Question 1 of 7: Definitions and Discussion

Question 1: Definitions and Discussion (20 marks — 4 each)

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. Five traffic-engineering terms, four marks each. Four of the five are qualitative; the peak-hour factor carries arithmetic, so an illustrative approach count is supplied below (15-minute counts of 380, 420, 465 and 400 veh on one approach) so the definition is anchored to a number.

Find. A definition and a discussion of the engineering purpose of each term, with the discriminating feature of each contrasted pair made explicit.

Approach. Define each term, then say what design or analysis decision it feeds, because a four-mark “define and discuss” part is only half answered by a definition.

1. Part (a) — Circular versus spiral curves. A circular (simple) horizontal curve has a single constant radius $R$, so its curvature $1/R$ is zero along the tangent and jumps instantaneously to $1/R$ at the point of curvature. A vehicle following it would need an instantaneous lateral acceleration of $v^{2}/R$, which no driver can deliver, so drivers steer their own transition and encroach on the adjacent lane. A spiral (clothoid, or Euler spiral) is a transition curve whose radius decreases in inverse proportion to distance travelled, $R\,\ell = \text{constant}$, so curvature grows linearly from zero at the tangent-to-spiral point (TS) to $1/R$ at the spiral-to-curve point (SC). The spiral therefore gives a natural steering path, allows the superelevation runoff to be developed over the spiral length instead of on the tangent or inside the circular arc, and provides a smooth rate of change of lateral acceleration. The Shortt–Smirnoff relation sizes it, $L_{s} = 0.0214\,V^{3}/(R\,C)$ with $V$ in km/h, $R$ in m and $C$ the comfort rate of change of lateral acceleration (about $0.3\ \text{m/s}^{3}$); TAC and AASHTO in practice make $L_{s}$ at least the superelevation-runoff length. Spirals are warranted on sharper curves at higher speeds and on rail alignments, and are generally omitted where $R$ is large enough that the runoff can be developed on the tangent.
2. Part (b) — Peak hour factor. The peak hour factor measures how uniformly demand is spread within the peak hour. It is the ratio of the hourly volume to the peak rate of flow implied by the busiest 15-minute period, $$\mathrm{PHF} = \frac{V_{60}}{4\,V_{15,\max}}$$ Substituting the illustrative counts, $V_{60} = 380 + 420 + 465 + 400 = 1665$ veh/h and $4\,V_{15,\max} = 4(465) = 1860$ veh/h, so $$\mathrm{PHF} = \frac{1665}{1860} = \boxed{0.895}$$ It is bounded by $0.25$ (all traffic in one quarter-hour) and $1.00$ (perfectly uniform); real approaches fall between about 0.80 in small communities and 0.98 on congested urban freeways. Its purpose is to convert a measured or forecast hourly volume into the equivalent rate of flow $v = V/\mathrm{PHF}$ that capacity and level-of-service procedures require, because a facility that is adequate for the average hour can still break down in its busiest quarter-hour. Here $v = 1665/0.895 = 1860$ veh/h, i.e. the design must accommodate 1860 veh/h, not 1665.
3. Part (c) — Pedestrian clearance time. The pedestrian clearance interval is the flashing “don't walk” period that follows the steady WALK indication. It must be long enough for a pedestrian who legally leaves the kerb at the very end of WALK to reach the far kerb (or a median refuge) before conflicting vehicles are released. It is sized as crossing distance divided by an assumed walking speed, $$\mathrm{PCT} = \frac{L}{S_{p}}$$ with $L$ the crossing length and $S_{p}$ the design walking speed — MUTCDC and TAC practice uses $1.2\ \text{m/s}$ generally and $1.0\ \text{m/s}$ where older pedestrians, schools or hospitals dominate. It is distinct from the walk interval (the perception–reaction and start-up allowance, typically a 7 s minimum), and the two together give the minimum pedestrian green requirement often written $G_{p} = 3.2 + L/S_{p} + 0.27N$ for a crossing carrying $N$ pedestrians per cycle. Under-provision is a direct safety and liability exposure; over-provision inflates the cycle length and therefore vehicle delay at every other approach.
4. Part (d) — Gap acceptance behaviour. Gap acceptance describes how a driver waiting on a minor movement — a stop-controlled side street, a permitted left turn, a roundabout entry, a merge — judges the headways in the conflicting stream and decides which one is safe to use. The behaviour is characterised by two parameters: the critical gap $t_{c}$, the gap that would be accepted by half the driver population (a driver rejects all gaps smaller than $t_{c}$ and accepts all larger ones), and the follow-up headway $t_{f}$, the headway between successive queued vehicles using the same large gap. Because the conflicting headways are approximately negative-exponentially distributed, the potential capacity of the minor movement follows $$c_{p} = v_{c}\,\frac{e^{-v_{c}t_{c}/3600}}{1 - e^{-v_{c}t_{f}/3600}}$$ with $v_{c}$ the conflicting flow in veh/h. This is the basis of the HCM two-way-stop and roundabout capacity procedures, and it explains why unsignalised capacity collapses non-linearly as the major flow rises: the supply of acceptable gaps decays exponentially. Gap acceptance is not a constant — it varies with driver age, approach grade, number of conflicting lanes and waiting time (impatient drivers accept shorter gaps), which is why signalisation warrants exist at all.
5. Part (e) — Effective green versus displayed green. The displayed green $G$ is what the driver sees: the duration of the green indication. The effective green $g$ is the portion of the cycle during which vehicles actually discharge at the saturation flow rate. They differ by lost time. At the start of green the first few vehicles accelerate from rest, so the departure headways exceed the saturation headway for roughly the first four vehicles — this is start-up lost time $\ell_{1}$, about 2 s. At the end, part of the yellow and all-red is used productively and the remainder is not, giving clearance lost time $\ell_{2}$. Hence $$g = G + Y + AR - t_{L}, \qquad t_{L} = \ell_{1} + \ell_{2}$$ so effective green may be shorter or longer than the displayed green depending on how much of the change interval is used. Capacity is always computed on effective green, $c = s\,(g/C)$, never on displayed green; using $G$ instead of $g$ typically overstates approach capacity by 4–8 per cent on a short cycle.