Question 4 of 7: Horizontal curve, spiral length and clear zone on an urban expressway

Question 4: Horizontal curve, spiral length and clear zone on an urban expressway (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 3 km circular section of a six-lane divided urban expressway.

Find. Whether 650 m satisfies the minimum radius at the design speed; the spiral length to recommend; and the clear-zone width on the outside of the curve.

Approach. Test the radius against the point-mass equation at the design speed, evaluate all three TAC spiral-length criteria and take the largest, then read the clear zone from the straight-section table and multiply by the horizontal-curve adjustment factor for the outside of the curve.

1. Test the radius against the point-mass equation. The minimum radius for a design speed V, superelevation e and side friction factor f_s is $$R_{min} = \frac{V^{2}}{127\,(e + f_s)} = \frac{100^{2}}{127\,(0.06 + 0.11)} = \boxed{463.2\ \text{m}}$$ The built radius of 650 m comfortably exceeds this, so on curvature grounds the section is safe at 100 km/h.
2. Check the friction actually demanded. The margin is best expressed as the side friction the vehicle really calls on: $$f_{demand} = \frac{V^{2}}{127\,R} - e = \frac{10\,000}{127(650)} - 0.06 = 0.1211 - 0.06 = \boxed{0.061}$$ which is only 56 % of the 0.11 available, so the curve is comfortably within the design envelope. Inverting the same relation, the radius would support $$V_{max} = \sqrt{127\,R\,(e + f_s)} = \sqrt{127(650)(0.17)} = 118.4\ \text{km/h}$$ so the geometry has roughly 18 km/h of speed reserve. The radius is safe.
3. Apply the superelevation-runoff criterion for the spiral. On a divided highway each roadway is rotated about its median edge, so all three lanes of one carriageway change cross-slope. With a relative slope of 0.0044 m/m for design speeds of 100 km/h and above, $$L_{s,runoff} = \frac{w\,n\,e}{\text{relative slope}} = \frac{3.75(3)(0.06)}{0.0044} = \boxed{153.4\ \text{m}}$$
4. Apply the travel-time criterion. A driver should spend at least two seconds on the transition, which is independent of the radius: $$L_{s,time} = \frac{V}{1.8} = \frac{100}{1.8} = \boxed{55.6\ \text{m}}$$
5. Apply the comfort (rate of change of lateral acceleration) criterion. Using the appendix relations $A^{2} = 0.03577\,V^{3}$ and $A^{2} = R\,L_s$, $$L_{s,comfort} = \frac{0.03577\,V^{3}}{R} = \frac{0.03577(100)^{3}}{650} = \frac{35\,770}{650} = \boxed{55.0\ \text{m}}$$
6. Select the spiral length. The governing criterion is the largest of the three, and on a wide divided cross section the superelevation runoff always wins because it scales with the number of lanes rotated: $$L_s = \max(153.4,\ 55.6,\ 55.0) = 153.4\ \text{m} \quad\Longrightarrow\quad \boxed{L_s = 155\ \text{m recommended}}$$ Rounding up to a 5 m increment keeps the stationing tidy. Note that the comfort criterion, which governs on two-lane roads of the same speed, is here less than half the runoff requirement.
7. Read the straight-section clear zone. The design speed is 100 km/h, the AADT of 111,200 falls in the over 6000 row, and a 5:1 fill slope sits in the 5:1 to 4:1 foreslope column. The appendix table on page 16 gives a straight-section clear zone of 11.0 to 13.5 m.
8. Apply the horizontal-curve adjustment. The adjustment factor applies to the outside of curves only. For a design speed of 100 km/h the tabulated factors are 1.2 at R = 700 m and 1.3 at R = 600 m; taking the conservative tabulated value below the built radius, K_cz = 1.3, so $$CZ = 1.3 \times (11.0\ \text{to}\ 13.5) = \boxed{14.3\ \text{to}\ 17.6\ \text{m}}$$ Recommend a clear zone of at least 14.5 m from the edge of the travelled way on the outside of the curve, measured over the 5:1 recoverable foreslope. The asterisked table entry warns that clear zones of this width may not be practical in an urban corridor; where the full width cannot be obtained, the alternative is to remove or make breakaway every fixed object inside it and, failing that, to shield the hazard with a barrier whose own deflection and working width then have to be accommodated.