Question 6 of 7: Curve Types and Crest Vertical Curve Design
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations, December 2016 — 98-Civ-B10 Traffic Engineering. 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 paper’s own grading scheme: Q1 and Q2 and Q5 are (a) to (e) at 4 marks each, Q3 and Q4 are single 20-mark questions, Q6 is (a) 6 marks with (b) and (c) 7 marks each, and Q7 is (a) to (h) at 2.5 marks each. The paper states that a total of five solutions is required and that only the first five in the answer book will be marked. All seven questions are solved here, because this set is a study resource rather than a sitting. The paper also permits assumptions — “Any data required, but not given, can be assumed” — and every assumption made below is stated explicitly where it is used.
Reference texts. Garber, N. J. & Hoel, L. A., Traffic and Highway Engineering, 5th ed. (Cengage) — the core reference for this exam code; Transportation Association of Canada, Geometric Design Guide for Canadian Roads (TAC GDG); Institute of Transportation Engineers (Canadian District), Canadian Capacity Guide for Signalized Intersections, 3rd ed.; Transportation Research Board, Highway Capacity Manual (HCM) — pedestrian crossing-time model; AASHTO, A Policy on Geometric Design of Highways and Streets (the “Green Book”, 2001 edition — the source of the stopping-sight-distance table printed on page 4 of this paper); Transportation Association of Canada, Manual of Uniform Traffic Control Devices for Canada (MUTCDC). Canadian practice governs wherever the paper does not name a specific standard.
Question 6: Curve Types and Crest Vertical Curve Design (20 marks — 6 + 7 + 7)
Part (a) — the three curve types. A vertical curve lies in the profile: it is the parabolic transition that joins two tangent grades in the vertical plane, and it is described by its algebraic grade change $A = |G_1 - G_2|$, its length $L$, and the rate of vertical curvature $K = L/A$ (the horizontal distance needed for a one-per-cent change of grade). Crest curves are governed by sight distance over the hump; sag curves are governed at night by headlight throw and by rider comfort. A horizontal curve lies in plan: it is the circular arc that joins two straight tangents in the horizontal plane, defined by its radius $R$ or degree of curve, its deflection angle, and the superelevation and side friction that together resist the centripetal demand through $R = V^2/[127(e + f)]$. Its governing constraint is lateral stability plus sight distance around any obstruction on the inside of the curve.
A spiral curve is neither of these but a transition between them in plan — a clothoid whose radius decreases uniformly from infinity at the tangent to the radius $R$ of the circular curve at its far end, so that the curvature, and therefore the lateral acceleration demanded of the driver, is introduced gradually instead of instantaneously. Its purposes are to give a natural steering path, to provide a rational length over which to run out the superelevation and the pavement widening, and to improve the appearance of the alignment. Its defining property is that the product of radius and distance along the spiral is constant, $R\,\ell = A_s^2$, and TAC sets the minimum spiral length from the superelevation runoff requirement and from a comfort limit on the rate of change of lateral acceleration.
Given. For part (b): $G_1 = +4.0\%$, $G_2 = -3.0\%$, design speed 80 km/h, driver eye height $h_1 = 1{,}060$ mm and object height $h_2 = 100$ mm. For part (c): $L = 150$ m, $G_1 = +4.5\%$, $G_2 = -2.5\%$, $h_1 = 1{,}080$ mm and $h_2 = 600$ mm. Stopping sight distances are read from the AASHTO (2001) table printed on page 4 of this paper.
Find. (b) the minimum crest length for adequate stopping sight distance; (c) the highest design speed the existing 150 m crest can support.
Crest vertical curve controlled by stopping sight distance. The sight line from the driver’s eye to the object grazes the pavement surface at the crest; the case shown is S < L.
Approach. Take the design stopping sight distance from the printed AASHTO table, apply the crest-curve sight-distance relation for the sight distance contained within the curve, and confirm that the assumed case is the valid one; in part (c) invert the same relation for $S$ and then read the speed back off the table.
Cross-check the printed table before using it. The AASHTO metric relations are $d = 0.278\,V\,t + 0.039\,V^2/a$ with $t = 2.5$ s and $a = 3.4$ m/s², the values stated in the note beneath the table. At 80 km/h these give $0.278(80)(2.5) = 55.6$ m and $0.039(80^2)/3.4 = 73.4$ m for a calculated total of 129.0 m, matching the printed row exactly, so the design value of $S = 130$ m may be used with confidence.
Part (b) — set up the crest relation. The algebraic grade change is $A = |{+}4.0 - ({-}3.0)| = 7.0\%$. For a sight distance contained entirely within the curve ($S < L$), $$L = \frac{A\,S^2}{100\left(\sqrt{2h_1} + \sqrt{2h_2}\right)^2}$$ Evaluating the height term with $h_1 = 1.060$ m and $h_2 = 0.100$ m, $$100\left(\sqrt{2(1.060)} + \sqrt{2(0.100)}\right)^2 = 100\,(1.4560 + 0.4472)^2 = 362.2$$ which replaces the familiar constant 658 that applies only to the standard 1,080 mm / 600 mm heights.
Evaluate the minimum length. Substituting $A = 7.0$ and $S = 130$ m, $$L = \frac{7.0\,(130)^2}{362.2} = \frac{118{,}300}{362.2} = \boxed{326.6\ \text{m}}$$ The assumption $S < L$ holds, since $130 < 326.6$, so this is the governing case. For completeness the alternative $S > L$ form gives $L = 2S - 200(\sqrt{h_1}+\sqrt{h_2})^2/A = 208.3$ m, which contradicts its own premise ($130 \not> 208.3$) and is therefore rejected. In practice the curve would be built at 330 m, rounded up to a convenient station length.
Note why the curve is so long. The 100 mm object height is the reason. Lowering the object from the standard 600 mm to 100 mm cuts the height constant from 658 to 362, which is an 82 per cent increase in required length for the same speed and grade change. Designing for small objects — debris, a fallen load, a pothole edge — is expensive, and it is the reason AASHTO and TAC both settled on a 600 mm object as the general criterion.
Part (c) — invert the relation for the sight distance available. Here the heights are the standard pair, so the constant returns to $100(\sqrt{2(1.080)} + \sqrt{2(0.600)})^2 = 658.0$, and the grade change is $A = |{+}4.5 - ({-}2.5)| = 7.0\%$. Assuming $S < L$ and solving $L = A S^2/658$ for $S$, $$S = \sqrt{\frac{658.0\,L}{A}} = \sqrt{\frac{658.0 \times 150}{7.0}} = \sqrt{14{,}100} = \boxed{118.7\ \text{m}}$$ and the premise is satisfied, $118.7 < 150$.
Read the design speed off the printed table. The curve supplies 118.7 m of stopping sight distance. From the AASHTO table the design stopping sight distance is 105 m at 70 km/h and 130 m at 80 km/h, so 80 km/h cannot be supported and $$\boxed{V = 70\ \text{km/h}}$$ is the highest standard design speed for which this curve provides ample stopping sight distance. Interpolating the calculated column, the true limit is about 76 km/h, but design speeds are selected in 10 km/h increments, so 70 km/h is the answer to post.
