1)
Stiffness matrix for element 1 same as element 2
[ k ] 1 o r 2 = ∣ 1000 − 1000 − 1000 1000 ∣ N / m m [k]^{1or2}=\begin{vmatrix}
1000 & -1000 \\
-1000 & 1000
\end{vmatrix}N/mm [ k ] 1 or 2 = ∣ ∣ 1000 − 1000 − 1000 1000 ∣ ∣ N / mm
Stiffness matrix for element 3
[ k ] 3 = ∣ 500 − 500 − 500 500 ∣ N / m m [k]^3=\begin{vmatrix}
500 & -500 \\
-500 & 500
\end{vmatrix}N/mm [ k ] 3 = ∣ ∣ 500 − 500 − 500 500 ∣ ∣ N / mm
Stiffness matrix for element 4
[ k ] 1 = ∣ 200 − 200 − 200 200 ∣ N / m m [k]^1=\begin{vmatrix}
200 & -200 \\
-200 & 200
\end{vmatrix}N/mm [ k ] 1 = ∣ ∣ 200 − 200 − 200 200 ∣ ∣ N / mm
Global stiffness matrix
[ k ] = ∣ 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 ∣ N / m m [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 [ k ] = ∣ ∣ 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 ∣ ∣ N / mm
Let u 1 , u 2 , u 3 , u 4 , u 5 u_1,u_2,u_3,u_4,u_5 u 1 , u 2 , u 3 , u 4 , u 5 be the displacement at the nodes, then the stiffness equation becomes;
∣ 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 ∣ ∣ u 1 u 2 u 3 u 4 u 5 ∣ \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} ∣ ∣ 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 ∣ ∣ ∣ ∣ u 1 u 2 u 3 u 4 u 5 ∣ ∣
= ∣ R 1 500 0 300 0 ∣ =\begin{vmatrix}
R_1 \\
500 \\
0 \\
300 \\
0
\end{vmatrix} = ∣ ∣ R 1 500 0 300 0 ∣ ∣
2)
∣ − 1000 2000 − 1000 0 − 1000 1500 0 0 − 500 ∣ ∣ u 2 u 3 u 4 ∣ = ∣ 500 0 300 ∣ \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} ∣ ∣ − 1000 0 0 2000 − 1000 0 − 1000 1500 − 500 ∣ ∣ ∣ ∣ u 2 u 3 u 4 ∣ ∣ = ∣ ∣ 500 0 300 ∣ ∣
u 2 = 1.7 m m u_2=1.7mm u 2 = 1.7 mm
u 3 = 0.9 m m u_3=0.9mm u 3 = 0.9 mm
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