Steel Structures 6 (2006) 71-91 www.kssc.or.kr Stability Analysis and Design of Steel Building Frames Using the 2005 AISC Specification Donald W. White1,*, Andrea E. Surovek2, Bulent N. Alemdar3, Ching-Jen Chang4, Yoon Duk Kim4 and Garret H. Kuchenbecker5 1Structural Engineering, Mechanics and Materials, Georgia Institute of Technology, Atlanta, GA 30332-03551, USA 2Civil and Environmental Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA 3RAM International, 2744 Loker Avenue West, Carlsbad, CA, 92010, USA 4Structural Engineering, Mechanics and Materials, Georgia Institute of Technology, Atlanta, GA, USA 5Hermanson Egge Engineering, Rapid City, SD, USA Abstract The 2005 AISC Specification reflects the latest advances in the stability analysis and design of structural steel buildings. The new Specification defines the general requirements for stability analysis and design and gives engineers the freedom to select or devise their own methods within these constraints. It also provides several specific procedures. This paper first gives an overview of the elastic analysis and design procedures in AISC (2005) as well as specific second-order distributed plasticity methods upon which, in part, these procedures are based. The relationship between the AISC elastic provisions and the refined inelastic methods is explained. Secondly, the paper highlights one interpretation of the AISC inelastic analysis and design provisions that greatly facilitates the application of elastic-plastic hinge methods of analysis. The paper closes by presenting four basic examples selected to illustrate key characteristics of each of the methods. Keywords: Advanced analysis, distributed plasticity analysis, direct analysis, effective length, plastic hinge analysis 1. Introduction fabrication and erection tolerances) relative to the ideal configuration of the structure. Chapter C of the AISC (2005a) Specification, Stability The above Chapter C statement gives engineers the Analysis and Design, states that any analysis and design freedom to select or devise methods that are best suited method that addresses the following effects on the overall for the various structure types encountered in practice. It stability of the structure and its elements is permitted: allows for innovation as long as there is proper 1.Flexural, axial and shear deformations (in members, consideration of the physical effects that influence the connections and other components), structural response. 2.Reduction in stiffness (and corresponding increases Since the 1961 AISC Specification, when the column in deformations) due to residual stresses and material effective length concept was first introduced by AISC, yielding, 3.P-∆ effects, which are the effects of axial loads P acting through the relative transverse displacements of the member ends ∆ (see Fig. 1), 4.P-δ effects, which for initially straight members loaded by bending in a single plane, are due to the member axial load acting through the transverse bending displacements relative to the member chord (see Fig. 1), and 5.P-∆ and P-δ effects, which are caused by the o o member axial loads acting through unavoidable initial ∆ and δ geometric imperfections (within o o *Corresponding author Tel: +404-894-5839, Fax: +404-894-2278 Figure 1. Second-order P-∆ and P-δ effects (from White E-mail: dwhite@ce.gatech.edu and Kim (2006)). 72 Donald W. White et al. American analysis and design methods have addressed all incipient elastic buckling (considering the interaction of of the above effects in some fashion whenever they are the member with the rest of the structure). The parameter deemed to have an important influence on the structural F' is calculated by dividing this buckling stress by the e response. Also, there has always been implicit recognition factor of safety for column elastic buckling, 23/12=1.92. that engineers can use their professional judgment to The axial stress F' is determined typically using the e disregard specific effects (e.g., member shear deformations, equation connection deformations, etc.) whenever they are considered negligible. Member yielding, residual stress effects, and 12π2E F '=---------------------------- (3) geometric imperfection effects traditionally have been e 23(KL ⁄r )2 addressed in the formulation of member design resistances, b b and, with minor exceptions, have not been considered in where KL /r is the column effective slenderness ratio b b the analysis. In some cases, engineers have included a in the plane of bending, and K is the effective length nominal out-of-plumbness effect in the analysis of gravity factor associated with the above buckling solution. Also, load combinations, particularly if the geometry and loading this K factor is used typically in calculating the column are