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

Question 6 of 7: Crest Vertical Curve — Geometry, Sight Distance and Grade

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

Notes on this paper

Paper format. 98-Civ-B7 Highway Engineering, National Examinations May 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 on the last page confirms 20 marks per question: Q1 (a) and (b) 10 marks each; Q2 (a) through (e) 4 marks each; Q3 (a) to (j) 2 marks each; Q4 (a) and (b) 10 marks each; Q5 (a) and (b) 10 marks each; Q6 (a) through (e) 4 marks each; Q7 20 marks. All seven printed questions are worked here, 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 not given but required may be assumed, and that assumptions should be recorded with the answer — several questions below need that licence, and each assumption is flagged where it is made.

Reference texts. N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed. (geometric design, sight distance, vertical curves, earthwork, pavement design); AASHTO, Guide for Design of Pavement Structures (1993) (rigid and flexible thickness design, reliability, drainage and load-transfer coefficients); Transportation Association of Canada, Geometric Design Guide for Canadian Roads (Canadian design-domain values for sight distance and vertical curvature); Asphalt Institute, Mix Design Methods MS-2, 7th ed. (mixture volumetrics, VMA, VFA, absorbed binder); B.M. Das, Principles of Geotechnical Engineering, 9th ed. (compaction, Proctor testing, zero-air-voids line, CBR); M.S. Mamlouk and J.P. Zaniewski, Materials for Civil and Construction Engineers, 4th ed. (concrete and asphalt materials); A.M. Neville, Properties of Concrete, 5th ed., and CSA A23.1 (air entrainment, curing, joints in concrete pavement).

Check — assumptions carried through this paper. Three items are not supplied by the exam and are assumed under the paper’s own Note 2 (“any data, not given but required, can be assumed”), each stated again at the point of use: (i) Question 5 gives the mass of the Proctor mould but not its volume, so the ASTM D698 / AASHTO T99 standard 101.6 mm mould volume of 944 cm³ is used; (ii) Question 6 does not name a design speed, so the available stopping sight distance is computed from the Canadian/AASHTO eye and object heights of 1.08 m and 0.60 m; (iii) Question 7 lists the modulus of subgrade reaction as “1.0 MPa”, which is dimensionally incomplete — it is read as 1.0 MPa/m and the sensitivity of the answer to that reading is reported with the result.

Question 6: Crest Vertical Curve — Geometry, Sight Distance and Grade (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. An equal-tangent parabolic crest curve joining a rising 4 % grade to a falling 3 % grade, with the intersection point and the curve length both fixed.

Given data
Item	Value
Approach grade, g₁	+4 %
Departure grade, g₂	−3 %
Station and elevation of PVI	15+00 (chainage 1500 m), 400.00 m
Length of vertical curve, L	1000 m
Station interval	100 m

Find. The station and elevation of the PVC and PVT; the curve elevation at every 100 m station; the stopping sight distance the curve actually provides; the station and elevation of the high point; and the rate of change of grade.

Figure 6.1 — The crest curve, its two tangents, and the high point at station 15+71.43. Dots mark the 100 m station elevations computed in part (b).

Approach. An equal-tangent parabola is fully described by its PVC elevation, the entering grade and the constant second derivative A/(100L); every part of the question is then a substitution into the single elevation equation, except the sight distance, which comes from the standard crest formula with Canadian eye and object heights.

(a) Locate the PVC and PVT. The tangents are equal, so each is half the curve length either side of the PVI: $$\text{PVC}=1500-\tfrac{1000}{2}=1000\ \text{m}=\text{station }10{+}00$$ $$\text{PVT}=1500+\tfrac{1000}{2}=2000\ \text{m}=\text{station }20{+}00$$ Their elevations follow along the tangent grades: $$E_{PVC}=400.00-0.04(500)=\boxed{380.00\ \text{m}},\qquad E_{PVT}=400.00-0.03(500)=\boxed{385.00\ \text{m}}$$
(a) Write the curve equation. With $x$ measured from the PVC in metres and grades in percent, $$E(x)=E_{PVC}+\frac{g_1}{100}x+\frac{g_2-g_1}{200\,L}x^2 =380.00+0.04x-3.5\times 10^{-5}x^2$$ because $A=g_2-g_1=-7$ % and $A/(200L)=-7/200\,000$.
(b) Elevations at 100 m intervals. Substituting $x=0,100,\dots,1000$ gives the profile below; as a check, $E(1000)=385.00$ m reproduces the PVT elevation found independently in step 1.

