In this lecture, we'll cover:

• Why element stiffness matrices must be expressed in a common global reference frame before assembly
• The relationship between local and global coordinate systems for truss (bar) elements
• Derivation of the displacement transformation matrix using simple geometry
• Transformation of forces between local and global reference frames
• Construction of the element stiffness matrix in global coordinates

In this lecture, we focus on the essential step of transforming element quantities from their local coordinate systems into a common global reference frame before assembling the overall structure stiffness matrix. We begin by considering a simple truss in which each member has its own local axes, while the structure as a whole is defined in a global $(x_g, y_g)$ system. Through a geometric argument, we derive the transformation relationship between local and global coordinates, introducing the angle $\theta$ between the two systems and expressing the transformation in matrix form.

We then extend this idea to both displacements and forces, showing that displacements transform using a transformation matrix $[\mathbf{T}]$, while forces transform using its transpose. By combining these relationships with the familiar local force–displacement equation for a bar element, we derive the key result: the global element stiffness matrix is obtained from the local stiffness matrix via,

We see that this global stiffness matrix incorporates the member’s orientation through sine and cosine terms, making it suitable for assembling into the overall structural stiffness matrix in the direct stiffness method.

Next up:

With the theoretical tools now in place, the next lecture introduces Section 4, where we apply the direct stiffness method to a complete truss example for the first time.