In 1678, Robert Hooke set down the basis for modern finite element stress analysis with Hooke's law. An elastic body extends or compresses in proportion to the force acting on it (or the stress in it). Mathematically:
F=kx
Hooke proved the equation by using weights to stretch wires hanging from the ceiling.
Imagine that a coffee cup is sitting on a table. It is broken down into 2,000 little tetrahedral elements. Each element has four corners, or nodes. All nodes on the bottom of the coffee cup are fixed (all translations are constrained), so they cannot move. Press down on just one node near the top of the cup.
That one node moves a little because all materials have some elasticity. F = kx describes the movement for that element, except that other elements are in the way. In fact, as the force is transmitted through the first element, it spreads out to other nodes.
In the finite element method, an element stiffness formulation step occurs. A stiffness (k) is created for the relationship between every node on each element. Every node is connected to every other node on the element by a spring. It behaves according to Hooke's law. We reduce the coffee cup to a large system of springs. A value for the translation (x) and force (F) is determined for each node by the formula F = kx.
The individual equations are assembled into a matrix and solved simultaneously through various numerical methods. The results are the relative nodal displacements throughout the model, the material strain that the displacements represent, and the resultant stress.
Stress determination is possible because the force at each node and the geometry and stiffness of all elements are known. The raw stress data is then resolved into the following values to facilitate results evaluation: