16-Civ-B10 Traffic Engineering · May 2015

Question 6 of 7: Service Volumes on an Urban Freeway and Level of Service on a Rural Two-Lane Highway

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

Notes on this paper

Paper format. 98-Civ-B10 Traffic Engineering, National Examinations, May 2015. Three-hour duration, OPEN BOOK, any non-communicating calculator permitted. Seven questions, all of equal value (20 marks each) with the mark split printed in the grading scheme; the paper requires a total of five solutions. All seven are worked below, because this document is a study resource rather than an exam script.

Reference texts. Garber, N. J. and Hoel, L. A., Traffic and Highway Engineering — Ch. 4 (traffic-engineering studies: volume counts, peak-hour factor, spot-speed studies), Ch. 6 (fundamental principles of traffic flow: the Greenshields model, deterministic and stochastic queueing, the moving-observer method) and Ch. 8 (intersection control: saturation flow, change and clearance intervals). Transportation Research Board, Highway Capacity Manual — basic-freeway-segment and two-lane-highway level-of-service criteria and the lane-width, lateral-clearance and heavy-vehicle adjustment factors. Transportation Association of Canada, Geometric Design Guide for Canadian Roads and the Manual of Uniform Traffic Control Devices for Canada — Canadian lane widths, crosswalk practice and clearance-interval policy. Institute of Transportation Engineers, Traffic Engineering Handbook — queueing applied to parking and toll facilities.

Check — declared assumptions, invoked under the paper’s own NOTE 1 and NOTE 2 (“any data required, but not given, can be assumed”, with a clear statement of the assumption). This paper is open-book and deliberately withholds several standard table values. The following are used consistently throughout and each is restated where it is first applied:

Bus (heavy-vehicle) passenger-car equivalent on the level intersection approaches of Question 1: E_B = 2.0.
Two-phase signal operation at the Question 1 intersection (one phase for the north–south pair, one for the east–west pair), with no start-up lost time given, so the whole intergreen is taken as lost time.
Maximum service flow rates and volume-to-capacity ratios for basic freeway segments at a 100 km/h design speed: 700/1000/1450/1800/2000 pc/h/lane is the 110 km/h family, so the 100 km/h family 600/1000/1450/1800/2000 pc/h/lane (v/c = 0.30/0.50/0.72/0.90/1.00) is used, with an ideal capacity of 2000 pc/h/lane.
Two-lane-highway ideal capacity 2800 pc/h both directions, with level-terrain v/c ratios 0.15/0.27/0.43/0.64/1.00 at 0 % no-passing zones, and a 50/50 directional split (f_d = 1.00).
Passenger-car equivalents on specific grades: E_T = 5 for trucks on both the 4 % / 1.0 km grade and the 3 % / 1.5 km grade, E_R = 3 for recreational vehicles on the 4 % grade, E_B = 1.6 for intercity buses on grades of 4 % or flatter, and E_T = 2 for trucks in general level terrain.

Different editions of the capacity manual tabulate these cells slightly differently. Where an alternative reading changes an answer materially, the alternative is computed and stated alongside; the method and the arithmetic are what carry the marks.

Question 6: Service Volumes on an Urban Freeway and Level of Service on a Rural Two-Lane Highway (2 × 10 = 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.

Part (a) — urban six-lane freeway	Value
Lanes per direction	N = 3, each 3.5 m wide
Right shoulder / left shoulder	1.5 m / 0.5 m, so obstructions both sides
Grade	3 %, 1.5 km long
Trucks / intercity buses	6 % / 3 %
Peak hour factor	0.90
Average highway speed	100 km/h

Part (b): rural two-lane highway; 3.75 m lanes with 3 m shoulders; level terrain over a long section; ideal alignment at an average highway speed of 120 km/h; 100 % passing opportunity, so no no-passing zones; 6 % trucks; design hourly volume 1900 veh/h.

Find. (a) The hourly service volume the freeway can carry at each of the five levels of service. (b) Which level of service the two-lane highway actually delivers at its design hourly volume.

Approach. Both parts use the same skeleton: an ideal capacity, a volume-to-capacity ratio per level of service, and multiplicative adjustments for geometry and heavy vehicles. Part (a) asks the forward question (find the volume for a given level of service) and additionally converts a flow rate to an hourly volume through the peak hour factor; part (b) asks the inverse question (find the level of service for a given volume).

