Finite elements¶
The strut-and-tie model is an idealisation: it assumes where compression flows and where tension appears, and designs from that. Finite element analysis solves the cap as a three-dimensional solid and shows the stress field that actually develops.
It does not replace the strut model — it replaces faith in it.
The mesh¶
PCO builds the mesh over the real polygon of the cap, with two element types:
| Element | When to use it |
|---|---|
| Hexahedral | Caps with regular geometry. Converges faster and with fewer elements for the same accuracy |
| Tetrahedral | Geometries that hexahedra do not fill well — irregular arrangements, cut-outs, many columns |
Start with hexahedral
For the ordinary cap — rectangular, piles in a regular arrangement — the hexahedron gives a better result with a coarser mesh. The tetrahedron is the way out when the geometry cannot reasonably be divided into hexahedra.
What to read in the result¶
The analysis returns the stress field in the solid. Three readings matter:
Where compression really flows. The principal compressive stress trajectories draw the real struts. Comparing them with the assumed struts shows whether the truss model represents that cap — in a deep, well proportioned cap they coincide; in a shallow, wide cap, compression spreads in a way no simple truss reproduces.
Where tension appears. The idealised tie is a bar at the bottom. In the solid, tension occupies a region, and the height of that region tells you whether concentrating all the reinforcement at the bottom is appropriate or whether it needs to go higher.
Whether any node is overloaded. The analytical node check uses idealised areas. The solid shows the real concentration.
Node check¶
PCO uses the FEM result to check the stresses at the compression nodes — under the column and over the piles — against the NBR 6118 limits.
Without FEM solved, the check is analytical
Checking the nodes against the stress field requires the solved model. Without it, the program falls back on the analytical check, with the idealised areas of the strut model.
The distinction is not academic: the analytical check may pass a cap that the stress field fails, because it assumes a uniform distribution where there is concentration. If the cap is critical, run the FEM.
Pile reactions¶
The finite element model also provides the distribution of reactions among the piles — which, in a cap with several columns or eccentric loading, is not the one returned by the linear distribution formula.
Since the model is linear, the load combination can be solved case by case and the reactions enveloped afterwards. See Load combinations.
Limits of validity¶
Range of application
- The model is linear elastic. The concrete does not crack, the steel does not yield, there is no plasticity. Close to failure, the real field redistributes in a way the linear model does not capture — and it is precisely that redistribution that underpins the strut model.
- Therefore: linear FEM does not replace the strut-and-tie model. It checks assumptions and locates concentrations; the design still comes from the truss model, which is what the code supports.
- The piles enter as supports. Their real stiffness — which the SP line programs calculate — changes the distribution of reactions in a statically indeterminate cap.
- Mesh refinement influences the result in regions of concentration. A peak stress at a re-entrant corner grows with refinement and does not converge: it is a geometric singularity, not a real force.