1. 3.4.6 Vertical Curves, p ~ p.3-163
You learned how to lay out a vertical curve, PVC, PVI, and PVT for given grades in CE113 Surveying and reviewed in CE361 Intro to Transportation Engineering. If you forgot the concept, please review Chapter 21 of your survey book (by McCormac?).
Be familiar with the steps for determining the length of the vertical curve
Know that the criteria for determining the length of crest and sag curves are different
Know how to lay out a vertical curve (Review it by yourself)

2. Criteria for determining the length of the vertical curve, p.3-149
The main criteria are:
Provision of stopping sight distance (as defined by AASHTO)
Adequate drainage
Comfort in operation
Pleasant appearance
For both crest and sag vertical curves
For sag curves only
Roller coaster and hidden dip effects need to be avoided.
When a vertical curve ahead is too small, the road may appear sharply bent.

3. Crest vertical curves,
Fig. 3-41

4. Two cases of SSD on crest vertical curves
SSD > Length of V-curve
SSD < Length of V-curve
Fig. 3-42
Eq. 3-44
Eq. 3-43

5. Derivation of crest vertical curve length formulas: S > L
Let g represent the difference between the gradient of the sight line and the gradient G1. Then, A – g will be the difference between the gradient of the sight line and the gradient G2. g A g
(A - g)
To find the slope of the sight line that will make S a minimum, set dS/dg = 0.
Substitute g in S equation above and get
See Appendix of this course notes file.

6. Derivation of crest vertical curve length formulas: S < L
Use the basic offset property of the parabolic curve, y = ax2. As long as the point of interest is within the parabola, we can use this. We know the mid-curve offset E = AL/800 and this happens when x = L/2. So if you have the curve offset y = H1 with x = S1, we have: E
and
Solve for S1 and S2, and sum them to get S.

7. Graphical representation of minimum crest vertical lengths (Fig
Graphical representation of minimum crest vertical lengths (Fig. 3-43: Design Controls for Crest VC)

8. Sag vertical curves (p.3-157)
The minimum length of sag vertical curves is controlled by (1) sight distance provided by the headlight (at night: during the day you can see the vehicles in the opposite direction), (2) rider comfort, (3) control of drainage, and (4) general appearance.
H = 2 ft (headlight height), β = 1-degree (divergence angle)

9. Two cases of SSD on sag vertical curves (p.3-158)
For S > L
For S < L
H = 2 ft, the height of the headlight above the ground

10. Graphical representation of minimum sag vertical lengths (Fig
Graphical representation of minimum sag vertical lengths (Fig. 3-44: Design Controls for Sag VC, p.3-159)

11. 3 other criteria for sag vertical curves (p.3-160)
Comfort criteria (minimum length), usually 50% of the SSD requirement:
u = design speed, mph
Drainage criteria (maximum length within which a grade must be established) when curbs are used (Know how to read the Drainage maximum line in Figure 3-44):
A minimum grade of 0.3% must be provided within 50 ft of the level point of the curve. (K=167)
General appearance (minimum length):
L = 100A

12. Another K value…
The minimum lengths of the crest and sag curves which are computed based on stopping sight distance (S < L cases) can be expressed like:
L = KA
To make it easier to get the value from the minimum curve length tables or charts.
Sag vertical curves:
Crest vertical curves:

13. K-value for crest vertical curves (p.3-155)
a. Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K=L/A
Table 3-34

14. K-value for PSD (rare case, though) (p.3-156)

15. Change in GB2011, because PSD criteria changed (p.3-157)

16. K-value for sag vertical curves, (p.3-161)
a. Rate of vertical curvature, K, is the length of curve per percent algebraic difference in intersecting grades (A). K=L/A
Tab 3-36

17. Sight Distance at Undercrossings, p.3-161
Sight distance on the highway through a grade separation should be at least as long as the minimum stopping sight distance and preferably longer.
(Fig. 3-45)

18. Why do we use the truck driver’s eye height here?
h1 = 8 ft for a truck driver, h2 = 2.0 ft for the taillights of a vehicle.
Why do we use the truck driver’s eye height here?
p through p.3-163

19. See p through p.3-163

20. General controls for vertical alignment, p.3-163 ~p.3-164
A smooth gradeline with gradual changes should be sought.
The “roller-coaster” or “hidden-dip” type of profile should be avoided.
Undulating gradelines should be avoided.
A “broken-back” gradelines should be avoided.
On longer grades, it is recommended to have the steepest grade at the bottom and flatten the grades near the top of the ascent or to break the sustained grade by short intervals of flatter grade instead of providing a uniform sustained grade.
Where at-grade intersections occur, reduce the grade through the intersection.
Sag vertical curves should be avoided in cuts unless adequate drainage can be provided.

21. Appendix