16-Civ-A6 Highway Design, Construction, and Maintenance · December 2018

Question 4 of 5: Crest vertical curve — reconstructing a rear-end collision

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

Question 4: Crest vertical curve — reconstructing a rear-end collision (25 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. A crest curve joining $+2\ \%$ to $-3\ \%$ over 300 m on a highway designed for 80 km/h; a vehicle travelling at 90 km/h struck a stalled vehicle at 10 km/h; the driver's eye height is 1.38 m, the stalled vehicle stands 1.10 m, and test runs give a friction coefficient of 0.32.

Find. (a) whether the sight distance available on this curve was in fact insufficient, and what the collision itself implies about the driver's response; (b) the other factors that plausibly contributed.

Approach. Compute the sight distance the curve actually provides for the specific eye and object heights measured in this crash, compare it with the stopping sight distance required at both the travel speed and the design speed, and then close the loop by using the impact speed to back-calculate the perception-reaction time the driver must have taken. The last step is what turns an inequality into a finding.

Part (a) — testing the driver's claim

1. Establish the geometry. The algebraic difference in grade and the rate of vertical curvature are $$A = |G_2 - G_1| = |-3 - (+2)| = 5\ \% \qquad K = \frac{L}{A} = \frac{300}{5} = 60\ \text{m per percent}$$
2. Compute the sight distance the curve provides for these particular heights. The crest relation on appendix page 9 is $K = S^{2}/[200(\sqrt{H}+\sqrt{h})^{2}]$, valid while the sight line lies wholly within the curve. Inverting it for $S$ with the measured heights — not the standard design heights — gives $$S = \sqrt{\frac{200\,L\,(\sqrt{H}+\sqrt{h})^{2}}{A}} = \sqrt{\frac{200(300)(\sqrt{1.38}+\sqrt{1.10})^{2}}{5}}$$ $$S = \sqrt{\frac{60\,000\,(1.1747 + 1.0488)^{2}}{5}} = \sqrt{\frac{60\,000(4.9441)}{5}} = \sqrt{59\,330} = \boxed{S = 243.6\ \text{m available}}$$ The result is 243.6 m against a 300 m curve, so $S \lt L$ and the branch of the formula used is the correct one. It is worth noting how much the object height matters: repeating the calculation with the standard design values $H = 1.08$ m and $h = 0.60$ m returns only 198.7 m. Using the design heights would understate the sight line by 18 %, because a stalled car is a far taller object than the 0.60 m hazard the design criterion is written around.
3. Compute the stopping sight distance the driver required. With the standard perception-reaction time of 2.5 s (the paper does not state one; this is the AASHTO and TAC design value, assumed under Note 2) and the measured friction of 0.32, on a level basis $$SSD_{90} = 0.278\,V\,t + \frac{V^{2}}{254(f+G)} = 0.278(90)(2.5) + \frac{90^{2}}{254(0.32)} = 62.6 + 99.7 = 162.2\ \text{m}$$ Taking the worst grade the vehicle could have been on, the $-3\ \%$ departure leg, lengthens this to $$SSD_{90,\,-3\%} = 62.6 + \frac{8100}{254(0.32-0.03)} = 62.6 + 110.0 = 172.5\ \text{m}$$ At the highway's design speed of 80 km/h the requirement is smaller still, $SSD_{80} = 55.6 + 78.7 = 134.3$ m.
4. Compare, and dispose of the claim. Setting the three numbers side by side, $$243.6\ \text{m available} \;\gt\; 172.5\ \text{m needed at 90 km/h on the worst grade} \;\gt\; 134.3\ \text{m needed at the design speed}$$ The curve provided 41 % more sight distance than the driver needed even at an illegal speed on the steepest part of the curve, and 81 % more than the design criterion demands. The claim is not supported.
5. Confirm that the curve itself complies with the design standard. This is a separate question from the crash and is worth answering explicitly, because it is what an expert would be asked in court. At the design speed of 80 km/h, using the standard design heights, the required rate of vertical curvature is $$K_{req} = \frac{SSD_{80}^{2}}{200(\sqrt{1.08}+\sqrt{0.60})^{2}} = \frac{134.3^{2}}{200(3.2899)} = 27.43 \quad\Rightarrow\quad L_{req} = K_{req}A = 137.1\ \text{m}$$ The curve as built is 300 m long, more than twice the minimum. Geometrically, the section is generous rather than deficient.
6. Use the impact speed to find what the driver actually did. This is the decisive step. The distance needed to brake from 90 km/h down to the 10 km/h impact speed, at the measured friction, is $$d_{brake} = \frac{V_0^{2} - V_t^{2}}{254(f+G)} = \frac{90^{2} - 10^{2}}{254(0.32)} = \frac{8000}{81.28} = 98.4\ \text{m}$$ The stalled vehicle first became visible 243.6 m away, so the distance covered before the brakes took effect was $243.6 - 98.4 = 145.2$ m, travelled at 90 km/h (25 m/s): $$t_{pr} = \frac{145.2}{25.0} = \boxed{5.81\ \text{s of perception-reaction time implied}}$$ On the $-3\ \%$ leg the braking distance rises to 108.6 m and the implied reaction time falls to 5.40 s. Either way the driver took more than twice the 2.5 s design value and roughly four times the 1.0–1.5 s an alert driver achieves in controlled tests.
7. State the conclusion. The available sight distance was not the cause. Had the driver reacted within the 2.5 s the geometry is designed around, braking would have begun 181 m before the stalled vehicle against the 98 m required, and the vehicle would have stopped roughly 80 m short of the collision. The claim is rejected, and the reconstruction instead points to a delayed response, compounded by a travel speed 10 km/h above the design speed of the highway.

