16-Civ-B10 Traffic Engineering · December 2018

Question 2 of 7: Curves (6 + 7 + 7 marks)

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

Notes on this paper

Paper format. National Examinations, December 2018, 16-Civ-B10 Traffic Engineering. Three-hour duration, OPEN BOOK (any non-communicating calculator permitted). Seven questions on four pages, all of equal value at 20 marks each; the paper requires five solutions and marks only the first five as they appear in the answer book. Because the set is a study resource, all seven questions are solved here. The paper’s own Note 1 invites a clear statement of any assumptions made and Note 2 permits any required-but-not-given data to be assumed; every assumption used below is stated explicitly where it is introduced. Page 2 reproduces the AASHTO 2001 metric stopping-sight-distance table, which Question 2 is built around.

Reference texts.

Garber, N.J. and Hoel, L.A., Traffic and Highway Engineering, 5th ed., Cengage — Ch. 4 (traffic-engineering studies, moving-vehicle method), Ch. 6 (fundamental principles of traffic flow and queueing), Ch. 8 (intersection control and signal timing), Ch. 15 (geometric design of highways, vertical curves).
Transportation Research Board, Highway Capacity Manual — signalised-intersection capacity, saturation flow, lane-group definition and the pedestrian-interval (minimum green) method.
AASHTO, A Policy on Geometric Design of Highways and Streets, 2001 metric edition — stopping sight distance and crest vertical curves. The SSD table printed on page 2 of this paper is AASHTO 2001, Table 3-1.
Transportation Association of Canada, Geometric Design Guide for Canadian Roads — the Canadian design-controls document, and the governing reference for Canadian practice.
Transportation Association of Canada, Manual of Uniform Traffic Control Devices for Canada (MUTCDC) — signal displays, actuated control, pedestrian intervals and the Canadian signal-warrant procedure.
Webster, F.V. and Cobbe, B.M., Traffic Signals, Road Research Technical Paper No. 56, HMSO — the optimum-cycle and average-delay relations used in Questions 3 and 7.

Canadian context. These are Engineers Canada national examinations, so the answers are framed for Canadian practice: the TAC Geometric Design Guide for Canadian Roads and the MUTCDC are the governing documents, and metric design controls are used throughout. Question 1(e) names the US MUTCD explicitly, so its eight warrants are answered as printed; the Canadian equivalent procedure is noted alongside, which is polish rather than a correction.

Question 2: Curves (6 + 7 + 7 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. The AASHTO 2001 metric stopping-sight-distance table printed on page 2 of the paper, together with its stated basis of a 2.5 s brake reaction time and a 3.4 m/s² deceleration rate.

Extract of the AASHTO 2001 metric SSD table printed on page 2 (the rows this question uses)
Design speed (km/h)	Brake reaction distance (m)	Braking distance on level (m)	Calculated SSD (m)	Design SSD (m)
50	34.8	28.7	63.5	65
80	55.6	73.4	129.0	130
90	62.6	92.9	155.5	160
100	69.5	114.7	184.2	185
110	76.5	138.8	215.3	220
120	83.4	165.2	248.6	250

Before using the table at all, it is worth reproducing it, because an open-book table printed on an examination page is exactly where a mis-printed digit hides. The published metric relations are

$$d_{brake} = 0.278\,V t,\qquad d_{stop} = \frac{0.039\,V^{2}}{a},\qquad \text{SSD} = d_{brake} + d_{stop}$$

with $t = 2.5$ s and $a = 3.4$ m/s$^2$ as the paper’s own note states. At 100 km/h these give $0.278(100)(2.5) = 69.50$ m and $0.039(100)^2/3.4 = 114.71$ m, summing to 184.21 m against the printed 184.2 m; at 120 km/h they give 83.40 m, 165.18 m and 248.58 m against the printed 248.6 m. Every row of the table reproduces to within 0.05 m, so the table is sound and the Design column is simply the Calculated column rounded up to the next 5 m.

Find. (a) the distinction between vertical, horizontal and spiral curves; (b) the design speed a given 265 m crest supports; (c) the minimum crest length for 120 km/h with a low object.

Part (a) — vertical, horizontal and spiral curves. The three differ in the plane in which they act and in whether their curvature is constant.

