Doubly-Reinforced Cracked Moment of Inertia Icr (ACI 318-19)
The cracked moment of inertia for a section with compression steel (a doubly-reinforced section).
Example
You enter
- Section width b (in) 12
- Effective depth d to tension steel (in) 17.5
- Tension steel area As (in²) 3
- Concrete strength f'c (psi) 4000
- Compression steel area As' (in², 0 if none) 1
- Compression steel depth d' (in) 2.5
You get
- Cracked inertia Icr 4129 in^4
- Neutral-axis depth kd 6.35 in
- Modular ratio n 8.04 (Ec 3605 ksi)
Details, formula, and sources
Once the section cracks the tension concrete is ignored and it becomes a transformed area: compression concrete b above the neutral axis, tension steel n As at d, and compression steel (n-1) As' at d' (the n-1 crediting the concrete it displaces). The neutral axis c solves b c^2/2 + (n-1) As'(c - d') = n As(d - c), and Icr = b c^3/3 + (n-1) As'(c - d')^2 + n As(d - c)^2, with n = Es/Ec and Ec = 57000 sqrt(f'c). A 12 x 20 in beam, d 17.5, As 3.0 in^2, f'c 4000 with 1.0 in^2 of compression steel at d' 2.5 gives n 8.04, kd 6.35 in, and Icr 4,129 in^4; drop the compression steel and it returns exactly 4,017 in^4 - the singly-reinforced value for this beam, so the two agree in the limit. Compression steel raises the neutral axis and stiffens the cracked section, which is why top bars cut long-term deflection. Feed this Icr into the effective moment of inertia (or a deflection check) for a doubly-reinforced beam. Rectangular section, elastic cracked analysis. A design aid; the engineer of record governs.
Neutral axis c: b c^2/2 + (n-1) As'(c - d') = n As(d - c). Icr = b c^3/3 + (n-1) As'(c - d')^2 + n As(d - c)^2. n = Es/Ec, Ec = 57000 sqrt(f'c), Es = 29,000,000 psi. As' = 0 reduces to the singly-reinforced Icr.
ACI 318-19 §19.2.2.1 modulus Ec = 57000 sqrt(f'c) with the standard elastic cracked-transformed-section analysis of a doubly-reinforced rectangular beam (Wight & MacGregor, Reinforced Concrete; PCA Notes on ACI 318), by name; verified against the singly-reinforced limit (As' = 0) 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. Width, depth, steel areas, and f'c are the user's inputs.
Estimate. AHJ and licensed professional govern.
Field names used by the API: b_in, d_in, as_in2, fc_psi, asp_in2, dp_in, icr_in4, kd_in, n
- Neutral axis b c^2/2 + (n-1) As'(c - d') = n As(d - c)elastic cracked transformed section
- Cracked inertia Icr = b c^3/3 + (n-1) As'(c - d')^2 + n As(d - c)^2; n = Es/Ec, Ec = 57000 sqrt(f'c)ACI 318-19 §19.2.2.1 + mechanics
- Limit / scope As' = 0 reduces to the singly-reinforced Icr; rectangular section only (a T-beam is separate)scope of this tile