Concrete deflection
Immediate and long-term deflections of a continuous line of nonprestressed concrete beams or one-way slabs by ACI 318-19 24.2: cracked sections at midspan and at every support, Ie by the 2019 expression of Table 24.2.3.5 at three service load levels in sequence (D, D + sustained L, D + L), the Ie of each span by the method you choose, the time-dependent factor λΔ and the limits of Table 24.2.2. Concrete beam checks the deflection of the member it designs with Ie at midspan; this type is the detailed, multi-span version for a beam whose bars are known.
Inputs of the concrete deflection #
- Materials: f'c, the concrete type and density, Ec by 19.2.2.1.
- Section: rectangular or T (the effective flange of 6.3.2 is yours); the negative regions as the web rectangle (the usual simplification) or as the T with its flange in tension; bar centroids from the top and bottom faces.
- Spans and reinforcement: one to eight spans, left to right, each with its bottom bars and the top bars at its right support; the top bars at the left end; compression bars at midspan for ρ'; pinned or fixed ends.
- Service loads: superimposed dead load, self-weight, live load (for flat roofs the largest of Lr, S and R), the sustained part of the live load (yours), live load on every span or the checkerboard pattern of each span.
- Method and limits: Ie at midspan (24.2.3.7), the average of the positive and negative sections (24.2.3.6), the ACI 435 weighted average (0.70 Iem + 0.15 (Ie1 + Ie2); end spans 0.85 Iem + 0.15 Ie,cont), or Ie varying along the beam with the moment; ξ of Table 24.2.4.1.3; the part of the time-dependent deflection that occurs before attachment (footnote 2 of Table 24.2.2, 0 by default); the limits as span / n.
Auto-design varies the depth h (8 to 60 in.); the spans, loads, bars, widths, the method and the limits are kept.
Checks performed #
| Check | Basis |
|---|---|
| Section properties | Ig, Mcr = fr Ig / yt (Eq. 24.2.3.5), Icr of the cracked transformed section |
| Minimum depth (note) | Table 9.3.1.1 (beams) or 7.3.1.1 (slabs), with the fy and lightweight modifications |
| Immediate live load deflection, each span | Δ(D + L) − Δ(D) against ℓ/n (Table 24.2.2) |
| Deflection after attachment, each span | (1 − before) λΔ Δ(D + sustained L) + ΔL against ℓ/n (24.2.4.1, Table 24.2.2) |
The moments of each level come from an elastic analysis with Ig; the deflections from a second analysis with the stiffness EcIe of each span (or of each element for the varying method). The drawings show the beam elevation with its loads and the immediate deflection under D + L.
Limitations #
- Prestressed members, two-way slabs, cantilevers and haunched members are not covered.
- The moments are those of a continuous beam on knife-edge supports: support rotations through the columns (a frame) are not modelled.
- The bars are taken over the whole region they are entered for; cut-offs are not modelled.
Verification examples #
Verification status: 14 values compared in 4 examples (2 published, 2 hand calculations from the cited equations): 14 within tolerance, 0 recorded deviations. Thin: only 2 published examples. Code basis: the provisions it uses were read in the target edition.
| Example | What it checks | Published | ILX Calcs |
|---|---|---|---|
| StructurePoint DE-CNB-ACI14, Table 3 | Negative region 16 × 24 in., 7 #8 at 22 in. | Ig 18,432 in.⁴, Mcr 60.72 kip-ft, kd 8.42 in., Icr 10,897 in.⁴ | same |
| Same, Table 4 | T section, 120 × 8 in. flange | Ig 39,684 in.⁴, Mcr+ 89.77, Mcr− 240.36 kip-ft | same |
The example's Ie and deflections use ACI 318-14 (Branson) and a frame with columns, so they are not compared. Hand calculations check a simple span, uncracked (5wL⁴/384EcIg) and cracked (Ie of Table 24.2.3.5, λΔ = 2.0).