16-Civ-B7 Transportation Planning and Engineering · December 2013

Question 2 of 7

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 2013 — a three-hour open-book examination; any non-communicating calculator is permitted. The cover page states that a total of five solutions is required, that only the first five as they appear in the answer book will be marked, and that all questions are of equal value. The grading scheme printed on page 1 confirms 20 marks per question, split as: Q1 20; Q2 20; Q3 (a) 15 and (b) 5; Q4 (a) 8 and (b) 12; Q5 (a) 8 and (b) 12; Q6 20; Q7 (a) 8 and (b) 12. All seven printed questions are worked below, because this set is a study resource rather than a timed attempt; on exam day a candidate submits only the first five, in order. The paper also states that any data required but not given may be assumed and that assumptions should be recorded with the answer — several questions need that licence, and every assumption is flagged where it is made.

Reference texts. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed. (sight distance, vertical and horizontal alignment, traffic stream models, earthwork); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (Canadian design-domain values for stopping sight distance, perception-reaction time and deceleration); AASHTO, A Policy on Geometric Design of Highways and Streets (the tabulated metric stopping sight distances); AASHTO, Guide for Design of Pavement Structures (1993) (rigid pavement thickness, reliability, drainage and load-transfer coefficients); Asphalt Institute, Mix Design Methods MS-2 (gradation charts, the 0.45 power chart, aggregate blending); M.S. Mamlouk and J.P. Zaniewski, Materials for Civil and Construction Engineers, 4th ed. (aggregate moisture states, sieve analysis); Transportation Association of Canada, Pavement Asset Design and Management Guide (Canadian pavement design practice).

Check — assumptions carried through this paper. Four inputs the exam does not supply are assumed under its own Note 2 (“any data required, but not given, can be assumed”), and each is restated at the point of use: (i) Question 2 needs a stopping-sight-distance basis — a 2.5 s perception-reaction time and a 3.4 m/s² deceleration, the TAC and AASHTO design values, giving the tabulated 185 m at 100 km/h; (ii) Question 2 also needs to know whether the 600 m radius is to the road centreline — it is taken as the centreline, and Step 5 shows the alternative reading changes the answer by 0.02 m; (iii) Question 6 does not say whether the transverse joints are dowelled — dowels are assumed, giving a load-transfer coefficient J = 3.2, with the undowelled case quantified in a callout; (iv) Question 6 gives a drainage description rather than a coefficient, so C_d = 1.00 is read from the AASHTO table, again with the alternative quantified.

Question 2 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. A simple circular curve on a two-lane rural highway, with the radius quoted to the road centreline:

Given data — horizontal curve sight line
Quantity	Symbol	Value
Radius to the road centreline	$R$	600 m
Lane width	$w$	3.5 m
Number of lanes	—	2 (one each way)
Design speed	$V$	100 km/h
Perception-reaction time (assumed)	$t$	2.5 s
Deceleration rate (assumed)	$a$	3.4 m/s²

Find. The lateral distance, measured from the inside edge of the inside lane, over which sight obstructions must be removed so that a driver in the inside lane always has the full stopping sight distance available.

Sight line across the inside of the horizontal curve. The clearance is measured from the inside edge of the inside lane, not from the vehicle path.

Approach. Compute the stopping sight distance for 100 km/h, treat that distance as an arc travelled along the driver’s own path (the centre of the inside lane), find the middle ordinate of the corresponding chord, and then convert that ordinate — which is measured from the vehicle path — into a clearance measured from the inside edge of the lane by deducting half a lane width.

