In this lecture, we'll cover:

• Deriving the two fundamental finite element equations: the element stiffness matrix and the element force vector
• Applying the Principle of Minimum Potential Energy (PMPE) to a 3D finite element
• Defining configurations in terms of nodal displacements and degrees of freedom
• Constructing expressions for strain energy and work done by applied loads
• Formulating the total potential energy expression as a precursor to differentiation

In this lecture, we begin the derivation of the two key finite element equations that will allow us to compute the element stiffness matrix and the equivalent force vector for a general element. We base the derivation on the Principle of Minimum Potential Energy, which states that, of all possible displacement configurations, the true configuration minimises the total potential energy. We interpret a configuration as a set of nodal displacements, and we illustrate this using a three-dimensional, eight-node element with 24 translational degrees of freedom.

We then construct the total potential energy expression by defining it as the strain energy stored in the element minus the work done by applied loads. We derive the general integral form of the strain energy in terms of stress and strain, introduce the constitutive (material) matrix to express stress in terms of strain, and formulate the work done by both body forces and surface forces. Bringing these components together gives us an expression for total potential energy in terms of general displacements within the element. We conclude by recognising that, in the next lecture, we must express this energy purely in terms of nodal displacements so that we can differentiate it, set the result equal to zero, and obtain the governing finite element equations.

Next up:

In the next lecture, we complete the derivation by introducing the displacement interpolation and strain–displacement matrices, and arrive at the general finite element equilibrium equation.