symmetric. Strictly speaking, this is not necessary for axial resistance F . If desired, F'=F/1.92 and F can be a e e a the in-plane strength assessment of beam-columns in the determined directly from the buckling analysis model. prior AISC Specifications. However, this practice is The term C in Eqs. (1) and (2) is discussed subsequently. m necessary to determine P-∆ effects on bracing forces, AISC (1986) LRFD was the first American Specification o beam moments, connection moments, and in-plane moments to refer explicitly to the calculation of second-order for checking the out-of-plane resistance of beam-columns. moments from a structural analysis. This specification Also, plastic design procedures incorporate material introduced the following two-equation format for the yielding into the analysis, typically by the use of idealized beam-column resistance: plastic hinge models in adequately braced compact- section members. However, geometric imperfection and P M P --------u------+--------u----≤1.0 for -------u---<0.2 (4a) residual stress effects traditionally are not included in this 2φ P φ M φ P c n b n c n type of analysis. The 1961 AISC Specification, and other AISC P 8 M P -------u---+-----------u----≤1.0 for -------u---≥0.2 (4b) Specifications up until 1986, relied strictly on the φ P 9φ M φ P c n b n c n structural analysis only for calculation of geometrically linear (first-order) forces and moments in the idealized where M is defined as the maximum second-order u geometrically-perfect, nominally-elastic (or elastic-plastic) elastic moment along the member length. AISC (1986) structure. For elastic analysis and design, second-order states that M may be determined from a second-order u (P-∆ and P-δ) effects were addressed very discreetly via elastic analysis using factored loads. However, it also the following beam-column strength interaction equation: provides a procedure that uses amplification factors to calculate the second-order elastic moments from a first- f C f order elastic analysis. This procedure is in fact an ---a--+-----------m-----b--------≤1.0 (1) F ⎛ f ⎞ approximate second-order analysis. The moments M in a ⎝1–-F---a--'⎠Fb Eqs. (4) are the second-order elastic moments in uan e idealized model of the structure assuming initially perfect The expression geometry and completely linear elastic material response. In all the AISC Specifications from AISC (1961) C through AISC (1989) ASD and AISC (1999) LRFD, the ----------m--------=AF (2) ⎛ f ⎞ influence of geometric imperfections and residual stresses ⎝1–-F---a--'⎠ was addressed solely within the calculation of the e member resistances (F and F in ASD and P and M in a b n n in Eq. (1) amplifies the member flexural stresses f to LRFD). The new direct analysis provisions in AISC b account for the P-∆ and P-δ effects. (2005a) recognize that specific advantages can be Generally, Eq. (2) gives only a coarse approximation of realized by moving an appropriate nominal consideration the true second-order effects. LeMessurier (1977) and of these effects out of the resistance side and into the others later addressed better ways to determine the structural analysis side of the design equations. By amplified bending stresses in general rectangular frames, incorporating an appropriate nominal consideration of all within the context of Allowable Stress Design (ASD). these effects in the analysis, the resistance side of the However, even today (2005), engineers using the AISC design equations is greatly simplified, and the accuracy of (1989) ASD provisions can apply Eq. (2) in ways that the design checks is generally improved. These attributes significantly under-estimate the physical second-order are discussed further in the subsequent sections. effects in certain cases. The accuracy hinges largely on All of the above analysis and design procedures are the proper determination of F, the member axial stress at based inherently on the use of second-order analysis e Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 73 (first-order elastic analysis with amplifiers being one type as 0.85 in Eqs. (1) and (2) for frames subject to joint of second-order elastic analysis). Elastic analysis does not translation. This C value typically underestimates the m include the consideration of the member resistances in sidesway amplification effects (Salmon and Johnson, 1996). itself. Therefore, all of the above elastic methods must Nevertheless, the ASD moment amplifier summarized in include member resistance equations. However, the Eq. (2) is still conservative in many practical cases. This method of analysis and the equations for checking the is because the predominant second-order effects are often member resistances are inextricably linked. Changes in associated solely with the structure