Overview on The Design for Stability Chapter in the AISC Specifications
ABSTRACT
In the AISC specifications of chapter C, the design for stability, the manual provides a valuable explanation about how to design a member or a whole structure when considering stability. This paper’s review, then, will focus on requirements that need to be considered in the design for stability. To account for such requirements AISC specifications offer three methods: (1) direct analysis method, (2) effective length factor, and (3) firstorder analysis. Throughout this paper, I will consider the three methods; however, the first method will be more detailed.
 INTRODUCTION
Galambos (1998) defined the Stability as “The capacity of a compression member, element, or frame to remain in position and support load, even if forced slightly out of line or position by an added lateral force”.
To understand the behavior and design of metal structures effectively, an engineer needs a fundamental understanding of structural stability. In addition to structures designed with other building materials, steel structures are largely governed by the stability limits. All major international design requirements include provisions based on stability theory. (Galambos 2008).
Instability problems are often catastrophic and most frequently occur during erection. For example, a number of large steel box girder bridges collapsed in the late 1960s and early 1970s, leading to many deaths among erection personnel. (Galambos 2008).
 GENERAL STABILITY REQUIREMENTS
Carter (2013) gives good explanation for the general stability requirements. Chapter (C) “Stability Analysis and Design” provides that the design of the structure for stability as a whole and each of its element should consider all of the following: (1) flexural, shear, and axial member deformations – these are the all other component and connection deformations that contribute to displacements of the structure; (2) secondorder effects – these are the increases that occur in moments and forces due to displacements of the structure caused by the loads, containing both P effects(displacement due to sidesway of the structure or the member). (3) geometric imperfection – these are the initial outofstraightness of the members and the initial outofplumbness of the structure; (4) stiffness reductions due to inelasticity – these are the effects of residual stresses; and, (5) variability in component and system stiffness – these are the effects of variations in material and crosssectional properties of members, as well as the other effects generally accounted for in the resistance factors (LRFD) and safety factors(ASD).
The specification states expressly that any method of analysis and design that accounts all the required effects is allowable, and then presents certain specific approaches that consider for the last four of the listed effects (P∆ effects, Pδ effects, geometric imperfections, and inelasticity). (Nair 2007)
2.1. DIRECT ANALYSIS METHOD
The direct analysis method (DAM) is essentially considering three issues, Freund (2012): (1) impacts of initial geometric imperfections. (2) secondOrder effects – axialdisplacement moments P∆ and P δ (as shown below figure 1). (3) Effects of material nonlinearity – inelasticity due to residual stresses.
Figure 1 Pd and PD Sketch
DAM is applicable to all types of structures and all sidesway amplification values (_{2nd} order/_{1st} order). Moreover, it’s more accurate than effective length method and there is no need to calculate the effective stiffness factor k. (Landis 2016).
(Hewitt 2008) explained Stepbystep the process to use the Direct Analysis Method: (1) make a model of the lateral frame being analyzed, containing the leaning columns.
(2) minimize stiffness of lateral framing members in the model. a) when a braced frame is modeled with pinned connections, you should modify the modulus of elasticity in the model to 0.8E. b) when a braced frame modeled with rigid connections, the modulus of elasticity in the model should be modified to 0.8τbE. Another way for applying τb, a notional load can be added 0.001 times the gravity load to the notional load that will be discussed in step 3. Although, this could require an iteration in analysis when the size of member change and the inequality reverses. This inequality will be frequently satisfied by moment frames by permitting τb to equal 1.
(3) All load combinations should be applied by notional loads equal to 0.002 times the gravity load (LRFD or 1.6 ASD) for each story. (By AISC Specifications, notional loads should be added to load combinations in which the notional load is greater than the lateral load on the frame.
Therefore, notional loads generally can be ignored in all but the gravity only load combinations. Yet, if a designer wants to make the design process simpler, it is always conservative to take into account the notional loads).
(4) Create a secondorder analysis for the structure under applied loads, either by doing full direct, explicit secondorder analysis of the structure. Or using the amplified firstorder analysis method (B1/B2).
(5) To resist the forces you just determined, design the members using AISC specification using AISC specifications and reset the modulus of elasticity to E to (29,000 ksi). Finally (6) check the limits for seismic design and wind.
