16-Civ-A6 Highway Design, Construction, and Maintenance · December 2018

Question 2 of 5: Horizontal curve adequacy, clear zone and spiral length

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

Notes on this paper

Paper format. National Examinations, December 2018 — 16-Civ-A6 Highway Design, Construction, and Maintenance. Three-hour, closed-book paper with a ten-page appendix. Five questions are printed and four solutions constitute a complete paper; all questions carry equal value (25 % each) and the marks for sub-questions are shown in brackets. Note 1 invites the candidate to state any interpretation assumed, and Note 2 permits any required data that is not given to be assumed. All five questions are worked here, because the set is a study resource rather than an examination attempt.

Reference texts. AASHTO, Guide for Design of Pavement Structures, 1993 — Part II Ch. 2 (flexible pavements) and Ch. 3 (rigid pavements); Figures 2.5–2.7, 3.1, 3.3, 3.6, 3.7 and Tables 2.4–2.6 are reproduced on appendix pages 1–8. Garber, N.J. and Hoel, L.A., Traffic and Highway Engineering, 5th ed. — Ch. 3 (driver characteristics, stopping sight distance), Ch. 15 (geometric design), Ch. 20 (flexible and rigid pavement design). AASHTO, A Policy on Geometric Design of Highways and Streets (Green Book), 7th ed. — Ch. 3. Transportation Association of Canada, Geometric Design Guide for Canadian Roads — Ch. 1.2 (design controls), Ch. 2.1 (sight distance), Ch. 3.2 (horizontal alignment and spirals), Ch. 4 (roadside design and clear zones). AASHTO, Roadside Design Guide, 4th ed. — Ch. 3 (clear zone). Huang, Y.H., Pavement Analysis and Design, 2nd ed. — Ch. 11 and 12.

Check — chart readings and assumed data. Three of the five questions are solved from nomographs and tables reproduced on the appendix pages. Each chart reading used here was taken from the printed appendix chart and is quoted in the step where it is used, so that a reader working from a different print can substitute their own reading. Two items are genuinely not supplied by the paper and are assumed under Note 2: the percentage of time the pavement structure is near saturation in Question 1 (needed for the drainage coefficients $m_2$, $m_3$), and AASHTO Figure 3.4, the rigid-foundation correction needed in Question 3(b), which the ten-page appendix does not include. Both are flagged where they arise and the sensitivity of the answer to each is quantified.

Question 2: Horizontal curve adequacy, clear zone and spiral length (25 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 2 km curved section of four-lane divided urban expressway, 3.75 m lanes, posted at 100 km/h, on a 600 m centreline radius, carrying 45,200 veh/day, mainly in cut with a 3:1 back-slope in the clear zone.

Design inputs, with the appendix values they lead to
Quantity	Value	Source
Centreline radius $R$	600 m	question
Posted speed $V$	100 km/h	question
Design AADT	45,200 veh/day	question
Lane width, lanes per direction	3.75 m, two	question
Side slope in the clear zone	3:1 cut	question
Coefficient of side friction at 100 km/h	0.11	appendix page 10
Maximum superelevation, urban conditions	$e = 0.06$	assumed, TAC/AASHTO
Relative slope for superelevation runoff	0.004	question

Find. (a) whether a 600 m radius is sufficient at the posted speed; (b) the clear zone distance to be provided, inside and outside the curve; (c) the spiral transition length to be recommended at each end of the circular curve.

Figure 2 — plan of the 600 m curve showing the spiral transitions and the clear zone widths required inside and outside the curve.

Approach. Compare the supplied radius with the minimum radius from the point-mass equation at the posted speed and at the next two design-speed increments, which simultaneously answers part (a) and establishes the design speed to be used in part (c). Read the clear zone from the appendix table for the traffic and side-slope combination, then apply the curve correction factor. Finally compute all three spiral-length criteria and adopt the largest.

