State Of The Industry Dam Full

Stability Analysis and Design

Fastrak Building Designer

Do you know what your design software is up to? A closer look at the new stability analysis and design requirements in the 21st century

 CSC Inc., January 2009

Prepared by

State of the Industry

Jason R Ericksen, SE Technical Manager, CSC Inc.

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Disclaimer Do you know what your software is up to? A closer look at the new stability analysis and design requirements in the 21st century. Although CSC Inc. takes great care to ensure that any data, information, advice or recommendations it may give either in this publication or elsewhere are accurate, no liability or responsibility of any kind, including liability for negligence, howsoever and from whatsoever cause arising from or related to their use, is accepted by CSC Inc., its servants or agents.

Figure 1: Complex buildings with sloping members, hard-to-define floors, and nonorthogonal framing are commonplace today. [Screenshot from Fastrak Building Designer. Model courtesy of Fisher Engineering.]

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SUMMARY The 2005 AISC Specification for Structural Steel Buildings brings significant changes to the way stability analysis and design is required to be carried out. For the first time, the specification has something to say about the analysis you and your design software is performing. Do you know how your favorite software package has implemented the new requirements? Does it simply meet the minimum code requirements using old methods intended more for hand calculations or does your software offer you the most accurate, reliable and flexible tools available? AISC STABILITY ANALYSIS AND DESIGN REQUIREMENTS The design of all structures for stability must consider: 1. flexural, shear, and axial deformations of members 2. all component and connection deformation that contribute to the lateral displacement of the structure 3. second-order effects (both P-∆ and P-δ) 4. geometric imperfections 5. member stiffness reductions due to residual stresses The first three requirements are covered by the structural analysis and the first two are included in nearly all analysis packages. Several methods to account for #4 and #5 (including provisions relating to #3) are presented in the specification. One of these, the Direct Analysis Method, is given as the most general and most accurate. DIRECT ANALYSIS METHOD AISC developed the Direct Analysis Method (DAM) as a solution to meeting the stability analysis and design requirements in a modern way that is most suitable for implementation in analysis and design software. This method is not limited in its application, applies to all buildings, and is the most general and accurate approach provided. The requirements for the DAM include: ♦

Second-Order analysis: A second-order analysis which considers both P-∆ and P-δ effects is required.

♦

Initial imperfections: The effects of initial imperfections of the structure geometry are considered by applying notional loads, which are lateral loads proportional to the gravity loads applied at each framing level.

♦

Inelasticity: The axial and flexural stiffnesses of members that contribute to the stability of the structure are required to be reduced. This is to account for the effects of residual stresses that lead to inelastic softening before the members reach their design strength.

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Stability Analysis and Design

♦

Fastrak Building Designer

Effective Length Factor: And now we come to the best part (at least one of the best parts), K=1.0! Setting K to be 1.0 can be allowed because the effects for which it was meant to compensate (initial imperfections and inelasticity) have already been accounted for in the method.

SECOND-ORDER ANALYSIS The changes in the AISC Specification are, in part, in recognition that buildings have changed over the years. Buildings were once (in most cases) straight forward, rectilinear structures with multiple redundancy and permanent or semi-permanent substantial walls within the framing. Now, modern buildings are more just a skeleton of steel so that the frame has to work a lot harder. Even in buildings with regular bays, discrete frames of differing types with irregular layouts are commonly used to meet architectural requirements, as seen in the structure in Figure 2. Therefore, stability analysis is more critical and any second-order effects have to be determined more accurately.

Figure 2: Combinations of moment frames and braced frames with irregular layouts are commonly used to meet architectural requirements, even in buildings with regular bays. [Screenshot from Fastrak Building Designer. Training example.]

A second-order analysis is required for all structures that implement the Direct Analysis Method and can take the form of a rigorous analysis or approximate methods like the amplified first-order method (the B1, B2 method) – which is really intended more for hand calculations and other approximations. The amplified first-order method certainly works well for some buildings, generally those that are simple, regular structures. However, to allow for the greatest range of structure geometry with the highest degree of accuracy in determining internal forces, the best course

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of action is to take advantage of a rigorous second-order analysis. It would be difficult to apply the approximate methods to the structure in Figure 3. Even smaller projects, with relatively small budgets, often have complex framing requirements that will benefit from a rigorous second-order analysis.

