In this lecture, we'll cover:

• How to reduce the primary stiffness matrix to the structure stiffness matrix
• How to impose known (restrained) displacements using the standard modification procedure
• How to identify restrained degrees of freedom from support conditions
• How to form the reduced stiffness matrix by removing modified rows and columns

In this lecture, we focus on reducing the full force–displacement relationship down to the structure stiffness matrix by correctly applying boundary conditions. We begin with the primary stiffness matrix and identify the restrained degrees of freedom associated with the supports. In this case, we determine which global degrees of freedom correspond to pinned supports and therefore have known (zero) displacements.

We then apply the standard modification procedure: placing a 1 on the diagonal entry corresponding to each restrained degree of freedom, setting all other terms in the same row and column to 0, and adjusting the corresponding entries in the force vector. This process effectively imposes the known displacements into the system of simultaneous equations. Finally, we consider two equivalent representations: retaining the full modified matrix with the inserted 1s and 0s, or reducing it by removing the associated rows and columns to obtain a smaller structure stiffness matrix. In doing so, we move from the original 12 $\times$ 12 system to a reduced 8 $\times$ 8 system, ready for solution.

Next up:

In the next lecture, we solve the reduced system for the unknown nodal displacements of the 8-bar truss.