A worked example — a single-column spread footing
A spread footing has no piles to collect the reaction into neat points: the ground pushes back as a distributed pressure, and the column drives a concentrated load into the middle of it. That is a D-region, and the strut-and-tie model handles it directly.
AStrutTie 2020 Technical Book, Example 02-2.3 (one-column spread foundation).
Given
- Footing 8,000 × 7,000 × 2,000 mm; one column 2,500 × 2,500 mm
- fck = 35 MPa · fy = 400 MPa
- Load case ST1O05: V = 9,625 kN · H = 125 kN · M = 14,670 kN·m
- Ground: vertical subgrade modulus 30,000 kN/m² · horizontal 10,000 kN/m² (bearing pressure computed automatically)
- Reinforcement: top and bottom 69-#8 · vertical shear 18-#6 @400 · horizontal 18-#6 @600
The truss the template builds
10 nodes, 17 elements, 3 supports, 2 bearing plates. Top chord nodes 1–5, bottom chord 6–10; the ground pressure enters as five distributed load elements along the base. Node offsets: 100 mm cover top and bottom, nodes 100 mm inside the column face, three nodes under the column.
Member forces
Compression negative, tension positive — the sign is the model telling you which members are struts and which are ties.
| Element | Force (kN) | Element | Force (kN) |
|---|---|---|---|
| E3 | −1,767.3 | E12 | −6,275.7 |
| E7 | +939.6 | E14 | +316.4 |
| E8 | −2,515.1 | E15 | +1,695.8 |
| E9 | +3,308.2 | E16 | +4,711.4 |
| E10 | −5,567.7 | E17 | +4,736.8 |
| E11 | −2,730.3 | E6 | −379.6 |
Ties
ACI 318 §23.7Provided 69-#8 = 34,962.90 mm², so the governing tie (Tie 17) uses about 45% of the steel present — O.K. with margin. Shear ties: Tie 7 (939.58 kN) and Tie 9 (3,308.25 kN), vertical, 18-#6, weff = 1,150.0 mm, sh = 400.0 mm, φFn = 4,425.0 kN.
| Tie | Fu (kN) | As,req (mm²) |
|---|---|---|
| Tie 14 | 316.37 | 1,054.55 |
| Tie 15 | 1,695.84 | 5,652.80 |
| Tie 16 | 4,711.37 | 15,704.58 |
| Tie 17 | 4,736.76 | 15,789.21 |
Struts
ACI 318 §23.4S12 is the governing strut — 146.5 mm required against 217.0 mm provided. S10 is the tightest ratio: 130.0 of 192.0 mm.
| Strut | θ | Fu (kN) | wreq (mm) | wprov (mm) |
|---|---|---|---|---|
| S3 | 0.0° | 1,767.3 | 41.3 | 200.0 |
| S6 | 40.6° | 379.6 | 8.9 | 217.0 |
| S8 | 57.4° | 2,515.1 | 58.7 | 372.0 |
| S10 | 57.4° | 5,567.7 | 130.0 | 192.0 |
| S11 | 90.0° | 2,730.3 | 63.7 | 100.0 |
| S12 | 40.6° | 6,275.7 | 146.5 | 217.0 |
Nodal zones
ACI 318 §23.9Node 3 is CCT (βn = 0.80) with C-2 288.2, C-3 1,767.3, C-8 2,515.1 kN and T-9 3,308.2 kN. Nodes 7, 8 and 9 are CTT (βn = 0.60) — the ties framing in drop the usable strength by a quarter against CCT, and by 40% against a pure-compression node. All faces O.K.
The same three checks again, on a footing with no piles at all. Governing tie 4,737 kN, governing strut 146.5 mm of 217.0 mm — and the nodes, not the members, are where the ties cost you strength. How we verify →