Effective moment of inertia for members with axial loads

[Agent666]

Hi all

In the process of updating//checking a spreadsheet a colleague wrote and I've noticed something which I wonder if someone has come across before, or can offer some advice. I think I know the answer but just want some confirmation that I have not gone mad!

Calculation relates to determining the effective section properties with a member carrying moment and axial load.

Now I can determine the cracking moment (M_cr) by using the typical equation f_r * Ig / y_t, but with the modification that its (f_r + N/A_g)*Ig / y_t when there is an axial load present. This could be tension or compression. Code equations are generally setup for beams with only flexure.

We can then calculate the cracked moment of inertia (I_cr) using normal means based on a calculated neutral axis depth and a given reinforcement arrangement. Keeping in mind that when there is an axial load the neutral axis depth is deeper than say the same member with no axial load.

This results in some configurations where the actual moment M_a at the serviceability limit state is obviously less than the M_cr, suggesting that under combined moment and axial load that its uncracked. Calculation for I_cr (due to the increased neutral axis depth) results in an I_cr value larger than Ig (bit nonsensical).

The point that's up for grabs is I believe I_cr should be limited to Ig as a maximum at this step before going on to calculate I_e (code doesn't explicitly say this because this is never an issue for members with no axial load). I_e equation is then limited to the upper bound of I_g
Agree/disagree?

So essentially the equation for I_e should really be:-
=MIN(I_g,(M_cr/M_a)^3*I_g+(1-(M_cr/M_a)^3)*MIN(I_g,I_cr)) instead of
=MIN(I_g,(M_cr/M_a)^3*I_g+(1-(M_cr/M_a)^3)*I_cr)

 

[JAE]

I've usually calculated Icr only as a bending-only member which is usually much lower than Ig, not considering the change in neutral axis due to axial loads.

This may or may not be technically correct - but in design of axial-bending members, the lower Icr is the lower Ie will end up being for the various P-M conditions.

This will increase the second order effects in your subsequent analysis and be a conservative design.

ACI uses Ig as the gross section moment of inertia neglecting the reinforcing so to answer your question directly I'd say yes, you could limit Icr to a maximum of Ig.

But I like to stay on the conservative side with this as the Ie value is only a rough approximation anyway of the member stiffness.


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[Agent666]

I also typically ignored the axial load in the past or never gave it a second thought, but in checking this spreadsheet I got to thinking that it's not strictly correct to do this.. Which lead to a question about how to allow for it in an Ie sense.

 

[JAE]

Just to be clear - I don't include the axial loads in deriving Icr.
But I do include the axial loads in deriving Mcr.



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[techiestruc]

If you want to intensively compute the effective moment of inertia and centroid of a section with combined axial and moment. You may refer to Concrete Bridge Practice by Raina (Chapter 25).

I have used this reference to determine more approximately the stress acting on the extreme tension reinforcements which is used to compute the cracking width based on EN. Ignoring the axial effects (compression) especially on columns underestimates the capacity of section to resist cracking. As for tension, it overestimates the section cracking resistance.

 

[Agent666]

Thanks I will try find a copy.

The issue is something that might be conservative for the member in question might not be conservative for other elements its connected to if they have a different stiffness, so a reasonable estimate of the stiffness is really required knowing what we know (i.e. axial load and moment).

The problem I see with your method JAE in an effort to add some conservatism is that when a member is in tension it is unlikely to be conservative as janssenbrian notes.

 

[JAE]

I wasn't talking about tension members.

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[rapt]

If Ma is less than Mcr then it is uncracked and you do not use Ie, you use Ig.

Ie can never be greater than Ig. Where is the problem? You cannot create stiffness greater than the gross by applying an axial compression.

The code writers probably did not think of saying Ie is limited to Ig for fear of making themselves look stupid.

 

[Agent666]

Thanks, I've always been on board with all that. The codes I regularly deal with (NZS3101, ACI318, etc) state that Ie is never taken greater than Ig.

I wasn't picking up the fact that if Ma was less than Mcr, then that's the end of the calculation, end of story (Ie=Ig). Going on the calculate Icr after this just results in nonsensical answers.

I did a few sensitivity checks using JAE's method, and when neglecting the axial load in my 'test' columns for the calculation of Icr but accounting for it in the Mcr calc, there was generally a 5-20% reduction in the effective moment of inertia depending on the level of axial load and magnitude of the bending moment (closer to the 5-10% end of things in most cases). So its a pretty reasonable simplification, and simplifies the calculations a little.

Again thanks for all your input!

 

[rapt]

Agent66,

Doubts have been expressed about the overall equation, especially at stress levels just below and above the cracking moment. This equation predicts no sudden reduction in stiffness on formation of the first crack and is un-conservative until the moment is significantly higher than the cracking moment.

To account for this, the Australian code never allows Ie to be greater than about .6Ig for RC members!

The other thing that is not considered is that reinforcement itself within the concrete induces axial tension due to shrinkage restraint. The Australian code has picked up on this and includes a tension stress component from restraint. This can vary from .5MPa to 1.5 or 2MPa depending on how heavily reinforced the section is.