16-Civ-B7 Transportation Planning and Engineering · December 2015

Question 4 of 8: Minimum Radius and Length of a Freeway Exit Ramp

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 98-Civ-B7 Highway Engineering, National Examinations, December 2015. Three hours, open book, any non-communicating calculator. Eight questions of equal value (20 marks each); five solutions constitute a complete paper and only the first five in the answer book are marked. Note 1 invites the candidate to state any assumption made about an ambiguous input, and Note 2 permits any datum that is required but not given to be assumed. All eight questions are solved here, because the set is a study resource rather than a timed attempt.

Reference texts. Garber & Hoel, Traffic and Highway Engineering, 5th ed. (geometric design, sight distance, pavement design); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (superelevation and spiral tables — the paper's Table 2.1.2.5 is TAC page 2.1.2.12); AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book) for runoff distribution and relative-gradient limits; AASHTO, Guide for Design of Pavement Structures (1993) for the flexible pavement equation and layer/drainage coefficients; Asphalt Institute MS-2, Asphalt Mix Design Methods and Mamlouk & Zaniewski, Materials for Civil and Construction Engineers, for mixture volumetrics and binder grading.

Check: assumptions carried through this paper. Under the paper's own Note 2 the following values are assumed and stated where used: the AASHTO maximum relative gradient (0.50 % at 80 km/h) and the 70 % / 30 % split of superelevation runoff either side of the PC for two lanes rotated (Question 2); a truck factor of 0.52 for all trucks on a rural Interstate and a lane-distribution factor of 0.70 for three lanes in one direction (Question 6); and a downhill 2 % ramp grade in Question 4, since the freeway is elevated above the local street. Each is flagged again at the point of use with the sensitivity of the answer to it.

Question 4: Minimum Radius and Length of a Freeway Exit Ramp (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Ramp design data
Item	Value
Maximum freeway speed	100 km/h
Sign to exit gore	100 m
Coefficient of friction (longitudinal and side)	0.20
Perception-reaction time	2.5 s
Maximum superelevation on the ramp	$e_{max} = 0.08$
Freeway grade / ramp grade	0 % / 2 %
Sign letter height, legibility index	100 mm; 15 m per 25 mm

Find. The minimum horizontal radius of the ramp, and the minimum ramp length between the gore and the stop line at the local street.

The deceleration sequence. The driver first reads the exit sign 60 m upstream of it, reacts for 2.5 s, and then has the remainder of the 160 m to slow from 100 km/h to the ramp entry speed.

Approach. The legibility distance of the sign fixes where the manoeuvre begins; subtracting the perception-reaction distance leaves the braking distance available on the freeway, which fixes the speed at the ramp nose. That speed governs the minimum radius through the point-mass curve equation and the ramp length through a second stopping-sight-distance calculation on the 2 % ramp grade.

