A worked example — a two-column pier footing on piles
A footing carrying two columns on a 21-pile group is a D-region twice over: the load spreads from each column into the cap, and the pile reactions come back up as concentrated forces. Beam theory does not describe either. Here is the strut-and-tie design, end to end.
AStrutTie 2020 Technical Book, Example 02-2.2 (two-column pile foundation). Every figure below is taken from that published source.
Given
- Footing 10,900 × 4,500 × 1,500 mm; two columns 1,500 × 1,500 mm at 6,400 mm centres
- Piles: φ508 steel pipe, L = 10,000 mm — 7 across × 3 deep = 21 piles
- fck = 24 MPa · fy = 300 MPa
- Column 1: V = 5,700 kN · H = 900 kN · M = 5,300 kN·m
- Column 2: V = 5,300 kN · H = 1,000 kN · M = 5,200 kN·m
- Load case LC01 = 1.00 L1 + 1.00 L2
Model the footing upside down
In a pile-cap model the column reactions are the supports and the pile reactions are the applied loads — the structure is analysed the way the forces actually flow. AStrutTie builds this from the template: 22 nodes, 41 elements, 6 supports (three under each column), 11 bearing plates.
| STM setting | Value |
|---|---|
| Nodes under each column (A) | 3 |
| Node offset from column face (B) | 100 mm |
| Top cover (C) · bottom cover (D) | 100 mm · 200 mm |
| Diagonal height/width ratio (F) | 0.8 |
| Reinforcement | top 22-#7 · bottom 43-#8 · vertical shear 10-#6 @400 · horizontal 6-#6 @400 |
Optimize, then check the sign of every diagonal
The first optimization run left element E28 in tension at +45.6 kN — a diagonal that was supposed to be a strut. That is not a rounding artefact; it means the assumed truss does not match the flow. The direction was corrected and the model re-analysed (ordinary analysis, not optimization) to give the final result.
Ties — required steel
ACI 318 §23.7All O.K. Shear ties: Tie 19 (1,299.05 kN) and Tie 23 (272.37 kN), both vertical, 10-#6 with weff = 950.0 mm and sh = 400.0 mm giving φFn = 1,523.1 kN.
| Tie | Fu (kN) | As,req (mm²) | Provided |
|---|---|---|---|
| Tie 5 (top) | 119.41 | 530.73 | 22-#7 → 8,534.90 mm² |
| Tie 32 | 875.60 | 3,891.57 | 43-#8 → 21,788.53 mm² |
| Tie 34 | 1,147.02 | 5,097.88 | 43-#8 |
| Tie 38 | 206.20 | 916.43 | 43-#8 |
| Tie 40 | 1,612.49 | 7,166.64 | 43-#8 |
Struts — required width
ACI 318 §23.4Note S28: it is prismatic, so βs = 1.00 and its required width collapses to 34.6 mm — while S3 with the same order of force needs 133.3 mm at βs = 0.40. The strut condition, not the force alone, sets the width.
| Strut | βs | θ | Fu (kN) | wreq (mm) | wprov (mm) |
|---|---|---|---|---|---|
| S3 | 0.40 | 0.0° | 3,671.8 | 133.3 | 200.0 |
| S18 | 0.40 | 51.6° | 3,317.5 | 120.5 | 203.0 |
| S24 | 0.40 | 51.6° | 2,695.3 | 97.9 | 203.0 |
| S28 | 1.00 | 61.6° | 2,382.8 | 34.6 | 183.0 |
| S20 | 0.40 | 36.9° | 2,165.1 | 78.6 | 250.0 |
Nodal zones
ACI 318 §23.9Every node is classified and checked face by face. Node 3 is CCC (βn = 1.00) carrying C-2 604.2, C-3 3,671.8 and C-15 1,032.6 kN; Node 5 is CTT (βn = 0.60) with C-4 1,612.7, T-5 119.4, T-19 1,299.1 and C-20 2,165.1 kN; Node 7 is CTT with five members framing in. All faces O.K.
Twenty-one piles, two columns, 41 truss elements — and the design comes down to the same three checks, each printed with the clause and factors it used. The tie that governs is Tie 40 at 1,612 kN; the strut that governs is S3 needing 133 mm of the 200 mm available. How we verify →