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Plane stress at a point in a body is defined by principal stresses 3s and s. The ratio of the normal stress to the maximum shear stress on the plane of maximum shear stress is
  • a)
    1
  • b)
    2
  • c)
    3
  • d)
    4
Correct answer is option 'B'. Can you explain this answer?
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Plane stress at a point in a body is defined by principal stresses 3s ...
Explanation:

Given, principal stresses are 3s and s.

Let the plane on which maximum shear stress occurs be inclined at an angle θ with the principal plane of stress σ = 3s.

Then, the normal stress on the maximum shear plane is given by:
σ_n = (3s + s cos 2θ)/2 = (3s + s (2 cos^2 θ - 1))/2

σ_n = (3s + s (2 sin^2 θ))/2

σ_n = (3s + s (1 - cos^2 θ))/2

σ_n = (4s - s cos^2 θ)/2

σ_n = 2s (2 - cos^2 θ)/2

σ_n = s (2 - cos^2 θ)


Also, the maximum shear stress on the plane of maximum shear stress is given by:
τ_max = (3s - s)/2 = s


Therefore, the ratio of normal stress to maximum shear stress is given by:
σ_n / τ_max = s (2 - cos^2 θ) / s = 2 - cos^2 θ


As the maximum value of cos^2 θ is 1, the minimum value of the ratio is obtained when cos^2 θ = 1/2.

Therefore, the minimum value of the ratio is:
2 - cos^2 θ = 2 - (1/2) = 1.5


Thus, the ratio of normal stress to maximum shear stress on the plane of maximum shear stress is always greater than 1 and less than 2. Hence, the correct answer is option B, 2.
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Plane stress at a point in a body is defined by principal stresses 3s ...
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Plane stress at a point in a body is defined by principal stresses 3s and s. The ratio of the normal stress to the maximum shear stress on the plane of maximum shear stress isa)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?
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