In this lecture, we'll cover:

• How to label nodes and members in preparation for stiffness matrix formulation
• How to define and interpret local and global degrees of freedom
• How to determine the orientation angle of a member
• How to use the orientation angle to construct each element’s global stiffness matrix

In this lecture, we focus on determining the element stiffness matrices for a simple structure. We begin by carefully labelling the nodes and members, as this forms the foundation for everything that follows. We then assign degrees of freedom at each node with respect to a common global reference frame. For each member, we distinguish clearly between local and global coordinate systems, noting how the local axes are aligned along and perpendicular to the member, while the global axes remain fixed.

We then concentrate on the crucial step of finding each member’s angle of orientation. We place node i at the origin of the global coordinate system and measure the angle anti-clockwise from the positive global x-axis to the member’s local x-axis. We see that when the local and global axes coincide, the angle is zero, but for inclined members we must determine the angle from the structure’s geometry. Once this orientation angle is known, we substitute it into the standard formulation to obtain the element’s global stiffness matrix. The key takeaway is that consistent measurement of this angle for every member enables us to compute the correct global stiffness matrix for each element in the structure.

Next up:

In the next lecture, we take the individual element stiffness matrices and assemble them into the primary stiffness matrix for the whole structure.