Fundamentals of Analysis and Design for Stability
Webinar Date: January 2016
The recording of this webinar is available at: Fundamentals of Analysis and Design for Stability
The slides from the presentation can be downloaded from here: Analysis and Design for Stability - Slides.pdf
The process of designing for strength and for deflection and drift are generally well understood, but the issue of Stability is less understood and often ignored. What can potentially cause a structure to be unstable? How can those instabilities be discerned and addressed in the design of a structure? In this webinar the fundamentals of Stability, including P-delta, out-of-plumbness, and member imperfections were presented, with an explanation of how Stability or a lack thereof can impact both the member design forces and the structural drift and deflection. Strategies and techniques for addressing stability issues through 1st- and 2nd-order analysis, amplification of design forces, and reduction of member capacity are presented. Code requirements related to stability are often obscure and the intent is unclear; several modern building codes and specifications are highlighted and compared, including ASCE 7-10, AISC 360-10 Direct Analysis Method, ACI 318-14, Eurocode EN 1993-1-1:2005, and Australia AS 4100-98. It is shown how RAM Frame and other software can be used to consider and satisfy these stability demands.
The following questions were submitted during the presentation of the webinar.
Questions and Answers:
Why is it necessary to perform a 2nd-order analysis? We use a 1st-order analysis for most common buildings.
Historically it was difficult to include P-delta and the various 2nd-order effects in the analysis of a structure (that was often done by hand or charts). When the equations for calculating member capacity were established in the various specifications they were calibrated (i.e., generally conservative) to be appropriate for the analyses being performed at the time. The ability to perform more sophisticated analyses has become more accessible through computer automation, so therefore the capacity equations can be made more precise. Recognizing this, the AISC 360 specification committee developed more refined capacity equations based on the assumption that a more sophisticated analysis would be performed to calculate the member forces. This resulted in the development of the requirements for the Direct Analysis Method, which describes in detail what needs to be done with the analysis so that the capacity equations are valid. The Effective Length Method and the First-Order Analysis Method are given in Appendix 7 of AISC 360-10 as acceptable alternatives to the Direct Analysis Method if the P-delta effects are small. This means that the capacity equations of Chapters D through K are only valid if you have performed an acceptable analysis. If the analysis hasn’t accounted for the stability effects, the capacity equations will be unconservative. This is true for all modern steel and concrete design specifications, not just AISC 360.
What quick method you do recommend to determine if P-delta is critical or not? Or should we go straight to performing a P-delta analysis?
For large P-delta (P-D), Appendix 8 of AISC 360-10 gives the method for calculating the B2 factor, a “multiplier to account for P-D effects”. It is calculated for each story, and is a function of the total vertical load, the story height, the story drift, and the horizontal force that produced that drift. This can be calculated by performing a 1st-order analysis. If the value of B2 is 1.0 (or less), P- D is not critical; otherwise B2 will indicate what impact P-D has. For example, if B2 is 1.2, P-D increases the design forces due to the lateral forces by 20%. Similarly, Appendix 8 gives a method for determining the impact of small P-delta (P-d), with a B1 factor.
ASCE 7-10 Section 12.8.7 defines a Stability Coefficient, q, for which a value less than 0.10 indicates that the analysis does not need to include P-delta. Although appropriate for other structures, I recommend that this not be used to determine that a P-delta analysis is not necessary for a steel structure; AISC 360 requires that P-delta be considered, and as explained above, the AISC member capacity equations are not valid unless P-delta has been accounted for.
Having said all of this, I recommend that you always include P-delta in your analysis and avoid the effort required to determine if it is significant or not. Most software readily accommodates this.
For analysis of steel structures as per AISC 360-10, should both P-D and P-d effects be considered simultaneously or can any one be done individually?
AISC 360-10 requires that both be considered simultaneously. They can be handled using analysis methods, or the B1 and B2 factors, or a mix of those.
In contrast, Australia AS 4100 gives a simplified approach using the larger of the two, but permits a more detailed approach (similar to AISC 360) using both.
How is the magnitude of B2 calculated?
B2 is given by Equation (A-8-6) in Appendix 8 of AISC 360-10. It is calculated for each story; all of the members on a story have the same B2 value. To calculate this value it is necessary to first perform a 1st-order analysis on the structure. From that analysis determine the interstory drift, DH (which is the displacement of the story being considered minus the displacement of the story below). Determine the total vertical load, Pstory, imposed on all members, not just those on Frame members. H is the story shear produced by the lateral forces used to compute the drift. From these, B2 can be calculated.
