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Geometric Design: Professional Practice

 
Sag Vertical Curves

The following excerpt was taken from the 1994 edition of AASHTO's A Policy on Geometric Design of Highways and Streets (pp. 288-292).

At least four different criteria for establishing lengths of sag vertical curves are recognized to some extent. These are (1) headlight sight distance, (2) rider comfort, (3) drainage control, and (4) a rule-of-thumb for general appearance.

Headlight sight distance has been used directly by some authorities and for the most part is the basis for determining the length of sight distance used herein. When a vehicle traverses a sag vertical curve at night, the portion of highway lighted ahead is dependent on the position of the headlights and the direction of the light beam. General use is being given to a headlight height of 600 mm and a 1° upward divergence of the light beam from the longitudinal axis of the vehicle. The upward spread of the light beam provides some additional visible length but this is generally ignored.

The following formulas show the S, L, and A relation, using S as the distance   between the vehicle and point where the 1° angle of light ray intersects the   surface of the roadway: (The equations below include the H and B terms which were not explicitly shown by AASHTO in its discussion.)

When S is less than L,

 L=(A*S^2)/(200*(H + S*tan(B)))

 When S is greater than L,

 L=2*S - (200*(H + S*tan(B)))/A

Where:
L = length of sag vertical curve, m;
S = light beam distance, m; and
A = algebraic difference in grades, percent.
H = height of headlights above the roadway, m
B = divergence angle of headlight beam, degrees

For overall safety on highways, a sag vertical curve should be long enough so that the light beam distance is nearly the same as the stopping sight distance. . . .

The comfort effect of change in vertical direction is greater on sag than on crest vertical curves because gravitational and centrifugal forces are combining rather than opposing forces. Comfort due to change in vertical direction is not measured readily because it is affected appreciably by vehicle body suspension, tire flexibility, mass carried, and other factors. The limited attempts at such measurements have led to the broad conclusion that riding is comfortable on sag vertical curves when the centrifugal acceleration does not exceed 0.3 m/s2. The general expression for such a criterion is:

L = (AV2)/395

where L and A are the same as in previous formulas, and V is the design speed, km/h.

The length of vertical curve required to satisfy this comfort factor at the various design speeds is only about 50 percent of that required to satisfy the headlight sight distance requirement for the normal range of design conditions.

Drainage effects the design of vertical curves . . . where curbed sections are used. An approximate criterion for sag vertical curves is the same as that expressed for crest conditions, that is, providing a minimum grade of 0.30 percent within 15 m of the level point.

For general appearance, some use formerly was made of a rule-of-thumb for length of sag vertical curves wherein the minimum value of L is 30A. . . . This approximation is a generalized control for small or intermediate values of A. Compared with headlight sight distance, it corresponds to a design speed between 70 and 80 km/h. On high-type highways longer curves are deemed appropriate to improve appearance.

From the preceding it is evident that design controls for sag vertical curves differ from those for crests, and separate design values are needed. The headlight sight distance basis appears to be the most logical for general use, and the values determined for stopping sight distances are within the limits recognized in current practice. . . .