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A piece of material is subjected, to two perpendicular tensile stresses of 70 MPa and 10 MPa. The magnitude of the resultant stress on a plane in which the maximum shear stress occurs is
- a)70 MPa
- b)60 MPa
- c)50 MPa
- d)10 MPa
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
A piece of material is subjected, to two perpendicular tensile stresse...
Given data:
Perpendicular tensile stresses, σ₁ = 70 MPa and σ₂ = 10 MPa
To find:
Resultant stress on a plane in which maximum shear stress occurs.
Solution:
Let the plane on which maximum shear stress occurs be inclined at an angle of θ with the horizontal.
The normal stress on this plane can be given as:
σₓ = (σ₁ + σ₂)/2 + (σ₁ - σ₂)/2 cos2θ
The shear stress on this plane can be given as:
τₓy = (σ₁ - σ₂)/2 sin2θ
To find the maximum shear stress, we need to differentiate the shear stress equation with respect to θ and equate it to zero.
dτₓy/dθ = (σ₁ - σ₂)/2 cos2θ = 0
cos2θ = 0
θ = 45°
Substituting θ = 45° in the above equations, we get:
σₓ = (70 + 10)/2 + (70 - 10)/2 cos²45°
σₓ = 40 MPa
τₓy = (70 - 10)/2 sin2×45°
τₓy = 30 MPa
The magnitude of the resultant stress can be calculated using the following formula:
σR = √(σₓ² + 4τₓy²)
σR = √(40² + 4×30²)
σR = 50 MPa
Therefore, the correct option is (c) 50 MPa.
Perpendicular tensile stresses, σ₁ = 70 MPa and σ₂ = 10 MPa
To find:
Resultant stress on a plane in which maximum shear stress occurs.
Solution:
Let the plane on which maximum shear stress occurs be inclined at an angle of θ with the horizontal.
The normal stress on this plane can be given as:
σₓ = (σ₁ + σ₂)/2 + (σ₁ - σ₂)/2 cos2θ
The shear stress on this plane can be given as:
τₓy = (σ₁ - σ₂)/2 sin2θ
To find the maximum shear stress, we need to differentiate the shear stress equation with respect to θ and equate it to zero.
dτₓy/dθ = (σ₁ - σ₂)/2 cos2θ = 0
cos2θ = 0
θ = 45°
Substituting θ = 45° in the above equations, we get:
σₓ = (70 + 10)/2 + (70 - 10)/2 cos²45°
σₓ = 40 MPa
τₓy = (70 - 10)/2 sin2×45°
τₓy = 30 MPa
The magnitude of the resultant stress can be calculated using the following formula:
σR = √(σₓ² + 4τₓy²)
σR = √(40² + 4×30²)
σR = 50 MPa
Therefore, the correct option is (c) 50 MPa.
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Community Answer
A piece of material is subjected, to two perpendicular tensile stresse...
The maximum shear stress = (70-10)/2 = 30. At maximum shear plane normal stress = sigma(Avg)= (70+10)/2= 40 . So the resultant stress is equal to √[30^2+40^2] = 50 Mpa. By drawing Mohr's circle you can clearly understand
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