An unsymmetrical vertical curve is a curve in which the horizontal distance from the PVI to the PVC is different from the horizontal distance between the PVI and the PVT. In other words, l1 does NOT equal l2. Unsymmetrical curves are sometimes described as having unequal tangents and are referred to as dog legs. Figure 11-19 shows an unsymmetrical curve with a horizontal distance of 400 feet on the left and a horizontal distance of 200 feet on the right of the PVI. The gradient of the tangent at the PVC is 4 percent; the gradient of the tangent at the PVT is +6 percent. Note that the curve is in a dip.

As an example, lets assume you are given the following values:

Elevation at the PVI is 332.68

Station at the PVI is 42 + 00

l1 is 400 feet

l2 is 200 feet

g1 is 4%

g2 is +6%

To calculate the grade elevations on the curve to the nearest hundredth foot, use figure 11-20 as an example. Figure 11-20 shows the computations. Set four 100-foot stations on the left side of the PVI (between the PVI and the PVC). Set four 50-foot stations on the right side of the PVl (between the PVI and the PVT). The procedure for solving an unsymmetrical curve problem is essentially the same as that used in solving a symmetrical curve. There are, however, important differences you should be cautioned about.

First, you use a different formula for the calculation of the middle vertical offset at the PVI. For an unsymmetrical curve, the formula is as follows:

Figure 11-19.Unsymmetrical vertical curve.

Figure 11-20.Table of computations of elevations on an unsymmetrical vertical curve.

In this example, then, the middle vertical offset at the PVI is calculated in the following manner:

e = [(4 x 2)/2(4 + 2)] x [(+6) - (4)] = 6.67 feet.

Second, you are cautioned that the check on your computations by the use of second difference does NOT work out the same way for unsymmetrical curves as for a symmetrical curve. The second difference will not check for the differences that span the PVI. The reason is that an unsymmetrical curve is really two parabolas, one on each side of the PVI, having a common POVC opposite the PVI; however, the second difference will check out back, and ahead, of the first station on each side of the PVI. Third, the turning point is not necessarily above or below the tangent with the lesser slope. The  horizontal location is found by the use of one of two formulas as follows:

from the PVC

or from the PVT

The procedure is to estimate on which side of the PVI the turning point is located and then use the proper formula to find its location. If the formula indicates that the turning point is on the opposite side of the PVI, you must use the other formula to determine the correct location; for example, you estimate that the turning point is between the PVC and PVI for the curve in figure 11-19. Solving the formula:

xt= (l1)² (g1)/2e xt= [(4)²

However, Station 42 + 80 is between the PVI and PVT; therefore, use the formula:

xt= (l2)² (g2)//2e xt= [(2)²

Station 42 + 20 is the correct location of the turning point. The elevation of the POVT, the amount of the offset, and the elevation on the curve is determined as previously explained.