Figure shows a cluster of four springs. Calculate the deflection of each spring when a force of 2000N is applieed. Model the springs as 1-D element.

Subject: Finite Element Analysis

Topic: One Dimensional Problems

Difficulty: Medium

a) Number of elements and nodes,

b) Element matrix Equation is given by,

$k\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}\begin{Bmatrix} u_1^e\\ u_2^e \end{Bmatrix}\begin{Bmatrix} F_1^e \\ F_2^e \end{Bmatrix} $

For element 1, k=25
$25\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}\begin{Bmatrix} u_1^1\\ u_2^1 \end{Bmatrix}\begin{Bmatrix} F_1^1 \\ F_2^2 \end{Bmatrix} $

For element 2, k=75
$75\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}\begin{Bmatrix} u_1^2\\ u_2^2 \end{Bmatrix}\begin{Bmatrix} F_2^1 \\ F_2^2 \end{Bmatrix} $

For element 3, k=22
$20\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}\begin{Bmatrix} u_1^3\\ u_2^3 \end{Bmatrix}\begin{Bmatrix} F_1^3 \\ F_2^3 \end{Bmatrix} $

For element 4, k=50
$50\begin{bmatrix} 1&-1\\ -1&1 \end{bmatrix}\begin{Bmatrix} u_1^4\\ u_2^4 \end{Bmatrix}\begin{Bmatrix} F_1^4 \\ F_2^4 \end{Bmatrix} \\ $

c) Global Matrix Equation,
$ \begin{bmatrix} 25+75+50&-25-75&-50\\ -25-75&25+75+20&-20 \\ -50&-20&20+50 \end{bmatrix}\begin{Bmatrix} u_1\\ u_2 \\u_3 \end{Bmatrix}=\begin{Bmatrix} F_1 \\ F_2\\F_3 \end{Bmatrix}$

$\therefore \begin{bmatrix} 150&-100&-50\\ -100&120&-20 \\ -50&-20&70 \end{bmatrix}\begin{Bmatrix} u_1\\ u_2 \\u_3 \end{Bmatrix}=\begin{Bmatrix} F_1 \\ F_2\\F_3 \end{Bmatrix} $

d) Imposing boundary condition,
$u_1=0\,\,,\,\,F_3=2000N$
$F_2=0\hspace{3cm}$ -for balancing,

$ \begin{bmatrix} 150&-100&-50\\ -100&120&-20 \\ -50&-20&70 \end{bmatrix}\begin{Bmatrix} 0\\ u_2 \\ u_3 \end{Bmatrix}=\begin{Bmatrix} F_1 \\ 0 \\ F_3 \end{Bmatrix} \\ $

e) Frame the equation,
$-100u_2-50u_3=F_1\hspace{3cm}$ -(1)
$120u_2-20u_3=0\hspace{3.7cm}$ -(2)
$-20u_2+70u_3=2000\hspace{3cm}$ -(3)

Solving equation (2) and (3) we get,
$\therefore \,\,\,u_2=5mm\hspace{2cm}u_3-30mm$
Substituting above values in equation (1), we get,
$-100*5-50*20=F_1$
$\therefore \,\,\,F_1=-2000\,N$

f) Verification
$\sum F_x=F_1+F_2+F_3=-2000+0+2000=0$
Hence verified.