sidesway. Equation the analysis calculation of the required strengths (e.g., f (1) applies a single amplifier indiscriminately to the total a and f C /(1−f /F') in Eq. (1) or P and M in Eqs. (4)) flexural stresses from both sidesway and non-sway b m a e u u can lead to simplifications in the calculation of the displacements. The amplification factor procedure in member resistances (typically F in Eq. (1) or P in Eqs. AISC (1999) LRFD and AISC (2005a) is more accurate, a n (4)). Specifically, if the structural analysis is configured but involves a cumbersome subdivision of the analysis to provide an appropriate representation of the internal into separate no-translation (nt) and lateral-translation (lt) member forces, the in-plane resistance of the structure parts. Kuchenbecker et al. (2004) and White et al. (2005a can be checked entirely on a cross-section by cross- & b) outline an amplified first-order elastic analysis section basis. approach that provides good accuracy for rectangular AISC (2005a) adopts the equations from AISC (1999) framing. This approach avoids the above cumbersome LRFD as a base representation of the beam-column attributes of the AISC (1999 & 2005a) amplification factor resistances for all of its analysis and design procedures. procedure. However, the notation is generalized such that the In many cases, Eqs. (5) provide a more liberal equations apply to both ASD and LRFD: characterization of the beam-column resistances than the multiple beam-column strength curves in AISC ASD P 8M P (1989). Also, AISC (2005a) provides the following -----r+---------r≤1.0 for -----r≥0.2 (5b, AISC H1-1a)1 P 9M P alternative interaction equation to characterize the out-of- c c c plane flexural-torsional buckling resistance of doubly- P 8M P symmetric I-section members subjected to axial compression ------r--+---------r≤1.0 for -----r<0.2 (5a, AISC H1-1b) 2P 9M P and major-axis bending moment: c c c The terms in these equations are defined as follows: --P---r--+⎝⎛--M-----r-⎠⎞2≤1.0 (6, AISC H1-2) P =the required axial compressive strength, determined P M in ArSD by analyzing the structure under 1.6 times the co cx ASD load combinations and then dividing the results by where P is the out-of-plane column strength and M co cx 1.6, or determined in LRFD by analyzing the structure is the flexural strength with respect to lateral-torsional under the LRFD load combinations. buckling. White and Kim (2006) provide a detailed M =the required flexural strength, determined in ASD discussion of the background to Eq. (6) and explain that r by analyzing the structure under 1.6 times the ASD load this equation should be applied only for doubly-symmetric combinations and then dividing the results by 1.6, or compact I-section members. determined in LRFD by analyzing the structure under the In the subsequent developments, it is useful to consider LRFD load combinations. the characterization of separate in-plane and out-of-plane P =the allowableordesign axial compressive strength, beam-column resistances. This can be accomplished by c givenby P /Ω in ASD or by φP inLRFD,whereP is using Eqs. (5) with different definitions of P and M, or n c c n n c c the nominal compressive resistance determined in accordance by using Eqs. (5) to characterize the in-plane strength and withChapterE. Eq. (6), where it is applicable, to characterize the out-of- M =the allowable or design flexural strength, given by plane strength. The in-plane resistance is estimated with c M /Ω in ASD or by φ M in LRFD, where M is the Eqs. (5) by neglecting: (1) out-of-plane flexural, torsional n b b n n corresponding nominal resistance determined in accordance or flexural-torsional buckling in the calculation of P and c with Chapter F. (2) lateral-torsional buckling in the calculation of M. The c φ and φ =resistance factors for axial compression and out-of-plane resistance is estimated with Eqs. (5) by using c b bending, both equal to 0.9. P =P and using the governing resistance from Chapter c co Ω and Ω =factors of safety for axial compression and F for M (White and Kim, 2006). c b c bending, both equal to 1.67. For checking the in-plane and out-of-plane strength of The above 1.6 factor for ASD is smaller than the general I-section members, Eqs. (5) must be used. However, column safety factor of 1.92 in the AISC ASD (1989) for doubly-symmetric compact I-section members subjected amplification of the flexural stresses (see Eqs. (1) through to in-plane major-axis bending and large axial loads, Eq. (3)). However, ASD-H1 also states that C shall be taken (6) provides a more liberal assessment of the out-of-plane m 1The AISC (2005a) equation numbers are denoted by “AISC” followed by the equation number. 