In this lecture, we'll cover:

• How to reduce the primary stiffness matrix to the structure stiffness matrix
• How to impose boundary conditions by modifying the stiffness matrix and force vector
• How known zero displacements are enforced using rows and columns of the matrix
• Why placing ones and zeros in specific locations is logically consistent with the governing equations

In this lecture, we focus on reducing the primary stiffness matrix to the structure stiffness matrix by imposing the boundary conditions. We begin with the full force–displacement relationship for a six-degree-of-freedom system and identify which displacements and reactions are present. We then determine which degrees of freedom are constrained to zero due to supports and explain how these known displacements are incorporated directly into the system of equations.

We see that boundary conditions are imposed by placing a 1 on the diagonal of the stiffness matrix corresponding to each known zero displacement, setting all other entries in the associated rows and columns to 0, and inserting 0 in the corresponding positions of the force vector. We explore why this procedure works: zeroing the rows enforces the prescribed displacements explicitly (e.g. ensuring a displacement equals zero), while zeroing the columns prevents these constrained degrees of freedom from influencing other equilibrium equations. By working through specific rows of the modified system, we clarify that this process is not arbitrary but a logical and consistent way to enforce supports and reduce the system to a solvable form.

Next up:

In the next lecture, we solve the reduced system of equations to determine the unknown nodal displacements.