Question

In the spring system as shown in the figure below, each spring constant is ki=k2=1000 N/mm, k3=500 N/mm, k4=200 N/mm, and the applied load is F2=500 N, F300 N. (1) Find the stiffness equation of the spring system (KU=F). (2) Find the displacement of node 2 and node 3

Answer 1

1)

Stiffness matrix for element 1 same as element 2

$[k]^{1or2}=\begin{vmatrix} 1000 & -1000 \\ -1000 & 1000 \end{vmatrix}N/mm$

Stiffness matrix for element 3

$[k]^3=\begin{vmatrix} 500 & -500 \\ -500 & 500 \end{vmatrix}N/mm$

Stiffness matrix for element 4

$[k]^1=\begin{vmatrix} 200 & -200 \\ -200 & 200 \end{vmatrix}N/mm$

Global stiffness matrix

$[k]=\begin{vmatrix} 1000 & -1000 & 0 & 0 & 0\\ -1000 & 2000 & -1000 & 0 & 0\\ 0 & -1000 & 1500 & 500 & 0\\ 0 & 0 & -500 & 700 & -200\\ 0 & 0 & 0 & -200 & 200 \end{vmatrix}N/mm$

Let $u_1,u_2,u_3,u_4,u_5$ be the displacement at the nodes, then the stiffness equation becomes;

$\begin{vmatrix} 1000 & -1000 & 0 & 0 & 0\\ -1000 & 2000 & -1000 & 0 & 0\\ 0 & -1000 & 1500 & 500 & 0\\ 0 & 0 & -500 & 700 & -200\\ 0 & 0 & 0 & -200 & 200 \end{vmatrix}\begin{vmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ \end{vmatrix}$

$=\begin{vmatrix} R_1 \\ 500 \\ 0 \\ 300 \\ 0 \end{vmatrix}$

2)

$\begin{vmatrix} -1000 & 2000 & -1000 \\ 0 & -1000 & 1500 \\ 0 & 0 & -500 \\ \end{vmatrix}\begin{vmatrix} u_2 \\ u_3 \\ u_4 \\ \end{vmatrix}=\begin{vmatrix} 500 \\ 0 \\ 300 \end{vmatrix}$

$u_2=1.7mm$

$u_3=0.9mm$