74 Donald W. White et al. flexural-torsional buckling strength. AISC (2005a) allows the engineer to neglect out-of-plane moments whenever M /M is smaller than 0.05 in the out-of-plane direction, ry cy where M and M are the required moment and the ry cy corresponding resistance in this direction. Otherwise, an extended form of Eqs. (5) must be used that includes the out-of-plane bending. 2. Overview of Stability Analysis and Design Procedures 2.1. Design by distributed plasticity analysis This section discusses the requirements that must be satisfied for strength design using a refined second-order inelastic frame analysis in which the spread of yielding is tracked explicitly through the cross-section and along the member length. This type of analysis is referred to in this Figure 2. Lehigh (Galambos and Ketter 1959) residual paper as a distributed plasticity analysis. stress pattern. For hot-rolled compact I-section members, ASCE (1997), Deierlein (2003), and Surovek-Maleck and White (2003 & 2004) have shown that distributed plasticity analysis closely replicates the in-plane AISC LRFD beam-column strengths based on an exact inelastic effective length, for a comprehensive range of end conditions, when the following nominal geometric imperfections, residual stresses and material idealizations are included in the analysis: •A sinusoidal or parabolic out-of-straightness with a maximum amplitude of δ =L/1000, where L is the o unsupported length in the plane of bending. •An out-of-plumbness of ∆ =L/500, the maximum o tolerance specified in the AISC (2005b) Code of Standard Practice. •The Lehigh (Galambos and Ketter, 1959) residual stress pattern shown in Fig. 2. Figure 3. Example cantilever beam-columns. •An elastic-perfectly plastic material stress-strain response. These results are not surprising, since Eqs. (5) were the plane of bending. Martinez-Garcia (2002), Surovek- originally developed in part based on calibration to results Maleck and White (2003), and Deierlein (2003) summarize from this type of analysis (ASCE, 1997; Surovek-Maleck the results from other more comprehensive studies. and White, 2004). Since the AISC (2005a) analysis and If the above distributed plasticity analysis is to be used design procedures do not make any distinction between in an AISC (2005a) LRFD context, the resistance factors hot-rolled and general built-up members, the above φ =φ =0.9 must be included. One way of doing this is c b nominal geometric imperfections and residual stresses are to determine the nominal beam-column strength curves as sufficient to capture the Specification requirements for all shown in Fig. 4, and then to multiply both the abscissa compact I-section member types. Figure 3 defines two and the ordinate by φ =φ =0.9 to obtain the final c b simple cases that illustrate the above findings. These are member design resistance. However, identical results are W10×60 cantilever beam-columns with L/r=40 subjected obtained if both the yield strength F and the elastic y to axial and transverse loads at their free ends. In the first modulus E are factored by 0.9. If only the yield strength case, the member is subjected to major-axis bending, F is factored by 0.9, the design strengths are overestimated y while in the second case it is subjected to minor-axis for highly slender columns that fail by elastic buckling. bending. Figure 4 compares the nominal strength interaction The factoring of both E and F by 0.9 up front is y curves obtained from the above type of distributed preferred, since this approach facilitates the general plasticity analysis to the corresponding AISC (2005a) inelastic analysis and design of structural systems. There beam-column strength curve. The AISC curve is obtained is no straightforward way of applying distributed plasticity using the AISC (2005a) effective length method with K= analysis, or any other form of inelastic analysis, in the 2, and is the same for both the minor and major-axis context of ASD. In as such, AISC (2005a) disallows the bending examples, since their L/r values are the same in use of inelastic analysis for this approach. Most of the Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 75 values producing the smallest total resistance should be selected. For structures in which the postbuckling response is asymmetric, with one direction corresponding to an unstable postbuckling response and the other corresponding to a stable one, the direction associated with the unstable postbuckling response gives the smallest total resistance. Although they can be programmed, the above out-of- straightness considerations involve a level of complexity that many engineers would find unacceptable. The αP < r 0.15P rule allows the engineer to disregard these eL considerations in many practical situations. The appropriate direction for the out-of-plumbness is typically much easier to