Establish the stopping sight distance for the design speed. The standard two-term expression adds the distance covered during perception-reaction to the braking distance: $$S = 0.278\,Vt + \frac{V^2}{254\,(a/9.81)}$$ $$S = 0.278(100)(2.5) + \frac{100^2}{254\,(3.4/9.81)} = 69.5 + 113.6 = 183.1\text{ m}$$ Both TAC and AASHTO round this up to the tabulated design value, so $$S = \boxed{185\text{ m}}$$ is carried forward. (Using the computed 183.1 m instead changes the final clearance by about 0.11 m, so the choice is not critical.)
Identify the radius of the path the driver actually follows. Sight distance is measured along the centre of the inside lane, not along the road centreline. Because the inside lane lies half a lane width inside the centreline, $$R_v = R - \frac{w}{2} = 600 - \frac{3.5}{2} = 598.25\text{ m}$$
Compute the middle ordinate of the sight-line chord. The line of sight is the chord joining the driver and the object, both on the vehicle path; the sight obstruction must be cleared back to that chord. For an arc of length $S$ on a circle of radius $R_v$, the half angle subtended is $28.65\,S/R_v$ degrees, and $$M = R_v\left[1 - \cos\!\left(\frac{28.65\,S}{R_v}\right)\right]$$ $$M = 598.25\left[1 - \cos\!\left(\frac{28.65 \times 185}{598.25}\right)\right] = 598.25\,\bigl[1 - \cos(8.8596^\circ)\bigr]$$ $$M = 598.25\,(0.0119345) = \boxed{7.14\text{ m}}$$ The constant 28.65 is simply $90/\pi$, which converts the half-arc $S/2R_v$ from radians into degrees.
Convert the middle ordinate into a clearance measured from the lane edge. $M$ is measured inward from the vehicle path, which is itself $w/2 = 1.75$ m inside the lane’s inner edge, so the width that must actually be cleared beyond the pavement is $$m = M - \frac{w}{2} = 7.14 - 1.75 = \boxed{5.39\text{ m}}$$ Rounding up for construction, the clear-sight (daylight) zone should extend 5.4 m from the inside edge of the inside lane, and nothing higher than the 0.6 m object height — barriers, sign supports, guide rail, vegetation, cut slopes or noise walls — may stand inside it.
Test the assumption made about the radius. The question does not say whether the 600 m applies to the centreline or to the inside lane. If instead the 600 m is already the vehicle-path radius, then $M = 600\,[1 - \cos(28.65 \times 185/600)] = 7.12$ m and the clearance becomes 5.37 m, a difference of only 0.02 m. The answer is therefore insensitive to that ambiguity, and 5.4 m is safe under either reading.
Confirm that the geometry the formula assumes is the geometry that exists. The middle-ordinate expression assumes the whole sight line lies within the circular curve, which requires the curve to be at least $S = 185$ m long. At 600 m radius that is a central angle of $185/600$ radians, about $17.7^\circ$; any curve with a smaller deflection would place the driver or the object on the tangent, where the required clearance is less and a graphical or two-part solution is needed. The 5.39 m result should therefore be quoted together with the condition that the curve subtends at least about $18^\circ$.

Check — values assumed, not given. The exam supplies only speed, radius and lane width, so the perception-reaction time (2.5 s) and the deceleration rate (3.4 m/s²) are taken from TAC and AASHTO practice for rural highways under the paper’s Note 2. A 0.6 m object height and a 1.08 m eye height are implied by the same standards; they do not enter the arithmetic but they do define what counts as an obstruction inside the 5.4 m zone. If a provincial standard imposed a longer sight distance — say 210 m for a wet-weather or truck-braking criterion — the clearance would grow roughly with $S^2$, to about 7.5 m.

Final results — Question 2
Quantity	Symbol	Value
Stopping sight distance at 100 km/h	$S$	185 m (183.1 m computed)
Radius of the vehicle path (centre of inside lane)	$R_v$	598.25 m
Half angle subtended by the sight chord	—	$8.8596^\circ$
Middle ordinate from the vehicle path	$M$	7.14 m
Clearance required from the inside edge of the inside lane	$m$	5.39 m, use 5.4 m

Topic. Horizontal sight-line offset (middle ordinate) and the clear-sight or daylight zone on the inside of a horizontal curve.

Key relations. $S = 0.278Vt + V^2/[254(a/9.81)]$; $M = R_v[1 - \cos(28.65\,S/R_v)]$ with the angle in degrees, or equivalently $M = R_v[1 - \cos(S/2R_v)]$ with the angle in radians; inverted, $S = (R_v/28.65)\cos^{-1}[(R_v - M)/R_v]$; and the clearance from the lane edge is $m = M - w/2$.

Why this works. Sight distance on a curve is limited not by the pavement but by whatever stands on the inside of it, because the line of sight is a straight chord while the road bends away from it. The maximum separation between chord and arc is the middle ordinate, so clearing that width guarantees the sight line is unobstructed everywhere along the curve. The relation is pure circle geometry: the chord subtends a half angle $S/2R_v$ at the centre, and the sagitta of that chord is $R_v(1 - \cos)$ of it. Nothing about vehicle dynamics enters once $S$ is fixed.

Common pitfall. The most frequent error is computing $M$ on the centreline radius and reporting it directly, which is wrong twice over — the sight line belongs to the driver in the inside lane, and the answer is asked for from the lane edge, not from the vehicle path. Deducting the half lane width is the step most often forgotten, and it is worth 1.75 m out of 7.14 m here, a quarter of the answer. A second trap is applying the formula when the curve is shorter than the sight distance, in which case the chord runs off onto the tangents and the true clearance is smaller.

Source. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Chapter 3, horizontal alignment and sight distance on curves; AASHTO, A Policy on Geometric Design of Highways and Streets, stopping sight distance tables and the horizontal sightline offset; Transportation Association of Canada, Geometric Design Guide for Canadian Roads.

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