Finite element basics

The finite element method breaks a structure into small pieces, the elements, joined at nodes. Within each element the displacement is assumed to follow a simple shape, and from that assumption the element's stiffness matrix is derived. Assembling all the element matrices gives a global stiffness matrix K linking nodal forces F to nodal displacements u, so the whole analysis reduces to solving K u = F.

The simplest element is a bar carrying axial force, with stiffness A E over L and two degrees of freedom. Beam elements add rotations and bending stiffness; plate and shell elements model slabs, walls and tanks; solid elements model foundations, dams and soil. Boundary conditions are applied by fixing the restrained degrees of freedom, and loads by placing forces on nodes, with distributed loads converted to equivalent nodal loads.

Accuracy depends on the element type and the mesh. A coarse mesh with simple elements is usually too stiff, giving deflections that are too small. Refining the mesh, or using higher-order elements, makes results converge towards the true solution; a mesh convergence check, re-running with a finer mesh until key results stop changing, is basic good practice. Stress concentrations at re-entrant corners and point supports never fully converge and must be interpreted, not read blindly.

Software makes finite element analysis routine, which makes checking more important, not less. Equilibrium of reactions against applied loads, a hand estimate of the main moment or deflection, and inspection of the deformed shape catch most modelling errors. Units, local axes, release conditions and load directions are the commonest sources of mistakes.