Subject: Finite Element Analysis

Topic: One Dimensional Problems

Difficulty: Medium

a) Number of elements and nodes.

b) Elemental matrix,
$\begin{bmatrix}K\end{bmatrix}^e=\frac{GJ}{h_e}\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}$

$For\,\, element \,\,1,\,\,d=100mm \,\,,\,\, h_e=450mm$
$J=\frac{\pi}{32}d^4=\frac{\pi}{32}*100^4=9.817*10^6$
$\frac{GJ}{h_e}=\frac{1*10^5*9.817*10^6}{450}=2181.7*10^6$
$\begin{bmatrix} K \end{bmatrix}^1=10^6\begin{bmatrix} 2181.7&-2181.7\\ -2181.7&2181.7 \end{bmatrix}$

$For\,\, element \,\,2,\,\,d=80mm \,\,,\,\, h_e=400mm$
$J=\frac{\pi}{32}d^4=\frac{\pi}{32}*80^4=4.02*10^6$
$\frac{GJ}{h_e}=\frac{1*10^5*4.02*10^6}{400}=1005.3*10^6$
$\begin{bmatrix} K \end{bmatrix}^2=10^6\begin{bmatrix} 1005.3&-1005.3\\ -1005.3&1005.3 \end{bmatrix}$

$For\,\, element \,\,3,\,\,d=50mm \,\,,\,\, h_e=500mm$
$J=\frac{\pi}{32}d^4=\frac{\pi}{32}*50^4=0.613*10^6$
$\frac{GJ}{h_e}=\frac{1*10^5*0.613*10^6}{500}=122.7*10^6$
$\begin{bmatrix} K \end{bmatrix}^3=10^6\begin{bmatrix} 122.7&-122.7\\ …