In this lecture, we'll cover:

• Solving for nodal displacements by reducing the global force vector and applying the inverse stiffness matrix
• Programmatically reconstructing the full global displacement vector including restrained degrees of freedom
• Computing reaction forces using the primary stiffness matrix
• Calculating axial member forces using transformation matrices and a loop over all members

In this lecture, we focus on completing the solution phase of our structural analysis. We begin by solving for the unknown nodal displacements, carefully reducing the global force vector to remove entries associated with restrained degrees of freedom and ensuring the vector has the correct orientation. We then perform the matrix multiplication with the inverse structure stiffness matrix to obtain the reduced displacement vector.

Next, we rebuild the full global displacement vector programmatically. Instead of manually inserting zero displacements at restrained degrees of freedom, we use a loop to step through all degrees of freedom, inserting zeros where restraints exist and sequentially placing the calculated displacement values elsewhere. With this reconstructed global displacement vector, we compute the reaction forces by multiplying it by the primary stiffness matrix.

Finally, we calculate the axial forces in each member. Using a loop over all members, we extract geometric properties, form the transformation matrix, convert global displacements to local coordinates, and apply the standard axial force relationship. By structuring the process programmatically, we ensure the procedure can be applied efficiently to every member in the structure.

Next up:

In the next lecture, we automate the plotting of undeformed and deformed shapes so that a visualisation is generated directly from the input data.