Curve elevations at 100 m stations
Station	x from PVC (m)	Elevation (m)
10+00 (PVC)	0	380.000
11+00	100	383.650
12+00	200	386.600
13+00	300	388.850
14+00	400	390.400
15+00	500	391.250
15+71.43 (high point)	571.43	391.429
16+00	600	391.400
17+00	700	390.850
18+00	800	389.600
19+00	900	387.650
20+00 (PVT)	1000	385.000

(c) Available stopping sight distance. For a crest curve with the sight line contained within the curve, the standard relation is $$L=\frac{A\,S^2}{100\left(\sqrt{2h_1}+\sqrt{2h_2}\right)^2}=\frac{A\,S^2}{658}$$ where the constant 658 follows from the driver eye height $h_1=1.08$ m and the object height $h_2=0.60$ m used by both TAC and AASHTO. Rearranging for the sight distance the curve actually delivers, $$S=\sqrt{\frac{658\,L}{A}}=\sqrt{\frac{658\times 1000}{7}}=\sqrt{94\,000} =\boxed{306.6\ \text{m}}$$ The assumption $S<L$ is satisfied (306.6 m against 1000 m), so the formula used is the correct branch. A sight distance of 307 m exceeds the stopping requirement for every Canadian rural design speed up to and beyond 120 km/h, so this curve is generous.
(d) Station and elevation of zero grade. The grade at a distance $x$ from the PVC is the derivative of the elevation equation, and the high point is where it vanishes: $$\frac{dE}{dx}=\frac{g_1}{100}+\frac{A}{100\,L}x=0 \ \Longrightarrow\ x=\frac{g_1 L}{|A|}=\frac{4\times 1000}{7}=571.43\ \text{m}$$ $$\text{Station}=1000+571.43=\boxed{15{+}71.43}$$ $$E=380.00+0.04(571.43)-3.5\times 10^{-5}(571.43)^2=\boxed{391.429\ \text{m}}$$ The high point sits 71 m beyond the PVI station, on the flatter side, which is always the case when the entering grade is the steeper of the two.
(e) Rate of change of grade. The parabola has a constant second derivative, so $$r=\frac{A}{L}=\frac{-7\ \%}{1000\ \text{m}}=\boxed{-0.007\ \%\ \text{per metre}} =-0.7\ \%\ \text{per 100 m station}$$ Equivalently the curvature is expressed as $K=L/|A|=1000/7=142.9$ m per percent of grade change, which is the form Canadian design tables use.

The five answers describe one object consistently: a very long, very flat crest whose K value of 143 is close to double the K = 74 that a 110 km/h Canadian rural highway requires for stopping sight distance (S = 220 m), and still half again the K = 95 required at 120 km/h. The 1000 m length is therefore not driven by sight distance at all — about 515 m would serve the 110 km/h control and 665 m the 120 km/h one — so it must come from drainage, from appearance, or from matching an existing profile. That is worth noticing because a crest this flat drains poorly near the high point: the longitudinal gradient is flatter than 0.5 % from chainage 1500 to 1643, a 143 m stretch, and the designer would need to check that the cross-fall and the gutter grade still move water away.

Quantity	Value
(a) PVC	Station 10+00, elevation 380.00 m
(a) PVT	Station 20+00, elevation 385.00 m
(b) Elevations at 100 m stations	380.000, 383.650, 386.600, 388.850, 390.400, 391.250, 391.400, 390.850, 389.600, 387.650, 385.000 m
(c) Available stopping sight distance	S = 306.6 m (S < L, so the correct branch)
(d) High point (zero grade)	Station 15+71.43, elevation 391.429 m
(e) Rate of change of grade	−0.007 % per metre (−0.7 % per station); K = 142.9 m/%

Concept & Theory

Topic: Equal-tangent parabolic vertical curves — geometry, turning point, rate of change of grade and crest sight distance.

Key relations: E(x) = E_PVC + (g₁/100)x + Ax²/(200L) with A = g₂ − g₁ in percent; turning point at x = g₁L/|A|; r = A/L; K = L/|A|; crest sight distance L = AS²/658 for S < L, and L = 2S − 658/A for S > L, with h₁ = 1.08 m and h₂ = 0.60 m.

Why this works: A parabola is chosen for vertical curves precisely because its grade varies linearly with distance, which makes the rate of change of grade constant and the ride smooth, and because its offsets from the tangent grow as the square of the distance, which makes elevations easy to compute by hand. Setting the first derivative to zero locates the crest, and because the grade falls linearly from g₁ to g₂ across the curve, the turning point divides the length in the ratio of the two grades. The sight-distance formula comes from writing the offsets of the eye and the object below the curve and requiring the sight line to be tangent to the crest; the constant 658 is simply 100 times the square of (√2h₁ + √2h₂).

Common pitfall: Using the wrong sight-distance branch. The two formulas cross where S = L, and applying the S < L version when the sight distance exceeds the curve length (or vice versa) gives a plausible-looking but wrong answer; the assumption must be stated and then verified, as in step 4. A second trap is measuring x from the PVI rather than the PVC, which corrupts every elevation in part (b), and a third is taking A as 1 % instead of 7 % by ignoring the signs of the grades.

Source: N.J. Garber and L.A. Hoel, Traffic and Highway Engineering, 5th ed., Ch. 15 (vertical alignment and sight distance); AASHTO, A Policy on Geometric Design of Highways and Streets, 7th ed., Ch. 3; Transportation Association of Canada, Geometric Design Guide for Canadian Roads (K values and design-domain sight distances).

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