Write a note on mohr's circle of stresses. What is the importance of this circle?
Answers
Mohr's circle is a graphical representation of transformation equations for plane stress. This graphical representation is extremely useful because it enables you to visualize the relationships between the normal and shear stresses acting on various inclined planes at a point in a stressed body........
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Mohr's circle, named after Christian Otto Mohr, is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
After performing a stress analysis on a material body assumed as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point.
The abscissa, {\displaystyle \sigma _{\mathrm {n} }} \sigma _{{\mathrm {n}}}, and ordinate, {\displaystyle \tau _{\mathrm {n} }} \tau _{{\mathrm {n}}}, of each point on the circle, are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the state of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress element.
Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. Mohr's contribution extended the use of this representation for both two- and three-dimensional stresses and developed a failure criterion based on the stress circle.[1]
Alternative graphical methods for the representation of the stress state at a point include the Lamé's stress ellipsoid and Cauchy's stress quadric.
The Mohr circle can be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.