Part (b) — other contributing factors

A reconstruction that ends at "the driver was slow" is incomplete; the useful question is what would cause a five-second response on a road with a clear sight line. The factors below are the ones an investigator would pursue, grouped by the element of the road-vehicle-driver system they belong to.

Driver factors. Distraction is the leading candidate: a five-second gap is close to the median duration of a mobile-telephone interaction or a glance away from the road. Fatigue, alcohol or drug impairment and medical incapacity all extend perception-reaction time in the same way and would be tested for. Expectancy matters too — a driver on a rural highway does not expect a stationary vehicle in the travelled lane, and detecting an unexpected object takes measurably longer than detecting an anticipated one. Finally, the vehicle was travelling at 90 km/h where the design speed is 80 km/h, which shortens every available time margin by about 11 %.

Conditions and conspicuity. A stationary vehicle presents almost no relative motion cue, so it is intrinsically hard to detect; whether its hazard lights were on, whether a warning triangle had been deployed, and whether it was fully or partly in the travelled lane are all material. Ambient conditions matter in the same way: at night the sight distance is set by headlight throw rather than by the crest geometry, and rain, fog, low sun on the crest or glare from oncoming headlights would all delay detection. A vehicle ahead that swerved late around the obstruction would have masked it until the last moment.

Pavement and vehicle. The 0.32 friction coefficient comes from test runs made after the event; if the surface was wet, contaminated, polished or rutted at the time, the achievable deceleration would have been lower and the braking distance longer than the 98 m calculated. Worn tyres, an under-inflated or unevenly loaded vehicle, poorly maintained brakes or the absence of anti-lock braking would have the same effect, as would a heavy load. Any of these would reduce the implied reaction time somewhat, though not enough to bring it within the design value: even at $f = 0.20$ the braking distance is only 157 m and the implied reaction time is still 3.4 s.

Roadway and roadside. Whether a shoulder of usable width was available for the disabled vehicle to pull clear is a direct design question, and its absence would be a genuine contributing factor rather than a driver failing. The same applies to the absence of advance warning or emergency-response measures, to the horizontal alignment through the crest (a combined horizontal and vertical curve can hide an object that the vertical geometry alone would reveal), and to whether roadside vegetation or a barrier on the inside of any accompanying horizontal curve intruded on the sight line. These should be checked before the file is closed, because they are the factors within the road authority's control.