Question 7 of 7: Curves: sight distance and crest vertical curve design

Question 7 — Curves: sight distance and crest vertical curve design (a) 6 marks, (b) and (c) 7 marks each — 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.

(a) Perception–reaction time and braking distance

Perception–reaction time is the interval between the instant a hazard first becomes visible to a driver and the instant the driver's braking input begins to act on the vehicle. It is conventionally decomposed into the four PIEV stages: perception (the stimulus registers), identification or intellection (the driver recognises what it is), evaluation (a course of action is chosen — brake, steer, or both), and volition (the muscular action is executed). Because it is a human rather than a mechanical property, it varies widely between drivers and with alertness, expectancy, age, complexity of the situation and lighting; measured values for simple expected events run from about 0.5 s to well over 2 s. Design must accommodate the slow tail of that distribution rather than its mean, and AASHTO and the TAC Geometric Design Guide both adopt $t = 2.5$ s for stopping-sight-distance calculations — a value near the 90th percentile for an unexpected hazard. During this interval the vehicle continues at its initial speed, covering the brake reaction distance

which for 60 km/h is $0.278(60)(2.5) = 41.7$ m — exactly the value printed in the table on the paper.

Braking distance is the distance the vehicle travels from the onset of effective braking until it comes to rest. It follows from equating the initial kinetic energy to the work done by the retarding force. Expressed through a deceleration rate $a$,

and expressed instead through a coefficient of friction $f$ on a grade $G$ (as a decimal, positive upgrade),

AASHTO 2001 adopts $a = 3.4$ m/s$^2$ — a comfortable deceleration that most drivers can achieve on wet pavement without loss of control, rather than the maximum the tyres could deliver. At 60 km/h, $d_2 = 0.039(60)^2/3.4 = 41.3$ m, again matching the printed table. The stopping sight distance is the sum:

Reproducing three rows of the printed table from these two relations (20 km/h → 13.9 / 4.6 / 18.5 m; 60 km/h → 41.7 / 41.3 / 83.0 m; 100 km/h → 69.5 / 114.7 / 184.2 m) confirms that the table on the paper is internally consistent, which is worth doing before relying on any value from it. The distinction between the two braking-distance forms matters: the deceleration form is the one keyed to this table, and the friction form should be used only where a question supplies a coefficient of friction.

(b) Minimum crest curve length at 60 km/h with non-standard sight heights

Approach. On a crest the sight obstruction is the pavement surface itself, so the minimum length follows from the standard AASHTO crest relations. Two cases exist depending on whether the sight distance is shorter or longer than the curve; compute the $S \lt L$ case first and verify the inequality it assumes.

