Dapped-end beam
The reduced-depth end (nib) of a precast beam or double-tee stem that bears on a ledge or corbel, designed for the five failure modes of the PCI Design Handbook (8th Ed., 5.6) with the ACI 318-19 provisions they rest on (shear friction 22.9, shear 22.5, bearing 22.8, development 25.4) and the ASCE 7-22 2.3.1 combinations. The PCI Design Handbook is not on disk: its equations are built from knowledge and marked "to confirm against the code text" on screen and in the report until the handbook is read.
Inputs of the dapped end #
- Materials: f'c, reinforcement grade, concrete type (λ).
- Beam and nib: web width b (one stem of a double tee), full depth H, nib depth hn, nib projection ℓp from the dap face, clear cover.
- Bearing: distance av from the dap face to the bearing center, pad length and width.
- Reinforcement: the nib top bars As, the horizontal closed ties Ah (within 2/3 d below As), the hanger stirrups Ash next to the dap face and the distance from the dap face to their centroid (it sets the shear span a = av + that distance), the nib stirrups Av, the beam stirrups beyond the dap and the full-depth d.
- Shear-friction coefficient: the code's μ = 1.4λ for monolithic concrete (default) or the handbook's effective μe = 1000 λ b hn μ / Vu ≤ 3.4 (the engineer's choice).
- Loads: per load type, unfactored, the reaction V and the horizontal tension N at the bearing; Nu is taken as at least 0.2 Vu unless the engineer clears that box (provision made for the restraint force otherwise).
Auto-design chooses the lightest As (size, number), then the numbers of hanger stirrups, ties and nib stirrups; the section, nib depth, bearing and loads stay the engineer's. See Auto-design.
Checks performed #
| Check | Basis |
|---|---|
| (1) Flexure and axial tension of the nib: As ≥ [Vu (a/d) + Nu (hn/d)]/(φ fy) | PCI 5.6 (to confirm), φ = 0.75 |
| (2) Direct shear at the dap face: As ≥ 2/3 Avf + An, Avf = Vu/(φ fy μ), An = Nu/(φ fy); Vu ≤ φ Vn,max of Table 22.9.4.4 | PCI 5.6, ACI 318-19 22.9 |
| Horizontal ties Ah ≥ 0.5 (As − An) | PCI 5.6 (to confirm) |
| (3) Hanger reinforcement at the re-entrant corner: Ash ≥ Vu/(φ fy) | PCI 5.6 (to confirm) |
| (4) Diagonal tension in the nib: Vu ≤ φ(Av fy + Ah fy + 2λ√f'c b d), Av ≥ (Vu/φ − 2λ√f'c b d)/(2 fy) | PCI 5.6 (to confirm) |
| (5) Shear of the full-depth section: Vu ≤ φ(2λ√f'c b d + Av fy d/s), stirrup spacing ≤ d/2 and 24 in. | ACI 318-19 22.5, 9.7.6.2.2 |
| Bearing on the nib: Vu ≤ 0.65 × 0.85 f'c A1 | ACI 318-19 22.8 |
| Clear spacing of As | ACI 318-19 25.2.1 |
| Nib depth about H/2 or more; a/d ≤ 1 (advisories) | PCI 5.6 (to confirm) |
| Development lengths of As, Ah, Ash (reported) | ACI 318-19 25.4.2.4, Ktr = 0 |
The drawings show the end elevation (the nib, bearing pad and reaction, the bar groups) and the section through the nib.
Limitations #
- The strut-and-tie alternative (ACI 318-19 Chapter 23) is not built in this release; the checks follow the handbook's failure modes only.
- The PCI equations, the 0.2 Vu minimum, the μe limit of 3.4 and the H/2 and a/d ≤ 1 limits are from knowledge of the handbook (to confirm against the code text).
- Rectangular webs; flanges, prestress within the nib, inclined hanger bars and the bearing pad design are not covered. The anchorage details (welded cross bar or plate at the end of As, hooks of the hangers) are the engineer's; the program reports the development lengths.
Verification examples #
Verification status: 11 values compared in 2 examples (0 published, 2 hand calculations from the cited equations): 11 within tolerance, 0 recorded deviations. No published example: every value is a hand calculation. That checks the program against this reading of the code, not the reading itself. Code basis: written without the code document on hand, and confirmed only in part through published examples.
Every value is a closed-form hand calculation of the equations above (backend/tests/test_verify_dapped_end.py): the five failure modes of the default beam, bearing, and the PCI effective shear-friction coefficient. No published dapped-end example is on disk (the PCI Design Handbook and the PCI connections manual are listed in research/DOCUMENTS_NEEDED.md).