Instantly calculate maximum bending moment for simply supported, cantilever, fixed-fixed, and propped concrete beams — with support reactions in lb·ft and kip·ft. ACI 318-19 compliant.
Select your beam type and loading condition to calculate M_max, reactions & governing formula.
Reactions at each end; max moment at midspan for UDL
UDL applies equally along the full beam span
Clear span between supports (center-to-center)
UDL in lb/ft includes dead + live load
In structural engineering, bending moment is the internal rotational force that develops within a concrete beam when external loads are applied. It is measured as force × distance — expressed in lb·ft or kip·ft in US practice. Bending moment is the primary design driver for flexural reinforcement in concrete beams: the larger the moment, the more tension steel (rebar) is required below the neutral axis to keep the beam from fracturing. Per ACI 318-19, every reinforced concrete beam must have sufficient moment capacity (φMn ≥ Mu) to resist the factored design moment under ASCE 7 load combinations.
ACI 318-19 requires that the design moment capacity φMn ≥ factored moment Mu, where φ = 0.90 for tension-controlled sections (most beams). The nominal moment capacity Mn = As × fy × (d − a/2), where a is the equivalent stress block depth = As·fy / (0.85·f'c·b). For most residential and commercial concrete beams in the USA, Grade 60 rebar (fy = 60,000 psi) and f'c = 4,000 psi are standard.
The most common concrete beam condition — supported at both ends by a pin and roller. The maximum bending moment occurs at midspan for uniform loads. Used for floor beams, grade beams, and lintels. M_max = wL²/8 for UDL or PL/4 for a midspan point load. Reactions are equal at both supports for symmetric loading.
Fixed at one end with the other end free — common for balconies, retaining walls, overhanging slabs, and cantilevered roof structures. The maximum bending moment occurs at the fixed support and equals wL²/2 for UDL or PL for an end point load. Cantilevers require top steel (negative moment reinforcement) at the fixed support.
Both ends fully restrained (fixed-fixed) or one fixed with one pinned (propped cantilever) — common in continuous concrete frames. Fixing the ends reduces midspan moment but introduces negative moments at supports. A fixed-fixed beam with UDL has M_max = wL²/12 at each support vs. wL²/8 for simply supported — a 33% reduction.
Calculating the maximum bending moment in a concrete beam involves three steps: (1) determine your beam's support conditions and span length, (2) identify the applied loads (dead load, live load, self-weight per ASCE 7), and (3) apply the governing bending moment formula from ACI 318 or classical beam theory. For statically determinate beams (simply supported and cantilever), the calculation is direct. For statically indeterminate beams (fixed-fixed, propped), the moment distribution depends on end fixity and stiffness.
The bending moment formulas above use service loads. For ACI 318-19 structural design, you must apply ASCE 7 load factors to get the factored design moment Mu. The governing combo is typically Mu = 1.2D + 1.6L, where D = dead load (lb/ft) and L = live load (lb/ft). Then confirm φMn ≥ Mu with φ = 0.90. Never design rebar using unfactored service moments.
Quick-reference bending moment and reaction formulas for all four common concrete beam configurations used in US residential and commercial construction. Values align with ACI 318-19, classical structural analysis, and regional US concrete design practice. Use these to cross-check your calculator results.
| Beam Type | Load Type | M_max Formula | M_max Location | Reaction (Left) | Reaction (Right) | Moment Type |
|---|---|---|---|---|---|---|
| Simply Supported | Full UDL (w) | wL² / 8 |
Midspan (L/2) | wL / 2 | wL / 2 | Positive |
| Simply Supported | Point Load at midspan | PL / 4 |
Midspan (L/2) | P / 2 | P / 2 | Positive |
| Simply Supported | Point Load at distance a | Pab / L |
At load (dist. a) | Pb / L | Pa / L | Positive |
| Cantilever | Full UDL (w) | wL² / 2 |
Fixed support | wL (fixed) | Free end = 0 | Negative |
| Cantilever | Point Load at free end | P × L |
Fixed support | P (fixed) | Free end = 0 | Negative |
| Fixed-Fixed | Full UDL (w) | wL² / 12 |
Both fixed ends | wL / 2 | wL / 2 | Negative (ends) |
| Fixed-Fixed | Full UDL — midspan | wL² / 24 |
Midspan | wL / 2 | wL / 2 | Positive (mid) |
| Propped Cantilever | Full UDL (w) | 9wL² / 128 |
3L/8 from fixed end | 5wL/8 (fixed) | 3wL/8 (pin) | Mixed |
After calculating M_max using this concrete bending moment calculator, the next step is verifying that the concrete section has sufficient capacity. Per ACI 318-19 Section 22.3, the nominal flexural strength Mn of a rectangular beam with tension reinforcement is calculated using the equivalent rectangular stress block (Whitney stress block). The factored design moment Mu must not exceed the design flexural strength φMn, where φ = 0.90 for tension-controlled sections. For concrete beams in freeze-thaw regions like Utah or Oregon, minimum cover and steel detailing requirements are even more critical.
Multiply service loads by ASCE 7 load factors before computing Mu. The governing load combo for most beams is Mu = 1.2D + 1.6L. Use the factored total load w_u (lb/ft) in the bending moment formula. Never compare unfactored M_max directly to rebar capacity.
ACI 318 nominal capacity: Mn = As·fy·(d − a/2) where a = As·fy / (0.85·f'c·b). For a typical 12"×24" beam with 3 #7 bars (As = 1.80 in²), f'c = 4,000 psi, fy = 60 ksi, and d = 21.5": Mn ≈ 178 kip·ft. The design capacity is φMn = 0.90 × 178 = 160 kip·ft.
The ACI 318-19 acceptance criterion is simply φMn ≥ Mu. If the demand exceeds capacity, increase beam depth, add more tension steel, or use higher-strength concrete (f'c). ACI 318 also requires checking minimum steel ratio ρ_min = max(3√f'c/fy, 200/fy) to prevent brittle failure.
Official design codes and technical references for concrete beam bending moment analysis in the USA.
The American Concrete Institute's ACI 318-19 is the definitive US code for reinforced concrete design — covering flexural strength, load factors, minimum reinforcement, beam sizing, and all bending moment design requirements for residential and commercial construction.
Visit ACIASCE 7-22 (Minimum Design Loads and Associated Criteria) provides the US standard load combinations — including the 1.2D + 1.6L factored load combo — used to compute the factored design moment Mu for ACI 318 concrete beam design across all US states.
Visit ASCEThe Portland Cement Association (PCA) publishes comprehensive design guides for reinforced concrete beams, slabs, and columns — including worked examples for bending moment calculations, reinforcement detailing, and ACI 318 code compliance for US engineers.
Visit PCA