Establish the superelevation to be assumed. The appendix supplies the side-friction factors but not $e_{max}$. On an urban expressway, where snow and ice removal, intersecting side streets and slow-moving vehicles all limit cross-slope, both the TAC guide and the Green Book cap the superelevation at $$e_{max} = 0.06$$ This is the one assumption in part (a); Step 3 shows the verdict does not turn on it.
Compute the minimum radius at the posted speed. The point-mass relation printed on appendix page 9, in the metric form $R = V^{2}/[127(e+\mu)]$ with $V$ in km/h, gives at 100 km/h with $\mu = 0.11$ $$R_{min} = \frac{100^{2}}{127\,(0.06 + 0.11)} = \frac{10\,000}{21.59} = 463.2\ \text{m}$$ The supplied 600 m therefore exceeds the minimum by 30 %, and $$\boxed{R = 600\ \text{m} \;\gt\; R_{min} = 463\ \text{m} \;\Rightarrow\; \text{the curve is adequate at 100 km/h}}$$
Confirm the verdict from the friction actually demanded. A second, independent statement of the same check is to invert the relation and ask how much side friction a vehicle travelling at the posted speed on this curve actually needs: $$\mu_{demanded} = \frac{V^{2}}{127R} - e = \frac{10\,000}{127(600)} - 0.06 = 0.1312 - 0.06 = 0.0712$$ That is 65 % of the 0.11 available, so a comfortable margin remains even if the superelevation were built at 0.04 rather than 0.06 (the demand would then be 0.0912, still below 0.11). The curve is safe on both readings.
Check the next design-speed increments, because part (c) needs a design speed. Repeating the calculation upward: $$R_{min}(110) = \frac{110^{2}}{127(0.06+0.09)} = 635.2\ \text{m} \qquad R_{min}(120) = \frac{120^{2}}{127(0.06+0.07)} = 872.2\ \text{m}$$ Both exceed the 600 m provided. The section is therefore adequate at 100 km/h and inadequate at 110 km/h or above, so its design speed is the posted 100 km/h.
Read the clear zone for the straight-section condition. Enter the appendix clear-zone table at a design speed of 100 km/h and a design ADT greater than 6,000 veh/day (45,200 is far above the highest band), on the 3:1 cut-slope column, which returns 6.0–6.5 m. Given the very high volume, the upper end of the band is appropriate: $$L_{C,straight} = 6.5\ \text{m}$$
Apply the horizontal-curve correction. The note beneath the correction table states that the adjustment is applied to the outside of curves only, and that curves flatter than 900 m need no adjustment. At $R = 600$ m and a design speed of 100 km/h the factor is 1.3, so $$L_{C,outside} = 6.5 \times 1.3 = \boxed{8.45\ \text{m on the outside, 6.5 m on the inside}}$$ The distinction matters on the ground: because this section is in cut, the outside of the curve is where the back-slope rises, so it is the outside 8.45 m that must be kept free of rigid hazards or protected by barrier, while the median side needs only the unadjusted 6.5 m.
Compute the superelevation-runoff spiral length. Each carriageway of a divided highway is rotated about its own centreline, so the width raised is the two-lane carriageway width $w = 2(3.75) = 7.5$ m and, with the relative slope $s$ the question directs, the sheet's relation $L_s = we/(2s)$ gives $$L_{s,runoff} = \frac{7.5 \times 0.06}{2 \times 0.004} = \frac{0.45}{0.008} = 56.25\ \text{m}$$
Compute the two-second travel-time criterion. A driver should spend at least about two seconds within the transition, which fixes a length independent of the radius: $$L_{s,time} = \frac{2V(1000)}{3600} = \frac{2(100)(1000)}{3600} = 55.56\ \text{m}$$
Compute the comfort (rate of change of lateral acceleration) criterion. The spiral parameter follows from the sheet's relation $A = \sqrt{0.03577V^{3}}$ and the spiral identity $A^{2} = RL_s$: $$A = \sqrt{0.03577(100)^{3}} = \sqrt{35\,770} = 189.13 \qquad L_{s,comfort} = \frac{A^{2}}{R} = \frac{35\,770}{600} = 59.62\ \text{m}$$
Adopt the governing length. The three criteria must all be satisfied, so the largest governs: $$L_s = \max(56.25,\ 55.56,\ 59.62) = 59.62\ \text{m} \;\Rightarrow\; \boxed{L_s = 60\ \text{m at each end of the circular curve}}$$ Comfort governs, but only just — the three criteria fall within 7 % of one another, which is the signature of an alignment whose radius, speed and cross-slope are mutually consistent.

