Branson’s Stress Ratio
The most common method for determining an effective moment of inertia in concrete members is Branson’s Formula:
Ie = (Mcr/Ma)4 Ig + [1 – (Mcr/Ma)4] Icr
where= | ||
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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:
Ie = (fcr/fa)4 Ig + [1 – (fcr/fa)4] Icr
where= | ||
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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 (fcr/fa)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].