Super-elevation Rate Calculation for Horizontal Curve

This solution explains how to calculate the required rate of super-elevation for a horizontal curve on a national highway, given the design speed and the radius of the curve.

Understanding Super-elevation

Super-elevation, also known as banking of the road, is the process of raising the outer edge of the pavement with respect to the inner edge on horizontal curves. The primary purpose of super-elevation is to counteract the centrifugal force acting on vehicles moving along the curve. This helps in reducing the tendency of vehicles to skid outwards and provides a smoother, safer passage.

Relevant Formula for Super-elevation Design

In highway engineering, particularly following Indian Roads Congress (IRC) standards, the relationship between super-elevation ($e$), the coefficient of lateral friction ($f$), the design speed ($V$), and the radius of the horizontal curve ($R$) is typically represented by the following equilibrium equation:

$$e + f = \frac{V^2}{127R}$$

Where:

• $e$ = rate of super-elevation (dimensionless)
• $f$ = allowable lateral friction coefficient (dimensionless)
• $V$ = design speed in kmph
• $R$ = radius of the horizontal curve in meters

IRC guidelines also specify maximum limits for both super-elevation ($e_{max}$) and the lateral friction coefficient ($f_{allowable}$) based on the design speed and road type.

Given Parameters

From the question, we have the following values:

• Radius of the horizontal curve, $R = 500 \text{ m}$
• Design speed, $V = 65 \text{ kmph}$

Standard Design Limits (IRC Guidelines)

For national highways, the standard design limits are generally:

• Maximum allowable super-elevation rate, $e_{max} = 0.07$ (or 7%)
• Allowable lateral friction coefficient for $V = 65 \text{ kmph}$, $f_{allowable} = 0.15$

Step-by-Step Calculation

1. Calculate the required centrifugal force resistance value:

   Using the formula $e + f = \frac{V^2}{127R}$, we first calculate the term $\frac{V^2}{127R}$. This value represents the total horizontal force resistance needed, combining super-elevation and friction.

   $$ \frac{V^2}{127R} = \frac{(65 \text{ kmph})^2}{127 \times 500 \text{ m}} $$

   $$ \frac{V^2}{127R} = \frac{4225}{63500} $$

   $$ \frac{V^2}{127R} \approx 0.066535 $$

2. Determine the designed super-elevation rate ($e$):

   The design principle is to provide a super-elevation rate $e$ such that $e \le e_{max}$ and the remaining force is balanced by friction, where the required friction $f_{req} = \frac{V^2}{127R} - e$ does not exceed $f_{allowable}$.

   In many design scenarios, if the calculated value $\frac{V^2}{127R}$ (which represents the required $e+f$) is less than or equal to the maximum allowable super-elevation ($e_{max}$), the designed super-elevation $e$ is set equal to this calculated value (assuming minimal friction is needed or relied upon).

   Here, the calculated value is approximately $0.066535$.

   We compare this value to the maximum allowable super-elevation:

   Calculated value $\approx 0.066535$

   Maximum allowable $e_{max} = 0.07$

   Since $0.066535 \le 0.07$, the designed super-elevation rate ($e$) can be set to this value.

3. Select the closest option:

   The calculated super-elevation rate is approximately $0.066535$. Comparing this with the given options:

   • $0.07$
   • $0.066$
   • $0.06$
   • $0.05$

   The value $0.066535$ is closest to $0.066$. Therefore, the rate of super-elevation for the given horizontal curve is $0.066$.

Conclusion

Based on the calculations using standard highway design formulas and parameters, the required rate of super-elevation for a horizontal curve with a radius of $500 \text{ m}$ at a design speed of $65 \text{ kmph}$ is approximately $0.0665$, which is best represented by the option $0.066$.

Final Answer: The final answer is $0.066$