Sight Distance and Crest Vertical Curves: 16-Civ-B10 Traffic Engineering, Undated paper, Q5

Question 5: Sight Distance and Crest Vertical Curves (6 + 7 + 7 = 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.

Part (a) — What sight distance is, and two types that matter. Sight distance is the length of roadway ahead that is continuously visible to a driver, measured along the travelled path from a stated driver eye height to a stated target height. It is a geometric property of the alignment, not of the vehicle: it is limited by the crest of a vertical curve cutting the line of sight against the pavement, by an obstruction inside a horizontal curve (a rock face, a bridge pier, a noise wall, roadside vegetation), or by a structure overhead on a sag curve at night. Because every driver action — stopping, passing, changing lanes, choosing a path through an intersection — requires the driver to see the conflict far enough ahead to perceive it, decide and execute, sight distance is the criterion that most often controls geometric design.

The two types most important in traffic engineering are:

Stopping sight distance (SSD) is the distance required for a driver travelling at the design speed to perceive an object in the travelled way, react, and brake to a complete stop before reaching it. It is the sum of the distance covered during the perception-reaction interval and the braking distance:

$$\mathrm{SSD}=0.278\,V t + \frac{V^{2}}{254\bigl(\frac{a}{9.81}\pm G\bigr)}\;=\;0.278\,V t + 0.039\,\frac{V^{2}}{a}$$

with $V$ in km/h, $t = 2.5$ s and $a = 3.4$ m/s², exactly as noted under the table printed on this paper. SSD is the minimum acceptable sight distance everywhere on a road; a length of road that does not provide it is a design deficiency, not a compromise.

Passing (overtaking) sight distance (PSD) is the distance a driver on a two-lane, two-way road needs to see in order to complete an overtaking manoeuvre in the opposing lane, including the perception and initial manoeuvre, the time occupying the opposing lane, a clearance interval on return, and the distance travelled by an opposing vehicle in the same period. PSD is several times SSD at the same speed — roughly 670 m at 100 km/h against 185 m — so it is not provided continuously; instead, a design target is that an adequate proportion of the route offers marked passing zones, and where it is not available the centreline is marked as a no-passing barrier line. Two further types are worth naming: decision sight distance, used where a driver must make a complex or unexpected choice (an interchange exit, a lane drop, a toll plaza), which is longer than SSD because the reaction interval is extended; and intersection sight distance, the clear sight triangle a driver on a minor approach needs in order to accept a gap.

Part (b) — Given. Entering grade $g_1 = +3\%$, departing grade $g_2 = -1\%$, design speed $V = 100$ km/h, driver eye height $h_1 = 1050$ mm $= 1.05$ m, object height $h_2 = 150$ mm $= 0.15$ m. From the AASHTO table printed on the paper, the design stopping sight distance at 100 km/h is $S = 185$ m (calculated value 184.2 m).

Find. The minimum length $L$ of the crest curve that provides that stopping sight distance.

Question 5(b): crest vertical curve joining +3% to -1% (A = 4%). The minimum length that provides 185 m of stopping sight distance for a 1050 mm eye and a 150 mm object is 343.3 m (K = 85.8).

Approach. Compute the algebraic grade difference, form the sight-height constant, apply the $S < L$ branch of the crest-curve relation, verify the branch assumption, and check the result against the comfort-and-appearance minimum.

Part (b) — Algebraic difference in grades. $$A=\lvert g_1-g_2\rvert=\lvert (+3)-(-1)\rvert=4\%$$ A positive $A$ with $g_1 > g_2$ confirms this is a crest.
Form the sight-height constant. The two branches of the crest relation share one constant, because $\bigl(\sqrt{2h_1}+\sqrt{2h_2}\bigr)^{2} = 2\bigl(\sqrt{h_1}+\sqrt{h_2}\bigr)^{2}$, so $$k=200\bigl(\sqrt{h_1}+\sqrt{h_2}\bigr)^{2}=200\bigl(\sqrt{1.05}+\sqrt{0.15}\bigr)^{2}=200(1.41199)^{2}=398.75$$ Computing it once and reusing it for both branch tests removes the commonest arithmetic slip in this family.
Apply the $S < L$ branch. For a sight line that lies entirely on the curve, $$L=\frac{A\,S^{2}}{k}=\frac{4(185)^{2}}{398.75}=\frac{136\,900}{398.75}=\boxed{343.3\ \text{m}}$$
Verify the branch and the comfort minimum. The assumption $S < L$ requires $185 < 343.3$, which holds, so the branch is correct and no switch to $L = 2S - k/A$ is needed. The comfort-and-appearance minimum $L_{\min} = 0.6V = 60$ m is far below the sight-distance answer and does not govern. The resulting rate of vertical curvature is $K = L/A = 343.3/4 = 85.8$ m per percent.
Cross-check the method against the published AASHTO K-table. Because $K = L/A = S^{2}/k$ identically, recomputing the same problem with the standard heights (1080 mm and 600 mm, for which $k = 200(\sqrt{1.08}+\sqrt{0.60})^{2} = 658.0$) gives $K = 185^{2}/658.0 = 52.0$, against the published AASHTO metric crest $K = 52$ at 100 km/h. The match to three significant figures validates the branch choice, the arithmetic and the reading of the printed SSD table in one line — and it quantifies what the low object costs: the standard-height curve would need only $L = 208.1$ m, so designing for a 150 mm object lengthens the curve by 65%.

