JNTU B.Tech III MECHANICAL WORKING Previous Question Paper (September - 2021 - R16)

Strain at a point is defined as the state of deformation in the immediate vicinity of a specific point within a body. Since deformation varies from point to point, it is characterized by the change in length and shape of an infinitesimal element surrounding that point. In a three-dimensional state, the strain at a point is represented by a symmetric second-order strain tensor consisting of six independent components: three normal strains (εx, εy, εz) representing changes in length per unit length, and three shear strains (γxy, γyz, γzx) representing changes in the angles between originally perpendicular lines.

Mathematically, the normal strain is defined as ε = ΔL/L, and the shear strain is defined as the change in the right angle between two line elements. At any point, these components can be transformed to different coordinate systems using strain transformation equations, and the principal strains represent the maximum and minimum normal strains at that point, where shear strain components vanish.

Mohr's circle is a graphical representation of the transformation equations for plane stress, used to determine the normal and shear stresses acting on any plane at a point. The circle is plotted on a coordinate system where the horizontal axis represents normal stress (σ) and the vertical axis represents shear stress (τ).

To construct the circle for a state of stress with known σx, σy, and τxy:

1. Locate two points on the σ-τ plane: (σx, τxy) and (σy, -τxy).
2. Draw a line connecting these points; the intersection of this line with the σ-axis is the center of the circle, located at ((σx + σy)/2, 0).
3. Draw the circle with this center and a radius R = √(((σx - σy)/2)² + τxy²).

The points where the circle intersects the σ-axis represent the principal stresses (σ1 and σ2), where shear stress is zero. The highest point on the circle gives the maximum in-plane shear stress (τmax = R). For example, if a material is subjected to σx = 100 MPa, σy = 40 MPa, and τxy = 30 MPa, the center is at (70, 0) and the radius is √((30)² + (30)²) ≈ 42.4 MPa, yielding principal stresses of 112.4 MPa and 27.6 MPa.