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To solve this problem involving a suspended cable supporting a water-filled pipe, let’s analyze each question step-by-step.

### Given Data:
- **Length of cable (L):** 60 m
- **Weight of pipe (w):** 6.5 kN/m
- **Allowable tensile load (for parts 1 and 3):** Given as either 2560 kN (part 1) or 2000 kN (part 3)
- **Cable sag (y):** Different values given per part
- **Cable diameter (d):** 75 mm (for part 2)

### Problem Breakdown:
1. **Maximum Sag Calculation at Midpoint** (given allowable tensile load of 2560 kN)
   
   For a uniformly loaded cable, the relationship between the horizontal tension \( H \), the cable load \( w \), the span \( L \), and the sag \( y \) at the midpoint is:
   \[
   H = \frac{wL^2}{8y}
   \]
   Given that the maximum tension \( T \) in the cable is at the endpoints, we use:
   \[
   T = \sqrt{H^2 + (wy)^2}
   \]
   Set \( T = 2560 \, \text{kN} \) to find \( y \), the maximum sag.

2. **Maximum Axial Stress** (given sag of 3 m and diameter of 75 mm)
   
   With a specified sag \( y = 3 \, \text{m} \), the horizontal component \( H \) can be calculated using:
   \[
   H = \frac{wL^2}{8y}
   \]
   Then, we calculate the maximum tension \( T \) and determine the stress \( \sigma \) using:
   \[
   \sigma = \frac{T}{A}
   \]
   where \( A = \frac{\pi d^2}{4} \) is the cross-sectional area of the cable.

3. **Additional Load Capacity** (given sag of 2 m and allowable tensile load of 2000 kN)
   
   Here, \( y = 2 \, \text{m} \) and \( T = 2000 \, \text{kN} \) are provided. We calculate the existing load’s tension and then determine how much additional load could be applied before reaching the tensile limit.

Let me know if you would like a detailed solution to each part or if you have any specific questions about the calculations! 

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### Follow-Up Questions
1. How does the sag of a cable affect the overall tension in suspension structures?
2. What is the significance of the maximum tensile load in engineering designs for cables?
3. How would changing the weight of the pipe per unit length affect the cable sag?
4. Can this problem be solved using a parabolic approximation instead of a catenary model, and how accurate would it be?
5. How would using a material with a higher tensile strength alter the maximum sag?

### Tip
When dealing with suspended cables under uniform loads, always remember that increasing the sag reduces the horizontal tension, allowing the structure to support more load without exceeding tensile limits.

The water supply pipe shown is suspended from a cable using a series of close and equally spaced hangers. The length of the pipe is supported by the cable is 60 m. The total weight of the pipe filled with water is 6.5 kN/m. 1) What is the maximum sag at the lowest point of the cable which occurs at mid-length if the allowable tensile load in the cable is 2560 kN? 2) The sag of the cable at mid-length is 3 m. Find the maximum axial stress (MPa) if the cable diameter is 75 mm. 3) The sag of the cable at mid-length is 2 m. If the allowable tensile load in the cable is 2000 kN, how much additional load can the cable carry?

Learn how to analyze a suspended cable supporting a water-filled pipe, including calculating the maximum sag and axial stress. This problem explores the relationship between sag and tensile load, stress analysis with a given diameter, and the additional load capacity for various sag conditions. Suitable for Civil Engineering students.

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