If required to use both ASCE-7 and AISC, which one governs P-Delta method?
Section 12.8.7 of ASCE 7-10 addresses requirements for P-D and stability. A Stability Coefficient, q, is defined, with limits on what is allowed (qmax). That requirement must be satisfied even if the structure is being designed under AISC requirements. Section 12.8.7 of ASCE 7-10 also gives conditions under which it is not necessary to consider P-D effects. However, AISC 360 always requires that P-D effects be considered, so in my opinion a P-D analysis should always be performed. One could argue, perhaps correctly, that they “considered” P-D by relying on Section 12.8.7 of ASCE 7-10 and determined that it is not necessary to include P-D effects in the design.
Do you need to use K values in addition to a P-Delta Analysis to capture all stability issues?
If your analysis and design conform to the requirements of the Direct Analysis Method given in AISC 360-10 Chapter C (including P-delta, stiffness reduction, notional loads, etc.) or to the requirements of the First-Order Analysis Method given in AISC 360-10 Appendix 7, then you are permitted to use an effective length factor, K, of 1.0, even for moment frames. If you are using the Effective Length Method given in Appendix 7 you must determine and apply the effective length factor, K.
How does the RAM Structural System combine the reduced stiffness due to geometric non-linearity (the Geometric Stiffness Method’s stiffness reduction for P-D) and the residual stresses (Direct Analysis Method’s 0.8 and tb stiffness reductions)?
The program applies both simultaneously. For example if the Geometric Stiffness Method determined that the stiffness needed to be reduce by a factor of, say, 0.95 for the P-D analysis, the program would reduce the stiffness by multiplying the stiffness by (0.95)(0.8tb). This is appropriate since each is accounting for different effects that occur simultaneously.
Can response spectrum be combined with the AISC 360 Direct Analysis Method?
Yes, it can, but not all software is capable of doing that. The Direct Analysis Method’s 0.8 and tb stiffness reduction factors can be applied to the stiffnesses of the members in the response spectrum analysis, and the Notional Loads can be combined with those results if required. The difficult comes in the requirement to consider P-delta. If the software uses an iterative method for P-delta it cannot be included in the response spectrum analysis because an iterative analysis and a response spectrum analysis are incompatible. So if the software performs an iterative analysis for P-delta, the response spectrum results will not include P-delta and will therefore not conform to the requirements of the Direct Analysis Method. On the other hand, as explained in the webinar, the Geometric Stiffness Method for P-delta analysis involves stiffness reductions (instead of iterations) to get to the same result. These stiffness reductions can be applied to the analysis model that is used in the response spectrum analysis, and the result is that the response spectrum analysis results include the P-delta effects.
The recording of a webinar previously presented, A Practical Approach to Using the Response Spectra Analysis Method, is available for viewing free on-demand. It can be found at:
A Practical Approach to Using the Response Spectra Analysis Method
When using reduced stiffness the analysis model will experience higher deflection so do the final deflection values need to be reduced?
The 0.8 and tb stiffness reduction required for member design by the AISC 360 Direct Analysis Method should not be applied to the analysis model when analyzing for the story drifts. They should only be applied to the analysis model when determining member design forces. This requires two separate analyses. See the Commentary on page 16.1-280 of the AISC Steel Construction Manual 14th Edition. Also, see the wiki on Bentley Communities titled ASCE 7, AISC 360, and the Direct Analysis Method in the RAM Structural System.
It was written specifically for the RAM Structural System, but the methodology and concepts are applicable regardless of the software used.
In the case of steel frames, lateral bracing contributes to the stability of the structure. How is the software taking account of the imperfections in the connections and the bracing itself?
The nature of connection imperfections is not well documented nor easy to quantify. Consequently it isn’t commonly explicitly modeled in the analysis of most structures. It is usually ignored. If it is felt that for a given condition the effect might be significant, and if the effect of the connection imperfection can be quantified, connection springs can be assigned to the member ends. For the braces themselves, if the Direct Analysis Method is being used, the 0.8 stiffness reduction is applied to the brace stiffness.
How do we calculate first order drift and second order drift?
