In this lecture, we will cover the following:

• How to reduce the global force vector and solve for the nodal displacements
• How to reconstruct the full global displacement vector by reinserting zeros at restrained degrees of freedom
• How to calculate the reactions and confirm vertical force equilibrium for the structure
• How to extract the maximum displacement in each direction and identify the node at which it occurs

In this lecture we finally solve our system of equations. We start by taking a deep copy of the force vector and deleting the rows that correspond to restrained degrees of freedom, so that its size matches that of the reduced structure stiffness matrix. We then invert that stiffness matrix and multiply it by the reduced force vector to obtain the vector of unknown displacements.

To prepare for calculating reactions, we build a full-size global displacement vector by stepping through every degree of freedom and inserting either a zero (if it is restrained) or the next calculated displacement. Multiplying the primary stiffness matrix by this reconstructed global displacement vector gives us a global force vector containing the reactions, although we also need to add back in the self-weight contributions for elements directly connected to restrained nodes, since those rows were deleted earlier.

To check our work, we compute the total applied vertical load by summing the $z$ components of the equivalent nodal forces, and compare it with the sum of the vertical reactions extracted from the global force vector. Confirming vertical force equilibrium giving us strong confidence that everything is functioning correctly.

We then extract the $u$, $v$ and $w$ components of displacement for every node, calculate the resultant magnitude and identify the maximum in each direction along with the node and coordinates at which it occurs.

Next up

In the next lecture, we will turn these raw numerical results into a plot we can more easily interpret.