ACI 318 versions prior to 318-19 referenced Branson’s Formula for calculating the effective moment of inertia of concrete members. In ACI 318-19, Branson’s equation is used for prestressed concrete members; however, Bischoff’s equation (see ACI 318-19 Eq 24.2.3.5(b)) is referenced for calculating the effective moment of inertia of nonprestressed members.

Bischoff’s equation is not implemented in RAM Concept. Branson’s equation is used for the calculation of the effective moment of inertia for all ACI 318 versions, including ACI 318-19.

Bischoff’s equation is:

I_e = (M_cr/M_a)⁴ I_g + [1 – (M_cr/M_a)⁴] I_cr

where

Ie	=	the effective moment of inertia
Ig	=	the gross concrete moment of inertia
Icr	=	the cracked concrete moment of inertia
Mcr	=	the gross cross section cracking moment
Ma	=	the applied moment

As Branson’s formula does not consider axial forces which may be present (especially in post-tensioned structures), we have modified it to consider axial forces:

I_e = (f_cr/f_a)⁴ I_g + [1 – (f_cr/f_a)⁴] I_cr

where

fcr	=	the concrete flexural tensile strength
fa	=	the cross-section tensile fiber stress (based on gross section properties)

If there is no axial force, then this formulation is identical to Branson’s formula. If there are axial forces, this formulation is a reasonable (but not theoretically identical) extrapolation of Branson’s formula.

We call the value (f_cr/f_a)⁴ "Branson’s Stress Ratio". Its value is always limited to be less than or equal to 1.0.

Note that you will more commonly see Branson’s formula used with a power of 3 instead of 4. The power of 3 is appropriate when a peak moment in a span is used to determine an effective moment of inertia for the entire span. The power of 4 is appropriate for determining a local effective moment of inertia using local section forces [Branson, Dan E., "Instantaneous and Time-Dependent Deflections of Simple and Continuous Reinforced Concrete Beams", Report #7, Part 1, Alabama Highway Research Department, Bureau of Public Roads, August 1963, pp.1-78].