Drift is the difference between the lateral deflection at one level and that of the level below. For example, if the Third story deflects 1.0” horizontally and the Second story deflects 0.75” horizontally, the Third story drift is 1.0 – 0.75 = 25”. When referring to 1st-order drift vs. 2nd-order drift, what is usually meant is the drift without the P-delta effects vs the drift with P-delta effects (since P-delta is usually the largest contributor to the 2nd-order effects). So to get the 1st-order drift perform the analysis without the P-delta analysis, and to get the 2nd order drift perform the analysis with the P-delta analysis.
Can the AISC 360 Direct Analysis Method be used in the concrete columns?
The requirements of the Direct Analysis Method have been calibrated to produce member design forces appropriate for use with the steel member capacities given in Chapters D through K. They would not necessarily be appropriate for the design of concrete members. If a model was a mix of steel and concrete members it would be appropriate however to apply either the stiffness reduction factors or else cracked factors (e.g., Table 126.96.36.199.1(a) of ACI 318-14) to the concrete members for the analysis to determine the design forces used for the design of the steel members. ACI 318 and other concrete codes have their own specific requirements for consideration of stability effects in the analysis used to determine the member forces in the design of concrete members. In a mixed model the analysis used for the AISC 360 Direct Analysis Method would probably be acceptable for use in designing the concrete columns, but it would be necessary to determine and use the Effective Length Factor, k, on the concrete columns in Sway Frames; you couldn’t automatically use K=1 as you can for steel columns, because the concrete code doesn’t permit it.
For notional loads per AISC 360-10, when do you use 0.003 vs 0.002?
The basic value of 0.002 is required as specified in Section C2.2b(1). Section C2.3 give requirements for the reduction of member stiffness, including the application of a tb factor. This factor is a function of the axial load in the member, which means that the tb factor can only be calculated after the analysis is complete, but since the tb factor impacts the analysis results the application of the correct tb factor requires iterations of analysis, calculating new tb factors, and reanalyzing. To avoid this the Specification permits that, in lieu of applying the tb factor, an additional notional load of 0.001 times the gravity loads can be applied (for a total of 0.003). Note that tb is usually 1.0, which means it has no impact on the stiffness reduction. When using the RAM Structural System it is recommended that 0.002 be specified for the Notional Loads and that tb be set to 1.0. The AISC 360 Direct Analysis Validation report will indicate if tb should have been less than 1.0 for any members, and the engineer can take the necessary steps by either specifying a smaller tb, specifying the larger Notional Load, or by increasing the size of those members that require a smaller tb such that with the larger size tb can be 1.0. The latter approach is usually the most economical approach.
When using LRFD design what are the load factors to be used for Notional Loads?
The Notional Loads use the same factors as the associated gravity load case. For example, if the load factor on Dead Load is 1.4 the factor on the Dead Load Notional Load is 1.4.
Do I need to include Notional Loads and Seismic Loads in the same combinations?
AISC 360 only requires that Notional Loads be included in the gravity-only combinations unless the ratio of the 2nd order drift to the 1st order drift is greater than 1.7. If that ratio is greater than 1.7 (which is unusual) you are required to include the Notional Loads in all combinations.
Eurocode appears to require that Notional Loads be included in all combinations.
Australia AS 4100 requires that Notional Loads only be included with the gravity-only load combinations, not with wind and seismic combinations.
How does ACI 318-14 address out-of-plumbness?
ACI 318 doesn’t directly address the issue of out-of-plumbness. It doesn’t require that it be modelled or that Notional Loads be applied.
How does Eurocode 1993 address out-of-plumbness and other stability criteria?
This was discussed briefly in the webinar, but in particular see Section 5 Structural Analysis. In particular, Section 5.2.1 Effects of deformed geometry of the structure indicates when an analysis that considers the 2nd-order effects is required or not (based on the value of acr). When required, Section 5.2.2 Structural stability of frames indicates the acceptable methodologies. Section 5.3 Imperfections identifies some of the imperfections that should be considered. Out-of-plumbness (“initial sway imperfection”) is dealt with by using Notional Loads, defined in Section 5.3.2(3) and Section 5.3.2(7). Out-of-straightness (“initial bow imperfection” can be dealt with by applying member notional loads as illustrated in Figure 5.4, but the use of the equations in Chapter 6 provide a more practical way of accounting for this effect.
Is it OK to apply the 10 percent of the gravity loads as the component for notional loads? I see that the range from 0.002 to 0.003 is allowed per AISC so why is 10 percent used?