A vertical curve lies in the profile — the vertical plane containing the centreline. It joins two tangent grades $g_1$ and $g_2$ and is a simple parabola, because a parabola gives a constant rate of change of grade and therefore a constant vertical acceleration at constant speed. It is a crest when $g_1 > g_2$ and a sag when $g_1 < g_2$. The design controls are the length $L$, the algebraic difference in grades $A = |g_2 - g_1|$ and the rate of vertical curvature $K = L/A$, the length per one per cent of grade change. Crests are governed by sight distance (the road surface itself hides the object); sags are governed by headlight illumination, comfort, drainage and appearance.
A horizontal curve lies in plan. It is a circular arc of constant radius $R$ joining two tangents that intersect at a deflection angle $\Delta$, with tangent length $T = R\tan(\Delta/2)$, arc length $L_c = R\Delta$ (radians), external distance and middle ordinate following from the same geometry. Its design control is the balance between superelevation $e$ and side friction $f_s$ against centripetal demand, $R_{min} = V^2 / [127(e + f_s)]$ in metric units, plus sight distance around the inside of the curve.
A spiral (transition, or clothoid) curve also lies in plan, but its curvature is not constant: it varies linearly with distance along the curve, from zero at the tangent to $1/R$ at the circular curve. Its purpose is to spread the onset of centripetal acceleration over a finite length instead of applying it as a step at a tangent-to-curve point, which lets the driver steer a natural path and gives a rational length over which to run off the superelevation. Its minimum length follows from a limiting rate of change of lateral acceleration, $L_s = V^3/(46.7\,C R)$ in metric form, or from the superelevation runoff length, whichever governs.

So the ordering is: horizontal and spiral curves control the plan alignment and are about lateral acceleration; the vertical curve controls the profile and is about sight distance and comfort in the vertical plane.

Part (b) — what design speed does the given 265 m crest support?

Question 2(b). The 265 m crest joining +2 per cent to −2 per cent. With a 1080 mm driver eye and a 600 mm object the sight line grazes the crest with both ends on the curve, so the S < L branch applies.

Approach. Invert the crest-curve sight-distance formula for the available $S$, then read the largest tabulated design speed whose required SSD does not exceed it.

Algebraic difference in grades. The crest joins a rising tangent to a falling one, so the two grades subtract with opposite signs: $$A = |g_2 - g_1| = |-2 - (+2)| = \boxed{4\ \text{per cent}}$$
Form the branch constant once. The two AASHTO crest branches read $L = A S^{2}/\left[100\left(\sqrt{2h_1}+\sqrt{2h_2}\right)^{2}\right]$ for $S < L$ and $L = 2S - 200\left(\sqrt{h_1}+\sqrt{h_2}\right)^{2}/A$ for $S > L$. Because $\left(\sqrt{2h_1}+\sqrt{2h_2}\right)^{2} = 2\left(\sqrt{h_1}+\sqrt{h_2}\right)^{2}$, the denominator of the first and the numerator of the second are the same number, so it is computed once: $$k = 100\left(\sqrt{2(1.080)}+\sqrt{2(0.600)}\right)^{2} = 100\left(1.4697 + 1.0954\right)^{2} = 657.99$$
Solve the S < L branch for the available sight distance. Rearranging $L = A S^{2}/k$ gives $S = \sqrt{Lk/A}$, so $$S = \sqrt{\frac{(265)(657.99)}{4}} = \sqrt{43\,591.3} = \boxed{208.79\ \text{m}}$$
Confirm the branch. The formula chosen is only valid if the sight line really does begin and end on the curve. Here $S = 208.79$ m against $L = 265$ m, so $S < L$ holds and the branch is self-consistent; no second trial is needed.
Read the design speed from the table. The available 208.79 m must cover the required SSD. The Design column gives 185 m at 100 km/h and 220 m at 110 km/h, so 100 km/h is met with 23.79 m to spare while 110 km/h falls short by 11.21 m. The Calculated column (184.2 m and 215.3 m) gives the same verdict, which is a free check that the answer does not hinge on which column is read: $$V_{design} = \boxed{100\ \text{km/h}}$$
Cross-check on the K value. The rate of vertical curvature is $K = L/A = 265/4 = 66.25$ m per one per cent. The published AASHTO metric crest $K$ list for the standard heights reads 52 at 100 km/h and 74 at 110 km/h, so $K = 66.25$ again lands on 100 km/h. This is not a coincidence: for the standard heights $K = L/A = S^{2}/k$ identically, so the two tables must agree, and their agreeing confirms both the arithmetic and the reading of the table.
Comfort and appearance. AASHTO also imposes a minimum length of about $0.6V = 0.6(100) = 60$ m for comfort and appearance. At 265 m the given curve is far above it, so sight distance is the only governing criterion.

The continuous root of the SSD relation is worth quoting because it shows how much margin the answer really has: solving $0.695V + 0.011471V^{2} = 208.79$ gives $V = 107.98$ km/h. The curve would comfortably serve 108 km/h, but design speeds are adopted in standard increments, so the next lower increment — 100 km/h — is the answer.

Part (c) — minimum crest length at 120 km/h for a 300 mm object.

Question 2(c). The +1 per cent to −1 per cent crest at 120 km/h. Halving the object height from the standard 600 mm to 300 mm lowers the far end of the sight line and lengthens the required curve by a third.

Approach. Take the required SSD for 120 km/h from the printed table, form the branch constant for the non-standard 1050/300 mm heights, and solve the $S < L$ branch for $L$.

