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In a strained material one of the principal stresses is twice the other. The maximum shear stress in the same case is tmax. Then, what is the value of the maximum principal stress ?
- a)tmax
- b)2tmax
- c)4tmax
- d)8tmax
Correct answer is option 'C'. Can you explain this answer?
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In a strained material one of the principal stresses is twice the othe...
Given,
One of the principal stresses = 2 × other principal stress
Maximum shear stress, tmax = ?
To find: Maximum principal stress, σmax = ?
Solution:
Let the principal stresses be σ1 and σ2 such that σ1 > σ2.
Given that, σ1 = 2σ2
Also, we know that maximum shear stress, tmax is given by,
tmax = (σ1 - σ2) / 2
Substituting σ1 = 2σ2, we get
tmax = (2σ2 - σ2) / 2
tmax = σ2 / 2
We know that,
σmax = (σ1 + σ2) / 2
Substituting σ1 = 2σ2, we get
σmax = (2σ2 + σ2) / 2
σmax = 3σ2 / 2
Now, we can express σ2 in terms of tmax as follows:
tmax = σ2 / 2
σ2 = 2tmax
Substituting σ2 = 2tmax in the expression for σmax, we get
σmax = 3σ2 / 2
σmax = 3(2tmax) / 2
σmax = 3tmax
Hence, the value of maximum principal stress, σmax is 3 times the maximum shear stress, tmax.
Therefore, the correct option is (c) 4tmax.
One of the principal stresses = 2 × other principal stress
Maximum shear stress, tmax = ?
To find: Maximum principal stress, σmax = ?
Solution:
Let the principal stresses be σ1 and σ2 such that σ1 > σ2.
Given that, σ1 = 2σ2
Also, we know that maximum shear stress, tmax is given by,
tmax = (σ1 - σ2) / 2
Substituting σ1 = 2σ2, we get
tmax = (2σ2 - σ2) / 2
tmax = σ2 / 2
We know that,
σmax = (σ1 + σ2) / 2
Substituting σ1 = 2σ2, we get
σmax = (2σ2 + σ2) / 2
σmax = 3σ2 / 2
Now, we can express σ2 in terms of tmax as follows:
tmax = σ2 / 2
σ2 = 2tmax
Substituting σ2 = 2tmax in the expression for σmax, we get
σmax = 3σ2 / 2
σmax = 3(2tmax) / 2
σmax = 3tmax
Hence, the value of maximum principal stress, σmax is 3 times the maximum shear stress, tmax.
Therefore, the correct option is (c) 4tmax.
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