Doubly-Reinforced Concrete Beam Flexural Capacity (ACI 318-19)
When a beam's depth is limited and the tension steel outruns a singly-reinforced section.
Example
You enter
- Beam width b (in) 14
- Effective depth d (in) 22
- Compression-steel depth d' (in) 2
- Tension steel As (in²) 8
- Compression steel A's (in²) 3
- Concrete strength f'c (psi) 4000
- Steel yield fy (psi) 60000
You get
- A (in) 6.303
- C (in) 7.415
- Nominal Mn 771.2 kip-ft
- Design phi Mn 694.1 kip-ft
Checked against published worked examples from: University of Memphis.
Details, formula, and sources
ScopeThe designer adds compression steel and the capacity gains a second couple A's fy (d - d'). ACI equilibrium gives a = (As - A's) fy / (0.85 f'c b) and Mn = the singly-reinforced couple plus the steel-to-steel couple. A 14x24 beam with As 8, A's 3 in^2 reaches 771 kip-ft (694 design) - above what the section could singly - and the compression steel both raises capacity and shrinks the stress block. Both layers assumed to yield (flagged if not). A design aid, not the engineer of record's stamped design.
Formula or tablea = (As - A's) fy / (0.85 f'c b); c = a/beta1; eps's = 0.003 (c - d')/c; eps_t = 0.003 (d - c)/c; Mn = (As - A's) fy (d - a/2) + A's fy (d - d'); phi Mn (phi = 0.90 tension-controlled).
Source and editionThe ACI 318-19 doubly-reinforced rectangular-beam flexural capacity with the compression-steel yield check and the tension-controlled strength-reduction factor, by name.
Free accessACI 318 is readable free through the ACI online reading room at concrete.org; the flexural provisions are in the published code.
What governsEstimate. AHJ and licensed professional govern.
API field namesb_in, d_in, dp_in, as_in2, asp_in2, fc_psi, fy_psi, a_in, c_in, mn_kipft, phi_mn_kipft
- Two couples the singly-reinforced concrete couple plus the compression-steel-to-tension-steel couple A's fy (d - d')ACI 318-19 22.2
- Both yield both steel layers are assumed at fy; the compression-steel strain is checked and flaggedACI 318-19 22.2.1.1
- Tension-controlled phi = 0.90 at epsilon_t >= epsilon_ty + 0.003 (the 318-19 limit, not the fixed 0.005); transition belowACI 318-19 21.2.2