Mumbai University > Mechanical Engineering > Sem 6 > Finite Element Analysis

Marks: 5M

Year: Dec2016

For conservative structural systems , of all the kinematically admissible deformations , those corresponding to the equilibrium state extremize (i.e minimize or maximize) the total potential energy. If the extremum is a mimimum,the equilibrium state is stable.

The potential energy of an elastic body is defined as,

$\pi$ = strain energy (U) - potential energy of loading (M)

For above spring system,

$\therefore$ Equation of total potential energy.

$\pi$ = $\frac{1}{2} k,u_2^2 + \frac{1}{2}k_2(u_3-u_2)^2-F.u_3$

Obtain the equilibrium equations by minimizing the potential energy.

$\frac{\partial \pi}{\partial u_2}=k_1,u_2 +k_2(u_3-u_2)(-1)=0 $

$\therefore (k_1+k_2)u_2-k_2 u_3=0 \hspace{2cm} --------- (1) $

$\frac{\partial \pi}{\partial u_3}=k_2(u_3-u_2)-f=0$

$-k_2u_2+k_2u_3=f \hspace{2cm}-------(2)$

In matrix from, equations 1 and 2,

$\begin{bmatrix} \ k_1+k_2 & -k_2 \\ \ -k_2 & k_2 \\ \end{bmatrix}$ $\begin{Bmatrix} \ u_2 \\ \ u_4 \\ \end{Bmatrix}$ = $\begin{Bmatrix} \ o \\ \ f \\ \end{Bmatrix}$