Let's work through the given problem step by step. We'll start with the information provided:

• Degree of curve, $D = 12^\circ$
• Superelevation, $e = 0.08$ (This is the rate of the roadway banking)
• The road is on a level grade.

We need to determine:

1. Radius of the curve
2. Current maximum safe speed
3. Minimum distance from the proposed structure to the centerline of the curve

Part (a): Radius of the Curve

The radius $R$ of a simple circular curve can be calculated using the following formula:

$R = \frac{5729.578}{D}$

Where:

• $D$ is the degree of curvature.

Substitute $D = 12^\circ$ into the formula:

$R = \frac{5729.578}{12} \approx 477.47 \text{ feet}$

Thus, the radius of the curve is approximately 477 feet (rounded to the nearest foot).

Part (b): Maximum Safe Speed

To calculate the maximum safe speed for a curve, we use the following formula from the AASHTO guidelines (American Association of State Highway and Transportation Officials):

$V = \sqrt{15 \cdot R \cdot (e + f)}$

Where:

• $V$ = maximum safe speed (in miles per hour),
• $R$ = radius of the curve (in feet),
• $e = 0.08$ (superelevation),
• $f = 0.15$ (lateral friction factor for highways, typical value).

Substituting the known values:

$V = \sqrt{15 \cdot 477 \cdot (0.08 + 0.15)} = \sqrt{15 \cdot 477 \cdot 0.23}$

Let’s calculate that:

$V = \sqrt{15 \cdot 477 \cdot 0.23} = \sqrt{1646.55} \approx 40.57 \text{ mph}$

Thus, the current maximum safe speed is approximately 41 mph.

Part (c): Minimum Distance to the Proposed Structure

To maintain the maximum safe speed without reduction, the structure must be located far enough from the curve's centerline so that it doesn't interfere with the driving dynamics. A common standard is that the minimum distance from the centerline is at least the stopping sight distance (SSD) or clear zone for the design speed.

The stopping sight distance is calculated using the following simplified formula for highways:

$SSD = 1.47 \cdot V \cdot t + \frac{V^2}{30 \cdot (f + a)}$

Where:

• $V$ is the design speed in mph,
• $t$ is the reaction time (typically 2.5 seconds),
• $f$ is the friction factor (0.15),
• $a$ is the deceleration rate (assumed to be 11.2 ft/sec² for stopping).

Substituting the design speed $V = 41$ mph:

$SSD = 1.47 \cdot 41 \cdot 2.5 + \frac{41^2}{30 \cdot (0.15 + 11.2)}$

Let’s calculate the stopping sight distance:

$SSD = 1.47 \cdot 41 \cdot 2.5 + \frac{41^2}{30 \cdot 11.35} = 150.675 + \frac{1681}{340.5} = 150.675 + 4.94 \approx 155.61 \text{ feet}$

Thus, the minimum distance from the centerline to the proposed structure should be approximately 156 feet to avoid reducing the current maximum safe speed of 41 mph.

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Summary of Results:

• (a) Radius of the curve: 477 feet
• (b) Current maximum safe speed: 41 mph
• (c) Minimum distance to the structure: 156 feet

Would you like further details or have any questions?

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Here are five related questions to expand this topic:

1. How does increasing the degree of curve affect the radius?
2. What would be the maximum safe speed if the superelevation was increased to 0.1?
3. How does road friction influence the maximum safe speed?
4. What is the impact of reducing the reaction time in stopping sight distance calculations?
5. How is the clear zone determined in highway design?

Tip: A larger radius allows for higher speeds on curved roads, which is critical for highway safety and design.