In this lecture, we'll cover the following:

• Applying the principle of virtual work to a finite element
• Deriving the element stiffness matrix from internal and external virtual work
• Formulating internal virtual work using strain and stress relationships
• Expressing external virtual work from distributed and nodal loads
• Converting distributed loads into an equivalent nodal force vector

In this lecture, we focus on uncovering the origin of the element stiffness matrix and the equivalent nodal force vector by applying the principle of virtual work. We begin by equating internal and external virtual work, constructing each expression in turn. The internal virtual work is formulated as an integral of virtual strains and real stresses over the element volume, which we then transform into an area-based expression using generalised strains and stress resultants. Along the way, we make key substitutions involving transformation matrices and relationships between strains, stresses, and stress resultants.

We then turn to the external virtual work, separating contributions from distributed loads and point loads at nodes. By introducing shape functions and the strain–displacement matrix, we rewrite all terms in terms of nodal displacements. This allows us to systematically reduce the full virtual work expression into the familiar finite element form, where the element stiffness matrix emerges naturally as an integral of $\bf{B}^T \bf{D} \bf{B}$, and the equivalent nodal force vector arises from distributed loading. In doing so, we see clearly how the standard equation relating stiffness, displacement, and force is fundamentally rooted in virtual work principles.

Next up

Next, we will introduce Gauss quadrature, the numerical integration technique used to evaluate the stiffness matrix integrals in practice.