Figure 3: Complex buildings make it difficult to apply approximate second-order analysis methods. Today, even relatively small projects often have complex framing. [Screenshot from Fastrak Building Designer. Model courtesy of Scott Wilson.]

FASTRAK BUILDING DESIGNER Fastrak Building Designer implements the Direct Analysis Method fully, using a rigorous second-order analysis. CSC chose this method because it gives the user access to the most powerful analysis and design solutions. It allows for the widest range of building structures with accurate and reliable results. The AISC requirements listed as 1 through 5 at the start of the document are all met by Fastrak Building Designer. The second order analysis included in Fastrak Building Designer accounts for both P-Δ and P-δ effects. The Chen and Lui ‘two-step iterative’ second-order analysis method was chosen to be included in Fastrak Building Designer for several reasons. The main reason is that since the geometric (stress) stiffness is used in the method, there are no significant limitations on its use or applicability for most normal building structures. CONCLUSIONS The approach used by the Direct Analysis Method of putting as much as possible into the analysis without making it impractical (i.e. requiring full FE for the whole building) allows for accurate determination of required forces that can be compared to accurate strengths as determined from the AISC specification. The 2005 AISC Specification with the Direct

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Analysis Method marks a significant step forward for steel design. Fastrak Building Designer takes advantage of all these developments (including a rigorous second-order analysis capturing both P-∆ and P-δ effects) giving the designer increased flexibility, accuracy and reliability. Now you know what Fastrak Building Designer does for stability analysis and design. I will repeat my earlier questions. Do you know how your favorite software package has implemented the new requirements? Does it take full advantage of the Direct Analysis Method and the capabilities AISC now allows in the second-order analysis?

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INTRODUCTION

The 2005 AISC Specification for Structural Steel Buildings (AISC Specification) brings significant changes to the way stability analysis and design is required to be carried out. For the first time, the specification has something to say about the analysis you and your design software is performing. Do you know how your favorite software package has implemented the new requirements? Does it simply meet the minimum code requirements using old methods intended more for hand calculations or does your software offer you the most accurate, reliable and flexible tools available? AISC developed the Direct Analysis Method (DAM) as a solution to meeting the stability analysis and design requirements in the specification in a modern way that is most suitable for implementation in analysis and design software. This method applies to all structural steel buildings and there are no limitations as long as the method is performed according to the specified requirements. Combined with today’s best analysis tools, including a rigorous second-order P-Delta analysis and a user-friendly graphics interface, Fastrak Building Designer implements the DAM to give designers the best solutions available. This paper discusses: ♦

the real effects the new stability requirements are meant to capture,

♦

the methods used prior to the 2005 AISC Specification and the associated problems,

♦

the requirements for stability analysis and design in the 2005 AISC Specification, including the requirements of the DAM,

♦

stability analysis and design of structural steel buildings with modern software,

♦

and the details of how Fastrak Building Designer implements the AISC requirements for analysis and the DAM

THE REAL WORLD Structural engineering often requires modeling real structures in the virtual world of computer software for efficiency in the analysis and design process or because the structure is too complex to treat by hand. A relatively common structure, such as shown in Figure 4, would be very difficult to analyze and design by hand. In the real world, structural steel buildings are not perfectly straight and plumb. They are constructed within the tolerances specified by the American Institute of Steel Construction (AISC) in the Code of Standard Practice for Steel Buildings and Bridges. This document allows for some initial imperfections; out-of-straightness of each member and out-of-plumbness of columns. So, a real building has initial geometrical imperfections.

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Figure 4: A building such as this would be very difficult to analyze and design by hand. [Screenshot from Fastrak Building Designer. Model courtesy of Fisher Engineering.]

Loads applied to a structure will act on the ‘real’ shape of the structure, including initial imperfections (both sidesway of the structure and curvature of the members) and deflections caused by lateral loads. This introduces flexure into axially loaded members, where the magnitude of the flexure is dependent on the axial load and the deformed shape. This causes second-order behavior known as the P-Delta effect. There are two P-Delta effects; ♦

P-∆ is a structure effect in which axial loads in columns act on the displacement of the ends of the member - the story displacements,

♦

P-δ a member effect in which axial loads act on the displacement between the ends of the member.

