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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 ist max. 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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Given: One principal stress (σ1) is twice the other (σ2), and the maximum shear stress is τmax.
To find: The value of the maximum principal stress.
Solution:
Let's assume that σ1 = 2σ2.
The stress tensor can be represented as:
[σ] = [σ1 0 0]
[0 σ2 0]
[0 0 σ3]
Where σ3 is the third principal stress (which is assumed to be zero in this case since we are dealing with a two-dimensional problem).
The maximum shear stress can be calculated as:
τmax = (σ1 - σ2)/2
Substituting σ1 = 2σ2 in the above equation, we get:
τmax = (2σ2 - σ2)/2 = σ2
Now, we know that the maximum shear stress is equal to the difference between the maximum and minimum principal stresses divided by 2:
τmax = (σ1 - σ2)/2
Substituting σ1 = 2σ2, we get:
τmax = (2σ2 - σ2)/2 = σ2
Rearranging the above equation, we get:
σ1 - σ2 = 2τmax
Substituting σ1 = 2σ2, we get:
2σ2 - σ2 = 2τmax
Simplifying the equation, we get:
σ2 = 2τmax
Therefore, the maximum principal stress (σ1) can be calculated as:
σ1 = 2σ2 = 2(2τmax) = 4τmax
Hence, the correct answer is option C, i.e., 4τmax.
To find: The value of the maximum principal stress.
Solution:
Let's assume that σ1 = 2σ2.
The stress tensor can be represented as:
[σ] = [σ1 0 0]
[0 σ2 0]
[0 0 σ3]
Where σ3 is the third principal stress (which is assumed to be zero in this case since we are dealing with a two-dimensional problem).
The maximum shear stress can be calculated as:
τmax = (σ1 - σ2)/2
Substituting σ1 = 2σ2 in the above equation, we get:
τmax = (2σ2 - σ2)/2 = σ2
Now, we know that the maximum shear stress is equal to the difference between the maximum and minimum principal stresses divided by 2:
τmax = (σ1 - σ2)/2
Substituting σ1 = 2σ2, we get:
τmax = (2σ2 - σ2)/2 = σ2
Rearranging the above equation, we get:
σ1 - σ2 = 2τmax
Substituting σ1 = 2σ2, we get:
2σ2 - σ2 = 2τmax
Simplifying the equation, we get:
σ2 = 2τmax
Therefore, the maximum principal stress (σ1) can be calculated as:
σ1 = 2σ2 = 2(2τmax) = 4τmax
Hence, the correct answer is option C, i.e., 4τmax.
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