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, the 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 and source
a = (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).
The ACI 318-19 doubly-reinforced rectangular-beam flexural capacity with the compression-steel yield check and the tension-controlled strength-reduction factor, by name.
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