Distance at which the sign becomes legible. At 15 m of legibility per 25 mm of letter height, $$d_{read} = \frac{100\ \text{mm}}{25\ \text{mm}}(15\ \text{m}) = 60\ \text{m}$$ The driver therefore begins to act 60 m before the sign, which is itself 100 m before the gore, giving a total manoeuvre distance of $60 + 100 = 160$ m.
Deduct the perception-reaction distance. At the full freeway speed, $$d_{PR} = 0.278\,V t = 0.278(100)(2.5) = 69.50\ \text{m}$$ so the distance left for braking on the level freeway is $$d_b = 160.00 - 69.50 = 90.50\ \text{m}$$
Speed reached at the ramp nose. The braking model between two speeds on a level grade is $$d_b = \frac{V_1^{2} - V_2^{2}}{254\,(f \pm G)} \ \Rightarrow\ V_2^{2} = 100^{2} - 254(0.20)(90.50) = 5402.6$$ $$\boxed{V_2 = 73.5\ \text{km/h}}$$ This is the design speed of the ramp: a vehicle cannot be assumed to enter it any slower, because nothing upstream compels further deceleration.
Minimum radius. With the ramp built to the maximum superelevation and using the same friction factor the question supplies, $$R_{min} = \frac{V_2^{2}}{127\,(e + f)} = \frac{5402.6}{127(0.08 + 0.20)} = \frac{5402.6}{35.56} = \boxed{151.9\ \text{m}}$$ A radius of 155 m would be adopted in practice, since ramp radii are rounded up to a convenient value.
Ramp length — reaction leg. The driver sees the stop sign on entering the ramp but still needs the same 2.5 s to respond: $$d_{PR} = 0.278(73.5)(2.5) = 51.08\ \text{m}$$
Ramp length — braking leg. The freeway is elevated above the local street, so the ramp falls at 2 % and gravity works against the brakes. Taking $G = -0.02$, $$d_b = \frac{V_2^{2}}{254\,(f + G)} = \frac{5402.6}{254(0.20 - 0.02)} = \frac{5402.6}{45.72} = 118.17\ \text{m}$$
Minimum ramp length. Summing the two legs, $$L_{ramp} = 51.08 + 118.17 = \boxed{169.3\ \text{m}}$$ measured from the gore to the stop line. Adopt 170 m. Had the ramp risen at 2 % instead, the same arithmetic gives 147.8 m, and on the level 157.4 m — so the downgrade assumption is the conservative one and adds about 12 m to the requirement.
Consistency check on the two answers. The 169.3 m ramp carries a 151.9 m radius through a deflection of the order of $L/R \approx 1.1$ radians, about 64°, which is a realistic quarter-cloverleaf geometry for the layout shown. The radius and the length are therefore mutually compatible; had the required length been shorter than the arc needed to turn the vehicle onto the local street, the geometry rather than the braking would govern.

Question 4 — final results
Quantity	Value
Sign legibility distance	60.0 m
Total distance available before the gore	160.0 m
Perception-reaction distance at 100 km/h	69.5 m
Braking distance available on the freeway	90.5 m
Speed at the ramp nose (ramp design speed)	73.5 km/h
Minimum ramp radius	151.9 m (adopt 155 m)
Reaction distance on the ramp	51.1 m
Braking distance on the 2 % downgrade	118.2 m
Minimum ramp length	169.3 m (adopt 170 m)

Check: sense of the ramp grade. The question states only "the ramp is on a 2% grade". Because the freeway is elevated and the ramp terminates at a local street beneath it, the ramp has been taken as a 2 % downgrade, which lengthens the braking distance. If the examiner intended an upgrade the required length falls to 147.8 m; adopting 170 m covers both readings.

Topic. Exit-ramp design controlled by sign legibility, deceleration and the point-mass curve equation.

Key relations. Legibility distance $=$ (letter height / index height) × index distance; $d_{PR} = 0.278Vt$; $d_b = (V_1^{2}-V_2^{2})/[254(f\pm G)]$; $R_{min} = V^{2}/[127(e+f)]$.

Why this works. A ramp is not designed to an arbitrary speed; it is designed to the speed a driver can actually have reached by the time the pavement turns, and that speed is set by how far ahead the exit becomes visible. The legibility index converts letter height into sight distance, which is why signing and geometry are designed together: a larger sign buys deceleration distance exactly as a longer deceleration lane does. Once the entry speed is known the ramp is two independent problems — a horizontal one solved by $e+f$, and a longitudinal one solved by the stopping model on the ramp's own grade.

Common pitfall. Omitting the 60 m legibility distance and starting the manoeuvre at the sign, which gives an entry speed of 84 km/h and a radius 30 % too large. The second pitfall is to omit the reaction leg on the ramp on the grounds that the stop sign is visible immediately — visibility starts the reaction, it does not remove it.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., §3.3 and §16.5 (ramp design); MUTCDC for sign legibility indices; AASHTO Green Book, Chapter 10 (interchange ramps).

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