Question

Length, width and thickness of a plate are 400 mm, 400 mm, and 30 mm, respectively. For the material of the plate, Young’s modulus of elasticity is 70 GPa, the Yield stress is 80 MPa and Poisson’s ratio is 0.33. When the plate is subjected to longitudinal tensile stress of 70 MPa, the increase in the volume (in mm3) of the plate is _______.

Answer 1

Concept:

When a rectangular prismatic member subjected to stresses (σ_x, σ_y, σ_z) in three mutually perpendicular axis, then the volumetric strain is given by:

\(\epsilon=\frac{dV}{V}=\left(\frac{σ_x\;+\;σ_x\;+\;σ_x}{E}\right)(1-2μ)\)

Calculation:

Given:

L × B × H = 400 mm × 400 mm × 30 mm, E = 70 GPa = 70 × 10³ MPa, σ_yt = 80 MPa, μ = 0.33, σ_x = 70 MPa.

\(\epsilon=\frac{dV}{V}=\left(\frac{σ_x\;+\;σ_x\;+\;σ_x}{E}\right)(1-2μ)\)

When σ_y and σ_z is zero, then

\(\epsilon=\frac{dV}{V}=\left(\frac{σ_x\;+\;σ_x\;+\;σ_x}{E}\right)(1-2μ)\)

\(\epsilon=\frac{dV}{V}=\frac{σ_x}{E}(1-2μ)\)

\(\epsilon=\frac{dV}{V}=\frac{70}{70\;\times\;10^3}[1-(2\times0.33)]\)

​\(\epsilon=\frac{dV}{V}=\frac{0.34}{1000}\)

\({dV}{}=\frac{0.34}{1000}\times​​V\)

\({dV}{}=\frac{0.34}{1000}\times​​400\times400\times30=1632\;mm^3\)