When should the Effective Moment of Inertia be used?

When calculating deflection? for concrete beams, if the Cracking Moment (Mcr) has been exceeded then the Gross Moment of Inertia (Ig) should be substituted with the Effective Moment of Inertia (Ie).

How do you calculate the effective Moment of Inertia?

• Calculate the Modular Ratio (n)

n = {E_s\over E_c}

where:
Es = The Modulus of Elasticity of Steel (e.g. 29,000 ksi)
Ec = The Modulus of Elasticity of Concrete (e.g. 3,500 ksi)
• Calculate the distance to the neutral axis (cs)

c_s = {\left(nA_s\over b\right)}{\left(\sqrt{1+{2bd\over nA_s}}-1\right)}

or,

c_s = {nρd}{\left(\sqrt{1+{2\over nρ}}-1\right)}

where:
n = modular ratio
As = Area of steel
b = base of the beam
d = depth to the rebar (not to be confused with the height of the beam)
ρ = The steel reinforcement ratio for the web \left(ρ = {A_s\over b_wd}\right)
• Calculate the Cracked Moment of Inertia (Icr)

I_{cr} = {bc^3_s\over 3}+nA_s(d-c_s)^2

• Calculate the Effective Moment of Inertia (Ie)[1]

I_{e} = {\left(M_{cr}\over M_a\right)^2I_g}+{\left(1-\left(M_{cr}\over M_a\right)^3\right)I_{cr}} \le I_g

where:
Mcr = The Cracking Moment.
Ma = The service moment acting on the beam.

References

1. American Concrete Institute, "ACI 318", 2005
• The effective moment of inertia is given by ACI 318 9.5.2.3

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