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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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A circular solid shaft is subjected to a bending moment of 400 kNm and...
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? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about 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? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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?.
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? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about 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? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 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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