Cable Horizontal Reaction Calculation

This solution details the calculation for determining the horizontal reaction at the supports of a cable carrying a load. We'll analyze the forces and geometry involved using standard engineering principles.

Identifying Given Parameters

The problem provides the following key information about the cable setup:

• Uniform load along the horizontal span: w = 10 kN/m
• Total horizontal span between supports: L = 100 m
• Central dip or sag of the cable: d = 8 m
• The supports are situated at the same horizontal level.

Theoretical Basis for Cable Analysis

For a flexible cable supporting a load that is uniformly distributed horizontally, the cable's shape approximates a parabola. The horizontal reaction, often denoted as H, represents the horizontal component of the tension within the cable. This component is constant along the entire length of the cable and acts parallel to the line connecting the supports.

Formula for Horizontal Reaction (H)

The relationship between the horizontal reaction (H), the uniform horizontal load (w), the horizontal span (L), and the central dip (d) for a cable supported at the same level is given by the formula:

$$H = \frac{w L^2}{8d}$$

This formula is derived from the equilibrium equations of the cable, specifically considering the segment from the center to one support.

Step-by-Step Calculation of Horizontal Reaction

We can calculate the horizontal reaction by substituting the given values into the formula:

1. Input the known values:
   • w = 10 kN/m
   • L = 100 m
   • d = 8 m
2. Substitute these values into the formula:

   $$H = \frac{(10 \text{ kN/m}) \times (100 \text{ m})^2}{8 \times (8 \text{ m})}$$

3. Calculate the square of the span:

   $$(100 \text{ m})^2 = 10000 \text{ m}^2$$

4. Calculate the product in the numerator:

   $$10 \text{ kN/m} \times 10000 \text{ m}^2 = 100000 \text{ kN} \cdot \text{m}$$

5. Calculate the product in the denominator:

   $$8 \times 8 \text{ m} = 64 \text{ m}$$

6. Perform the final division to find H:

   $$H = \frac{100000 \text{ kN} \cdot \text{m}}{64 \text{ m}}$$

   $$H = 1562.5 \text{ kN}$$

Final Result

The calculated horizontal reaction developed at the support is 1562.5 kN. This corresponds to the value presented in the options.