Check — the "0.004 at 120 km/h" instruction. Part (c) supplies a relative slope tagged to 120 km/h, yet Step 4 shows the 600 m radius cannot be driven at 120 km/h (it would need 872 m) or even at 110 km/h (635 m). The instruction is read here as fixing the numerical value of $s$ to be used — 0.004, which the appendix table associates with speeds of 100 km/h and above — while the design speed for the travel-time and comfort criteria remains the section's own 100 km/h. Using 120 km/h throughout would raise the two-second criterion to 66.7 m and the comfort criterion to $0.03577(120)^{3}/600 = 103.0$ m, returning $L_s = 103$ m, and would implicitly certify a speed the alignment does not support.

Question 2 — final results
Part	Quantity	Value
(a)	Minimum radius at 100 km/h	463.2 m — 600 m provided, adequate
(a)	Side friction demanded at 100 km/h	0.0712 against 0.11 available
(a)	Minimum radius at 110 / 120 km/h	635.2 m / 872.2 m — both exceed 600 m
(b)	Clear zone, straight-section value	6.5 m
(b)	Clear zone, outside of the curve	$6.5 \times 1.3 = 8.45$ m
(b)	Clear zone, inside of the curve	6.5 m (no adjustment)
(c)	Superelevation runoff criterion	56.25 m
(c)	Two-second travel-time criterion	55.56 m
(c)	Comfort criterion ($A = 189.13$)	59.62 m — governs
(c)	Recommended spiral length	60 m at each end

Topic. Horizontal alignment: minimum radius, roadside clear zone and spiral transitions.

Key relations. Point-mass equilibrium $v^{2}/(gR) = e + \mu$, in practical form $R = V^{2}/[127(e+\mu)]$; clear zone from the design-speed / ADT / side-slope table with the curve factor applied to the outside only; the three spiral criteria $L_s = we/(2s)$, $L_s = 2V(1000)/3600$ and $L_s = A^{2}/R$ with $A = \sqrt{0.03577V^{3}}$.

Why this works. The point-mass equation says that the lateral acceleration a vehicle experiences on a curve must be balanced by the component of gravity supplied by the superelevation plus the side friction the tyres can develop. The side friction factors tabulated for design are not the friction available at the limit of adhesion; they are the friction a driver will tolerate before feeling uncomfortable, which is why they fall as speed rises. The clear zone is a different idea entirely — it is the width in which an errant vehicle can recover, and it is widened on the outside of a curve because a vehicle that departs the road there leaves tangentially and travels further before stopping. The spiral is what reconciles the two: it introduces the curvature, and with it the superelevation, gradually, so that neither the steering input nor the cross-slope has to change discontinuously at the tangent point.

Common pitfall. Applying the curve correction factor to both sides of the clear zone; the printed note restricts it to the outside. The second common error is using the full pavement width of a divided highway in $L_s = we/(2s)$ — each carriageway is rotated about its own axis, so only the carriageway width enters.

Source. Transportation Association of Canada, Geometric Design Guide for Canadian Roads, Ch. 3.2 (horizontal alignment and spirals) and Ch. 4 (clear zone); AASHTO Green Book, 7th ed., Ch. 3; AASHTO Roadside Design Guide, Ch. 3.

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