I am not aware of this requirement in any of the codes. It appears to be rule-of-thumb to ensure that the structure is designed for some minimum lateral force. BS 5950-1:2000 has a similar requirement: Clause 188.8.131.52 Resistance to horizontal forces requires that “the horizontal component of the factored wind should not be taken as less than 1.0% of the factored dead load applied horizontally.” Otherwise, the various requirements for Notional Loads appear to be in the range of 0.2% to 0.5%, not anywhere near 10%.
You recommend using the Effective Length Method for braced frame buildings to take advantage of the fact that K already equals 1. It appears that RAM Frame is only set up for running the Direct Analysis Method. Is this correct? Do you have any suggestions for running the Effective Length Method for a braced frame building?
Typically columns in braced frames are non-sway, and therefore K=1.0 (or potentially less than 1.0, but I am unaware of engineers that use K values less than 1.0 for buildings). RAM Structural System assumes that if AISC 360 is selected as the design code, K=1.0. This is appropriate for both the Direct Analysis Method for all frames and for the Effective Length Method for braced frames, so RAM Frame is capable of doing either method. If you are going to use the Direct Analysis Method you need to select the option to Use Reduced Stiffnesses; if you are going to use the Effective Length Method you should not select that option. The program does not allow you to use the Effective Length Method for moment frames because it doesn’t calculate and apply the K factors (other than 1.0) when AISC 360 is selected as the design code; we have had very few requests for that, since the preferred method for designing moment frames is the Direct Analysis method.
How is analyzing load cases, and then super-positioning, any easier or quicker than analyzing combinations? You still need to combine all the cases into the 100's of combos to get results.
Analyzing a structure, with all of the matrix operations, etc., takes significantly longer than creating a load combination. Limiting the analysis to a few load cases and then combining the results into the 100’s of combinations is significantly faster than combining the applied forces into the 100’s of combinations and then analyzing the structure for all of those combinations. Perhaps more significant, analyzing a few load cases means that there is much less computation results to review, and you are able to see precisely what effect an individual load case had. This tends to get lost if you are only reviewing load combination results; how can you tell, for example, how much of that was caused by the Live Load? Furthermore, the mountain of data produced by analyzing all of those load combinations discourages the engineer from even trying to review the results. It is much more manageable to deal with a few load cases.
Sometimes during analysis I get an error message indicating that an instability was found, and the analysis stops. Which instability is this referring to, and how do I fix it so that it will analyze?
When this error occurs it is usually during the large P-delta analysis. If the sizes initially assigned to the frames are too small, the frames may be truly unstable. To solve this, assign larger sizes. In some cases it is the result of the members being pinned at the joints such that there is no stiffness about an axis, nothing restraining the members to prevent large displacements or rotations. To solve this, provide proper Fixity assignments that properly represent the real structure.
When such an error occurs it is often helpful to turn off the P-delta analysis and perform quick checks of drift and strength. Once these are satisfied the model will usually then successfully analyse without the instability errors.
Is there any implication on the 3D analysis if transfer beams are PT? These beams would be stiffer than reinforced concrete beams and hence the redistribution of forces would be different.
Yes, the higher stiffness of the prestressed transfer beams could (and almost certainly do) have a significant impact on the distribution of forces. The vertical deflection of the frames above caused by the deflection of the transfer girder has a tremendous influence on the stiffness of those frames. The true stiffness of those transfer beams should be modeled as accurately as possible.
RAM Structural System Questions and Answers
Does the RAM Structural System have the ability to optimize steel members?
The RAM Structural System optimizes the Gravity beams and columns (the floor and roof framing), but it does not optimize the Frame beams, columns and braces; the engineer must assign those sizes. The program then provides features to interactively try alternative sizes to quickly determine what the final size assignment should be.
How about concrete members? Can we also optimize?
Member sizes must be assigned, but the program optimizes the reinforcement based on Code requirements and criteria defined by the engineer, including ranges of bar sizes, bar configurations, etc.
Regarding the lateral displacement or drift, is the latest version of the software capable of checking the capacity of the members by just giving a lateral drift limit? If you just input a drift limit say 25mm based on the code tables, can the software optimize the structure?
No, this would require that the program be able to optimize the size of all of the members so that the frames were sufficiently stiff to only drift to the specified limit under the specified lateral loads. The program does, however, provide a report showing the drift at any points specified by the engineer. These values can be compared with the drift limits and then the engineer can resize members as necessary. There is also a specialized module that identifies which members should be resized to most effectively limit the drift. A recording of a very popular webinar previously presented, Building Drift: Understanding and Satisfying Code Requirements, is available for viewing on-demand; there is a discussion of drift and code requirements related to drift, followed by a demonstration of the programs capabilities regarding drift. It can be found at:
Building Drift: Understanding and Satisfying Code Requirements
How does the RAM Structural System perform stability analysis of non-typical structures (unlike a frame structures)?