Part (a) — establish the ideal per-lane rates for a 100 km/h facility. For a basic freeway segment at a 100 km/h average highway speed, the maximum service flow rates and their volume-to-capacity ratios against an ideal capacity of 2000 pc/h/lane are: $$\begin{array}{lccccc} \text{LOS} & A & B & C & D & E\\ v/c & 0.30 & 0.50 & 0.72 & 0.90 & 1.00\\ MSF \text{ (pc/h/ln)} & 600 & 1000 & 1450 & 1800 & 2000 \end{array}$$
Determine the lane-width and lateral-clearance factor for obstructions on both sides. Shoulders of 1.5 m and 0.5 m are both narrower than the 1.8 m at which a roadside object ceases to influence driver behaviour, so this is a both-sides-obstructed case, and the convention is to use the average clearance: $$\text{clearance} = \frac{1.5 + 0.5}{2} = 1.0 \text{ m}$$ For a six-lane freeway with obstructions on both sides the bracketing table entries are 0.98 (3.6 m lanes, 1.2 m), 0.96 (3.6 m lanes, 0.6 m), 0.94 (3.3 m lanes, 1.2 m) and 0.92 (3.3 m lanes, 0.6 m). Interpolating on clearance at 1.0 m, $$f_w(3.6\text{ m}) = 0.98 + \frac{1.0-1.2}{0.6-1.2}(0.96-0.98) = 0.9733, \qquad f_w(3.3\text{ m}) = 0.94 - 0.0067 = 0.9333$$ and then on lane width at 3.5 m, $$f_w = 0.9333 + \frac{3.5-3.3}{3.6-3.3}\,(0.9733-0.9333) = 0.9333 + 0.0267 = 0.960$$
Build the heavy-vehicle factor for the two heavy classes on the grade. On a 3 % grade 1.5 km long, trucks are badly affected but intercity buses — which have a far better power-to-weight ratio — are barely slowed, so the equivalents differ sharply: $E_T = 5$ for trucks and $E_B = 1.6$ for buses on grades of 4 % or flatter. Hence $$f_{HV} = \frac{1}{1 + P_T(E_T-1) + P_B(E_B-1)} = \frac{1}{1 + 0.06(4) + 0.03(0.6)}$$ $$f_{HV} = \frac{1}{1 + 0.24 + 0.018} = \frac{1}{1.258} = 0.795$$ Treating the buses as if they were trucks would give $f_{HV} = 0.735$ and understate the service volumes by about 8 %.
Convert flow rate to hourly volume through the peak hour factor. The maximum service flow rate is a peak-15-minute rate; the service volume is the full-hour total that produces it, and the two are linked by the peak hour factor: $$SV_i = MSF_i \times N \times f_w \times f_{HV} \times PHF$$ The common multiplier is $$3 \times 0.960 \times 0.795 \times 0.90 = 2.060$$
Evaluate the service volume at each level of service. Multiplying each ideal rate by 2.060, $$\begin{aligned} SV_A &= 600(2.060) = 1236 \text{ veh/h} & SV_B &= 1000(2.060) = 2060 \text{ veh/h}\\ SV_C &= 1450(2.060) = 2988 \text{ veh/h} & SV_D &= 1800(2.060) = 3709 \text{ veh/h}\\ SV_E &= 2000(2.060) = 4121 \text{ veh/h} & & \end{aligned}$$ $$\boxed{\;SV = 1236 \,/\, 2060 \,/\, 2988 \,/\, 3709 \,/\, 4121 \text{ veh/h at LOS A / B / C / D / E}\;}$$ Level of service E is by definition capacity, so this six-lane section can carry about 4121 veh/h per direction — only 69 % of the 6000 veh/h an ideal three-lane section would deliver, the loss being split roughly one part geometry, three parts heavy vehicles on the grade, and two parts peaking.

Question 6(a) — service volumes for the six-lane urban freeway at each level of service, after the 0.960 geometry factor, the 0.795 heavy-vehicle factor and the 0.90 peak hour factor are applied to the ideal per-lane rates.

Part (b) reverses the question: the volume is known and the level of service is the unknown, and the facility type changes from a freeway to a two-lane highway, where opposing traffic rather than lane capacity is the binding constraint.

