In this lecture, we'll cover the following:

• What shape functions are and why they are fundamental in finite element analysis.
• How shape functions act as interpolation (weighting) functions between nodal values.
• Use of natural (local) coordinates $(r, s)$ in a four-node quadrilateral element.
• Formulation of displacement within an element as a weighted sum of nodal values.
• Matrix representation of shape functions and nodal quantities.

In this lecture, we explore the concept of shape functions and see that they are simply interpolation tools used to estimate field variables within an element from known nodal values. We focus on a four-node quadrilateral element defined in natural coordinates, where each node has an associated displacement. By introducing shape functions as weighting factors dependent on these local coordinates, we show how any displacement within the element can be expressed as a weighted sum of the nodal displacements. A key property is demonstrated: at a given node, its corresponding shape function evaluates to one while all others evaluate to zero, ensuring exact recovery of nodal values.

We then extend this idea beyond a single displacement component to include multiple degrees of freedom, such as rotations, and express the formulation in a compact matrix form. This highlights how shape functions are systematically applied across all nodal quantities within an element. Overall, we see that shape functions provide a simple yet powerful framework for interpolating values within an element, forming the foundation for subsequent development of the strain–displacement matrix.

Next up

Building on these shape functions, the next lecture derives the strain–displacement matrix B, which formally links nodal displacements to element strains.