### When should the Effective Moment of Inertia be used?

When calculating deflection? for concrete beams, if the Cracking Moment (M_{cr}) has been exceeded then the Gross Moment of Inertia (I_{g}) should be substituted with the **Effective Moment of Inertia (I _{e}).**

### How do you calculate the effective Moment of Inertia?

- Calculate the Modular Ratio (n)

n = {E_s\over E_c}

*where:*

E

E

_{s}= The Modulus of Elasticity of Steel (e.g. 29,000 ksi)E

_{c}= The Modulus of Elasticity of Concrete (e.g. 3,500 ksi)- Calculate the distance to the neutral axis (c
_{s})

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

A

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)

A

_{s}= Area of steelb = 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 (I
_{cr})

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

- Calculate the Effective Moment of Inertia (I
_{e})^{[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:*

### References

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