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At a certain point in a structural member, there are perpendicular stresses 80 N/mm2 and 20 N/mm2, both tensile. What is the equivalent stress in simple tension, according to the maximum principal strain theory? (Poisson’s ratio = 0.25)
- a)Zero
- b)20 N/mm2
- c)60 N/mm2
- d)75 N/mm2
Correct answer is option 'D'. Can you explain this answer?
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To determine the equivalent stress in simple tension using the maximum principal strain theory, we need to consider the given perpendicular stresses and Poisson's ratio.
Given:
Perpendicular tensile stress 1 (σ1) = 80 N/mm^2
Perpendicular tensile stress 2 (σ2) = 20 N/mm^2
Poisson's ratio (ν) = 0.25
Step 1: Calculate the principal stresses
The principal stresses can be calculated using the formula:
σ1' = (σ1 + σ2) / 2 + √[((σ1 - σ2) / 2)^2 + τ^2]
σ2' = (σ1 + σ2) / 2 - √[((σ1 - σ2) / 2)^2 + τ^2]
where τ is the shear stress, which is assumed to be zero in this case since only perpendicular stresses are given.
Substituting the given values into the formula, we get:
σ1' = (80 + 20) / 2 + √[((80 - 20) / 2)^2 + 0^2]
= 50 + √[30^2]
= 50 + 30
= 80 N/mm^2
σ2' = (80 + 20) / 2 - √[((80 - 20) / 2)^2 + 0^2]
= 50 - √[30^2]
= 50 - 30
= 20 N/mm^2
Step 2: Calculate the equivalent stress
The equivalent stress (σe) can be calculated using the formula:
σe = √(σ1'^2 - σ1'σ2' + σ2'^2)
Substituting the calculated values into the formula, we get:
σe = √(80^2 - 80*20 + 20^2)
= √(6400 - 1600 + 400)
= √(4400)
≈ 66.33 N/mm^2
Therefore, the equivalent stress in simple tension, according to the maximum principal strain theory, is approximately 66.33 N/mm^2. Since none of the given options match this value, it seems there might be an error in the question or the answer choices.
Given:
Perpendicular tensile stress 1 (σ1) = 80 N/mm^2
Perpendicular tensile stress 2 (σ2) = 20 N/mm^2
Poisson's ratio (ν) = 0.25
Step 1: Calculate the principal stresses
The principal stresses can be calculated using the formula:
σ1' = (σ1 + σ2) / 2 + √[((σ1 - σ2) / 2)^2 + τ^2]
σ2' = (σ1 + σ2) / 2 - √[((σ1 - σ2) / 2)^2 + τ^2]
where τ is the shear stress, which is assumed to be zero in this case since only perpendicular stresses are given.
Substituting the given values into the formula, we get:
σ1' = (80 + 20) / 2 + √[((80 - 20) / 2)^2 + 0^2]
= 50 + √[30^2]
= 50 + 30
= 80 N/mm^2
σ2' = (80 + 20) / 2 - √[((80 - 20) / 2)^2 + 0^2]
= 50 - √[30^2]
= 50 - 30
= 20 N/mm^2
Step 2: Calculate the equivalent stress
The equivalent stress (σe) can be calculated using the formula:
σe = √(σ1'^2 - σ1'σ2' + σ2'^2)
Substituting the calculated values into the formula, we get:
σe = √(80^2 - 80*20 + 20^2)
= √(6400 - 1600 + 400)
= √(4400)
≈ 66.33 N/mm^2
Therefore, the equivalent stress in simple tension, according to the maximum principal strain theory, is approximately 66.33 N/mm^2. Since none of the given options match this value, it seems there might be an error in the question or the answer choices.
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At a certain point in a structural member, there are perpendicular stresses 80 N/mm2 and 20 N/mm2, both tensile. What is the equivalent stress in simple tension, according to the maximum principal strain theory? (Poisson’s ratio = 0.25)a)Zero b)20 N/mm2c)60 N/mm2 d)75 N/mm2Correct answer is option 'D'. Can you explain this answer?
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