 CALCULATION OF REQUIRED STRENGTHS
Using direct analysis method of design, the required strengths of components of a structure will specified from the analysis conforming to section 3.1. One of the requirements in the analysis is the impacts of initial imperfections of the structure geometry, and that is discussed in section 3.2.
3.1.GENERAL ANALYSIS RQUIREMENTS (Gallant 2014)
The analysis of a structure must consider the following requirements:
a) All deformations such as flexural, shear and axial member deformations and all connection deformations and other component.
b) The analysis should consider second order effects (PΔ and Pδ).
c) Initial imperfections which is due to erection tolerances and fabrication.
d) Material imperfections which is due to residual stresses imposed during fabrication.
e) Inelasticity which happens when member is stressed beyond elastic range. When material softening occurs the axial and flexural stiffness is reduced, and that can amplify secondorder effects.
3.2. CONSIDERATION OF INITIAL IMPERFECTION
The impacts of initial imperfections of the structure geometry can be accounted either by directly model the imperfections in the geometry or by applying notional loads.
3.2.1. DIRECT METHOD OF IMPERFECTION
The most important imperfections are where the members intersect (column/beam joints), where in most structures can be considered for by the outofplumbness of columns. Based on AISC Code of Standard Practice, the nodes need to be modeled displaced from the nominal locations based on the structure tolerances. Usually, the utmost outofplumbness limit for columns is L/500, where there could be other values control in irregular cases. The pattern of initial displacements should account for the most destabilizing effect (Gallant 2014).
3.2.2. USE OF NOTIONAL LOADS TO REPRESENT IMPERFECTION
Ericksen (2011) provides a complete discussion about “Using of Notional Loads”. Much of the following discussion is adopted from this reference
To consider for the impact of initial imperfections, the specifications allow the engineer to directly design the imperfections within the model. Due to loading and expected buckling modes of the structure the set of initial displacements designed need to be considered displacements. As a result of this, we need to model outofstraightness of each member and outofplumbness of the columns. This need to be achieved by a way that would catch the maximum destabilizing impact on the structure, that means leastwise four different displacements need to be applied in four standard directions at every level with a corresponding selected group of member outofstraightness to add to the impact of the outofplumbness.
Figure 2 Notional Loads
The notional loads as shown in figure 2 are horizontal loads added to the structure to consider for the impacts of geometric imperfections. Figure 2 explains a noncomplex version of the conception. The notional loads (Ni) are computed as a component of the gravity loads (Yi) and applied at every level. Directions included below to help you to compute the magnitude of notional loads, in which direction(s) they should be taken and where to place them on the model.
Magnitude
The notional loads (Ni) are 1/500 of the overall factored gravity loads (Yi) at every level, and this can be formed as: Ni=0.2%Yi. Pay attention that 0.2% is equal to 1/500, which is as specified in AISC Code of Standard Practice (see Figure 3) the maximum tolerance for outofplumbness in steel structures. Whenever the true outofplumbness of structure is known a smaller value can be used.
Gravity loads are described in AISC specifications as “Load such as that produced by dead and live loads, acting in the downward direction.” The estimate of (Yi) is the overall factored gravity load in every load combination. Thus, the estimate of notional loads will be changing from combination to combination. Likewise, for the overall factored gravity load that contains all gravity loads on a level, and that is not just for the vertical loads that supported by lateral framing members. As a consequence, the impact of leaning columns is included in notional loads. With ASD when using direct secondorder analysis, the notional loads and total gravity loads are multiplied by 1.6. Note this is the same factored which is required for the secondorder analysis. Therefore, the loads are not to be multiplied twice by 1.6.
Figure 3 AISC Code of Standard Practice Tolerance for Column OutofPlumbness
_____________
When to Apply
All combinations are required to have notional loads. For combinations having lateral loads, other lateral loads should include notional loads. Yet, if the secondorder drift to the firstorder drift is less than 1.5, notional loads should be added to gravity only combinations. This ratio 1.5 is applicable to structures that have been analyzed using unreduced stiffness of lateral members. If using reduced stiffness, the ratio is 1.7.
3.3. ADJUSTMENT TO STIFFNESS
The required strengths will be specified using 0.8 factor applied to all stiffnesses, EI and EA, that participate to the stability of the structure. To avert unplanned deformation and load redistribution, it is allowable to apply this reduction to all members in the structure. Nevertheless, if the program adjusts the elastic modulus E, you should be careful that E is been reduced only for the analysis calculations, not in the member capacity calculations (Gallant 2014).
