$K = frac{GJ}{L} \begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix}\\ k_1 = \frac{100 \times 10^3 \times \frac{\pi}{32} \times 100^4}{450} \begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix} = 10^6 \begin{bmatrix} \ 2181.7 & -2181.7 \\ \ -2181.7 & 2181.7 \\ \end{bmatrix}\\ k_2 = \frac{100 \times 10^3 \times \frac{\pi}{32} \times 80^4}{400} \begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix} = 10^6 \begin{bmatrix} \ 1005.3 & -1005.3 \\ \ -1005.3 & 1005.3 \\ \end{bmatrix}\\ k_3 = \frac{100 \times 10^3 \times \frac{\pi}{32} \times 50^4}{500} \begin{bmatrix} \ 1 & -1 \\ \ -1 & 1 \\ \end{bmatrix} = 10^6 \begin{bmatrix} \ 122.7 & -122.7 \\ \ -122.7 & 122.7 \\ \end{bmatrix}$

Global matrix equation:

$10^6 \begin{bmatrix} \ 2181.7 & -2181.7 & 0 & 0 \\ \ -2181.7 & 3187 & -1005.3 & 0 \\ \ 0 & -1005.3 & 1128 & -122.7 \\ \ 0 & 0 & -122.7 & 122.7 \\ \end{bmatrix} \begin{Bmatrix} \ \theta_1 \\ \ \theta_2 \\ \ \theta_3 \\ \ \theta_4 \\ \end{Bmatrix} \begin{Bmatrix} \ T_1 \\ T_2 \\ T_3 \\ T_4 \\ \end{Bmatrix}$

B.C:- $\theta_1 = \theta_4 = 0\\ T_2 = 3 \times 10^6 N-mm, T_3 = -2 \times 10^6 N-mm$

$10^6 \begin{bmatrix} \ 2181.7 & -2181.7 & 0 & 0 \\ \ -2181.7 & 3187 & -1005.3 & 0 \\ \ 0 & -1005.3 & 1128 & -122.7 \\ \ 0 & 0 & -122.7 & 122.7 \\ \end{bmatrix} \begin{Bmatrix} \ \theta \\ \ \theta_2 \\ \ \theta_3 \\ \ 0 \\ \end{Bmatrix} \begin{Bmatrix} \ T_1 \\ 3 \\ -2 \\ T_4 \\ \end{Bmatrix}$

$-218.7 \theta_2 = T_1\\ 3187 \theta_2 - 1005.3 \theta_3 = 3\\ -1005.3 \theta_2 + 1128 \theta_3 = -2\\ -122.7 \theta_3 = T_4$

$\theta_2 = 5.314 \times 10^{-4} rad\\ \theta_3 = -1.3 \times 10^{-3} rad\\ T_1 = -1.159 KN.m\\ T_4 = 0.159 KN.m\\ \sum T = -1.159 + 3 - 2 + 0.159 = 0$