A body is subjected to two normal strains of magnitude Ex=0.001 & Ey=0.002. Determine the normal and shearing strains on a plane inclined to 30° with the Ex strain
To determine the normal and shearing strains on a plane inclined to 30° with the Ex strain, we need to use the concept of principal strains. Principal strains are the maximum and minimum strains on a plane.
Given that Ex = 0.001, Ey = 0.002, and the plane is inclined to 30° with the Ex strain, we can use trigonometry to find the normal and shearing strains on this plane.
1. Calculate the angle between the principal plane and the x-axis:
The angle between the principal plane and the x-axis can be found by subtracting 30° from 90°, as the plane is inclined to 30° with the x-axis.
Angle = 90° - 30° = 60°
2. Calculate the principal normal and shearing strains:
The principal normal strain (E1) and principal shearing strain (γ) can be calculated using the following formulas:
E1 = (Ex + Ey) / 2 + [(Ex - Ey) / 2] * cos(2 * Angle)
γ = [(Ex - Ey) / 2] * sin(2 * Angle)
Substituting the given values into the formulas:
E1 = (0.001 + 0.002) / 2 + [(0.001 - 0.002) / 2] * cos(2 * 60°)
γ = [(0.001 - 0.002) / 2] * sin(2 * 60°)
Evaluating the equations:
E1 = 0.0015
γ = -0.000433
Therefore, the normal strain on the inclined plane is 0.0015, and the shearing strain is -0.000433 (negative sign indicates opposite direction).
Note: The shearing strain can be positive or negative, indicating the direction of shear deformation.