Deep beam (strut-and-tie)
Checks a deep beam by the strut-and-tie method of ACI 318-19 Chapter 23 with 9.9. Strength design, φ = 0.75. Two ready models: one point load, which is a transfer girder under a column and, seen upside down, a two-pile cap; and two symmetric point loads. You choose the model's node depths and the checks tell you if they work.
Inputs of the deep beam #
- Beam: b, h, the span between support centers, the load's distance, and the length past each support.
- Bearings: the length at the supports and under the loads, along the span.
- Top node depth ws and tie node depth wt: the model's geometry. The lever arm is h − ws/2 − wt/2. Try wt as the depth the bars take, and ws as what the top strut needs: it is yours to iterate.
- βc: Table 23.4.3(b): at a bearing up to the lesser of √(A2/A1) and 2. 1.0 is on the safe side.
- Depth d: for Eq. 9.9.2.1 and the grid's spacing. 0 takes h − wt/2.
- Loads: unfactored, at each point. Own weight is added as dead load at the points if you choose: a lumping that is your decision.
- Tie: bars, anchorage, cover. A headed or mechanical anchorage is your own check, reported as such.
- Distributed steel: bars on each face, both ways. If it meets Table 23.5.1, interior struts take βs = 0.75; if it falls short, βs = 0.4, unless you say the strut is laterally restrained (23.5.3) or you choose Eq. 23.4.4. 9.9.3.1 requires the grid anyway, so a shortfall still fails.
Auto-design varies the tie bars and the distributed grid. The beam, the bearings, the node depths and the loads stay yours.
Checks performed #
| Check | Basis |
|---|---|
| Truss forces | Solved by the library's stiffness method; statics in hand checks |
| Angle | 23.2.7: at least 25 degrees |
| Section limit | Eq. 9.9.2.1: Vu ≤ φ 10√f'c b d. Eq. 23.4.4 if you use it for βs |
| Diagonal struts | 23.4: fce = 0.85 βc βs f'c, at the narrower end; width lb sin θ + wt cos θ at a support, bearing sin θ + ws cos θ at a load |
| Top strut (two loads) | 23.4: βs = 1.0 |
| Nodes | 23.9: βn = 0.80 at the supports (one tie), 1.0 under the load; the bearing, tie and strut faces |
| Tie | Eq. 23.7.2: Ats ≥ T / (φ fy); 9.9.3.2 minimum |
| Tie anchorage | 23.8.3: straight or hooked, past the extended nodal zone |
| Distributed steel | 9.9.3.1: 0.0025 each way; 9.9.4.3: spacing at most d/5 and 12 in. |
Limitations #
- Two ready models and no truss of your own. Not offered: three-dimensional pile caps, corbels by strut-and-tie, openings, uniform loads as multiple panels, prestress, seismic 23.11.
- Strut widths use the bearing and node depth as drawn; a node with unequal stress faces is not reshaped.
- Unequal two-point loads have no direct truss here. Model them with Truss analysis and check the pieces by hand.
- The tie's centroid is taken at wt/2: fit the bars inside the depth yourself (23.8.1).
- λ enters only in Eq. 23.4.4, as in the code. Chapter 23 has no λ in fce.
Verification examples #
Verification status: 46 values compared in 7 examples (4 published, 3 hand calculations from the cited equations): 42 within tolerance, 4 recorded deviations. Code basis: the provisions it uses were read in the target edition.
| Example | Source | Values |
|---|---|---|
| Two point loads | ACI SP-17(14), Strut-and-Tie Example 1, pp. 660 to 667 (318-14) | Pu 456, S1 and T1 533 kips; Ast 11.8 in²; φ fce 2,550, 1,913, 2,040 psi; φ Fns 673 and 918; φ Fnn 734 kips |
| One point load | StructurePoint DE-STM, pp. 10 to 16 (318-14) | Pu 638.5, R 319, strut 603, tie 512 kips; angle 32 degrees; φ 10√f'c b d 512 kips; Ast 11.4 in² |
Three published values differ on purpose. SP-17(14) prints the limit on Vu with 0.85, an older factor; the code has 0.75. A strut without distributed steel is 0.6 in 318-14 and 0.4 in 318-19. StructurePoint's extended zone follows its own node depth from a stress; here the depth is what you enter. Hand checks: a load off center, βc, Eq. 23.4.4. Both examples are to ACI 318-14. ACI MNL-17(21) Chapter 14 comes after wave 7 and would confirm these to 318-19; SP-17(14) Examples 2 to 5 and the FHWA and TxDOT worked examples are not run.