Question 4 of 7: Exit-ramp curves and roadside sight obstruction

Question 4: Exit-ramp curves and roadside sight obstruction (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. Two identical single-lane exit ramps on a constant upgrade, each carrying a tight horizontal curve with a tree line inside it:

Find. Whether curvature and sight distance are both adequate at the posted 30 km/h, and what physical changes would make the ramps adequate at a 45 km/h design speed.

Approach. Two independent checks must both pass. Curvature is tested with the point-mass equation; sight distance is tested by converting the tree offset into an available sight distance through the middle-ordinate relation and comparing it with the stopping sight distance on the 8 % upgrade.

(a) Adequacy at the posted 30 km/h (10 marks)

1. Curvature check. With $e=0.06$ and $\mu=0.17$,$$R_{\min}=\dfrac{V^{2}}{127(e+\mu)}=\dfrac{30^{2}}{127(0.23)}=\dfrac{900}{29.21}=\boxed{30.8\ \text{m}}$$and $45\ \text{m}\gt 30.8\ \text{m}$, so the radius is generous at the posted speed.
2. Stopping sight distance required. The ramp climbs, so gravity assists the braking:$$\text{SSD}=0.278Vt_{pr}+\dfrac{V^{2}}{254(f+G)}=0.278(30)(2.5)+\dfrac{900}{254(0.38+0.08)}=20.85+7.70=\boxed{28.6\ \text{m}}$$
3. Sight distance the tree line leaves. Inverting the middle-ordinate relation given in the appendix, $M_{s}=R\left[1-\cos\left(\frac{28.65\,S}{R}\right)\right]$, gives$$S=\dfrac{R}{28.65}\arccos\!\left(1-\dfrac{M_{s}}{R}\right)$$For ramp #1, $\arccos(1-5.0/45)=\arccos(0.8889)=27.27^{\circ}$, so$$S_{1}=\dfrac{45}{28.65}(27.27)=\boxed{42.8\ \text{m}}$$and for ramp #2, $\arccos(1-4.3/45)=\arccos(0.9044)=25.27^{\circ}$, giving$$S_{2}=\dfrac{45}{28.65}(25.27)=\boxed{39.7\ \text{m}}$$
4. Compare. Both available distances exceed the 28.6 m demanded — by 50 % on ramp #1 and 39 % on ramp #2. Working the comparison the other way, the offset actually needed at 30 km/h is $M_{s}=45\left[1-\cos(28.65\times 28.6/45)\right]=2.26\ \text{m}$, against 4.3 m and 5.0 m available.

Answer. Yes. At the posted 30 km/h both ramps are adequate on both counts: the 45 m radius exceeds the 30.8 m minimum, and the tree lines leave 42.8 m and 39.7 m of sight distance against a requirement of 28.6 m. The commuters’ complaint is understandable — the trees genuinely restrict forward view — but the restriction is not yet a deficiency at the posted speed. It becomes one the moment the ramp is driven faster, which is the substance of part (b).

(b) Making the ramps adequate at 45 km/h (10 marks)

1. Curvature is the binding constraint. At 45 km/h$$R_{\min}=\dfrac{45^{2}}{127(0.06+0.17)}=\dfrac{2\,025}{29.21}=\boxed{69.3\ \text{m}}$$far above the existing 45 m. Put differently, holding $R=45$ m and $e=0.06$ would demand$$\mu_{\text{demand}}=\dfrac{45^{2}}{127(45)}-0.06=0.354-0.060=0.294$$against $0.17$ available — a 73 % overshoot that no realistic increase in superelevation can absorb (even $e=0.08$ leaves $R_{\min}=63.8\ \text{m}$). The curve must be reconstructed.
2. Sight distance required at 45 km/h.$$\text{SSD}=0.278(45)(2.5)+\dfrac{2\,025}{254(0.46)}=31.28+17.33=\boxed{48.6\ \text{m}}$$
3. Offset the existing 45 m curve would need.$$M_{s}=45\left[1-\cos\!\left(\dfrac{28.65\times 48.6}{45}\right)\right]=45\left[1-\cos(30.95^{\circ})\right]=\boxed{6.41\ \text{m}}$$greater than both the 5.0 m and 4.3 m offsets available, so on the present geometry the trees would also have to be cleared.
4. The radius increase solves both problems at once. Adopt $R_{v}=70\ \text{m}$, which just satisfies the curvature requirement ($69.3\ \text{m}$) with $e=0.06$ — the exact superelevation needed is $e=45^{2}/[127(70)]-0.17=0.058$. On a 70 m curve the sight-line offset falls to$$M_{s}=70\left[1-\cos\!\left(\dfrac{28.65\times 48.6}{70}\right)\right]=70\left[1-\cos(19.89^{\circ})\right]=\boxed{4.18\ \text{m}}$$which both existing tree lines already clear (4.3 m and 5.0 m). Flattening the curve therefore removes the sight-distance deficiency as a by-product, and no tree removal is strictly required — although ramp #2 retains only 0.12 m of margin, so the vegetation line should still be cut back to a maintained 5 m and kept there.

Recommended works package. Reconstruct both ramp curves to a minimum radius of 70 m (75 m would give a comfortable margin) and superelevate at 6 %, rotating the pavement about the ramp centreline with a spiral or tangent runoff. Establish a maintained sight-line envelope of at least 4.5 m from the inner-lane centreline and add it to the vegetation-management contract, since the offset the design relies on is a maintenance commitment, not a one-off clearing. Widen the ramp on the curve to accommodate off-tracking of design vehicles, and check the 8 % upgrade against the truck-crawl speed at the ramp terminal. Finally, support the geometry with traffic control: advisory speed signing at the new design speed, chevron alignment markers on the outside of each curve, an object marker on any residual roadside hazard, and profiled edge line marking. If reconstruction is not affordable, the ramps must stay posted at 30 km/h, where they are demonstrably adequate.