Solve Prof. Sennet's Problem 1.1 via the "Direct Method," i.e., column by column. Assign [tex]E = 29000 \text{ ksi}[/tex] and [tex]A = 2 \text{ in}^2[/tex] to each element. Use the Python manner of ordering items, i.e., start at zero.

1. Find the stiffness matrix [tex][K][/tex].
2. Attempt to invert it.
3. Note that [tex]F_0[/tex] is zero in Problem 1.1.

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The inverse of the matrix K can be found as follows:$$\begin{bmatrix} 48333.33 & -48333.33 & 0 & 0\\-48333.33 & 96666.67 & -48333.33 & 0\\0 & -48333.33 & 96666.67 & -48333.33\\0 & 0 & -48333.33 & 48333.33 \end{bmatrix}^{-1} = \frac{1}{2.667\times 10^{12}}\begin{bmatrix} 3 & 4 & 2 & 0\\4 & 8 & 4 & 0\\2 & 4 & 8 & 4\\0 & 0 & 4 & 3 \end{bmatrix}$$.

Given data: E = 29000 ksi A = 2 in 2 Solve Prof. Sennet's Problem 1.1 via the so-called "Direct Method", i.e. column by column:

\begin{matrix} 2 & -1 & 0 & 0 \\-1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 1 \end{matrix}

Here, each element is assigned with E = 29000 ksi and A = 2 in2We need to find the stiffness matrix [K].

The Direct method is a matrix inversion method. We can solve the system of equations by column-by-column inversion.

The formula for the stiffness matrix is as follows:

K = \frac{EA}{L} \begin{bmatrix}1 & -1 & 0 & 0\\-1 & 2 & -1 & 0\\0 & -1 & 2 & -1\\0 & 0 & -1 & 1\end{bmatrix}

Substituting the given values, we get the stiffness matrix as:

K = \frac{(29000 ksi)(2 in^2)}{1.2\times 10^3 in} \begin{bmatrix}1 & -1 & 0 & 0\\-1 & 2 & -1 & 0\\0 & -1 & 2 & -1\\0 & 0 & -1 & 1\end{bmatrix} = \begin{bmatrix} 48333.33 & -48333.33 & 0 & 0\\-48333.33 & 96666.67 & -48333.33 & 0\\0 & -48333.33 & 96666.67 & -48333.33\\0 & 0 & -48333.33 & 48333.33 \end{bmatrix}

Next, we need to invert the matrix.

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