Calculating Vertical Curve Length

The problem asks us to find the length of a vertical curve that connects two different grades. This is a common task in highway or railway design.

We are given the following information:

• Initial grade ($g_1$): $+0.4\%$
• Final grade ($g_2$): $-0.3\%$
• Rate of change of grades ($r$): $0.1\%$ per $30$ m
• Type of curve: Summit curve (indicated by a positive grade followed by a negative grade).

The length of the vertical curve ($L$) depends on the total change in grade and the allowable rate of change of grade.

Determine the Total Change in Grade (A)

The total change in grade, often denoted by $A$, is the algebraic difference between the final grade and the initial grade. For vertical curves, we usually use the absolute value of this difference.

Total change in grade $A = g_2 - g_1$.

Substituting the given values:

\(A = (-0.3\%) - (+0.4\%)\)

\(A = -0.3\% - 0.4\%\)

\(A = -0.7\%\)

The absolute value of the total change in grade is \(|A| = |-0.7\%| = 0.7\%\).

Relate Total Change in Grade to Curve Length and Rate of Change

The rate of change of grade is given as $0.1\%$ per $30$ m. This means that for every $30$ meters of horizontal length along the curve, the grade changes by $0.1\%$.

Let $L$ be the total length of the vertical curve in meters.

The total change in grade ($|A|$) over the entire length ($L$) is related to the rate of change. The rate of change is essentially \(\frac{\text{Total change in grade}}{\text{Total length}}\). The problem gives us a specific rate over a specific length segment.

We can set up a proportion or use the formula derived from the definition of the rate of change:

Rate of change per unit length = \(\frac{\text{Given rate of change}}{\text{Length over which the rate applies}}\)

Rate of change per meter = \(\frac{0.1\%}{30 \text{ m}}\)

The total change in grade ($|A|$) is achieved over the total length $L$ with this rate.

\(|A| = (\text{Rate of change per meter}) \times L\)

\(0.7\% = \left(\frac{0.1\%}{30 \text{ m}}\right) \times L\)

Now, we can solve for $L$:

\(L = \frac{0.7\% \times 30 \text{ m}}{0.1\%}\)

The percentage units cancel out:

\(L = \frac{0.7 \times 30}{0.1} \text{ m}\)

\(L = \frac{21}{0.1} \text{ m}\)

\(L = 210 \text{ m}\)

Summary of Calculation

The length of the vertical curve is found by dividing the total change in grade by the rate of change per unit length. In this case, the rate is given over a specific length, so we adjust accordingly.

\(L = \frac{\text{Total change in grade}}{|A|} \times \frac{\text{Length for rate}}{\text{Rate of change over that length}}\)

\(L = \frac{0.7\%}{0.1\%} \times 30 \text{ m}\)

\(L = 7 \times 30 \text{ m}\)

\(L = 210 \text{ m}\)

Thus, the length of the vertical curve connecting the two grades with the given rate of change is $210$ m.