Effective Moment of Inertia and Deflections of a Cracked Concrete Beam Updated

Continued from before ... (here)

But wait! ... This assumes that we are cracked clear up to within 0.88 in. of the top of the beam. Yeah, it does. But just looking at the beam we can see it's not! It's not that cracked at lower loads, ... and it's not that cracked from end to end. Yeah, and that's why the ACI Code gives us an equation for an Effective Moment of Inertia, I _e, because it's not all that cracked!

... I _e = I _g (M_cr / M_a)³ + I _cr [ 1 - (M_cr / M_a)³ ],

where,

M_a is the applied moment ... (in this case at the mid-span).

Note that if our applied moment is equal to the cracking moment, our beam is not very cracked, and I_e is essentially I_g. If the applied moment is way larger than the cracking moment, our I_e is much more like our I_cr. The above equation is an approximation. But it does indeed approximately reflect reality, at least approximately. For applied moments less than the cracking moment, the equation doesn't work, and I_e is taken to be I_g.

The ugly thing about the above is that our `I' value changes depending on the applied load.

Well, let's do it.

Calculated Deflections Using Effective Moment of Inertia

Okay, let's put in some values for P and calculate anticipated deflections. Our calculated deflection will be that additional to the deflection due to self weight. (That deflection happened before we could measure it.) But, the moment that the beam feels at any stage in loading (M_a) includes the moment from the self weight ... M = M _sw + P L / 4 = M _sw + 1.5 P, ... from before.

So, ...

... Δ _P = P L³ / 48 E I =

... Δ _P = P (6 x 12 in.)³ / (48 x 3,372,000 psi x I_e) = ...

... Δ _P = 0.00231 P / I_e ...

Here are some values ...

P < 967 lb ... use I_g, etc. (uncracked beam ... here)

P = 967 lb (P_cr) ... I_e = I_g = 115 in.⁴ ... Δ = 0.019 in. ... same as before; yeah.

P = 1000 ... I_e = 106 ... Δ = 0.022

P = 1200 ... I_e = 70 ... Δ = 0.039

P = 1400 ... I_e = 50 ... Δ = 0.064

P = 1600 ... I_e = 38 ... Δ = 0.096

P = 1800 ... I_e = 31 ... Δ = 0.135

P = 1825 lb ... I_e = 30 in.⁴ ... Δ = 0.140 in.

The reason I stopped at P = 1825 lb (above) is because that is the load at which the steel first yields.

The reason the values beyond, say, P = 1200 lb are in italics ... is because our internal concrete stresses (here) climb above about half the concrete compressive strength, the concrete will start becoming non-linear, ... less stiff, and in reality the deflections will probably be greater than those calculated.

As we bend the beam beyond steel yield we can really no longer use our cracked Moment of Inertia (or I_e) to calculate stress and deflection. In reality, we will likely be able to bend the beam more ... but the steel will not take on any more stress. The concrete will strain more, and take on some more load, and so the total load on the beam may increase. As the steel yields the beam will kind of `hinge' and the cracks grow wider than hairline. The total load on the beam will not increase necessarily much, but the deflection sure will. At the value of P = 1889 lb, the concrete will be crushing, also (theoretically). In an earlier lesson (here) we called that value P_y. In that lesson we called it the P that breaks the beam in bending. I should probably go back and change the wording in that lesson, to, more precisely, the load that breaks the beam in steel-yielding-and-concrete-crushing. You can use the procedure of that lesson to calculate P_y, or, more precisely, the method that follows as the lesson is continued (here).

But, before we look more closely, let's calculate our stiffness at steel yield.

Cracked Stiffness at Steel Yield

... k_y = P_y / Δ_y = 1889 lb / 0.14 in. = 13,500 lb/in. Way less stiff than the uncracked beam.

And, the stiffness will in reality be less than `that', since the concrete is non-linear.

References

Building Code Requirements for Structural Concrete, ACI 318, American Concrete Institute, P.O. Box 9094, Farmington hills, Michigan, 48333.

Strength of a Reinforced Concrete Beam, Jeff Filler, Associated Content.

Deflections of an Uncracked Reinforced Concrete Beam, Jeff Filler, Associated Content.

Cracked Section Neutral Axis and Moment of Inertia, Jeff Filler, Associated Content.

Internal Stresses in a Cracked Concrete Beam, Jeff Filler, Associated Content.

Design of Concrete Structures, Christian Meyer, Prentice Hall, 1996.