In this lecture, we'll cover:

• How to use the reduced force–displacement relationship after applying boundary conditions
• How to identify and solve for unknown nodal displacements
• How to invert the structure stiffness matrix and apply the force vector
• How external nodal forces are incorporated into the solution

In this lecture, we focus on solving for the unknown nodal displacements once the boundary conditions have been applied and the primary stiffness matrix has been reduced to the structure stiffness matrix. We begin with the force–displacement relationship and show how, after imposing the supports, the system reduces to two equations in two unknowns — the horizontal and vertical displacements at node two. This reduction makes the problem directly solvable.

We then solve for the unknown displacements by inverting the structure stiffness matrix and multiplying it by the corresponding force vector. We carefully substitute the known external forces, noting that the horizontal force at node two is zero and the vertical force is −150 kN. We also observe that material properties such as Young’s modulus and cross-sectional area can be left symbolic to simplify hand calculations. Ultimately, we see that once the stiffness matrix has been correctly assembled and reduced, the remaining steps are straightforward matrix operations, reinforcing that the most demanding part of the process lies in constructing the primary stiffness matrix in the first place.

Next up:

In the next lecture, we use the computed displacements and the primary stiffness matrix to determine the unknown support reactions.