The circle obtained from two dimensional stress system is known as ________

Question

Examoogle · Accepted Answer

The correct answer is Option B: Mohr circle. Mohr's circle is a graphical representation of the transformation of two-dimensional stress states at a point within a stressed material. It maps all possible normal stresses ($\sigma$) and shear stresses ($	au$) onto a coordinate plane, where the center and radius of the circle are determined by the principal stresses.

Step-by-Step Solution & Concepts:
Mohr's circle is a graphical representation of the transformation of two-dimensional stress states at a point within a stressed material. It maps all possible normal stresses ($\sigma$) and shear stresses ($	au$) onto a coordinate plane, where the center and radius of the circle are determined by the principal stresses.

Working Principle: Given a state of plane stress defined by $\sigma_x, \sigma_y,$ and $	au_{xy}$, the Mohr's circle is constructed with its center on the $\sigma$-axis at $\frac{\sigma_x + \sigma_y}{2}$. The radius represents the maximum shear stress, calculated as $R = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + 	au_{xy}^2}$. Any point on the circumference of the circle represents the stress components on a plane inclined at an angle $	heta$ from the reference planes.

Key Formulas: $\sigma_{avg} = \frac{\sigma_x + \sigma_y}{2}$ — Center of Mohr's circle on the normal stress axis; $R = \sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + 	au_{xy}^2}$ — Radius of Mohr's circle representing maximum shear stress

Key Points: The circle intersects the $\sigma$-axis at the values of the major and minor principal stresses.; The angle between two planes in the physical body is represented by twice that angle ($2	heta$) in the Mohr's circle.; Mohr's circle is vital for determining failure criteria like the Mohr-Coulomb failure envelope in soil mechanics.

Advantages: Provides a clear visual understanding of stress transformation.; Easily identifies maximum normal and shear stresses without complex trigonometric calculations.

Disadvantages: Limited to 2D stress representations (though it can be extended to 3D via three circles).; Requires accurate plotting for manual graphical solutions.

Applications: Analysis of shear strength of soils.; Calculation of principal stresses for failure analysis in structural components.

Additional Info: The circle is named after Christian Otto Mohr.. Option A is incorrect because 'Principal stress circle' is not a standard term, although the circle displays principal stresses.. Option C is incorrect as the circle represents both normal and shearing stresses, not just shearing.

Answer: B is correct — The graphical construction used to represent the state of stress at a point in a 2D system is universally known as Mohr's circle.

Exam Tip: Always remember that the angle in the circle is $2	heta$, meaning a $45^\circ$ rotation in the physical plane corresponds to a $90^\circ$ rotation on the Mohr's circle.

Examoogle · Answer

Option A: Principal stress circle

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Option C: Shearing stress circle

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Option D: None of the mentioned

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