Understanding the Degree of a Simple Circular Curve

The degree of a simple circular curve is a way to define how sharp or flat the curve is. There are two common definitions used in surveying and civil engineering:

• Arc Definition: The angle subtended at the center by an arc of standard length (usually 100 feet in the US or 20 meters in some systems).
• Chord Definition: The angle subtended at the center by a chord of standard length (usually 100 feet or 20 meters).

The question specifies a "specified chord length" of 20 m, which indicates that we should use the Chord Definition with a standard chord length \(L = 20\) m. The radius of the curve is given as \(R = 300\) m.

Calculating the Degree of Curve (Chord Definition)

According to the Chord Definition, the degree of curve \(D\) is the angle subtended at the center by a standard chord length \(L\). Consider a triangle formed by the center of the curve and the two ends of the chord. This is an isosceles triangle with two sides equal to the radius \(R\) and the base equal to the chord length \(L\). Dropping a perpendicular from the center to the midpoint of the chord bisects both the chord and the central angle \(D\). The angle formed at the center for half the chord is \(D/2\).

Using trigonometry in the right-angled triangle formed:

\(\sin\left(\frac{D}{2}\right) = \frac{\text{Half Chord Length}}{\text{Radius}}\)

\(\sin\left(\frac{D}{2}\right) = \frac{L/2}{R}\)

Step-by-Step Calculation

Given values:

• Radius, \(R = 300\) m
• Specified Chord Length, \(L = 20\) m

Substitute the values into the formula:

\(\sin\left(\frac{D}{2}\right) = \frac{20\text{ m}/2}{300\text{ m}}\)

\(\sin\left(\frac{D}{2}\right) = \frac{10\text{ m}}{300\text{ m}}\)

\(\sin\left(\frac{D}{2}\right) = \frac{1}{30}\)

To find \(D/2\), we take the arcsin (inverse sine) of both sides:

\(\frac{D}{2} = \arcsin\left(\frac{1}{30}\right)\)

Calculate the value using a calculator in degrees mode:

\(\frac{D}{2} \approx \arcsin(0.033333...)\)

\(\frac{D}{2} \approx 1.9086^\circ\)

Now, find \(D\) by multiplying by 2:

\(D \approx 2 \times 1.9086^\circ\)

\(D \approx 3.8172^\circ\)

Rounding to two decimal places, the degree of the curve is approximately \(3.82^\circ\).

Comparing with Options

The calculated degree of curve is approximately \(3.82^\circ\). Let's compare this with the given options:

• Option 1: 5.73°
• Option 2: 5.37°
• Option 3: 3.82°
• Option 4: 3.28°

The calculated value \(3.82^\circ\) matches Option 3.

Revision Table: Simple Circular Curve

Additional Information: Degree of Curve and Radius Relationship

The degree of curve is inversely proportional to the radius. A smaller radius means a sharper curve and a larger degree of curve. A larger radius means a flatter curve and a smaller degree of curve.

While the Chord Definition is used when a specified chord length is given, the Arc Definition is also common, especially with metric units and a standard arc length of 20m. Under the Arc Definition, the relationship is simpler:

\(D_\text{arc} = \frac{\text{Standard Arc Length}}{R} \times \frac{180^\circ}{\pi}\)

Or, if using a standard arc length of 20m:

\(D_\text{arc} \approx \frac{20}{R} \times 57.296\)

For a standard arc length of 100 feet (common in US practice):

\(D_\text{arc} = \frac{100}{R_\text{feet}} \times \frac{180^\circ}{\pi} \approx \frac{5729.58}{R_\text{feet}}\)

It's important to know which definition is being used, as they yield slightly different values for the same curve, especially for sharp curves (small radii). The problem mentioning "specified chord" clarifies the definition to use here.