Chapter 5.5, Problem 5E

Reminder Round all answers to two decimal places unless otherwise indicated.

Surveying Vertical Curves When a road is being built, it usually has straight sections, all with the same grade, that must be linked to each other by curves. (By this we mean curves up and down rather than side to side, which would be another matter.) It’s important that as the road changes from one grade to another, the rate of change of grade between the two be constant. The curve linking one grade to another grade is called a vertical curve.

Surveyors mark distances by means of stations that are $100\text{ feet}$ apart. To link a straight grade of $g_1$ to a straight grade of $g_2$, the elevations of the stations are given by

$y=\frac{g_2-g_1}{2L}{x}^2+g_{1}x+E-\frac{g_{1}L}{2}$.

Here $y$ is the elevation of the vertical curve in feet, $g_1$ and $g_2$ are percents, $L$ is the length of the vertical curve in hundreds of feet, $x$ is the number of the station, and $E$ is the elevation in feet of the intersection where the two grades would meet.(See Figure 5.72.) The station $x=0$ is the very beginning of the vertical curve, so the station $x=0$ lies where the straight section with grade $g_1$ meets the vertical curve. The last station of the vertical curve is $x=L$, which lies where the vertical curve meets the straight section with grade $g_2$.

Figure 5.72

Assume that the vertical curve you want to design goes over a slight rise, joining a straight section of grade $1.35\%$ to a straight section of grade $-1.75\%$. Assume that the length of the curve is to be $500\text{ feet}$ (so $L=5$) and that the elevation of the intersection is $1040.63\text{ feet}$.

a. What is the equation for the vertical curve described above? Don’t round the coefficients.

b. What are the elevations of the stations for the vertical curve?

c. Where is the highest point of the road on the vertical curve? (Give the distance along the vertical curve and the elevation.)