Algebraic difference in grades. $$A = |g_2 - g_1| = |-1 - (+1)| = \boxed{2\ \text{per cent}}$$
Required stopping sight distance. Straight from the Design column of the printed AASHTO table at 120 km/h, $$S = 250\ \text{m}$$ (the Calculated column gives 248.6 m; both are carried through below).
Branch constant for the stated heights. The driver eye is the standard 1050 mm value used in older editions, but the object is a 300 mm small object rather than the standard 600 mm: $$k = 100\left(\sqrt{2(1.050)}+\sqrt{2(0.300)}\right)^{2} = 100\left(1.4491 + 0.7746\right)^{2} = 494.50$$ Lowering the object halves nothing linearly — $k$ falls from 657.99 to 494.50, a factor of 1.3306, and it is that factor which drives the answer.
Solve for the minimum length. On the $S < L$ branch, $$L = \frac{A S^{2}}{k} = \frac{2(250)^{2}}{494.50} = \frac{125\,000}{494.50} = \boxed{252.78\ \text{m}}$$
Confirm the branch. $S = 250$ m against $L = 252.78$ m, so $S < L$ holds by 2.78 m. The branch is self-consistent, though only just — a slightly shorter SSD would push the problem onto the other branch, which is why the test is made rather than assumed.
The calculated-column alternative. Using the Calculated SSD of 248.6 m instead gives $L = 2(248.6)^{2}/494.50 = 249.96$ m, again with $S < L$. The two columns differ by only 2.82 m, or 1.1 per cent.
What the low object costs. With the standard 1080/600 mm heights the same problem would need $L = 2(250)^{2}/657.99 = 189.97$ m, and $K = 250^{2}/657.99 = 94.99$ — which reproduces the published AASHTO crest $K$ of 95 at 120 km/h exactly, confirming the whole method against an independent table. Designing for the 300 mm object therefore lengthens the curve from 189.97 m to 252.78 m, a penalty of $$\frac{252.78}{189.97} - 1 = \boxed{33.1\ \text{per cent}}$$
Comfort and appearance. $0.6V = 0.6(120) = 72$ m, far below 252.78 m, so sight distance governs. The resulting $K = 252.78/2 = 126.4$ is high but unremarkable for a 120 km/h freeway profile.

Check: which SSD column to use. The paper prints both a Calculated and a Design column and does not say which to use. The Design column is the one intended for design — it is the Calculated value rounded up to a 5 m increment — so it is adopted for the boxed answers, with the Calculated-column result quoted alongside in every part. The two never differ by more than about 1 per cent here, so no conclusion turns on the choice.

Question 2 — final results
Quantity	Symbol	Result
Part (b): algebraic difference in grades	$A$	4 per cent
Part (b): available stopping sight distance	$S$	208.79 m
Part (b): design speed supported	$V$	100 km/h (continuous root 107.98 km/h; adopt the next lower increment)
Part (b): rate of vertical curvature	$K = L/A$	66.25 (K table also gives 100 km/h)
Part (c): algebraic difference in grades	$A$	2 per cent
Part (c): required SSD at 120 km/h	$S$	250 m (Design column); 248.6 m Calculated
Part (c): branch constant for 1050/300 mm	$k$	494.50
Part (c): minimum crest length	$L$	252.78 m (249.96 m from the Calculated column)
Part (c): penalty of the 300 mm object	—	33.1 per cent longer than the 189.97 m needed with a 600 mm object

Topic. Crest vertical curves and stopping sight distance.

Key relations. $\text{SSD} = 0.278Vt + 0.039V^{2}/a$; crest curves $L = AS^{2}/k$ for $S < L$ and $L = 2S - k/A$ for $S > L$, with the same $k = 100(\sqrt{2h_1}+\sqrt{2h_2})^{2} = 200(\sqrt{h_1}+\sqrt{h_2})^{2}$; rate of vertical curvature $K = L/A$, and for the standard heights $K = S^{2}/k$.

Why this works. On a crest the pavement itself is the sight obstruction, so the sight line is a chord of the parabola from the driver’s eye to the object. Because a parabola’s offset from its tangent grows with the square of the distance from the tangent point, the geometry collapses into the two algebraic branches above, and the two heights enter only through their square roots. Halving the object height therefore has a much smaller than proportional effect — but a very much larger effect than intuition suggests, because it is the sum of square roots that is then squared.

Common pitfall. Computing the two branch constants separately and getting one of them wrong. They are identically equal; compute the number once, use it in both branch tests, and the commonest arithmetic slip in this family disappears. The second pitfall is not testing the branch: a trial length must be checked against $S$ before it is reported, and here (c) clears the test by only 2.78 m.

Source. AASHTO, A Policy on Geometric Design of Highways and Streets, 2001 metric ed., Ch. 3 (sight distance, Table 3-1) and Ch. 3 (crest vertical curves); Garber & Hoel, 5th ed., Ch. 15; TAC Geometric Design Guide for Canadian Roads, Part 2 (design controls and vertical alignment).

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