Continuous beam pattern loading and moment redistribution per ACI 318-19

In a continuous beam, live load on one span changes the moments in every other span. The worst-case moment at each section therefore comes from a specific live load arrangement, not from loading the whole beam uniformly. ACI 318-19 Cl 6.4.2 requires the designer to consider these arrangements, and Cl 6.6.5 then allows a limited redistribution of the resulting support moments. For the single-span flexure and shear checks that follow from these moments, see the beam flexural design spoke and the beam methodology hub.

Applicable clauses

Why pattern loading matters

Live load is not permanent: any span may be loaded or empty at a given time. The arrangement that maximizes one moment typically minimizes another:

• Maximum positive moment in a span: load that span and every second span beyond it; leave the neighbors unloaded. Unloaded neighbors rotate less at the supports, so the loaded span behaves more like a simply supported span.
• Maximum negative moment at an interior support: load the two spans adjacent to that support (and alternate spans beyond). Both spans rotate into the support and stiffen the hogging moment.

Analyzing only the case with live load on every span is unconservative for midspan: it underestimates the positive moment, because loaded neighbors restrain the span. For support moments, the uniform case can also fall short of the adjacent-span arrangement. A design based on the uniform case alone can leave a section under-reinforced for a load pattern that will occur in service.

ACI 318-19 Cl 6.4.2 arrangements

Cl 6.4.2 calls for live load arrangements that produce the critical effects. For a beam with factored dead load D on every span, the standard arrangements are:

1. All spans loaded: D + L on every span.
2. Alternate spans loaded (checkerboard): D on every span, L on every second span. Produces maximum positive moment in the loaded spans.
3. Adjacent spans loaded: D on every span, L on two adjacent spans (and alternates beyond). Produces maximum negative moment at the support between them.

In general each span is either carrying live load or not, so a beam with n spans has 2ⁿ arrangements. calcnote generates every one of them rather than trying to guess the critical set.

The count doubles with every added span, but each case is a small stiffness-method solve, so the cost stays small for typical beams of up to six spans (64 cases). A cantilever overhang counts as a span for this purpose, since its live load also changes the support moments.

Simplified method (Cl 6.4.3.2)

Cl 6.4.3.2 permits the designer to limit the study to two arrangements: factored live load on two adjacent spans, and factored live load on alternate spans, each with factored dead load on all spans. It is intended for members that are prismatic with uniformly distributed loads, where the two arrangements bound the critical effects.

calcnote applies a stricter test before collapsing the pattern set to a single all-spans-loaded case: no point loads, live-to-dead ratio at most 0.75 in every span, at least two spans, and longest span no more than 1.2 times the shortest span. If any condition fails, the full 2ⁿ set is solved.

Moment envelope

Each pattern produces its own moment diagram. The moment envelope takes, at every station along the beam, the largest positive moment (M⁺_max) and the most negative moment (M⁻_min) over all patterns. Design moments are read from the envelope: M_u for the bottom steel comes from the positive envelope and M_u for the top steel at supports comes from the negative envelope. No single load case produces both, which is exactly why one case is not enough.

Moment redistribution (Cl 6.6.5)

A well-detailed continuous beam yields at the supports before collapse, shedding moment to the spans. Cl 6.6.5 lets the designer use this by reducing the elastic negative moments at supports, provided the section is ductile enough to rotate:

• Condition: ε_t ≥ 0.0075 at the section where the moment is reduced (Cl 6.6.5.1).
• Limit: reduction of at most 1000ε_t percent, never more than 20 percent. ε_t = 0.0075 allows 7.5 percent; ε_t = 0.010 allows 10 percent; ε_t ≥ 0.020 allows the 20 percent cap.
• Equilibrium: the positive span moments must be increased so that static equilibrium of each span is preserved.

Redistribution is useful for balancing reinforcement between a congested support and a lightly reinforced midspan, or for shaving the top steel at a column where bars are hard to place. It comes with a ductility requirement: the section needs a low steel ratio (large ε_t) to provide the rotation capacity the assumption relies on.

What calcnote computes

1. Build the stiffness model from the span lengths, supports, and optional cantilevers.
2. Generate all 2ⁿ live load on/off patterns (or one all-loaded case if the skip conditions above hold).
3. Combine factored loads per pattern (1.2D + 1.6L where live load is on, 1.2D where it is off) and solve each case.
4. Build the positive and negative moment envelopes from all patterns.
5. If redistribution is enabled, compute ε_t from the provided A_s; reject redistribution if ε_t < 0.0075; otherwise limit the requested percentage to min(1000ε_t, 20) and adjust the support and span moments.
6. Report the governing design moments with a warning whenever the requested redistribution is reduced or refused.

Try pattern loading on your own beam: run a continuous beam calculation or see a demo result.

Sources

• ACI 318-19: Cl 6.4.2 (arrangement of live load), Cl 6.4.3.2 (simplified arrangements), Cl 6.6.5 (moment redistribution), Cl 6.6.5.1 (ε_t ≥ 0.0075 and the 1000ε_t percent limit).

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