Last reviewed 30 Sept 2026 · 6 min read
Mesh: how fine, how regular
The mesh is the division of the structure into elements. It is a compromise between accuracy (fine mesh) and cost (coarse mesh). Some principles:
- Mesh where the answer changes quickly — near supports, point loads, openings, re-entrant corners, connections, cracks and anchorages — with larger elements where stresses vary slowly.
- Element size to capture the behaviour: at least 3–4 elements through the thickness of a solid for bending; a few elements per half wavelength of a mode in dynamics; enough along a girder to capture the moving-load position.
- Element quality: avoid long, thin (high aspect ratio) or badly skewed elements; quadrilaterals should be near-rectangular, triangles near-equilateral; aspect ratio ideally under 3 and no more than about 5–10.
- Smooth transition between coarse and fine regions (no abrupt jumps of size, and the connection of different element types must respect the degrees of freedom — a shell to a beam, for instance, needs rigid links).
- Mesh compatibility: adjacent elements share nodes; free (duplicate) nodes leave a crack in the model.
Refine the mesh at least twice and plot the quantity of interest (deflection, stress at a point) against the element size. When further refinement changes the result by less than a few per cent, the mesh has converged. Stress at a singularity (a point load, a sharp re-entrant corner) does not converge — it keeps rising with refinement — so read stresses at a small distance away.
Use symmetry
If both geometry and loading are symmetric, model one half or a quarter with symmetry boundary conditions (no normal displacement, no rotation about the plane's axes). It cuts the model size and the time; for anti-symmetric loading the opposite conditions apply.
Boundary conditions
Boundary conditions are the supports and constraints that stop the structure moving as a rigid body and represent its real restraints.
- Essential (geometric, Dirichlet) conditions — prescribed displacements or rotations (fixed, pinned, roller). They are imposed on the displacement unknowns.
- Natural (Neumann) conditions — prescribed forces or tractions; they arise from the loading and are satisfied automatically in the solution.
| Support | Restrained |
|---|---|
| Fixed | All translations and rotations |
| Pinned | Translations, free to rotate |
| Roller | One translation (normal to the surface) |
| Spring | Elastic — stiffness in each direction |
| Symmetry plane | Translation normal to the plane and rotations about axes in the plane |
Real supports are rarely fully fixed or fully free. Use springs for:
- Bearings — with the actual stiffness (vertical, and horizontal for elastomeric bearings, so that the piers share the longitudinal force properly);
- Foundations and soil — vertical and horizontal spring constants (Winkler springs) for soil–structure interaction, especially for pile and well foundations and seismic analysis;
- Partially fixed connections.
Rigid links and constraints
- Rigid link (master–slave) connects two nodes so that they move together — for example a bearing offset between the girder centroid and the pier cap, or a shear connector between deck and girder.
- Constraint equations tie the degrees of freedom of different nodes (equal displacement, diaphragm behaviour).
- Rigid diaphragms in buildings force floor nodes to move together in-plane.
Avoiding an unstable model
A structure must have no rigid-body motion left (no mechanism). Warning signs are singular matrix, huge displacements, or "zero pivot". Common causes: a missing support, a hinge that makes a mechanism, an unconnected node, or contact that has not closed. Check the number of restraints against the degrees of freedom.
Loads
| Load | How it is applied |
|---|---|
| Concentrated (point) load | At a node — or between nodes, converted to equivalent nodal loads |
| Distributed line / area load | Element load; converted to nodal loads by the program |
| Self-weight | Gravity load on each element |
| Temperature | Uniform temperature change and gradient — as thermal strain |
| Prestress | Equivalent loads from tendon forces, or as tendon elements |
| Moving loads | Vehicle trains moved along lanes; influence lines / surfaces |
| Wind, seismic | Static equivalent loads or response spectra / time histories |
| Support settlement | Prescribed displacement at a support |
Equivalent (consistent) nodal loads
A distributed load acting on an element is replaced by nodal loads that do the same work on the shape functions. For a beam element of length under uniform load (the "fixed-end" forces):
For an axial bar with uniform load : at each end. These consistent loads for a cubic beam element give the exact nodal displacements for a uniform load; a lumped distribution (half the total load to each node) is used for a mass matrix and in simple codes.
A continuous girder is modelled with 2 m long beam elements. A uniform load = 30 kN/m on one element gives fixed-end forces
at each node and (opposite senses at the two ends).
These are added to the global load vector, after which the global equations are solved and the fixed-end forces are added back to the member forces.
Load cases and combinations
Each load is applied in its own load case, and combinations use the code factors: ultimate, serviceability and fatigue combinations. Moving-load analysis produces envelopes (maximum and minimum of each action). The order of the sign of the wind and seismic direction should be considered — both positive and negative.