1. State the assumption on the grade signs. The question gives the two grades as magnitudes ("5% grade" and "3% grade") without signs, while part (c) is explicit ("+3%" and "−3%"). A crest requires the grade to decrease through the curve, which both $+5\% \to +3\%$ and $+5\% \to -3\%$ satisfy. The reading adopted is the hilltop case that produces a genuine sight-distance control:
   $$G_1 = +5\%, \quad G_2 = -3\%, \qquad A = |G_1 - G_2| = 8\%$$
   The alternative reading is treated in the callout below.
2. Take the stopping sight distance from the printed table. At a design speed of 60 km/h the AASHTO 2001 metric table on the paper gives a design SSD of
   $$S = 85\ \text{m}$$
   (the calculated value is 83.0 m; the design column is the rounded value used for design, and is the one adopted here). Cross-checking against the relations of part (a): $0.278(60)(2.5) + 0.039(60)^2/3.4 = 41.7 + 41.3 = 83.0$ m, which matches the table's calculated column exactly.
3. Apply the crest curve relation for $S \lt L$. When the sight distance lies wholly within the curve,
   $$L = \frac{A\,S^{2}}{100\left(\sqrt{2h_1} + \sqrt{2h_2}\right)^{2}}$$
   With the non-standard heights $h_1 = 1.05$ m and $h_2 = 0.50$ m the denominator constant is
   $$100\left(\sqrt{2(1.05)} + \sqrt{2(0.50)}\right)^{2} = 100\left(1.4491 + 1.0000\right)^{2} = 100(2.4491)^{2} = 599.8$$
   so
   $$L = \frac{8 \times 85^{2}}{599.8} = \frac{57\,800}{599.8} = \boxed{96.4\ \text{m}}$$
4. Verify the case assumption and the alternative case. The result must satisfy the inequality it was derived under: $S = 85$ m $\lt L = 96.4$ m $\checkmark$, so the $S \lt L$ branch is the correct one. For completeness, the $S \gt L$ branch would give
   $$L = 2S - \frac{200\left(\sqrt{h_1} + \sqrt{h_2}\right)^{2}}{A} = 170 - \frac{200(1.0247 + 0.7071)^{2}}{8} = 170 - 75.0 = 95.0\ \text{m}$$
   which contradicts its own premise ($95.0 \gt 85$), confirming that only the first branch is admissible.
5. Express the answer as a design length and check it against the minimum-length controls. The corresponding rate of vertical curvature is
   $$K = \frac{L}{A} = \frac{96.4}{8} = 12.05\ \text{m per percent}$$
   which is consistent with the AASHTO/TAC tabulated $K = 11$ for a 60 km/h crest on standard sight heights — slightly larger here because the raised object height of 500 mm is offset by a driver eye height of 1050 mm that is lower than the 1080 mm standard. The comfort-and-appearance minimum, $L_{min} = 0.6V = 36$ m, is not binding. A designer would specify $L = 100$ m in construction documents; the computed minimum is 96.4 m.

(c) Design speed of a given 225 m crest curve

Approach. Invert the crest curve relation to obtain the sight distance the curve actually provides, then enter the printed table with that sight distance and read off the highest design speed it supports.

1. Form the standard-height constant. With $h_1 = 1.08$ m and $h_2 = 0.60$ m,
   $$100\left(\sqrt{2(1.08)} + \sqrt{2(0.60)}\right)^{2} = 100\left(1.4697 + 1.0954\right)^{2} = 100(2.5651)^{2} = 658$$
   which is exactly the constant 658 that appears in the AASHTO metric crest formula — a useful confirmation that the "standard heights" of the question are the AASHTO standard values.
2. Invert the $S \lt L$ crest relation for the available sight distance. Rearranging $L = A S^{2}/658$,
   $$S = \sqrt{\frac{658\,L}{A}} = \sqrt{\frac{658 \times 225}{6}} = \sqrt{24\,675} = \boxed{157.1\ \text{m}}$$
   The premise holds, $S = 157.1$ m $\lt L = 225$ m $\checkmark$, so the branch is correct. (Back-substituting $S = 157.08$ m into the $S \lt L$ formula returns $L = 225.0$ m, closing the inversion.)
3. Enter the printed table and read the design speed. The relevant rows of the AASHTO 2001 metric table are

   Design speed (km/h)	Calculated SSD (m)	Design SSD (m)
   70	104.9	105
   80	129.0	130
   90	155.5	160
   100	184.2	185

   The available 157.1 m exceeds the 130 m required at 80 km/h but falls 2.9 m short of the 160 m design value required at 90 km/h. Because design values are what govern a design, the answer is
   $$V_{design} = \boxed{80\ \text{km/h}}$$
4. Quantify the margin and the alternative reading. Solving the SSD relation of part (a) for the exact speed the curve supports,
   $$157.1 = 0.695\,V + 0.011471\,V^{2} \;\Longrightarrow\; V = 90.6\ \text{km/h}$$
   so on the table's calculated column (155.5 m at 90 km/h) the curve does support 90 km/h, with 1.6 m to spare. The curve therefore sits almost exactly at the 90 km/h threshold: it satisfies the calculated requirement but not the rounded design requirement. The conservative and standard answer is 80 km/h, with the note that a 90 km/h posting could be justified only by accepting the unrounded criterion — a 1.8 % margin, which is not a prudent basis for a safety control. Lengthening the curve to $L = 6(160)^{2}/658 = 233$ m would provide the full 160 m and make 90 km/h defensible.