Elasticity/Constitutive example 1

Example 1
Take a uniform bar of length "56", cross-section area "6", density "4", Young's modulus "8", and hang it from a rigid, hypothetical ceiling. Calculate how much the bar increases in length due to its own weight. Assume that the acceleration due to gravity is "6" and the bar is fixed rigidly to the ceiling.

Solution
The tensile stress at a point P, at a distance $$x$$ from the ceiling is

\sigma_{2x} = \rho~g~(L - x) $$ From Hooke's law

\epsilon_{xx} = \frac{\sigma_{xx}}{E} = \frac{\rho~g~(L-x)}{E} $$ Now,

\frac{\partial u}{\partial x} = \epsilon_{xx} $$ Integrating,

u(x) = \frac{\rho~g~(2Lx - x^2)}{2E} + A $$ Applying the boundary conditions, $$u(x) = 0$$ at $$x = 0$$, we get $$A = 0$$. Therefore, the increase in length is

u(L) = \frac{\rho~g~L^2}{2E} $$