Cantilevered Concrete beam - applied deflection versus moment

[fredPE]

I have a cantilevered concrete beam, and I want to know what the moment is at the face of the cantilever if I deflect the beam a certain number of inches. Or at the very least, I would like to know if the beam would fail at that amount of deflection.

This is actually an existing concrete wall that has a low roof tied into it at its mid height, and during a lateral event the upper roof will deflect a certain amount of inches. IF it were a steel beam, I would very easily determine the moment by using the elastic beam equations out of AISC Table 3-23. Trying to use those equations with concrete, with its variable and undetermined moment of inertia is a little more difficult. Basically, assuming an Ieff, calculating a point load/moment, and verifying the Ieff assumption in an iterative process ends up Not Converging (Ieff gets larger causing the moment to get smaller causes I eff to get even smaller, etc.). So what am I missing. I think the amount of deflection is such a high number that it is a no brainer that the wall will be in the inelastic range, but I need to prove it.

 

[KootK]

Interesting problem Fred. I don't know the answer.

I was thinking though:

Suppose the cantilevered wall deflects so much that it fails. Isn't it still stable? Wouldn't you just have one pin-ended wall from the ground to the low roof and a second pin-ended wall from the low roof to the high? Maybe it doesn't matter if you fail the wall.

Obviously you'd need to do some additional checks like:

1) Can the lateral system associated with the low roof handle the wall shear that would accompany an over strength moment failure?

2) Is your wall under reinforced to the extent that you can count on it failing by steel yielding rather than concrete failure.

3) Are there any weird P-delta effects to deal with?

I'm sure there are others that I haven't thought of...

 

[JAE]

fredPE - how are you calculating I_e?

Normally, per ACI, the I_e is based upon
M_cr - the cracking moment = f_rI_g/y

f_r is the max. tensile stress = 7.5 x sqrt(f'c).

With a wall, or other compressive member, you also have axial stress that affects the cracking moment.

So the cracking stress is really f_r + P/A

Substituting that into the M_cr equation you get
M_cr=((f_r+P/A) x Ig)/y

Thus, with additional axial compression, the moment required to crack the section goes up and this results in higher I_e values.

 

[BAretired]

Your question cannot be answered based on the information you have provided. Better you should post a sketch showing the boundary conditions.

BA

 

[msquared48]

Without knowing the resteel in the top of the cantilever, it is impossible to answer the question. Any bottom (compression) steel could also affect the answers.

Mike McCann
MMC Engineering

 

[demayeng]

Rather than trying to use an iterative process (which you are right that you will get a circular loop happening) start with a point load then work out deflection. Repeat for a range of point loads until you get the deflection you want.

To make it even easier use a concrete beam program that works out your Ieff automatically.

 

[miecz]

assuming an I_eff, calculating a point load/moment, and verifying the I_eff assumption in an iterative process ends up Not Converging (I_eff gets larger causing the moment to get smaller causes I_eff to get even smaller, etc.).

Is that true? It would seem that it should converge. I would expect a larger I_eff would cause the moment to get larger, which in turn would reduce the I_eff. This in turn should reduce the moment, which would result in a larger I_eff, and so on. Am I missing something?