Part (b) — adopt the two-lane-highway framework. On a two-lane highway the two directions share the same pavement, so capacity is quoted for both directions combined: 2800 pc/h under ideal conditions. The service flow rate at a given level of service is $$SF_i = 2800 \times (v/c)_i \times f_d \times f_w \times f_{HV}$$ For a long section in level terrain with 100 % passing opportunity (no no-passing zones), the volume-to-capacity ratios are 0.15, 0.27, 0.43, 0.64 and 1.00 for levels of service A through E.
Evaluate the three adjustment factors. The directional-distribution factor for a balanced flow is $f_d = 1.00$; no split is given, so a 50/50 division is assumed. The lanes are 3.75 m wide with 3 m shoulders — at or above ideal on both counts, and the alignment is stated to be ideal — so $$f_w = 1.00$$ For trucks in general level terrain (as opposed to a specific grade) the equivalent is $E_T = 2$, so with 6 % trucks and no other heavy classes, $$f_{HV} = \frac{1}{1 + 0.06(2-1)} = \frac{1}{1.06} = 0.943$$
Compute the service flow rate at each level of service. The adjusted capacity is $$2800 \times 1.00 \times 1.00 \times 0.943 = 2641.5 \text{ veh/h, both directions}$$ and multiplying by each volume-to-capacity ratio, $$\begin{aligned} SF_A &= 2641.5(0.15) = 396 \text{ veh/h} & SF_B &= 2641.5(0.27) = 713 \text{ veh/h}\\ SF_C &= 2641.5(0.43) = 1136 \text{ veh/h} & SF_D &= 2641.5(0.64) = 1691 \text{ veh/h}\\ SF_E &= 2641.5(1.00) = 2642 \text{ veh/h} & & \end{aligned}$$
Locate the design hourly volume among those bands and state the level of service. The design hourly volume of 1900 veh/h exceeds the level-of-service D threshold of 1691 veh/h but falls below capacity at 2642 veh/h, so $$SF_D = 1691 < 1900 < 2642 = SF_E \quad\Rightarrow\quad \boxed{\;\text{Level of service E}\;}$$ The operating volume-to-capacity ratio is $$\frac{v}{c} = \frac{1900}{2641.5} = 0.719$$ comfortably above the 0.64 that level of service D permits. Level of service E on a two-lane road means unstable flow: platoons are essentially continuous, passing demand vastly exceeds passing supply, and drivers spend nearly all their time following. That an ideal-alignment highway with 120 km/h average speed still fails at 1900 veh/h is the central lesson of two-lane analysis — the constraint is not pavement capacity but the diminishing availability of gaps in the opposing stream, and geometric excellence cannot buy much past a v/c of about 0.64.

Question 6 — results
Quantity	Result
(a) f_w (3.5 m lanes, 1.0 m average clearance, both sides)	0.960
(a) f_HV (E_T = 5, E_B = 1.6)	0.795
(a) Service volume, LOS A	1236 veh/h
(a) Service volume, LOS B	2060 veh/h
(a) Service volume, LOS C	2988 veh/h
(a) Service volume, LOS D	3709 veh/h
(a) Service volume, LOS E (capacity)	4121 veh/h
(b) Adjusted two-lane capacity (both directions)	2641.5 veh/h
(b) Service flow rates, LOS A–E	396 / 713 / 1136 / 1691 / 2642 veh/h
(b) Operating v/c at DHV = 1900 veh/h	0.719
(b) Level of service provided	E

Topic. Level-of-service analysis in both directions: service volumes for a multilane freeway, and the inverse problem on a two-lane two-way highway.

Key relations. Freeway: $SV_i = MSF_i \cdot N \cdot f_w \cdot f_{HV} \cdot PHF$, with $MSF_i = (v/c)_i \times 2000$ pc/h/ln. Two-lane highway: $SF_i = 2800 \cdot (v/c)_i \cdot f_d \cdot f_w \cdot f_{HV}$, capacity quoted for both directions combined. Heavy vehicles: $f_{HV} = 1/[1+\sum P_j(E_j-1)]$, with specific-grade equivalents on a stated grade and general-terrain equivalents otherwise. Both-sides obstruction: use the average lateral clearance.

Why this works. The two facility types share an algebraic skeleton but rest on completely different physics, and the contrast is the real content of this question. On a freeway, the levels of service are defined by density, and density maps monotonically onto flow through the speed–flow curve, so a volume-to-capacity ratio is a legitimate proxy for quality of service; the peak hour factor then converts the peak-15-minute rate that actually determines operating conditions into the hourly volume a designer forecasts. On a two-lane highway the opposing stream, not the pavement, is the constraint: quality of service is measured by percent time spent following, passing demand is met only from gaps in the opposing flow, and the volume-to-capacity ratios that delimit the levels of service are therefore far lower — 0.64 for level of service D against 0.90 on the freeway. This is why the ratios must be taken from the correct table for the correct terrain and no-passing percentage, and why a 1900 veh/h two-way volume that would be trivial for one freeway lane pushes a two-lane highway into unstable flow.

Common pitfall. Applying the peak hour factor in the wrong direction, or applying it in part (b) where a design hourly volume is already an hourly figure. Using specific-grade truck equivalents in part (b), where general level terrain is stated, or general-terrain equivalents in part (a), where a grade is given. Treating intercity buses as trucks in part (a), which understates every service volume by about 8 %. And on the two-lane highway, comparing the 1900 veh/h against a one-direction capacity: the 2800 pc/h ideal is for both directions combined, and halving it would place the section past capacity altogether.

Source. TRB Highway Capacity Manual, basic freeway segments and two-lane highways (level-of-service criteria, lane-width and lateral-clearance factors, directional-distribution and heavy-vehicle adjustments); Garber & Hoel, Traffic and Highway Engineering, Ch. 9 (capacity and level of service for freeways and two-lane highways).

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