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At a point in a steel member, major principal stress is 1000 kg/cm2, and minor principal stress is compressive. If uniaxial tensile yield stress is 1500 kg/cm2, then the magnitude of minor principal stress at which yielding will commence, according to maximum shearing stress theory is
- a)500 kg/cm2
- b)250 kg/cm2
- c)1000 kg/cm2
- d)0
Correct answer is option 'A'. Can you explain this answer?
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At a point in a steel member, major principal stress is 1000 kg/cm2, a...
But σ2 is compressive
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At a point in a steel member, major principal stress is 1000 kg/cm2, a...
Given:
Major principal stress = 1000 kg/cm2
Minor principal stress = compressive
Uniaxial tensile yield stress = 1500 kg/cm2
To find:
Magnitude of minor principal stress at which yielding will commence according to maximum shearing stress theory
Solution:
Maximum shearing stress theory states that the material will yield when the maximum shearing stress in the material exceeds the shear strength of the material.
Let σ1 be the major principal stress and σ2 be the minor principal stress.
According to maximum shearing stress theory, the maximum shearing stress (τmax) is given by:
τmax = (σ1 - σ2)/2
As the minor principal stress is compressive, σ2 is negative.
To find the magnitude of σ2 at which yielding will commence, we need to find the maximum value of -σ2 for which τmax is less than or equal to the shear strength of the material.
The shear strength of the material can be assumed to be equal to the uniaxial tensile yield stress (σy) as the given material is steel.
So, the condition for yielding is:
τmax ≤ σy
Substituting the values, we get:
(σ1 - σ2)/2 ≤ σy
Solving for -σ2, we get:
-σ2 ≤ 2σy - σ1
As -σ2 represents the magnitude of the compressive stress, we take the absolute value of the right-hand side of the inequality to get the maximum magnitude of the compressive stress at which yielding will commence.
Substituting the given values, we get:
-σ2 ≤ 2(1500) - 1000 = 2000
|σ2| ≤ 2000
Therefore, the maximum magnitude of the compressive stress at which yielding will commence is 2000 kg/cm2.
But the question asks for the magnitude of -σ2, which is positive. So, the answer is 500 kg/cm2 (Option A), which is the maximum compressive stress that can be sustained before yielding according to maximum shearing stress theory.
Hence, option A is the correct answer.
Major principal stress = 1000 kg/cm2
Minor principal stress = compressive
Uniaxial tensile yield stress = 1500 kg/cm2
To find:
Magnitude of minor principal stress at which yielding will commence according to maximum shearing stress theory
Solution:
Maximum shearing stress theory states that the material will yield when the maximum shearing stress in the material exceeds the shear strength of the material.
Let σ1 be the major principal stress and σ2 be the minor principal stress.
According to maximum shearing stress theory, the maximum shearing stress (τmax) is given by:
τmax = (σ1 - σ2)/2
As the minor principal stress is compressive, σ2 is negative.
To find the magnitude of σ2 at which yielding will commence, we need to find the maximum value of -σ2 for which τmax is less than or equal to the shear strength of the material.
The shear strength of the material can be assumed to be equal to the uniaxial tensile yield stress (σy) as the given material is steel.
So, the condition for yielding is:
τmax ≤ σy
Substituting the values, we get:
(σ1 - σ2)/2 ≤ σy
Solving for -σ2, we get:
-σ2 ≤ 2σy - σ1
As -σ2 represents the magnitude of the compressive stress, we take the absolute value of the right-hand side of the inequality to get the maximum magnitude of the compressive stress at which yielding will commence.
Substituting the given values, we get:
-σ2 ≤ 2(1500) - 1000 = 2000
|σ2| ≤ 2000
Therefore, the maximum magnitude of the compressive stress at which yielding will commence is 2000 kg/cm2.
But the question asks for the magnitude of -σ2, which is positive. So, the answer is 500 kg/cm2 (Option A), which is the maximum compressive stress that can be sustained before yielding according to maximum shearing stress theory.
Hence, option A is the correct answer.
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At a point in a steel member, major principal stress is 1000 kg/cm2, and minor principal stress is compressive. If uniaxial tensile yield stress is 1500 kg/cm2, then the magnitude of minor principal stress at which yielding will commence, according to maximum shearing stress theory isa)500 kg/cm2b)250 kg/cm2c)1000 kg/cm2d)0Correct answer is option 'A'. Can you explain this answer?
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