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

Question 3 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 3 20 marks — (a) 15, (b) 5

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. Two independent traffic-stream problems, both to be treated with the linear (Greenshields) speed-density model:

Given data — traffic stream measurements
Part	Quantity	Symbol	Value
(a)	Average time headway	$\bar{h}$	2.0 s/veh
(a)	Average space headway	$\bar{s}$	50 m/veh
(b)	Free-flow speed	$u_f$	100 km/h
(b)	Capacity	$q_{max}$	4000 veh/h
(b)	Observed hourly volume	$q$	2000 veh/h

Find. For part (a), the flow rate in vehicles per hour, the density in vehicles per kilometre and the average (space mean) speed in km/h; for part (b), the space mean speed corresponding to the observed volume.

Greenshields linear speed-density model (left) and the resulting flow-density parabola (right), showing the two densities that carry 2000 veh/h.

Approach. Part (a) needs only the reciprocal definitions of headway and the identity $q = uk$; part (b) needs the Greenshields parabola, whose capacity fixes the jam density, after which the observed flow gives a quadratic in speed with two roots — one on the uncongested branch and one on the congested branch.

Convert the average time headway into an hourly flow rate. Flow is the reciprocal of the mean time headway, scaled to an hour: $$q = \frac{3600}{\bar{h}} = \frac{3600\text{ s/h}}{2.0\text{ s/veh}} = \boxed{1800\text{ veh/h}}$$
Convert the average space headway into a density. Density is the reciprocal of the mean spacing, scaled to a kilometre: $$k = \frac{1000}{\bar{s}} = \frac{1000\text{ m/km}}{50\text{ m/veh}} = \boxed{20\text{ veh/km}}$$
Recover the average speed from the fundamental identity. Flow, density and space mean speed are linked by $q = u k$ for any traffic stream, so $$u = \frac{q}{k} = \frac{1800\text{ veh/h}}{20\text{ veh/km}} = \boxed{90\text{ km/h}}$$ The same number follows directly from the headways, $u = \bar{s}/\bar{h} = 50/2.0 = 25$ m/s $= 90$ km/h, which is a useful independent check. Note that the linear speed-density assumption is not actually needed for part (a) — the identity $q = uk$ holds for every traffic stream model; the assumption would only be required to place this state on a particular curve.
Fix the jam density from the stated capacity, for part (b). Under Greenshields the speed falls linearly from $u_f$ at zero density to zero at the jam density $k_j$, so $q = u_f k(1 - k/k_j)$, a parabola whose peak occurs at $k = k_j/2$ and $u = u_f/2$. Capacity is therefore $q_{max} = u_f k_j/4$, and $$k_j = \frac{4q_{max}}{u_f} = \frac{4(4000)}{100} = 160\text{ veh/km}$$
Solve the parabola for the density at the observed volume. Substituting $q = 2000$ veh/h, $$2000 = 100\,k\left(1 - \frac{k}{160}\right) \quad\Longrightarrow\quad k^2 - 160k + 3200 = 0$$ $$k = \frac{160 \pm \sqrt{160^2 - 4(3200)}}{2} = \frac{160 \pm 113.14}{2} = 23.43 \text{ or } 136.57 \text{ veh/km}$$
Convert each root into a space mean speed and choose the operating branch. Using $u = q/k$, $$u = \frac{2000}{23.43} = \boxed{85.4\text{ km/h}} \qquad\text{or}\qquad u = \frac{2000}{136.57} = 14.6\text{ km/h}$$ Equivalently, and more directly, $u = \tfrac{1}{2}u_f\left[1 \pm \sqrt{1 - q/q_{max}}\right] = 50(1 \pm 0.7071)$. A volume of 2000 veh/h is only half of capacity, and a highway operating below capacity sits on the uncongested (upper) branch, so the estimate is 85.4 km/h. The lower root, 14.6 km/h, is the physically real congested state that carries the same volume in stop-and-go conditions; it should be quoted and then set aside, because reporting a single root without acknowledging the second is the classic incomplete answer to this question.

The two parts are worth comparing. Part (a) describes a stream at 20 veh/km and 90 km/h; part (b) describes one at 23.4 veh/km and 85.4 km/h. Both are light, free-flowing conditions well below the 80 veh/km that would signal capacity operation on the part (b) facility, which is why the speeds sit close to the free-flow value in each case.

Final results — Question 3
Part	Quantity	Value
(a)(i)	Volume (flow rate), $q$	1800 veh/h
(a)(ii)	Density, $k$	20 veh/km
(a)(iii)	Average (space mean) speed, $u$	90 km/h
(b)	Jam density implied by capacity, $k_j$	160 veh/km
(b)	Density on the uncongested branch, $k$	23.4 veh/km
(b)	Space mean speed (uncongested)	85.4 km/h
(b)	Space mean speed (congested root, rejected)	14.6 km/h

Topic. The fundamental relations of an uninterrupted traffic stream, and the Greenshields linear speed-density model.

Key relations. $q = 3600/\bar{h}$; $k = 1000/\bar{s}$; $q = uk$ with $u$ the space mean speed; Greenshields $u = u_f(1 - k/k_j)$, hence $q = u_f k(1 - k/k_j)$; capacity $q_{max} = u_fk_j/4$ at $k = k_j/2$ and $u = u_f/2$; and $u = \tfrac{1}{2}u_f[1 \pm \sqrt{1 - q/q_{max}}]$.

Why this works. Headway and spacing are the microscopic descriptions of the same stream that flow and density describe macroscopically, so the reciprocals are exact rather than approximate. The identity $q = uk$ is a definition once $u$ is understood as the space mean speed — the harmonic-type average over distance rather than the time mean speed measured at a point — and this is why exam questions are careful to ask for the space mean speed. Greenshields then adds one empirical assumption, that speed falls off linearly with density, which turns the flow into a parabola in density and makes capacity, jam density and free-flow speed mutually determined: fix any two and the third follows.

Common pitfall. Reporting only one root in part (b) loses marks even when the arithmetic is right, because a parabola in density gives two states at every sub-capacity flow. A second pitfall is using the time mean speed in $q = uk$; the identity holds only for the space mean speed, and the time mean speed always exceeds it. A third is misreading capacity as jam density: at capacity the highway is half jammed and moving at half the free-flow speed, not stopped.

Source. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Chapter 6 (Fundamental Principles of Traffic Flow), Greenshields model and the speed-flow-density relationships.

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