxiii. Gives the Ripley and Always results and asks which is correct and whether the wrong one can be corrected -- cf Ripley above.

Phiip Cheung. Bowed rail problem. M500 161 (?? 1998) 9. ??NYS. Paul Terry, Martin S. Evans, Peter Fletcher, solvers and commentators. M500 163 (Aug 1998) 10-11. L = 1 mile, ΔL = 1 ft. Terry treats the bowed rail as circular and gets H = 44.49845 ft. Evans takes L = 1 nautical mile of 6000 ft and gets almost exactly H = 50 ft. Fletcher says it took 15 people to lift a 60ft length of rail, so if someone lifted the 1 mile rail to insert the extra foot, it would need about 1320 people to do the lifting.