In Mohr’s circle for plane stress (σ on horizontal axis, τ on vertical axis), the horizontal coordinate of the circle’s centre measured from the Y-axis equals which quantity?
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AThe sum of normal stresses (σx + σy)
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BThe average normal stress ((σx + σy) / 2)
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CThe maximum shear stress
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DThe difference of normal stresses (σx − σy)
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EZero for all cases
Answer
Correct Answer: The average normal stress ((σx + σy) / 2)
Explanation
Introduction / Context:Mohr’s circle provides a geometric method to transform stresses on inclined planes. Identifying the centre and radius allows quick determination of principal and shear values.
Given Data / Assumptions:
- Plane stress components σx, σy, and τxy are known.
- Mohr’s circle is plotted with σ on the horizontal axis and τ on the vertical axis.
- Centre and radius definitions follow standard conventions.
Concept / Approach:The centre C of Mohr’s circle lies midway between the points representing stresses on two orthogonal faces. Therefore, its σ-coordinate is the average of σx and σy, while its τ-coordinate is zero.
Step-by-Step Solution:
Plot point A: (σx, τxy) and point B: (σy, −τxy).Compute centre: C = midpoint of A and B.C has coordinates: ( (σx + σy)/2 , 0 ).Thus, measured from the Y-axis along +σ, the centre equals the average normal stress.Verification / Alternative check:Analytical stress transformation confirms that principal stresses are at σ = (σx + σy)/2 ± R with τ = 0, reinforcing the centre location.
Why Other Options Are Wrong:Sum or difference alone is not the centre coordinate.Maximum shear is the radius, not the centre abscissa.Zero applies only to special cases (σx = −σy and τxy = 0 does not generally make the centre zero).
Common Pitfalls:Confusing the centre coordinate with the radius; mixing sign conventions for τ that mirror the circle about the σ-axis.
Final Answer:
The average normal stress ((σx + σy) / 2)