Question 3 of 7: Expressway curve — radius, clear zone and spiral length

Question 3: Expressway curve — radius, clear zone and spiral length (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. A 1.5 km circular section on a four-lane divided urban expressway, in cut, carrying very heavy traffic:

Find. Whether 595 m is a sufficient radius at the posted speed, the clear-zone width the roadside should provide, and a recommended spiral transition length.

Approach. Test the radius against the point-mass equation at the posted speed, read the clear zone from the appendix table and correct it for curvature, then size the spiral by the three criteria the appendix supplies — superelevation runoff, minimum travel time, and rate of change of lateral acceleration — and take the largest.

(a) Is the radius sufficient? (8 marks)

1. Select the superelevation rate. Urban expressways in Canada are designed with a maximum superelevation of 6 % because slow-moving and stopped traffic must be accommodated; take $e=0.06$.
2. Minimum radius at the posted speed. The appendix gives the point-mass relation directly:$$R_{\min}=\dfrac{V^{2}}{127\,(e+\mu)}=\dfrac{100^{2}}{127(0.06+0.11)}=\dfrac{10\,000}{21.59}=\boxed{463.2\ \text{m}}$$The curve supplies $595\ \text{m}\gt 463.2\ \text{m}$, a 28 % margin.
3. Express the margin as a friction demand. A more informative check is to ask how much side friction the curve actually calls for:$$\mu_{\text{demand}}=\dfrac{V^{2}}{127R}-e=\dfrac{10\,000}{127(595)}-0.06=0.1323-0.06=\boxed{0.072}$$against $0.11$ available, so only about two thirds of the design side friction is mobilised. The curve is comfortably safe at 100 km/h, and remains adequate even at a reduced $e=0.04$ ($R_{\min}=524.9\ \text{m}$).
4. Where the limit lies. At the next design-speed increment the picture changes: with $\mu=0.09$ at 110 km/h, $R_{\min}=110^{2}/[127(0.06+0.09)]=635.2\ \text{m}\gt 595\ \text{m}$. The curve is therefore correctly matched to a 100 km/h design speed and should not be signed higher.

Answer. Yes — the radius is sufficient. It exceeds the 463 m minimum by 28 % and mobilises only 0.072 of the 0.11 side friction available, but it would not support a design speed of 110 km/h.

(b) Recommended clear zone (4 marks)

1. Enter the clear-zone table. Design speed 100 km/h, design ADT $92\,200\gt 6\,000$, cut slope $3\!:\!1$ gives a base clear zone of 6.0 to 6.5 m.
2. Apply the horizontal-curve correction. The correction table is entered at a radius of 595 m; both the 600 m and the 500 m rows give 1.3 in the 100 km/h column, so no interpolation is needed and$$K_{cz}=1.3$$The note beneath the table restricts the factor to the outside of the curve, where a departing vehicle runs away from the alignment.
3. Recommended widths.$$L_{cz,\text{outside}}=6.5\times 1.3=\boxed{8.45\ \text{m}}\qquad L_{cz,\text{inside}}=\boxed{6.5\ \text{m}}$$Specify 8.5 m clear on the outside of the curve and 6.5 m on the inside, measured from the edge of the travelled way. Because the section is in cut, the back-slope must be no steeper than 3:1 within that width and the ditch must be of a traversable rather than a V-shaped section; where the full width cannot be obtained, a semi-rigid barrier placed at its own working width is the alternative.

(c) Recommended spiral length (8 marks)

1. Superelevation runoff criterion. Each carriageway of a four-lane divided highway is two 3.75 m lanes rotated about its own centreline, so the rotated width is $w=7.5\ \text{m}$ and the appendix gives$$L_{s}=\dfrac{w\,e}{2s}=\dfrac{7.5\times 0.06}{2\times 0.004}=56.3\ \text{m}$$The stated $s=0.004$ is slightly flatter than the 0.0044 the appendix table lists for speeds above 100 km/h, so this figure is on the conservative side.
2. Minimum travel-time criterion. The appendix relation $A^{2}=L_{s}R=2RV(1000)/3600$ is the two-second rule in disguise, because the radius cancels:$$L_{s}=\dfrac{2V(1000)}{3600}=\dfrac{2(100)(1000)}{3600}=55.6\ \text{m}$$i.e. two seconds of travel at the design speed.
3. Comfort (rate of change of lateral acceleration) criterion. The spiral parameter for comfort is$$A=\sqrt{0.03577\,V^{3}}=\sqrt{0.03577(100)^{3}}=189.1\ \text{m}\qquad\Longrightarrow\qquad L_{s}=\dfrac{A^{2}}{R}=\dfrac{35\,770}{595}=60.1\ \text{m}$$
4. Governing length and recommendation. The three criteria give 56.3 m, 55.6 m and 60.1 m, so comfort governs:$$L_{s}=\max(56.3,\ 55.6,\ 60.1)=60.1\ \text{m}\quad\Longrightarrow\quad\boxed{L_{s}=60\ \text{m at each end}}$$Both spirals are laid in, giving a total transition length of 120 m out of the 1.5 km section and leaving about 1,380 m of full circular curve.