In this lecture, we'll cover:

• How to express the element potential energy in terms of nodal displacements rather than inter-nodal displacements
• The introduction and role of the displacement interpolation matrix $[\mathbf{H}]$, including its body $[\mathbf{H}_V]$ and surface $[\mathbf{H}_S]$ forms
• The introduction of the strain–displacement matrix $[\mathbf{B}]$ and how it relates strain to nodal displacements
• How to substitute these matrices into the potential energy expression
• How to differentiate the potential energy to obtain the finite element equilibrium equations
• The derivation of the general 3D element stiffness matrix $[\mathbf{K}_e]$ and force vector $[\mathbf{F}_e ]$

In this lecture, we focus on reformulating the element potential energy so that it is written entirely in terms of nodal displacements. To achieve this, we introduce two key matrices: the displacement interpolation matrix $[\mathbf{H}]$, which allows us to express displacement at any point in the element as a function of nodal displacements, and the strain–displacement matrix $[\mathbf{B}]$, which relates strain at any point to those same nodal displacements. By making systematic substitutions using these matrices, we convert the potential energy expression into a form that depends only on the nodal displacement vector.

We then differentiate the potential energy with respect to the nodal displacements and set the result equal to zero, applying the principle of minimum potential energy. Although the expression initially appears complex, we see that the differentiation reduces to a straightforward quadratic form. This process leads directly to the standard finite element equilibrium equation,

from which we identify the general 3D element stiffness matrix

and the corresponding force vector. These are the fundamental finite element equations, forming the basis for deriving stiffness matrices for specific elements in subsequent lectures.

Next up:

In the next lecture, we apply these general finite element equations to the specific case of a bar element, recovering the familiar stiffness matrix and grounding the theory in a concrete example.