In this section, we'll cover:

• How to construct the global stiffness matrix for each element using a transformation matrix
• How to assemble the primary stiffness matrix for the entire structure
• How to impose support (boundary) conditions to obtain the reduced structure stiffness matrix
• How to solve for nodal displacements, reaction forces and member forces in a step-by-step procedure
• Why the direct stiffness method is well suited to computational implementation

In this lecture, we set out a clear roadmap of the direct stiffness method as it will be applied throughout the remainder of the section. We begin by forming the global stiffness matrix for each element, transforming the local 4×4 element stiffness matrix into the global coordinate system using a transformation matrix that accounts for member orientation. We then assemble these element matrices into the primary stiffness matrix for the whole structure, whose size is determined by the total number of degrees of freedom (two per node for a pin-jointed structure).

We then impose support (boundary) conditions on the primary stiffness matrix to obtain the reduced structure stiffness matrix. By solving the system, $\{\mathbf{F}\} = [\mathbf{K}] \{\mathbf{U}\}$, we determine the unknown nodal displacements, recognising that inverting the stiffness matrix is the computationally intensive step and therefore well suited to a computer-based approach. With the nodal displacements known, we calculate reaction forces and finally determine member forces using the fundamental relationship that force equals stiffness times displacement. Overall, we see that the direct stiffness method is a structured, formula-driven procedure that systematically delivers displacements, reactions and internal member forces.

Next up:

In the next lecture, we begin the worked example by calculating the element stiffness matrices, starting with node and member labelling and determining orientation angles.