How Accurate is Your Lateral Analysis?

**Q and A**

Tech Talk Date: July 2016

A YouTube recording of the Tech Talk can be found at: https://www.youtube.com/watch?v=HKuRhO0MGcc

The recording and Q and A from a related webinar can be found here: Accuracy, Precision, and Practicality in Structural Analysis

The topic of this Tech Talk was accuracy of analysis and design methods. Various topics were discussed in relation to this topic. Most of the questions revolved around those topics.

Q: P-Delta effects represent a softer structure hence a longer period - why is the base shear larger?

A: In the presentation an example was presented of a structure analyzed using a response spectra analysis. The analysis was run first with P-delta included, and then again without P-delta. Between these two analyses the periods were different and the base shears were different, showing that ignoring P-delta when performing response spectra analysis (which is what is done when the iterative P-delta method is used) results in less accurate results. With P-delta considered, the periods were longer, as expected. And generally when the period is longer the resulting base shear is smaller; that is certainly the case for the equivalent lateral force method. However, for this model, the structure with the longer period had the greater base shear, and hence the question. This analysis was a response spectra analysis, so it seems that the longer period was more close to a resonant period, and resonance can cause an increase in the seismic forces and hence the increase in the base shear.

Q: Do the columns, walls, and slabs have any stiffness reduction factor before going to lateral analysis?

A: For concrete members ACI 318 requires that crack factors be applied in the analysis. In the RAM Structural System the user is given the ability to specify these values, and then they are applied to the members for the analysis. For steel members designed using the AISC Direct Analysis Method, a stiffness reduction is required. If this option is selected, steel member stiffnesses are reduced for the analysis.

Q: Is the factor B2 of 1.08 always the case or do I need to run both analysis and get the ratio?

A: The B2 factor is a function of the total gravity load on the structure and the total buckling strength of all of the columns providing stability to the structure. This is different for every structure, so it must be calculated. The B2 factor can be calculated approximately using Eq. (A-8-6) of AISC 360-10. This represents the ratio of the lateral translation including P-delta and the lateral translation excluding P-delta. That ratio could be calculated exactly by performing both analyses and taking the ratio. Remember though that the purpose of the B2 factor is to avoid the necessity of running a P-delta analysis, so if a P-delta analysis is run it is not necessary to apply the B2 factor.

Q: Is it reasonable to run an analysis that includes P-delta effects for all the other load cases that are part of the seismic combination and then combine the results with the response spectrum analysis of the lateral loads?

A: This may be a practical approach, but remember that it losses some accuracy. If the P-delta effects are large this may be considerably unconservative. You can get a sense of the extent of the error by calculating the B2 factor. That would be an approximation of the degree of error incurred by ignoring the P-delta effects for the seismic forces. For example, if the B2 factor is 1.20, ignoring P-delta in the response spectra analysis would result in an error of about 20% in the earthquake design moments used to design the member.

Q: STAAD.Pro vs. RAM frame, which software would you recommend, seeing as both are Bentley products?

A: The RAM Structural System is specifically for building structures. This allows for faster modeling and more specialized analysis, design, and reporting. Generally the RAM Structural System is preferred for buildings. STAAD has a wider selection of building codes, some of which aren’t available in the RAM Structural System. It is also general purpose, suitable for any structure, building, plant, tank or frame of virtually any configuration. Both programs have their strengths, so it depends on what is being analyzed and designed. With the Structural Enterprise License, a bundled license of STAAD and the RAM line of products, you can have both:

https://www.bentley.com/en/products/product-line/structural-analysis-software/structural-enterprise

https://www.bentley.com/~/asset/14/2716.ashx

Q: To consider a concrete slab as diaphragm, would you recommend to use plate element or solid meshing?

A: Using a plate element would be more efficient that a 3D ‘brick’ or solid element. For what we are trying to accomplish with the diaphragm model, which is to get the distribution of the lateral forces to the frames and to determine the diaphragm shear, the simpler element is certainly sufficient.

Q: Why is the geometric stiffness approach not appropriate for a truss or tower?

A: The geometric stiffness method that was implemented in the RAM Structural System was derived based on assumptions of conditions common in buildings such as distinct floors, floor diaphragms, reasonable correlation between a lumped mass model and the real structure, etc. The assumptions don’t hold true for a truss or a transmission tower. There may be derivations of the geometric stiffness method that would be appropriate for those types of structures, but I am not aware of them.