So, a real building experiences second-order effects. The question is, are they significant and if so, how do I best deal with them? The process used to create structural steel sections creates residual stresses in the cross section of each member, often as high as thirty percent of the yield stress (0.3Fy). Due to these internal stresses, portions of the section will begin to yield when the applied stress due to loading is as low as 0.7Fy. This is well before the yield strength of the section as a whole is reached. When steel yields the effective modulus of elasticity, E, of that portion of the section tends to zero. Therefore, as portions of the section yield, the overall stiffness of the section is reduced due to inelasticity. So, real steel sections experience ‘early’ inelasticity and a reduction in effective stiffness. What do these ‘real’ effects mean to you? Basically, if you allow for them properly then K=1.0!

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The tedious calculation of the effective length factor, which is based upon assumptions that are rarely satisfied in real structures, is no longer necessary. I will explain this apparent jump in logic by starting with the state of design before the 2005 AISC Specification, then discussing the motivation and basic changes in the recent specification, and follow up with how a new method for stability analysis, the Direct Analysis Method, has eliminated K > 1.0.

BACKGROUND BEFORE 2005 Before the 2005 AISC Specification, a second order analysis was required by the specification. One method, the so-called B1, B2 method of approximate second-order analysis, has been around for at least 15 years in the AISC Specification. Even when secondorder effects were considered, typical analyses neglected the global effects of geometric imperfections and inelasticity in the members. However, the member design equations did include the effects of geometric imperfections (out-of-straightness) and inelasticity of the members. To compensate for neglecting the global effects of initial imperfections and member inelasticity, an effective length factor, K>1.0, was used to determine the axial capacity of the member. This approach leads to the following potential problems: ♦

It understates the moments in columns and members that provide rotational restraint to the column because it neglects the global effects of imperfections and inelasticity. The use of the effective length factor will likely give an adequate column size, but the members around the column may have understated moments. The required forces determined from the analysis for the column base plate or beam-column connections may be significantly less than they should be.

♦

The K factor on the capacity side only affects the column design. Even with a second-order analysis, without initial imperfections included, load combinations without lateral loads will never indicate moments that would be amplified by P-∆ effects.

♦

Displacements at strength levels will be underestimated, including their effect on stability, because the analysis does not account for the stiffness reduction associated with inelasticity in the members.

♦

The effective length factor is tedious and difficult to calculate correctly. The alignment charts commonly used to determine the K factor are based on nine assumptions. These idealized conditions rarely exist in real structures. For example, one condition states that all columns in a story are assumed to buckle simultaneously!

♦

Finally, this method is often overly conservative due to the inaccuracy of calculating K factors and applying demand effects onto the capacity of the member.

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Stability Analysis and Design

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AFTER 2005 The 2005 AISC Specification requires you to consider all the ‘real’ (global and local) effects previously discussed (second-order effects – P-∆ and P-δ, initial imperfections – global and member, and inelasticity in members) thus eliminating the problems with the earlier method. Chapter B requires that the design of members and connections be consistent with the intended behavior and the assumptions made in the analysis. This is a pretty general statement. The details of how this affects stability design are covered in the requirements of Chapter C. AISC has recognized that analysis options are increasing due to technological advances. In response, the specification now directly addresses the close interaction of the analysis method and the design method. However, the specification allows designers to use any rational method of analysis as long as it is consistent with the design procedures in the specification. The specification sets constraints that must be met to use the member design chapters and leaves the rest to the designer (for the most part).

2005 AISC DESIGN REQUIREMENTS CHAPTER C: STABILITY ANALYSIS AND DESIGN The design of all structures must consider the influence of the following on the stability of the structure and its elements. 1. flexural, shear, and axial deformations of the elements 2. all component and connection deformation that contribute to the lateral displacement of the structure 3. second-order effects (both P-∆ and P-δ) 4. geometric imperfections 5. member stiffness reductions due to residual stresses The first three requirements are covered by the structural analysis and the first two are included in nearly all analysis packages (or neglected as is standard practice e.g. moment connections are always considered rigid). Three methods to account for #4 and #5 (including provisions relating to #3) are presented in the specification: ♦

Direct Analysis Method (DAM)

♦

Effective Length Method

♦

First-Order Analysis Method

However, any rational method of accounting for the listed effects is permitted.

