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In the case of the 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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In the case of the bi-axial state of normal stresses, the normal stre...
In the case of the bi-axial state of normal stresses:
Normal stress on an inclined plane:
Shear Stress on an inclined plane:
Normal stress on 45° plane:
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In the case of the bi-axial state of normal stresses, the normal stre...
Understanding Bi-Axial State of Normal Stresses
In a bi-axial state of normal stresses, we have two perpendicular normal stresses acting on a material. Let's denote these stresses as σ1 and σ2. The analysis of stresses on a plane inclined at 45° helps in understanding the resultant stress conditions.
Normal Stress on 45° Plane
To find the normal stress acting on a plane that is oriented at 45°, we utilize the transformation equations for normal and shear stresses. Specifically, the normal stress on this 45° plane can be derived from the following relationship:
- The normal stress (σ45) on a plane inclined at 45° is given by the average of the two principal stresses.
Formula for Normal Stress on 45° Plane
- The equation can be simplified to express σ45 as:
σ45 = (σ1 + σ2) / 2
This indicates that the normal stress on a 45° plane is essentially half the sum of the two principal stresses acting on the material.
Conclusion
Thus, the correct answer to the question regarding the normal stress on a 45° plane in a bi-axial state of normal stresses is:
- Option C: Half the sum of the normal stresses
This conclusion stems from the fundamental principles of stress transformation, highlighting how stresses behave under different orientations in materials subjected to loading. Understanding this concept is crucial for engineers and designers to ensure the integrity and safety of structures under various loading conditions.
In a bi-axial state of normal stresses, we have two perpendicular normal stresses acting on a material. Let's denote these stresses as σ1 and σ2. The analysis of stresses on a plane inclined at 45° helps in understanding the resultant stress conditions.
Normal Stress on 45° Plane
To find the normal stress acting on a plane that is oriented at 45°, we utilize the transformation equations for normal and shear stresses. Specifically, the normal stress on this 45° plane can be derived from the following relationship:
- The normal stress (σ45) on a plane inclined at 45° is given by the average of the two principal stresses.
Formula for Normal Stress on 45° Plane
- The equation can be simplified to express σ45 as:
σ45 = (σ1 + σ2) / 2
This indicates that the normal stress on a 45° plane is essentially half the sum of the two principal stresses acting on the material.
Conclusion
Thus, the correct answer to the question regarding the normal stress on a 45° plane in a bi-axial state of normal stresses is:
- Option C: Half the sum of the normal stresses
This conclusion stems from the fundamental principles of stress transformation, highlighting how stresses behave under different orientations in materials subjected to loading. Understanding this concept is crucial for engineers and designers to ensure the integrity and safety of structures under various loading conditions.
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In the case of the bi-axial state of normal stresses, the normal stress on 45° plane is equal toa)the sum of the normal stressesb)difference of the normal stressesc)half the sum of the normal stressesd)half the difference of the normal stressesCorrect answer is option 'C'. Can you explain this answer?
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