The state of stress at a point in a 2-D loading is such that the Mohr's circle is a point located at 175 MPa on the positive normal stress axis. The maximum and minimum principal stresses respectively, from Mohr's circle, are:
+175 MPa; +175 MPa
Understanding Mohr's Circle as a Point
Mohr's circle is a graphical representation used in mechanics of materials to illustrate the state of stress at a specific point within a material. It plots normal stress ($\sigma$) on the horizontal axis and shear stress ($\tau$) on the vertical axis. The principal stresses, which are the maximum and minimum normal stresses experienced at that point, are found where the circle intersects the horizontal normal stress axis.
Interpreting a Point Mohr's Circle
In this specific problem, the Mohr's circle is described as a single point located at 175 MPa on the positive normal stress axis. This indicates a very specific and simple stress state:
- Radius is Zero: When the Mohr's circle is just a point, it signifies that the radius of the circle is zero. The radius of Mohr's circle is related to the shear stresses and the difference between normal stresses. A zero radius implies that the shear stresses ($\tau_{xy}$) are zero, and the normal stresses on the perpendicular planes are equal ($\sigma_x = \sigma_y$).
- Center Location: The location of this single point on the normal stress axis represents the center of the Mohr's circle. The center's coordinate on the normal stress axis is given by the average normal stress, $\sigma_{avg} = \frac{\sigma_x + \sigma_y}{2}$.
Calculating Principal Stresses
Given that the Mohr's circle is a point at 175 MPa on the normal stress axis:
- The center of the circle is at $\sigma_{avg} = 175$ MPa.
- The radius of the circle is $R = 0$ MPa.
The principal stresses ($\sigma_1$ and $\sigma_2$) are calculated using the formulas:
$$ \sigma_1 = \sigma_{avg} + R $$ $$ \sigma_2 = \sigma_{avg} - R $$Substituting the values:
$$ \sigma_1 = 175 \text{ MPa} + 0 \text{ MPa} = 175 \text{ MPa} $$ $$ \sigma_2 = 175 \text{ MPa} - 0 \text{ MPa} = 175 \text{ MPa} $$This means that at this point, the stress is hydrostatic (equal in all directions in the 2D plane), with both the maximum and minimum principal stresses being 175 MPa.
Conclusion
Therefore, the maximum and minimum principal stresses, derived from a Mohr's circle that is a single point at 175 MPa on the normal stress axis, are both +175 MPa.
Important Questions from Principle Stress
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