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Stability Analysis and Design

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THE DIRECT ANALYSIS METHOD: APPENDIX 7 The DAM is currently in Appendix 7 but will very likely move to Chapter C as the default method in the 2010 AISC Specification. This method is not limited in its application, applies to all buildings, and is the most general and accurate approach of the three. The requirements for the DAM include: ♦

Second-Order analysis: A second-order analysis which considers both P-∆ and P-δ effects is required. This can be accomplished through a rigorous second-order analysis or by using the approximate method presented in the specification. Secondorder analysis will be discussed in more detail later. Note that the analysis must be carried out using load combinations (either ASD or LRFD). Because the level of second-order effects depends upon the size of the load (the ‘P’) and the amount of deformation (the ‘Delta’), the load cases cannot be analyzed separately then superimposed in a load combination. For the same reason, if ASD is used, the second-order analysis is performed using 1.6 times the ASD load combinations. The results of the analysis are then divided by 1.6 to get required strengths.

♦

Initial imperfections: The effects of initial imperfections of the structure geometry are considered by applying notional loads. Notional loads are lateral loads that are applied at each framing level and are proportional to the gravity loads. The loads are expressed in terms of the gravity loads applied at each level and assume a standard out-of-plumbness based on the AISC Code of Standard Practice requirements. (Ni = 0.002Yi). Notional loads are typically applied as minimum lateral loads, which has the effect that they are usually only applied to gravity load combinations. However, notional loads are required to be added to all load combinations in certain situations. The specification also allows the designer to directly model the imperfections for analysis in place of adding notional loads.

♦

Inelasticity: The axial and flexural stiffnesses of members that contribute to the stability of the structure are required to be reduced. This is to account for the effects of residual stresses that lead to inelastic softening before the members reach their design strength. The flexural stiffness is multiplied by a factor, τb (less than or equal to one), which is dependent on the value of the axial load in the member. To avoid the iteration required to calculate this factor, the specification allows the designer to use a value of τb = 1.0, if the notional loads are increased to Ni = 0.003Yi.

♦

Effective Length Factor: And now we come to the best part (at least one of the best parts), K=1.0! No more calculating K when determining the nominal strength of columns. Setting K to be 1.0 can be allowed because the effects for which it was meant to compensate (initial imperfections and inelasticity) have already been accounted for in the method. It should be noted that a K value less than one can be used if it is justified by analysis.

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Stability Analysis and Design

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OTHER, LIMITED METHODS The following methods are also allowed by the specification in some circumstances. ♦

The Effective length method (a.k.a. design by second-order analysis) is limited in use to where second-order effects are not excessive (where the ratio of second-order displacements to first-order does not exceed 1.5). A second-order analysis performed on nominal geometry using nominal member stiffnesses is required. Notional loads (very similar to those in the DAM) are required as well. K factors are used to determine the nominal axial strength of columns. This method is similar to what has been done in the past with the addition of notional loads.

♦

The First-order analysis method is limited in use to where second-order effects are not excessive (where the ratio of second-order displacements to first-order does not exceed 1.5) and where axial loads are low (required strength is less than half the yield strength of the member) for frame members. A first-order analysis is performed on the nominal geometry with nominal member stiffnesses. An additive notional load is required as well.

STABILITY D ESIGN WITH MODERN SOFTWARE The changes in the AISC Specification are, in part, in recognition that buildings have changed over the years. Buildings were once (in most cases) straight forward, rectilinear structures with multiple redundancy and permanent or semi-permanent substantial walls within the framing. Now, modern buildings are more just a skeleton of steel so that the frame has to work a lot harder. Even in buildings with regular bays, discrete frames of differing types with irregular layouts are commonly used to meet architectural requirements, as seen in the structure in Figure 5. Therefore, stability analysis is more critical and any second-order effects have to be determined more accurately.

Figure 5: Combinations of moment frames and braced frames with irregular layouts are commonly used to meet architectural requirements, even in buildings with regular bays. [Screenshot from Fastrak Building Designer. Training example.]