For members of structure whose demand compression strength is greater than half the yield strength, a factor to be added to apply flexural stiffness (EI):
_{ } = 4( _{ }⁄ _{ })[1− ( _{ }⁄ _{ })]
Where
= 1.0 (LRFD) or = 1.6 (ASD)
_{ } = required axial compression, kips (N)
_{ } = _{ } _{ }, axial yield strength, kips (N)
When a structure system includes members that fail by inelastic buckling, a factor 0.8 _{ }will be accounted for the inelastic buckling before getting to the design strength. The _{ }factor by itself is accounted for the stiffness reduction under compression load ( _{ }) which is higher than half of the yield load ( _{y}). The 0.8 factor is accounted for the extra loss of stiffness due to combined axial and bending compression. This could be explained as having an available system strength of 80% of the elastic stability limit. This is roughly the same as the available strength in elastic compression members in Chapter E in the AISC specifications:
$\mathit{\u0278Pn}=0.9\left(0.877\mathit{Pe}\right)=0.8\mathit{Pe}$
4.ALTERNATIVE METHODS OF DESIGN
Nair (2007) gives a short summary for the “Indirect Methods” for structures where the secondorder effects don’t have large effect (the ratio of _{2ndorder} /_{1storder }is below the specification), AISC specifications offer two alternatives methods to the Direct Analysis Method.
Effective Length Method. For this approach, all members should be analyzed using nominal elastic stiffness and the nominal geometry; use secondorder analysis (either by rigorous secondorder analysis or secondorder analysis amplified firstorder analysis) to determine the required member strength. all gravityonly load combinations contain the least lateral load at each frame level of 0.002 of the gravity load applied at that level. Column effective length, K, must be calculated using sidesway buckling analysis, unless we have braced frame (K=1) or the secondorder drift to the firstorder drift is less than 1.1.
AISC specifications does not allow to use effective length method when the ratio of secondorder drift to the firstorder drift is larger than 1.5. This is by reason of the effective length method notably underestimates the internal forces and moments in particular conditions when the limit is exceeded (White et al 2006).
In this method, the effect of residual stresses, P∆ and Pδ effects are specified completely by the calculation of Pni from the equations of column strength. These equations can be written in terms of Fei or KLi. Unfortunately, it requires considerable judgment and skills to select an appropriate subassembly buckling model. Therefore, there is broad zone of various buckling models and K factor equations. In particular conditions, simple differences in models can produce very different results (White et al 2006).
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FirstOrder Analysis Method. This approach is only applicable if the required compressive strength is lower than half of the yield strength in all members which flexural stiffnesses are applied to participate to the lateral stability of the structure. In this approach, the members are analyzed using nominal elastic stiffness and the nominal geometry; to determine the required member strength we use the first order analysis; all load combinations include an additional lateral load at each frame level of a magnitude based on the gravity load applied at that level and the lateral stiffness of the structure; to determine the nominal strength of compression members we assume K=1; to account for nonsway amplification beamcolumn moments must be modified (using a provided formula).
The alternative analysis methods and corresponding stability design requirements in the 2005 AISC specification are summarized in Table 1 below.
Direct Analysis Method 
Effective Length Method 
FirstOrder Analysis Method 

Specification reference 
Appendix 7 
Section C.2.2a 
Section C.2.2b 
Limits on applicability? 
No 
Yes 
No 
Type of analysis 
SecondOrder 
SecondOrder 
FirstOrder 
Member stiffness 
Reduced EI & EA 
Nominal EI & EA 
Nominal EI & EA 
Notional lateral load? 
Yes 
Yes 
Additional Lateral Load 
Column effective length 
K=1 
Sidesway buckling analysis 
K=1 
Table 1 Comparison of Analysis and Design Options adopted from Nair (2007).
5. CONCLUSION
In this paper three elastic methods of analysis are discussed.These methods are the direct analysis, the effective length, and the firstorder analysis. The direct analysis method allows for more simplicity in the design by canceling the need for effective length factor.Moreover,the direct analysis method is mostly applicable to all types of structures and all sidesway amplification values. On the other hand, the effective length method significantly underestimates the internal forces and moments that should be considered in certain cases.
References STYLE
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