Concrete Effective Moment of Inertia Ie (ACI 318-19 Bischoff)
The effective moment of inertia Ie that sets a reinforced concrete member's immediate deflection.
Example
You enter
- Section width b (in) 12
- Total depth h (in) 20
- Effective depth d to tension steel (in) 17.5
- Tension steel area As (in²) 3
- Concrete strength f'c (psi) 4000
- Service moment Ma (kip-ft) 60
- Lightweight factor lambda (1.0 NW, 0.75 LW) 1
You get
- Gross inertia Ig 8000 in^4
- Icr (in⁴) 4017
- Ie (in⁴) 4662
- Cracking moment Mcr 31.62 kip-ft
Details, formula, and sources
ACI 318-19 §24.2.3.5 replaced Branson's cubic with Bischoff's form: Ie = Ig when the service moment Ma <= (2/3)Mcr, otherwise Ie = Icr / [1 - (Mcr/Ma)^2 (1 - Icr/Ig)]. Mcr = fr Ig/yt with fr = 7.5 lambda sqrt(f'c); Icr is the cracked transformed section (n = Es/Ec, kd, Icr = b kd^3/3 + n As (d-kd)^2). A 12 x 20 in beam, d 17.5, As 3.0 in^2, f'c 4000 psi under a 60 kip-ft service moment gives Ig 8,000, Icr 4,017, Mcr 31.6 kip-ft, and Ie 4,662 in^4 (58% of Ig). The immediate deflection is then (a load coefficient) w L^4 / (Ec Ie), which feeds the long-term deflection. Bischoff predicts larger deflections than Branson, especially for lightly reinforced slabs. Singly-reinforced rectangular section. A design aid; the engineer of record governs.
Mcr = fr Ig/yt, fr = 7.5 lambda sqrt(f'c); Icr = b kd^3/3 + n As (d-kd)^2 with n = Es/Ec, kd = d(sqrt((rho n)^2 + 2 rho n) - rho n); Ie = Ig if Ma <= (2/3)Mcr, else Ie = Icr/[1 - (Mcr/Ma)^2 (1 - Icr/Ig)].
The ACI 318-19 24.2.3.5 effective moment of inertia (Bischoff form), which replaced the Branson equation of ACI 318-14, by name.
ACI 318 is readable free through the ACI online reading room at concrete.org; the 24.2.3.5 effective-inertia and 19.2.2/19.2.3 modulus provisions are in the published code. The Branson-to-Bischoff change is documented by StructurePoint/PCA.
Estimate. AHJ and licensed professional govern.
Field names used by the API: b_in, h_in, d_in, as_in2, fc_psi, ma_kipft, lambda, ig_in4, icr_in4, ie_in4, mcr_kipft
- Bischoff Ie Ie = Icr/[1 - (Mcr/Ma)^2 (1 - Icr/Ig)] for Ma > (2/3)Mcr; else Ie = IgACI 318-19 24.2.3.5
- Cracked section Icr = b kd^3/3 + n As (d-kd)^2, n = Es/Ec, Ec = 57000 sqrt(f'c)ACI 318-19 19.2.2.1 / transformed section
- Singly-reinforced rectangular compression steel and T-flanges neglected; those use the appropriate transformed Icrscope of this tile