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In the case of bi-axial state of normal stresses, the normal stress on 45° plane is equal to
- a)The sum of the normal stresses
- b)Difference of the normal stresses
- c)Half the sum of the normal stresses
- d)Half the difference of the normal stresses
Correct answer is option 'C'. Can you explain this answer?
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Bi-axial state of normal stresses refers to a situation where a body experiences two normal stresses in perpendicular directions. In this case, the normal stress on the 45-degree plane is equal to half the sum of the normal stresses.
Explanation:
Let us consider a body subjected to two normal stresses, σ1 and σ2 in perpendicular directions x and y, respectively. The stress on the 45-degree plane is represented by σ45. By using the equations of statics, we can determine the value of σ45.
The stress on the 45-degree plane can be resolved into two components, one along the x-direction and the other along the y-direction. The components of σ45 along x and y-direction are given by:
σx = (σ1 + σ2)/2cos45 = (σ1 + σ2)/√2
σy = (σ1 + σ2)/2sin45 = (σ1 + σ2)/√2
Since the stress on the 45-degree plane is perpendicular to both x and y-directions, it does not contribute to the shear stress. Therefore, the shear stress on the 45-degree plane is zero.
The normal stress on the 45-degree plane is given by:
σ45 = σx + σy = (σ1 + σ2)/√2 + (σ1 + σ2)/√2
σ45 = (σ1 + σ2)
Hence, the normal stress on the 45-degree plane is equal to half the sum of the normal stresses, i.e., σ45 = (σ1 + σ2)/2.
Conclusion:
In the bi-axial state of normal stresses, the normal stress on the 45-degree plane is equal to half the sum of the normal stresses. This is because the stress on the 45-degree plane can be resolved into two components along the x and y-directions, and the sum of these components gives the normal stress on the 45-degree plane.
Explanation:
Let us consider a body subjected to two normal stresses, σ1 and σ2 in perpendicular directions x and y, respectively. The stress on the 45-degree plane is represented by σ45. By using the equations of statics, we can determine the value of σ45.
The stress on the 45-degree plane can be resolved into two components, one along the x-direction and the other along the y-direction. The components of σ45 along x and y-direction are given by:
σx = (σ1 + σ2)/2cos45 = (σ1 + σ2)/√2
σy = (σ1 + σ2)/2sin45 = (σ1 + σ2)/√2
Since the stress on the 45-degree plane is perpendicular to both x and y-directions, it does not contribute to the shear stress. Therefore, the shear stress on the 45-degree plane is zero.
The normal stress on the 45-degree plane is given by:
σ45 = σx + σy = (σ1 + σ2)/√2 + (σ1 + σ2)/√2
σ45 = (σ1 + σ2)
Hence, the normal stress on the 45-degree plane is equal to half the sum of the normal stresses, i.e., σ45 = (σ1 + σ2)/2.
Conclusion:
In the bi-axial state of normal stresses, the normal stress on the 45-degree plane is equal to half the sum of the normal stresses. This is because the stress on the 45-degree plane can be resolved into two components along the x and y-directions, and the sum of these components gives the normal stress on the 45-degree plane.
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