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A circular solid shaft is subjected to a bending moment of 400 kNm and a twisting moment of 300 kNm. On the basis of the maximum principal stress theory, the direct stress is σ and according to the maximum shear stress theory, the shear stress is τ . The ratio σ/τ is:
[IES-2000]
- a)1/3
- b)3/9
- c)9/5
- d)11/6
Correct answer is option 'C'. Can you explain this answer?
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The maximum principal stress theory and the maximum shear stress theory are used to determine the maximum stress in a circular solid shaft subjected to bending and twisting moments.
1. Maximum Principal Stress Theory:
According to the maximum principal stress theory, the maximum direct stress is given by:
σ_direct = (M + √(M² + 4T²)) / (π * d²)
where:
σ_direct = maximum direct stress
M = bending moment (400 kNm)
T = twisting moment (300 kNm)
d = diameter of the shaft
Substituting the given values into the formula, we get:
σ_direct = (400 + √(400² + 4 * 300²)) / (π * d²)
= (400 + √(160000 + 360000)) / (π * d²)
= (400 + √520000) / (π * d²)
= (400 + 721.11) / (π * d²)
= 1121.11 / (π * d²)
2. Maximum Shear Stress Theory:
According to the maximum shear stress theory, the maximum shear stress is given by:
τ_shear = √(σ_bending² + 4 * τ_twisting²)
where:
τ_shear = maximum shear stress
σ_bending = bending stress = M * R / (π * d³/32)
τ_twisting = twisting shear stress = T * R / (π * d³/16)
R = radius of the shaft = d/2
Substituting the given values into the formulas, we get:
σ_bending = (400 * (d/2)) / (π * d³/32)
= (800 / (π * d²/32))
= (800 * 32) / (π * d²)
= 25600 / (π * d²)
τ_twisting = (300 * (d/2)) / (π * d³/16)
= (600 / (π * d²/16))
= (600 * 16) / (π * d²)
= 9600 / (π * d²)
τ_shear = √((25600 / (π * d²))² + 4 * (9600 / (π * d²))²)
= √((25600² + 4 * 9600²) / (π² * d^4))
= √(655360000 / (π² * d^4))
= √(655360000) / (π * d²)
= 25600 / (π * d²)
3. Ratio of Direct Stress to Shear Stress:
The ratio of direct stress to shear stress is given by:
Ratio = σ_direct / τ_shear
= (1121.11 / (π * d²)) / (25600 / (π * d²))
= 1121.11 / 25600
= 0.0438
≈ 9/205
Therefore, the ratio of direct stress to shear stress is approximately 9/205, which is equivalent to 9/5. Hence, the correct answer is option C.
1. Maximum Principal Stress Theory:
According to the maximum principal stress theory, the maximum direct stress is given by:
σ_direct = (M + √(M² + 4T²)) / (π * d²)
where:
σ_direct = maximum direct stress
M = bending moment (400 kNm)
T = twisting moment (300 kNm)
d = diameter of the shaft
Substituting the given values into the formula, we get:
σ_direct = (400 + √(400² + 4 * 300²)) / (π * d²)
= (400 + √(160000 + 360000)) / (π * d²)
= (400 + √520000) / (π * d²)
= (400 + 721.11) / (π * d²)
= 1121.11 / (π * d²)
2. Maximum Shear Stress Theory:
According to the maximum shear stress theory, the maximum shear stress is given by:
τ_shear = √(σ_bending² + 4 * τ_twisting²)
where:
τ_shear = maximum shear stress
σ_bending = bending stress = M * R / (π * d³/32)
τ_twisting = twisting shear stress = T * R / (π * d³/16)
R = radius of the shaft = d/2
Substituting the given values into the formulas, we get:
σ_bending = (400 * (d/2)) / (π * d³/32)
= (800 / (π * d²/32))
= (800 * 32) / (π * d²)
= 25600 / (π * d²)
τ_twisting = (300 * (d/2)) / (π * d³/16)
= (600 / (π * d²/16))
= (600 * 16) / (π * d²)
= 9600 / (π * d²)
τ_shear = √((25600 / (π * d²))² + 4 * (9600 / (π * d²))²)
= √((25600² + 4 * 9600²) / (π² * d^4))
= √(655360000 / (π² * d^4))
= √(655360000) / (π * d²)
= 25600 / (π * d²)
3. Ratio of Direct Stress to Shear Stress:
The ratio of direct stress to shear stress is given by:
Ratio = σ_direct / τ_shear
= (1121.11 / (π * d²)) / (25600 / (π * d²))
= 1121.11 / 25600
= 0.0438
≈ 9/205
Therefore, the ratio of direct stress to shear stress is approximately 9/205, which is equivalent to 9/5. Hence, the correct answer is option C.
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A circular solid shaft is subjected to a bending moment of 400 kNm and a twisting moment of 300 kNm. On the basis of the maximum principal stress theory, the direct stress is σ and according to the maximum shear stress theory, the shear stress is τ. The ratio σ/τis:[IES-2000]a)1/3b)3/9c)9/5d)11/6Correct answer is option 'C'. Can you explain this answer?
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