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THE DIRECT ANALYSIS METHOD A major advantage of the Direct Analysis Method (DAM) is that it more accurately captures the internal forces in the structure than the other methods, particularly for structures that are susceptible to instability. For example, where there are high gravity loads and low lateral loads. This is especially true when a rigorous second-order analysis is used in combination with the DAM. Neither of the alternative methods offers any real advantage over the DAM. The effective length method trades the stiffness reduction calculations for K calculations. Therefore, the method still suffers from the difficulty of calculating K correctly. The first-order analysis method requires large notional loads and the determination of a B1 factor that is applied to ALL moments, not just those due to lateral loads – so it is likely to be conservative. In any case, if a sophisticated software package is doing all the work, the DAM will give more accurate analyses and more efficient designs with little extra effort for the designer. SECOND-ORDER ANALYSIS A second-order analysis is required for all structures that implement the Direct Analysis Method and can take the form of a rigorous analysis or approximate methods like the amplified first-order method (the B1, B2 method) – which is really intended more for hand calculations and other approximations. Choosing a rigorous second-order analysis frees the analysis from the limitations of the amplified first-order method. These limitations include (but are not limited to) the following as discussed in the AISC Commentary on Chapter C: ♦

AISC recommends a rigorous second-order analysis when B1>1.2 in order to determine accurate internal forces.

♦

When there is significant amplification in a location where several members join, a second-order analysis is likely required in order to determine accurate distribution of forces at the joint.

♦

Complex geometry such as sloping beams and columns can create difficulties for applying the approximate method. Similarly, when floor levels are not readily identifiable or vary throughout the building, the approximate method is difficult to apply and may be inaccurate.

♦

Even when certain simple conditions are encountered, engineering judgment is required to distribute the magnified moments properly. For example, consider a beam with framing above and below. The B2 factor from the story above and below will (very likely) be different. The greater of the B2 factors can be used to amplify the beam moments, which can add significant conservatism. Alternately, the sum of the differences of the magnified moments and the first-order moments can be distributed to the members based on their relative stiffnesses. This is just one situation where the method requires some engineering judgment or extra calculations. Since software

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cannot be counted on to make engineering judgments, the approximate methods are difficult or complicated to implement in a reasonable manner. The amplified first-order method certainly works well for some buildings, generally those that are simple, regular structures. However, to allow for the greatest range of structure geometry with the highest degree of accuracy in determining internal forces, the best course of action is to take advantage of a rigorous second-order analysis. Even smaller projects, with relatively small budgets, often have complex framing requirements that will benefit from a rigorous second-order analysis. This type of complexity is demonstrated in the structure in Figure 6.

Figure 6: Complex buildings make it difficult to apply approximate second-order analysis methods. Today, even relatively small projects often have complex framing. [Screenshot from Fastrak Building Designer. Model courtesy of Scott Wilson.]

FASTRAK B UILDING DESIGNER STABILITY DESIGN IN FASTRAK BUILDING DESIGNER Fastrak Building Designer implements the Direct Analysis Method fully, using a rigorous second-order analysis. CSC chose this method because it gives the user access to the most powerful analysis and design solutions. It allows for the widest range of building structures with accurate and reliable results. The AISC requirements are implemented as described in the Table 1.

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AISC Required effect to be considered

Fastrak Building Designer Implementation

Flexural, shear, and axial deformations

A general 3D analysis is performed which considers all required deformations

All component and connection deformation that contribute to the lateral displacement of the structure

All moment connections are assumed to be fully restrained. As is typical of most analysis programs, it is rationalized that these deformations do not contribute significantly to the stability of the structure and are therefore not directly included in the analysis.

Second-order effects

The Chen and Lui rigorous second-order analysis is implemented. The two-cycle iterative method automates a two-pass analysis procedure during which nodal displacements are used to determine ‘stress stiffening’ in structural elements. The resulting matrix accommodates the P-∆ and P-δ effects as well as accounting for ‘stress stiffness’.

(both P-∆ and P-δ)

Geometric imperfections

Notional loads are calculated and applied at each beamcolumn intersection based on the gravity load for each appropriate load combination.

Member stiffness reductions due to residual stresses

All relevant members have reduced axial and flexural stiffnesses. The option to take τb = 1.0 and increase the notional load factor is exercised.

