Computations

In order to achieve a smooth change of direction when laying out vertical curves, the grade must be brought up through a series of elevations. The surveyor normally determines elevation for vertical curves for the beginning (point of vertical curvature or PVC), the end (point of vertical tangency or PVT), and all full stations. At times, the surveyor may desire additional points, but this will depend on construction requirements.

Length of Curve
The elevations are vertical offsets to the tangent (straightline design grade) elevations. Grades G₁ and G₂ are given as percentages of rise for 100 feet of horizontal distance. The surveyor identifies grades as plus or minus, depending on whether they are ascending or descending in the direction of the survey. The length of the vertical curve (L) is the horizontal distance (in 100-foot stations) from PVC to PVT. Usually, the curve extends ½ L stations on each side of the point of vertical intersection (PVI) and is most conveniently divided into full station increments.

A sag curve is illustrated in figure 20. The surveyor can derive the curve data as follows (with BV and CV
being the grade lines to be connected).

Determine values of G₁ and G₂, the original grades. To arrive at the minimum curve length (L) in stations, divide the algebraic difference of G₁ and G₂ (AG) by the rate of change (r), which is normally included in the design criteria. When the rate of change (r) is not given, use the following formulas to compute L:

If L does not come out to a whole number of stations from this formula, it is usually extended to the nearest whole number. Note that this reduces the rate of change. Thus, L = 4.8 stations would be extended to 5 stations, and the value of r computed from r = Δ G/L. These formulas are for road design only. The surveyor must use different formulas for railroad and airfield design.

Station Interval
Once the length of curve is determined, the surveyor selects an appropriate station interval (SI). The first factor to be considered is the terrain. The rougher the terrain, the smaller the station interval. The second consideration is to select an interval which will place a station at the center of the curve with the same number of stations on both sides of the curve. For example, a 300-foot curve could not be staked at 100- foot intervals but could be staked at 10-, 25-, 30-, 50-, or 75-foot intervals. The surveyor often uses the same intervals as those recommended for horizontal curves, that is 10, 25, 50, and 100 feet.

Since the PVI is the only fixed station, the next step is to compute the station value of the PVC, PVT, and all stations on the curve.

Other stations are determined by starting at the PVI, adding the SI, and continuing until the PVT is reached.

Tangent Elevations
Compute tangent elevations PVC, PVT, and all stations along the curve. Since the PVI is the fixed point on the tangents, the surveyor computes the station elevations as follows:

The surveyor may find the elevation of the stations along the back tangent as follows:

Elev of sta = Elev of PVC + (distance from the PVC x G1).

The elevation of the stations along the forward tangent is found as follows:

Elev of sta = Elev of PVI + (distance from the PVI x G2)

Vertical Maximum
The parabola bisects a line joining the PVI and the midpoint of the chord drawn between the PVC and PVT. In figure 19, line VE = DE and is referred to as the vertical maximum (Vm). The value of Vm is computed as follows: (L = length in 100-foot stations. In a 600-foot curve, L = 6.)

In practice, the surveyor should compute the value of Vm using both formulas, since working both provides a check on the Vm, the elevation of the PVC, and the elevation of the PVT.

Vertical Offset. The value of the vertical offset is the distance between the tangent line and the road grade. This value varies as the square of the distance from the PVC or PVT and is computed using the formula:

A parabolic curve presents a mirror image. This means that the second half of the curve is identical to the first half, and the offsets are the same for both sides of the curve.

Station Elevation. Next, the surveyor computes the elevation of the road grade at each of the stations along the curve. The elevation of the curve at any station is equal to the tangent elevation at that station plus or minus the vertical offset for that station, The sign of the offset depends upon the sign of Vm (plus for a sag curve and minus for a summit curve).

First and Second Differences. As a final step, the surveyor determines the values of the first and second differences. The first differences are the differences in elevation between successive stations along the curve, namely, the elevation of the second station minus the elevation of the first station, the elevation of the third station minus the elevation of the second, and so on. The second differences are the differences between the differences in elevation (the first differences), and they are computed in the same sequence as the first differences.

The surveyor must take great care to observe and record the algebraic sign of both the first and second differences. The second differences provide a check on the rate of change per station along the curve and a check on the computations. The second differences should all be equal. However, they may vary by one or two in the last decimal place due to rounding off in the computations. When this happens, they should form a pattern. If they vary too much and/or do not form a pattern, the surveyor has made an error in the computation.

Example: A vertical curve connects grade lines G₁ and G₂ (figure 19). The maximum allowable slope (r) is 2.5 percent. Grades G₁ and G₂ are found to be -10 and +5.

The vertical offsets for each station are computed as in figure 20. The first and second differences are determined as a check. Figure 21 illustrates the solution of a summit curve with offsets for 50-foot intervals.

High and Low Points
The surveyor uses the high or low point of a vertical curve to determine the direction and amount of runoff, in the case of summit curves, and to locate the low point for drainage.

When the tangent grades are equal, the high or low point will be at the center of the curve. When the tangent grades are both plus, the low point is at the PVC and the high point at the PVT. When both tangent grades are minus, the high point is at the PVC and the low point at the PVT. When unequal plus and minus tangent grades are encountered, the high or low point will fall on the side of the curve that has the flatter gradient.

Horizontal Distance. The surveyor determines the distance (x, expressed in stations) between the PVC or PVT and the high or low point by the following formula:

G is the flatter of the two gradients and L is the number of curve stations.

Vertical Distance. The surveyor computes the difference in elevation (y) between the PVC or PVT and the high or low point by the formula 

Example: From the curve in figure 21, G₁= + 3.2%, G₂ = – 1.6% L = 4 (400). Since G₂ is the flatter gradient, the high point will fall between the PVI and the PVT.

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