Each of the four vertical links has an 8 times 36-mm uniform rectangular cross section, and each of the four pins has a 16-mm diameter. Determine the maximum value of the average normal stress in the links connecting points B and D, points C and E.
The Correct Answer and Explanation is :
To determine the maximum average normal stress in the vertical links connecting points B and D, and points C and E, we analyze the forces acting on these links and their cross-sectional areas.
Given Data:
- Cross-sectional dimensions of each link: 8 mm (thickness) × 36 mm (width)
- Diameter of each pin: 16 mm
- Applied load, P: 20 kN
- Geometric distances: 0.4 m, 0.25 m, and 0.2 m (as per the provided diagram)
Analysis:
- Free-Body Diagram (FBD):
- Consider the member ABC as a free body.
- The applied load P = 20 kN acts vertically downward at point C.
- Reactions occur at points B and A due to the pins.
- Moment Equations:
- Taking moments about point A to find the force in link BD:
- Sum of moments about A: (0.4 m) × F_BD – (0.25 m + 0.4 m) × 20 kN = 0
- Solving for F_BD: F_BD = (0.65 m × 20 kN) / 0.4 m = 32.5 kN
- This indicates that link BD is in tension.
- Taking moments about point B to find the force in link CE:
- Sum of moments about B: (0.4 m) × F_CE – (0.25 m) × 20 kN = 0
- Solving for F_CE: F_CE = (0.25 m × 20 kN) / 0.4 m = 12.5 kN
- This indicates that link CE is in compression.
- Cross-Sectional Area Calculations:
- Link BD:
- Net area considering pin hole: (36 mm – 16 mm) × 8 mm = 20 mm × 8 mm = 160 mm²
- Since there are two parallel links, total net area = 2 × 160 mm² = 320 mm²
- Link CE:
- Gross area without considering pin hole: 36 mm × 8 mm = 288 mm²
- For two parallel links, total area = 2 × 288 mm² = 576 mm²
- Stress Calculations:
- Link BD (Tension):
- Average normal stress, σ_BD = Force / Area = 32.5 kN / 320 mm² = 32,500 N / 320 × 10⁻⁶ m² = 101.56 MPa
- Link CE (Compression):
- Average normal stress, σ_CE = Force / Area = 12.5 kN / 576 mm² = 12,500 N / 576 × 10⁻⁶ m² = 21.7 MPa
Conclusion:
- The maximum average normal stress occurs in the links connecting points B and D, with a value of approximately 101.56 MPa in tension.
- The links connecting points C and E experience an average normal stress of approximately 21.7 MPa in compression.
These calculations are consistent with standard structural analysis methods, ensuring the links’ design can withstand the applied loads without exceeding material stress limits.