Crest and sag vertical curve length on the PE Civil Transportation exam

A vertical-curve problem on the Transportation exam asks either where the road is (an elevation, the high or low point, the clearance under a structure) or how long the curve must be to provide a sight distance. The two halves live in different books. The Handbook's Vertical Design section prints the parabola: offsets, the turning point, the rate of change of grade and K = L/A. It prints no relation between curve length and sight distance. Those come from Chapter 3 of the AASHTO Green Book (7th ed., 2018), which is supplied with the exam.

What you actually need to own

• The parabola. Measure x from the PVC: y = ax², a = (g2 − g1)/2L. The grade changes at the constant rate r = (g2 − g1)/L, and the high or low point sits x_m = g1L/(g1 − g2) from the PVC. The offset at the PVI is E = AL/8 with grades as decimals, or AL/800 with A in percent.
• Name the sight line before you pick the formula.
  • Crest, stopping. A 3.50-ft eye over the hump to a 2.00-ft object: L = AS²/2,158 when S < L, L = 2S − 2,158/A when S > L. Passing uses a 3.50-ft object, and 2,158 becomes 2,800.
  • Sag, unlit. The headlight beam, 2 ft high and diverging 1° upward: L = AS²/(400 + 3.5S) when S < L, L = 2S − (400 + 3.5S)/A when S > L, with S the stopping sight distance. This is the criterion the Green Book recommends for general use.
  • Sag, comfort. Vertical acceleration held to about 1 ft/s²: L = AV²/46.5, V in mph. It runs about half the headlight length, so use it when the problem makes comfort the control.
  • Undercrossing. A truck driver's 8.0-ft eye to a 2.0-ft object, under a structure with clearance C: L = AS²/[800(C − 5)] when S < L, L = 2S − 800(C − 5)/A when S > L. The 5 is the average of the two heights; for passing, both are 3.50 ft.
• K = L/A, both ways. The Green Book tabulates a design K by design speed, separately for crest-stopping, crest-passing and sag. L = KA gives a length. K = L/A on an existing curve gives the highest design speed it supports. For flat gradients there is also a minimum length of about 3V ft (V in mph). Take the larger of KA and 3V.
• Drainage on curbed sections. A 0.30% grade within 50 ft of the level point corresponds to K = 167 ft per percent. On a curbed sag that is a maximum length; every other sag criterion sets a minimum. On a curbed crest a larger K is allowed, but drainage near the high point then needs special design.

Where people lose points

• Calling crest or sag from the signs of the grades. The change in grade decides it: −5.0% to −1.0% is a sag, though both grades fall.
• Skipping the S-versus-L check. Solve with one form, then compare the result with S. In one problem (S = 790 ft) the S > L form returns 1,130 ft, longer than S, so it's invalid and the S < L form's 1,388 ft stands. When lengthening a curve, check against the new length, not the existing one.
• The nearest K instead of the provided K. A curve with K = 184.6 that falls between two rows supports the lower row's speed, even when the upper row's K is closer.
• The wrong table or the wrong heights. The passing K in a stopping question, the sag K on a crest, a car's eye in a truck-driver question, or the full clearance in place of C − 5. Each gives a plausible number.
• Clearance at the wrong station. Under level girders, clearance is least where the road beneath the deck is highest: a deck edge, or the crest's high point if it falls under the deck. A sag's low point gives the most room, not the least.
• A from magnitudes. +4.0% to −1.0% is A = 5.0, not 3.0.

How to study it

Take one pair of grades and size the curve four ways: crest-stopping, unlit sag, comfort, and undercrossing at a given clearance. Name the sight line before each equation and check S against L every time. Then run K backwards on an existing curve. The plan-view half of the same alignment is horizontal curve elements and stationing and superelevation and runoff.

Part of the Vertical Design area of the PE Civil Transportation exam. → Start practicing free — the whole practice bank, free with an account.