Answer to Question #128722 in Mechanics | Relativity for Ariyo Emmanuel

Question

(1) A rectangular block of metal of 50 mm x 25 mm cross-section and 125 mm length carries a tensile load of 100 kN along its length, a compressive load of 1.0 MN on its 50 x 125 mm faces and a tensile load of 400 kN on its 25 x 125 mm faces. If E = 208 GN/m² and Poisson’s ratio = 0.3, Find (a) change in volume of bar (b) increase required in 1.0 MN load to produce no change in volume.

Answer 1

(a)

$\sigma_x=\frac{F_x}{a\cdot b}=\frac{100}{25\cdot50}=80(N/mm^2)$

$\sigma_y=\frac{F_y}{h\cdot b}=-\frac{1000}{125\cdot50}=-160(N/mm^2)$

$\sigma_z=\frac{F_y}{h\cdot a}=\frac{400}{125\cdot25}=128(N/mm^2)$

$V=125\cdot25\cdot50=156250(mm^3)$

$\frac{dV}{V}=\frac{(\sigma_x-\sigma_y+\sigma_z)(1-2u)}{E}=\frac{(80-160+128)(1-2\cdot0.3)}{208\cdot10^3}=$

$=9.23\cdot 10^{-5}$

$\Delta V=V\cdot 9.23\cdot 10^{-5}=$

$=156250\cdot9.23\cdot 10^{-5}=14.42(mm^3)$

(b)

$\sigma_x-\sigma_y+\sigma_z=0$

$\sigma_y\cdot h\cdot b=F_y\to$

$F_y=208\cdot125\cdot50=1300000(N)=1300(kN)$