Q: Is the geometric stiffness approach what is generally used in place of an iterative p-delta analysis? Since you can't combine an RSA with an iterative p-delta, isn't it the only option?

A: There are three common approaches: iterative, geometric stiffness, and moment magnification (using B2, for example). Each has its appropriate uses and conditions. The Iterative approach is often touted as the most robust, but that doesn’t account for the loss of accuracy resulting from ignoring P-delta effects. This can be remedied by applying the B2 factor, but then it must be recognized that the iterative method isn’t necessarily the most accurate approach. The geometric stiffness method generally works very well for buildings, and is what is implemented in the RAM Structural System, along with an option to use B2 instead.

Q: The axial forces that the geometric stiffness is based upon is usually based on a single load case, which STAAD recommends to be Dead plus some percentage of Live load. Doesn't this underestimate the P-Delta effects since we're excluding the sometimes large axial loads introduced by seismic/wind?

A: When calculating the geometric stiffness modifications per the geometric stiffness method it is only necessary to consider the net forces acting on the structure as a whole. If we assume that Wind and Seismic cases have no net vertical force on the structure as a whole (or that their net force is small compared to the gravity forces), it is only necessary to consider the Dead Load and some portion of the Live Load (the portion that is reasonably expected to occur concurrently with the design level winds or earthquakes) in the calculation of these modifications. The member stiffnesses are then modified for the analysis, and all of the load cases applied. Each individual member will now experience the P-delta effects of the large wind and earthquake overturning forces acting on the member that has been displaced the greater distance due to the geometric stiffness modification.

Note that wind uplift actually helps the structural stability, so ignoring wind uplift in the geometric stiffness calculations is conservative. The contribution of any vertical seismic forces are generally small and traditionally ignored in P-delta analysis. The increase in the column axial load and moments is dealt with by adding the Ev = 0.2S_{DS}D term to the load combinations.

Q: How does the geometric stiffness approach approximate a p-delta analysis?

A: There is more to it than can be explained here, but roughly, based on the gravity loads on the entire structure a stiffness reduction is determined such that the resulting lateral displacements from a single analysis would be the same as if the structure were analyzed iteratively. The key here is the way the stiffnesses are reduced. When analyzed, the softened structure deflects the same under a single analysis as the unstiffened structure would deflect under a series of iterative analyses. Since member forces are derived from the displacements, the two models will result in the same set of forces since they have the same set of displacements. Note that the geometric stiffness method is an approximate method, based on assumptions of an ideal structure, but it has been found to compare favorably with iterative methods in most cases (unless the structure diverges drastically from the assumptions of the ideal structure).

Q: Isn't the structure further softened by the axial loads induced by the seismic forces?

A: For every column that is in compression (which softens the stiffness of the column) there is another column that is in tension (which increases the stiffness of the column). The net effect on the structure is that the earthquake forces do not impact the stiffness used in the large P-delta analysis.

Q: Does STAAD offer any way to help us calculate the ASCE stability coefficient?

A: Although it doesn’t calculate the coefficient directly it does provide useful reports of the horizontal shear and displacement, values needed for the calculation of the stability coefficient.

Q: Is the building period the 1st mode shape or the average from all modes?

A: It is not the average. The modal period that is considered the building period for a given direction is the period that dominates the combined modes for that direction. In most structures the 1^{st} mode period is considered the building period in one direction and the 2^{nd} mode period is considered the building period in the orthogonal direction, but that is not always the case. Using the Periods and Modes report from RAM Structural System it can be easily determined:

Looking at the Modal Participation Factors we see that the 1^{st} mode is almost exclusively an X-direction mode (because of the large participation value compared to the others), so what is considered as the building period in the X-direction is 1.5857 seconds. Similarly, the 2^{nd} mode is almost exclusively a Y-direction mode, so the period in the Y-direction is 0.6080 seconds. The 3^{rd} mode is also an X-direction mode, but not as dominant as the 1^{st} mode, so it isn’t considered as the building period. The 4^{th} mode is a rotational mode. This can be seen by looking at the mode shapes: