On a section of B.G track, the ruling gradient is 1 in 200. If the track is laid in that place at a curve of 5°, then the allowable ruling gradient on the curve will be
1 in 333
Understanding Ruling Gradient and Gradient Compensation in Railways
The ruling gradient is the steepest gradient permitted on a section of railway track. It is a critical factor in determining the maximum load a locomotive can haul on that section. However, when a curve is introduced on a gradient, the resistance faced by the train increases due to the curve itself (curve resistance). To ensure that the total resistance on a curve located on a gradient does not exceed the resistance on the ruling gradient section (straight track), the gradient on the curve is reduced. This reduction in gradient on a curve is known as gradient compensation.
Gradient Compensation for Broad Gauge (B.G.) Track
For Broad Gauge (B.G.) track, the gradient compensation required on a curve is calculated based on the degree of the curve or its radius. The amount of compensation applied is the minimum of the values obtained from the following two criteria:
- Compensation based on degree of curve: $0.04\% \times \text{Degree of curve (in degrees)}$
- Compensation based on radius of curve: $70 / \text{Radius of curve (in meters)}$
This compensation is expressed as a percentage reduction in gradient. The allowable ruling gradient on a curve is then the original ruling gradient minus the calculated gradient compensation.
Calculating the Allowable Ruling Gradient on the Curve
Let's calculate the allowable ruling gradient for the given scenario: a B.G. track with a ruling gradient of 1 in 200 on which a 5° curve is laid.
Step 1: Determine the radius of the curve
For Broad Gauge track, the relationship between the radius (R) in meters and the degree of curve (D) is approximately given by $R = 1750 / D$.
Given degree of curve, $D = 5^\circ$.
Radius, $R = 1750 / 5 = 350$ meters.
Step 2: Calculate gradient compensation based on degree of curve
Compensation $= 0.04\% \times \text{Degree of curve}$
Compensation $= 0.04\% \times 5 = 0.2\%$.
Step 3: Calculate gradient compensation based on radius of curve
Compensation $= 70 / \text{Radius (in meters)}$
Compensation $= 70 / 350 = 0.2$. This value represents a gradient ratio. To convert it to a percentage, we multiply by 100.
Compensation (percentage) $= 0.2 \times 100\% = 20\%$.
Correction: The formula $70/R$ directly gives the compensation in percentage points for BG track. So, the compensation is $70 / 350 = 0.2\%$.
Step 4: Determine the gradient compensation to be applied
The compensation applied is the minimum of the values calculated in Step 2 and Step 3.
- From degree: $0.2\%$
- From radius: $0.2\%$
The minimum compensation is $0.2\%$.
Step 5: Convert the original ruling gradient to percentage
The original ruling gradient is 1 in 200. This means a rise of 1 unit vertically for every 200 units horizontally.
Ruling gradient (percentage) $= (1 / 200) \times 100\% = 0.5\%$.
Step 6: Calculate the allowable ruling gradient on the curve
Allowable ruling gradient on curve = Original ruling gradient - Gradient compensation
Allowable gradient (percentage) $= 0.5\% - 0.2\% = 0.3\%$.
Step 7: Convert the allowable percentage gradient back to '1 in X' format
An allowable gradient of $0.3\%$ means a rise of $0.3$ units for every 100 units horizontally.
Gradient ratio $= 0.3 / 100 = 3 / 1000$.
To express this as 1 in X, we have $1 / X = 3 / 1000$.
$X = 1000 / 3 \approx 333.33$.
So, the allowable ruling gradient on the curve is approximately 1 in 333.33, which is closest to 1 in 333.
Summary Table of Calculation Steps
| Parameter | Value / Calculation | Result |
|---|---|---|
| Original Ruling Gradient | 1 in 200 | $0.5\%$ |
| Degree of Curve (D) | $5^\circ$ | |
| Radius of Curve (R) for B.G. | $R = 1750 / D$ | 350 m |
| Compensation (Degree Method) | $0.04\% \times 5^\circ$ | $0.2\%$ |
| Compensation (Radius Method) | $70 / 350$ | $0.2\%$ |
| Minimum Compensation | min($0.2\%$, $0.2\%$) | $0.2\%$ |
| Allowable Gradient (Percentage) | $0.5\% - 0.2\%$ | $0.3\%$ |
| Allowable Gradient (1 in X) | 1 in $(100 / 0.3)$ | 1 in 333.33... (approx.) |
Based on the calculation, the allowable ruling gradient on the curve will be approximately 1 in 333. This accounts for the increased resistance caused by the curve, ensuring consistent train performance.
Revision Table: Key Concepts in Railway Gradients and Curves
| Concept | Description | Significance |
|---|---|---|
| Ruling Gradient | Steepest gradient governing maximum train load on a section. | Limits train weight and speed. |
| Momentum Gradient | A steeper gradient permitted if a train can gain sufficient speed on a flatter section before encountering it. | Allows steeper slopes where feasible. |
| Gradients in Station Yards | Ideally flat or very gentle (e.g., 1 in 1000) to prevent wagon roll-off. | Ensures safety and operational ease. |
| Curve Resistance | Extra resistance encountered by a train negotiating a curve. | Increases total resistance, potentially slowing down the train or requiring more power. |
| Gradient Compensation | Reduction in gradient on a curve to counteract curve resistance. | Maintains total resistance approximately equal to that on the ruling gradient on straight track. |
| Broad Gauge (B.G.) | Railway track gauge of 1676 mm. | Influences curve radius formulas and compensation values compared to other gauges. |
Additional Information: Factors Affecting Train Movement
The movement of a train is affected by various resistances that the locomotive must overcome. Understanding these resistances is key to railway design and operation, including the concept of gradient compensation. Some primary factors include:
- Train Resistance: This includes resistance from friction (at axles, between wheel and rail), air resistance, and flange friction on curves.
- Gradient Resistance: The component of the train's weight acting parallel to an inclined track. Moving uphill requires overcoming this resistance, while moving downhill can be assisted by it. It is proportional to the gradient.
- Curve Resistance: The additional resistance caused by the curvature of the track, resulting from phenomena like flange friction, slip, and unequal wheel diameters. This is the resistance that gradient compensation aims to nullify on a gradient.
- Starting Resistance: Higher resistance encountered when starting a stationary train.
- Inertial Resistance: Resistance due to changes in speed (acceleration or deceleration).
Gradient compensation specifically addresses the combined effect of gradient resistance and curve resistance to prevent the total resistance on a curved gradient from being significantly higher than on a straight ruling gradient section, thus standardizing the maximum hauling capacity across the route.
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