Free body of a spring element

• we are initially fucking with a a spring (element) being stretched out with a scalar force (tension) as seen in the left diagram below.
• then we need to understand the

https://www.youtube.com/watch?v=3AB1sHkCb3I (link for the above ^)

• in the next derivation we will use $\hat{f}{1x}^{1}$ and $\hat{f}{2x}^{1}$ instead of $f_1$ and $f_2$ so that we can later express the forces in the y axis for each node
• since k is a scalar value, we needed to define a sort of “translation” matrix to give correct directions for forces and displacements to interact
• it is obvious that this matrix is $\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$

Displacement fuction

• the directions of $\hat{f}{1x}^{1}$ and $\hat{f}{2x}^{1}$ have swapped. This should make sense after the first video above using the equilibrium law
• the displacement function is: assumed solution for the displacements of the nodes in the system. the simplest solution is polynomial. the order of the polynomial is determined by the number of DOFs
• rule of thumb: order of the polynomial = nb DOFs -1

• to find $c_1$ and $c_2$, we use the boundary conditions:

$N_1$ and $N_2$ would be our shape functions