Stability Coefficient (Theta)

[KenStructural]

Please give me some reference or somebody who can explain to me on how to compute for the stability coefficient (theta) per Section 12.8.7 of ASCE P-delta effect. The computation of total vertical load has no individual load factor, is these simply means service load combinations? and another thing which confuses me is the drift, if the drift is the maximum drift derived from modal combinations? Any reply is very much appreciated.

 

[JoshPlumSE]

The old SEAOC blue book is a good reference for this coefficient. Though I haven't read it in awhile and don't know if it addresses your specific questions. The NEHRP documents may be a good source as well.

The total vertical load used in this calculation is allowed be based on unfactored loads. After all, the definition of Px does say that "no individual load factor need exceed 1.0". Personally, I use factored loads which should always be conservative.

The lateral drift used in this calculation is based on 12.8.6. But, then it is divided by Cd and multiplied by Ie. Therefore, it is essentially the elastic drift of the structure.

I believe the idea behind the stability coefficient is two-fold:
1) It gives you a simple method for calculating the P-Delta effect by hand.
2) The code used a max value for this coefficient which is intended to insure post-event stability (hence the 1.0 load factor on vertical loads). The idea being that the damaged structure will retain some inelastic drift. But, that it should still be stable under the applied vertical loads. It may need to be repaired or torn down post event. But, it should not fall down on its own.

 

[johnbridge231]

JoshPlum is right. In simple words, the theta coefficient is the ratio between the second order and first order moments.

Analysis and Design of arbitrary cross sections
Reinforcement design to all major codes
Moment Curvature analysis

http://www.engissol.com/cross-section-analysis-design.html

 

[KenStructural]

JoshPlum, i have one last question for this, definitely i agree with the application of P-Delta effect on the structure, my question is if theta is greater than thetamax then the structure is potentially unstable, is this requirement taken care of the global stability of structure aside from overturning?

 

[JoshPlumSE]

The NEHRP commentary says this about the limit:

"The equation establishes the maximum stability coefficient permitted. The intent of this requirement is to protect
structures from the possibility of stability failures triggered by post-earthquake residual deformation."

So, yes the structure would be potentially unstable. Maybe not under normal applied loads, but unstable after it has sustained significant damage from a design level earthquake. My belief is that these seismic stability requirements will normally be more stringent than other code stability requirements (such as AISC chapter C). So, there is some justification to saying that overall global stability is satisfied if this provision is met.

That being said, I would always still follow all the individual code provisions (AISC Chapter C and ACI's P-Delta requirements and such) related to structure stability.... even if the seismic stability concerns are likely to govern.