The RAM Structural System is special-purpose software specifically for building structures. Hence the features associated with P-delta, Notional Loads, etc., are geared towards buildings. With more effort a non-building structure can be modeled in the program but the program’s method of performing P-delta analysis would not likely be appropriate. STAAD.Pro is more appropriate for that type of structure, and has a full set of features to satisfy the requirements for stability analysis.
How effectively does the RAM Structural System incorporate the various destabilizing parameters discussed in the presentation?
The program offers a robust set of analysis and design options and features, allowing full conformance of code requirements related to stability analysis. The program was written with an emphasis on satisfying the requirements of the building codes through practical analysis and design features that permit the engineer to satisfy those requirements and to produce safe and cost-effective designs while working productively.
When the gravity concrete members are optimized, does it affect the design of the slab system? Or instead transfer the forces in the slab system?
The RAM Structural System doesn’t design the slabs, but the slabs are the mechanism for transferring the loads to the beams. The beams can be specified as T-beams, in which case the program automatically determines the width of the concrete slab acting as the flange of the tee, and the beams are designed accordingly. A different program, RAM Concept, is for the design of floor slabs. In that program the interaction of the beams and slabs is considered in the design of the slab.
Can the software handle other types of slab i.e. slab decking?
Concrete fill on metal deck (for composite beam design), roof decking, and flat slabs with or without drop panels can all be modeled in the RAM Structural System. Decks and slabs can be specified to span either one-way or two-way.
STAAD Questions and Answers
There were several questions specific to STAAD that would require responses longer than could be accommodated here. Information on STAAD is available elsewhere on Bentley Communities. Recordings of past STAAD webinars is available here: http://pages.info.bentley.com/videos/
Information on STAAD training courses is available here: https://www.bentley.com/en/learn/for-users/training-programs
What is the difference between P-Delta and buckling analysis and how can we do buckling analysis using STAAD.Pro?
A P-delta analysis merely determines whether the structure is stable for the loads applied on it. A buckling analysis determines the amount by which the existing load should be magnified, or reduced, to determine the amount of load that the structure can withstand (and still be stable). STAAD has two solvers for doing Buckling analysis – a) The basic (or standard) solver, and, b) the advanced solver. See Section 184.108.40.206.1 of the STAAD Technical reference Manual for more detailed information on the methods used by these solvers.
RAM Elements Questions and Answers
What effects does RAM Elements take into account in its P-Delta analysis?
The second order analysis considers the effect of the lateral displacements (Deltas) at the end of the members in the determination of the member forces (i.e., P-D). The analysis does not consider the second order effects of the curvature of each member (i.e., P-d), the cracking and non-linearity of the material, the creep of the material or the time of application of loads. It uses an incremental-iterative procedure, where loads are incrementally applied to the structural model. Two methods are available for the analysis: The standard or full Newton Raphson method (NR) and the Modified Newton Raphson method (MNR). Note that the program calculates tangent stiffness matrix of the structural model at each iteration and the process is pushed further iteratively until a pre-defined convergence tolerance is satisfied (i.e. the equilibrium point is found). This procedure, in a strict sense, is a non-linear procedure.
Are there plans to incorporate the Direct Analysis Method into RAM Elements?
Although there is no single command that says to perform the Direct Analysis Method, and although the requirements are not automated, everything required for that method can already be accommodated by the program. Member stiffnesses can be modified by the 0.8 and tb factors, P-delta analysis can be performed, Notional Loads can be created and applied, and the appropriate combinations can be created. There are no plans at this time to further automate the process.
RAM Connection Questions and Answers
When will RAM Connection and Limcon be integrated?
It is gradually happening. The only three features left to port over to RAM Connection are the Canadian code, Australian code and tubular connections.
What is the difference between RAM Connection and Limcon?
Both are connection design programs. RAM Connection works standalone and integrated with STAAD.Pro, RAM Structural System, RAM Elements and soon STAAD(X). RAM Connection is available in English, Spanish and Chinese. Limcon DXF/detailing drawings only have simple frontal view. Limcon reports do not have detailed equations. Sections for a specific country must be designed with the same code (e.g. you cannot use US sections with Australian code).