AISI Combined Compression and Bending - Amplification Factor (Phi)

[P1ENG]

Per the commentary of C5.2.1, the alpha factor (1/[Φ])of the standard is a magnification factor. The commentary also states that this amplification factor is due to the applied moment and the secondary moment resulting from the applied axial load P and the deflection of the member. This sounds like P-[Δ] to me, but perhaps is P-[δ].

My problem is this: I have a RISA-3D model with P-[Δ] being considered. I am pulling member forces from the model to check in CFS, but the member is failing greatly because the axial load is about 60% of capacity, which makes the [Φ] factor 2.5 (1/40%). Therefore, if I had a member check using P/Pa + M/Ma (0.6 + 0.4) = 1.0, but per the specification I need to do P/Pa + [Φ]*M/Ma (0.6 + 2.5*0.4) = 1.6. This is a large hit to take when I think the amplification factor is only there to conservatively account for P-[Δ] affects, and I have the actual P-[Δ] forces.

In summary, AISI does not address second-order affects, buy my model is considering them. Therefore, can I ignore the amplification factor on the M/Ma ratio?

P = Applied axial compression
Pa = Allowable axial compression
M = Applied bending moment
Ma = Allowable bending moment
*Equations above are simplified.

Juston Fluckey, E.I.
Engineering Consultant

 

[ishvaaag]

When using eq. 4 in the attachment you see that the setup of the equation for the bending components is a usual one readily dealt with P-Delta (and P-delta for the assumed imperfection standing in the column strength formula). So for that flexural part of the equation I think you are right to assume no further amplification is needed; it would require a complete assessment of the specification to confirm, and I have not the time now for the thing.

You may operate conversely and disable P-Delta, then proceed with the amplification, for a safer approach.

Then, it is clear that the estimate of the axial capacity involves the factors in eqs. 5

 

[ishvaaag]

As usual, the P-Delta analysis must be established at the factored level for what above stated be true.

 

[P1ENG]

You may operate conversely and disable P-Delta, then proceed with the amplification, for a safer approach.

I did turn off P-[Δ] to see the effect, and the bending moment in the member decreased ~7%. Taking this lower value with the amplification factor would certainly help, but I was really hoping for a consensus from the community here that a second-order analysis would negate the need for the amplification factor which would help me the greatest.

Also, the 9th Ed. of the AISC used a similar approach to increasing the moment interaction, but in the 13th Ed. it was removed. Again, the 13th Ed. requires a second-order analysis, hence further support for my reasoning. I will be doing some research today in both the AISI and AISC.

Juston Fluckey, E.I.
Engineering Consultant

 

[RFreund]

Where is commentary of C5.2.1 from or what manual?

I believe that factor comes from and is explained in Theory of Elastic Stability by Timoshenko however I can't remember if that accounts form P-d (axial and bending) effects or P-D (axial and displacement).
However for using RISA and the 13th Edition see the "Direct Analysis Method for 13th Edition" here:

http://www.risa.com/d_techpapers.html

It will explain how they handle p-d and P-D

Or you could use one of the simplified approaches and do your second order analysis using B1 (p-d) and B2 (P-D).

EIT

 

[P1ENG]

-RFreund

The commentary came from AISI's Standard of the Specification Design of Cold-Formed Steel Structural Members. Also, thank you for the RISA link. I had reviewed the help section, but that paper was much more informative.

Juston Fluckey, E.I.
Engineering Consultant

 

[RFreund]

It appears that the Φ factor is for axial and bending moment. I say this because they say that this is accurate for a braced member. They then continue to say that it is conservative for an unbraced member subject to sidesway (P-DELTA)or in reverse curvature. They then modify the equation by Cm (=0.85 or 0.6-0.4M1/M2 . This seems sorta odd as it is just a constant reduction and does not account for how much sidesway. So you could reduce the Φ factor. Or you can use the appendix which which is similar to the DAM method in AISC 13 and what the RISA paper is referring to.

EIT

 

[P1ENG]

This is what I have written to accompany my calculations:

The 2007 AISI Standard: North American Specification for the Design of Cold-Formed Steel Structural Members provides a non-linear ASD method interaction equation for combined compressive axial load and bending in Equation C5.2.1-1. This equation contains an "α" variable applied to the moment terms of the interaction equation that is essentially a magnification factor to due to "the secondary moment resulting from the applied axial load and the deflection of the member", i.e. the amplified effect of the local-axis moment term is affected by the utilization of the local-axis compressive capacity of the column. Therefore, if a second-order analysis is performed, the resulting axial loads and bending moments would negate the need for the redundant affects of the amplification factor.

Further support of this argument comes from AISC's Specification for Structural Steel Buildings (ANSI/AISC 360-05) in the 13th Edition of the AISC Steel Construction Manual. In the ASD 9th Edition of the AISC Steel Construction Manual, the combined axial and bending interaction equation (Equation H1-1) had a similar amplification of the moment term as the AISI interaction equation. However, the interaction equation (H1-1a and H1-1b) of the ANSI/AISC 360-05 removed the amplification factor, making the equation linear. Review of the commentary for this section reveals that the amplification term was "The amplification of the interspan moment due to member deflection multiplied by the axial force (the P-δ effect)" and that the linear approach can be taken when the required second-order axial and flexural strengths are used.

RISA-3D has the ability to consider P-Δ but not P-δ, which is a shortcoming of most modeling software not specific to very slender structures. However, per the attached document released by RISA: Practical Analysis with the AISC 13th Edition, the P-δ analysis can safely be ignored under certain circumstances. The particular model under consideration contains multiple nodes along the length of the cold-formed members, partially accounting for P-δ as described in the document, and therefore Pyramid1, Inc. feels comfortable ignoring the remaining affects of P-δ.


Juston Fluckey, E.I.
Engineering Consultant

 

[P1ENG]

O.K., last response for those interested or stumbling upon this later.

Appendix 2 in the 2007 AISI covers the second-order analysis. Basically, it conforms to what I have written above, but has more specifics. Thanks to Bob Glauz at RSG software (CFS) for providing the reference. I was using both the 2001 and 2007 AISI specifications when trying to figure this out, and apparently overlooked that section in the table of contents.

Juston Fluckey, E.I.
Engineering Consultant