In this lecture, we'll cover:

• Applying the previously derived finite element stiffness equation to a simple bar element
• Constructing the displacement interpolation matrix for a two-degree-of-freedom bar
• Deriving the strain–displacement matrix from the interpolation functions
• Substituting into the general stiffness matrix expression and recovering the familiar bar stiffness matrix
• Placing the recent theory (principle of minimum potential energy and element equations) into practical context

In this lecture, we apply the general finite element stiffness matrix equation derived previously to a simple bar element that we already understand well. We begin by constructing the displacement interpolation matrix for a bar with two degrees of freedom, expressing the axial displacement as a linear polynomial with two unknowns. By enforcing the boundary conditions at each node, we determine the interpolation functions and write the displacement field in matrix form. We then obtain the strain–displacement matrix by differentiating the interpolation matrix with respect to position, recalling that axial strain is the derivative of displacement.

With the strain–displacement matrix and the material matrix (Young’s modulus for uniaxial stress) defined, we substitute these into the general stiffness matrix expression derived from the principle of minimum potential energy. By evaluating the integral over the element volume, we recover the familiar bar stiffness matrix,

In doing so, we reinforce how the abstract finite element formulation connects directly to results we already know, and we see that the key effort lies in constructing the displacement interpolation matrix, from which the remainder of the formulation follows systematically.

Next up:

In the next lecture, we consolidate the full derivation procedure with a step-by-step review and preview the need for coordinate transformations when analysing multi-member structures.