2D Truss Element Global stiffness matrix

(CS stands for Co-ordinate System)

observing the point p in the global and local CS
we can use the transformation matrix, $T$ and its inverse/transpose ( $T^T = T^{-1}$ ) to work out the local x and y co-ordinates from the global and the other way round

to get the global stiffness matrix for a single truss element, we need to expand our translation matrix to handle both x and y forces for both nodes of the element
←first we had this for a 1D element
now for 2D element, we need to handle displacements and forces in the y direction for both nodes meaning we need to have a 4x4 stiffness matrix
since we are dealing with a truss element:
- we assume any forces in the local y direction do not affect tension in the element
this means that we have the second and fourth row and columns as all zeros
we take the values from the 1D stiffness matrix

now we have the thee equations on the left
- hatted variables for local and non-hatted for global
we can then form the equation on the right and finally using the orthoganal matrix property:

with $C = sin(\theta)$ , $S = cos(\theta)$, and $\theta$ = the angle between local and global CS