Flanged (T-Beam) Cracked Moment of Inertia Icr (ACI 318-19)

Once cracked, the compression zone is the full flange width b down to the flange thickness hf, then only the web width bw below; the tension steel is transformed with n As at d. If the neutral axis lands in the flange (c <= hf) the section is a rectangle of width b, b c^2/2 = n As(d - c); if it falls in the web, bw c^2/2 + (b - bw) hf(c - hf/2) = n As(d - c) and Icr = bw c^3/3 + (b - bw)[hf^3/12 + hf(c - hf/2)^2] + n As(d - c)^2, with n = Es/Ec and Ec = 57000 sqrt(f'c). A 48 in flange, 4 in thick, 12 in web, d 17.5, As 6.0 in^2, f'c 4000 puts the neutral axis in the web at 5.08 in with Icr 9,528 in^4; set bw = b and it returns exactly the singly-reinforced 4,017 in^4, so the two agree in the rectangular limit. The wide compression flange raises the neutral axis and roughly doubles Icr over the bare web, which is why a slab acting with the beam sharply cuts deflection. Take the effective flange width b from the T-beam flange rules; feed this Icr into a deflection check. A design aid; the engineer of record governs.

Flange trial (rectangle b): b c^2/2 = n As(d - c), valid if c <= hf, Icr = b c^3/3 + n As(d - c)^2. Web case: bw c^2/2 + (b - bw) hf(c - hf/2) = n As(d - c), Icr = bw c^3/3 + (b - bw)[hf^3/12 + hf(c - hf/2)^2] + n As(d - c)^2. n = Es/Ec, Ec = 57000 sqrt(f'c). bw = b reduces to the singly-reinforced rectangular Icr.

ACI 318-19 §19.2.2.1 modulus Ec = 57000 sqrt(f'c) with the standard elastic cracked flanged (T-beam) transformed-section analysis (Wight & MacGregor, Reinforced Concrete; PCA Notes on ACI 318), by name; verified against the singly-reinforced rectangular limit (bw = b) of concrete-effective-inertia.

The cracked-transformed-section equations are standard mechanics of reinforced concrete; ACI 318 (the Ec definition) is readable free through the ACI online reading room at concrete.org. Flange width, flange thickness, web width, depth, steel area, and f'c are the user's inputs.

Estimate. AHJ and licensed professional govern.

Field names used by the API: b_in, hf_in, bw_in, d_in, as_in2, fc_psi, icr_in4, kd_in

• Neutral axis flange: b c^2/2 = n As(d - c); web: bw c^2/2 + (b - bw) hf(c - hf/2) = n As(d - c)elastic cracked transformed section
• Cracked inertia web: Icr = bw c^3/3 + (b - bw)[hf^3/12 + hf(c - hf/2)^2] + n As(d - c)^2; n = Es/Ec, Ec = 57000 sqrt(f'c)ACI 318-19 §19.2.2.1 + mechanics
• Limit / scope bw = b reduces to the singly-reinforced rectangular Icr; tension steel only (a doubly-reinforced T is separate)scope of this tile