In this lecture, we'll cover:

• How to break each element global stiffness matrix into four 2$\times$2 quadrants
• How to create a primary stiffness matrix template based on the number of nodes
• How each template entry represents a 2$\times$2 submatrix corresponding to an element quadrant
• How to map element global stiffness matrix quadrants into the correct locations in the primary stiffness matrix
• How overlapping contributions arise at shared nodes

In this lecture, we focus on how to construct the primary stiffness matrix by assembling the global stiffness matrices of individual elements. We begin by splitting each element’s global stiffness matrix into four 2$\times$2 quadrants, $[\mathbf{K}_{11}]$, $[\mathbf{K}_{12}]$, $[\mathbf{K}_{21}]$, $[\mathbf{K}_{22}]$. We then create a primary stiffness matrix template based on the number of nodes in the structure. Although the template is arranged according to nodes (for example, a 3$\times$3 layout for three nodes), each entry actually represents a 2$\times$2 submatrix, reflecting the degrees of freedom at each node. This means the completed primary stiffness matrix expands to the expected size based on the total number of degrees of freedom.

We then see how to systematically insert each quadrant of an element’s global stiffness matrix into the correct position within the template. The key idea is that the element’s start and end nodes determine which rows and columns of the primary stiffness matrix are involved. By identifying the intersection of the relevant node numbers, we can place each quadrant precisely. When multiple elements share a node, their contributions overlap in the corresponding region of the primary stiffness matrix, reflecting the physical interaction at that node. This structured mapping process is the essential takeaway from the lecture and underpins the full assembly of the structure’s stiffness matrix.

Next up:

In the next lecture, we impose boundary conditions on the primary stiffness matrix to obtain the reduced structure stiffness matrix, ready for solving.