Check — design versus calculated SSD. The AASHTO table printed on the paper gives both a calculated SSD (184.2 m) and a rounded design SSD (185 m) at 100 km/h. The design column is used, as is standard practice and consistent with the table's own presentation; the calculated column would give $L = 340.4$ m, a difference of 0.9% that changes nothing. Grades are taken as constant either side of the curve and the curve as equal-tangent, both standard.

Part (c) — Given. A crest curve of fixed length $L = 200$ m joining $g_1 = +5\%$ to $g_2 = -3\%$, with standard sight heights $h_1 = 1080$ mm and $h_2 = 600$ mm.

Find. The highest design speed for which this curve provides ample stopping sight distance.

Question 5(c): the given 200 m crest curve joining +5% to -3% (A = 8%, K = 25.0). With standard 1080 mm / 600 mm heights it supplies 128.3 m of stopping sight distance, which supports a 70 km/h design speed.

Part (c) — Grade difference and rate of curvature. $$A=\lvert 5-(-3)\rvert=8\%,\qquad K=\frac{L}{A}=\frac{200}{8}=25.0\ \text{m per percent}$$
Invert the crest relation for the available sight distance. With standard heights the constant is $k = 200(\sqrt{1.08}+\sqrt{0.60})^{2} = 658.0$, so assuming the $S < L$ branch, $$S=\sqrt{\frac{L\,k}{A}}=\sqrt{\frac{200(658.0)}{8}}=\sqrt{16\,450}=\boxed{128.3\ \text{m}}$$ The branch assumption $S < L$ requires $128.3 < 200$, which holds.
Read the design speed off the printed AASHTO table. The curve supplies 128.3 m of sight distance. The table's metric design SSD values are 105 m at 70 km/h and 130 m at 80 km/h. Since $105 \le 128.3 < 130$, the highest speed fully supported is $$\boxed{V=70\ \text{km/h}}$$ Equivalently in $K$-values: the curve provides $K = 25.0$, while AASHTO requires $K = 105^{2}/658.0 = 16.8$ at 70 km/h and $K = 130^{2}/658.0 = 25.7$ at 80 km/h — the published metric crest K-table gives 17 and 26 respectively, confirming both the constant and the reading.
Report how close 80 km/h is. The curve falls short of an 80 km/h design by only 1.7 m of sight distance, or 1.3%. To carry 80 km/h the curve would have to be lengthened to $$L=\frac{A\,S^{2}}{k}=\frac{8(130)^{2}}{658.0}=205.5\ \text{m}$$ — an extra 5.5 m, about 3% more earthwork on the vertical curve. That is worth stating in a design report: a marginal decision like this is normally resolved in favour of the higher design speed, since the cost is trivial and posting 70 km/h on a road that geometrically almost supports 80 invites non-compliance.

Part	Quantity	Symbol	Result
(a)	Two types of sight distance	—	Stopping sight distance; passing (overtaking) sight distance
(b)	Algebraic grade difference	$A$	4%
(b)	Design stopping sight distance at 100 km/h	$S$	185 m (AASHTO table)
(b)	Minimum crest curve length	$L$	343.3 m $(K = 85.8)$
(b)	Equivalent length at standard heights	$L_{\text{std}}$	208.1 m — the 150 mm object costs +65%
(c)	Algebraic grade difference	$A$	8%
(c)	Sight distance available on the 200 m curve	$S$	128.3 m $(K = 25.0)$
(c)	Design speed supported	$V$	70 km/h (80 km/h needs 130 m, i.e. $L = 205.5$ m)

Topic. Stopping sight distance and the design of crest vertical curves against a published SSD table.

Key relations. $\mathrm{SSD} = 0.278Vt + 0.039V^{2}/a$; crest curve, $S < L$: $L = AS^{2}/\bigl[100(\sqrt{2h_1}+\sqrt{2h_2})^{2}\bigr] = AS^{2}/k$; crest curve, $S > L$: $L = 2S - k/A$ with the same $k = 200(\sqrt{h_1}+\sqrt{h_2})^{2}$; rate of vertical curvature $K = L/A = S^{2}/k$; comfort minimum $L_{\min} = 0.6V$.

Why this works. On a crest the parabola cuts the line of sight from the driver's eye to the object, and the geometry of a parabola makes the offset from the tangent proportional to the square of the distance from the tangent point. Working that through gives a sight distance that depends on $A$, $L$ and the two heights only through the combination $(\sqrt{h_1}+\sqrt{h_2})^{2}$ — which is why a single constant $k$ serves both branches, and why halving the object height is so expensive: the length varies inversely with $k$, and $k$ falls faster than $h_2$ does. Expressing the answer as $K = L/A$ removes $A$ entirely and lets any curve be checked directly against a published table, which is the cheapest available validation of the whole calculation.

Common pitfall. Computing the two branch constants separately and getting one of them wrong — they are identically equal, so compute one. The second pitfall is failing to test the branch assumption after solving: an answer obtained from $S < L$ is meaningless if the result turns out to have $S > L$, and on flat grade differences that reversal is common. The third is mixing the friction form $V^{2}/[254(f+G)]$ with the deceleration form $0.039V^{2}/a$ — use the deceleration form for anything keyed to the AASHTO table, since it is the form that reproduces the printed rows.

Source. Garber & Hoel, Traffic and Highway Engineering, 5th ed., §3.3 (sight distance) and §3.5 (vertical curves); AASHTO, A Policy on Geometric Design of Highways and Streets (2001 SSD and crest K tables, reproduced on page 3 of this paper); TAC Geometric Design Guide for Canadian Roads, Ch. 2.

Question 5 of 7: Sight Distance and Crest Vertical Curves

Question 5: Sight Distance and Crest Vertical Curves (6 + 7 + 7 = 20 marks)