Table 1: AISC Requirements and Fastrak Building Designer Implementation

NOTIONAL LOADS AND STIFFNESS REDUCTIONS The notional loads are calculated automatically as 0.003 times the total factored gravity load supported by the column at the beam-column intersection. The 0.003 factor is used in place of 0.002 because τb is set to 1.0. (See the earlier discussion of member inelasticity for more information.) This avoids the need to iterate in order to determine τb (you need the axial load in the members to determine τb, which affects the stiffness, which in turn affects the load distribution in the structure and the axial load in a member). This also allows different materials to be used in members contributing to the stability of the structure. In Fastrak Building Designer, members of all materials have reduced stiffnesses in the stability analysis. In Fastrak Building Designer, the notional loads are automatically calculated and can be applied in the global X, global Y or both directions simultaneously. The designer can apply a factor to adjust the value in the load combination if so desired. The notional loads are calculated for each load combination based on the total factored gravity loads in that combination. The notional loads are additive in any load combination in which they are included. The designer has complete control over the load combinations in Fastrak which, in turn, controls when the notional loads are applied and the magnitude of the loads in each combination.

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LIVE LOAD REDUCTION Live load reductions are different for each member in the structure. For example, in one story of a column the reduction might be 30% while in the story below it might be 40%. Hence, it is not possible to apply a reduced live load with a single value into the analysis. Consequently, live load reduction tends to be handled as part of the design (of members) and not part of the analysis. In the DAM a further complication is encountered. The size of the axial force (‘P’) influences the level of P-Δ (and P-δ) effects and so (even if it were possible) it is not correct to reduce the live loads prior to carrying out the second-order analysis. In Fastrak Building Designer, the analysis is run with the full live load; therefore any secondorder effects include the effects of the full live load on stability. The axial loads and moments from the analysis are therefore based on the unreduced live load. During the design process, the axial live load affect on the column is reduced by the appropriate factor and used as the required axial force for design. This approach is conservative in that the full live load is used for stability, but the advantage of reducing the axial load is still captured during the design process. SECOND-ORDER ANALYSIS A rigorous second-order analysis is performed by Fastrak for stability. Choosing a rigorous second-order analysis frees Fastrak Building Designer from the limitations of the amplified first-order method. The Chen and Lui ‘two-step iterative’ second-order analysis method was chosen to be included in for several reasons. The main reason is that since the geometric (stress) stiffness is used in the method, there are no significant limitations on its use or applicability unless gross deformation occurs, in which case a full non-linear iterative solution is more appropriate. This increased level of sophistication would be required for cable and membrane type structures. For most normal building structures where the secondorder effects do not approach or exceed the first-order effects, such sophistication is not required. The Commentary to Appendix 7 contains two benchmark problems against which the analysis in Fastrak Building Designer has been validated. When the design process is completed, Fastrak Building Designer allows the designer to quickly and easily view the various bits of information that will help verify the solution. Fastrak automates the calculations of all the necessary parameters, but allows the user to view the items and, where appropriate, control them. In this manner, Fastrak relieves the designer of the labor intensive parts of the DAM aids understanding but still provides the flexibility to make changes and take responsibility for the design.

CONCLUSIONS Buildings have changed over the years from being (in most cases) straight forward, rectilinear structures with multiple redundancy and permanent or semi-permanent substantial walls within the framing. Now, modern buildings are more just a skeleton of steel so that the

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frame has to work a lot harder and any second-order effects have to be more accurately determined. Figure 7 illustrates such an irregular building. Older versions of the AISC Specifications were aimed at allowing for second-order effects using hand methods (most notably the B1, B2 method). These hand methods are not conducive to software implementation; nearly 35 years of experience producing structural analysis and design software at CSC has told us this. Like it or not, most design is carried out on computers these days, if for no other reason than efficiency. AISC recognized this and in the 2005 Specification has developed a more accurate rigorous method that lends itself to computer implementation, the Direct Analysis Method. In addition, the rigorous secondorder analysis lends itself best to computer implementation. In recognition that the Direct Analysis Method is the superior method, the 2010 AISC Specification is likely to move it to Chapter C as the default method and move the others to appendices. And let us not forget, the Direct Analysis Method gives us K = 1.0! Now you know what Fastrak Building Designer does for stability analysis and design. I will repeat my earlier questions. Do you know how your favorite software package has implemented the new requirements? Does it take full advantage of the Direct Analysis Method and the capabilities AISC now allows in the second-order analysis?

Figure 7: Today’s buildings (for the most part) are no longer straight-forward, rectilinear structures with multiple redundancy, therefore they require more accurate second-order analyses. [Screenshot from Fastrak Building Designer. Model courtesy of Robinson Group.]

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