In this lecture, we'll cover:

• Calculating member forces
• Interpreting tension and compression from the sign of the force
• Handling members aligned with the global axis versus inclined members
• Using a transformation matrix to convert global nodal displacements to local coordinates
• Reviewing the full set of steps in the direct stiffness method for truss analysis

In this lecture, we focus on determining the internal member forces of a truss using the direct stiffness method. We begin by applying the fundamental relationship $\{\mathbf{F}\} = [\mathbf{K}] \{\mathbf{U}\}$, rooted in Hooke’s law. For a member aligned with the global reference frame, we directly use the global nodal displacements to calculate the change in length and hence the axial force, interpreting the sign to distinguish between compression and tension.

We then consider an inclined member, where the nodal displacements are initially known in the global coordinate system but are required in the member’s local coordinate system. To achieve this, we use a transformation matrix to convert global displacements into local axial displacements before applying the same force–displacement relationship. With this, we determine the internal force and again interpret its sign physically.

Finally, we step back and review the complete procedure of the direct stiffness method for truss analysis. We see how individual member stiffness equations form element stiffness matrices, how these assemble into the global (primary) stiffness matrix, how boundary conditions are applied to obtain the reduced structure stiffness matrix, and how solving the resulting system of linear equations yields nodal displacements, reactions, and member forces. We emphasise that the method is systematic and repeatable, making it well suited for implementation in a programming-based approach.

Next up:

In the next lecture, we begin Section 5, where we translate the hand-calculated solution into Python code, taking our first step towards a computational truss solver.