Cracked Beam Neutral Axis and Moment of Inertia

Continued from before ... (here)

As we load the beam past the cracking moment (and P greater than P_cr) ... the beam will crack. It will probably crack near the mid-span of the beam (under the load point of P), near the bottom, and the cracks will be `hairline'. As the load is increased, the crack will probably `grow' (propagate upward, toward the neutral axis of the beam), and additional cracks will be observed, as we look away from the mid-span of the beam toward the supports. At mid-span the stresses in the beam are primarily flexural tension and compression; these stresses are horizontal; and thus the cracks will be vertical. As we move toward the supports, the shear stresses increase, and the cracks will tend to be more inclined. Right at the support, where flexural stresses are small, and shear stresses are the greatest, the cracks will tend to be inclined at 45 degrees - the so-called `diagonal tension' cracks associated with shear.

The Cracked Section and Transformed Section

In a so-called cracked section the concrete is assumed to act in compression only, and the reinforcement takes over in the role of carrying flexural tension. Analysis of such a section is traditionally accomplished with the so-called `transformed section' where the steel is transformed (in our minds) into a equivalent amount of concrete. This equivalent concrete is different than real concrete, however, in that it is capable of resisting tension. The reason we do this is that our deflection equations are cast in terms of single values of I and E (a single material), and a reinforced concrete section has two materials. So, we transform the steel into concrete, and then we're back to one material. Or we could transform the concrete to steel, and work with an all-steel section. But generally we do the former.

We do the transformation based on strain compatibility. Since steel is many times stiffer than concrete, for a certain amount of strain (or flex), anywhere in the beam, say at the location of the steel, it seems like there is many times more of it (steel). We use this idea to do the transformation.

The equivalent amount of steel transformed to concrete is ...

... n A_s ...

where

... n is the ratio of the Modulus of Elasticities of the two materials (steel over concrete) ...

... n = E_s / E_c.

E_s = 29,000,000 psi ... a relatively `constant' value (same regardless of steel strength) ...

E_c = 57,000 √f '_c psi ... (quite dependent on the strength of the concrete).

In our example, E_c = 3,372,000 psi.

So,

... n = 29,000,000 / 3,372,000 = 8.6.

So, in terms of stiffness, one # 3 bar of cross section 0.11 in.² acts like 8.6 x 0.11 in.² = 0.95 in.² of concrete.

In our example of 2 - # 3 bars, then, ...

... n A_s= 8.6 x 2 x 0.11 = 1.89 in.²of equivalent concrete.

Our transformed beam behaves like there is 1.89 in.² of concrete acting (in tension) at the location of the rebar.

Note: this is a lot less than the lower half of the section previously considered to act in tension (before it cracked) ... in our case ... 11 in. wide x 2.5 in. = 27.5 in.². In the lower part of the beam, then, we have 2 in.² vs 27.5 in.² ... our beam is going to be a whole lot less stiff. (Even though it is WAY stronger.)

Transformed Section Neutral Axis

The Moment of Inertia of this transformed section is found by assuming that the concrete in compression is stressed linearly from the neutral axis outward toward the extreme fiber in compression, and that the tension is carried entirely by the steel transformed into equivalent concrete. It gets a bit complicated, though, because since we `lost' material in the tension zone, the neutral axis moves upward. And we need to find the `new' location of the neutral axis. It is not impossible, however. If we call the distance from the top of the beam to the new neutral axis y _bar, then we can solve for it by the idea of centroids (or Principle of Moments) ... balancing the moment of the compression zone with the moment of the tension zone ...

... (y _bar / 2) (y _bar b) = (d - y _bar) n A_s ,

where,

... d = the effective depth = distance from top of beam down to rebar centerline.

In our example ...

... (y _bar / 2) (y _bar 11 in.) = (3.0 in. - y _bar) 1.89 in.²,

which I generally just solve by trial and error ...

... y _bar = 0.88 in.

Note how the neutral axis moved up quite significantly ... from ½ of 5 in. = 2.5 in., to less than an inch from the top of the beam. We have a lot less concrete acting in compression. We anticipate a lot less stiffness.

Transformed Section and New Moment of Inertia

Now that we have the location of the neutral axis, we can determine the Moment of Inertia of this new (transformed) section. To do so we will assume that the stresses still vary linearly from the neutral axis outward, which is okay as long as we have an `under-reinforced' beam (save for another discussion).

... I_cr = I of rectangle, compression + I of the n A_s in tension ...

... I_cr = (1/3) b y _bar³ + (d - y _bar)² n A_s

... I_cr = (1/3) (11 in.)(0.88 in.)³ + (3 - 0.88 in.)² (1.89 in.²)

... I_cr = 11.0 in.⁴.

Whoa! ... the cracked section has, in this case, not even one-tenth the moment of inertia of the uncracked section. We anticipate, then, that the beam will be a LOT less stiff (once it starts cracking).

Next we'll look at the deflections with the cracked section ... (here)

References

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

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

Effective Moment of Inertia and Deflections of a Cracked Reinforced Concrete Beam, Jeff Filler, Associated Content.