specify than the appropriate direction for out-of-straightness. For the overall assessment of building frames, it is sufficient in the vast majority of cases to specify the out-of-plumbness in the total net direction that the structure sways under the applied Figure 4. Nominal strength curves by distributed plasticity loadings. analysis versus the AISC (2005a) effective length method At the present time (2005), distributed plasticity analysis for the example W10×60 cantilever beam-columns. has been applied most commonly in research studies investigating the resistance of frames composed of subsequent discussions in this paper are phrased in the adequately braced compact-section members. However, context of LRFD. in design practice, the member strengths can be governed The distributed plasticity analysis solution captures the by out-of-plane buckling, flange or web local buckling, or in-plane design resistances of compact I-section members combinations of these strength limits. These limit states completely when it includes the above attributes. Therefore, cannot be captured by a planar distributed plasticity the in-plane beam, column and beam-column resistance analysis. Although some progress has been made on 3D checks are automatically satisfied for these member types distributed plasticity methods (e.g., see Pi and Trahair if the distributed plasticity analysis shows that the (1994), Izzudin and Smith (1996), Teh and Clarke (1998), structure supports the design loadings. The engineer does Battini and Pacoste (2002) and Nukala and White (2004) not need to perform any separate evaluation of the in- among others), the complexity and cost of the analysis is plane member resistances. The Australian Standard significantly greater. Furthermore, the appropriate handling AS4100 (SAA, 1990) was the first to explicitly permit of residual stresses, geometric imperfections, warping this type of analysis and design. This Standard coined the continuity at beam-to-column joints, local-overall member term advanced analysis, in a very specific context, to buckling interactions, restraint from and interaction with denote an analysis that supersedes the Specification member floor slabs, and other important attributes that can influence strength checks. Subsequently, various SSRC publications the 3D response have not been studied thoroughly at this (e.g., White and Chen (1993)) as well as other research time. In fact, many of these effects are considered in papers and reports have adopted this terminology. rather simplistic ways in ordinary design practices, e.g., In many practical steel building structures, member out-of-plane and spatial beam-column resistances are out-of-straightness has little to no effect on the frame based typically on the assumption of unrestrained warping resistance. White and Nukala (1997) explain that when at the member ends. Nevertheless, in frames subjected αP is less than P /7≅0.15P , where predominantly to in-plane loading, the authors assert that r eL eL α=1.6 for ASD or 1.0 for LFRD and distributed plasticity analysis can be utilized to determine P =π2EI/L2using the moment of inertia I in the plane an accurate estimate of the internal forces in the structure. eL of bending, These forces then can be checked against Specification out-of-straightness effects can be neglected generally in member resistance equations corresponding to any of the the distributed plasticity analysis (i.e., δ can be taken equal above 3D limit states not included in the analysis. o to zero). Otherwise, one must determine the appropriate Various analysis refinements are possible relative to the direction for the member out-of-straightness. Usually, the above procedures. For instance, a number of approaches strength is reduced the most due to member out-of- have been suggested for reducing the nominal out-of- straightness if δ is specified in the direction of the plumbness relative to base values, e.g., see the Commentary o member deformations (relative to its chord) due to the of AISC (2005), White et al. (2003) and CEN (2003). applied loads. However, in some cases, it is advisable to Also, other nominal residual stress distributions as well as specify the various member δ values in the pattern of the more complete stress-strain models (e.g., models including o fundamental buckling mode obtained from an eigenvalue strain-hardening) can be incorporated within the distributed buckling analysis. In these cases, the direction of the δ plasticity analysis. However, these considerations introduce o 76 Donald W. White et al. additional complexities that must be addressed. The applied at this level. Table 1 emphasizes the use of above procedures fully satisfy the base requirements of notional lateral loads. However, in general cases where the AISC (2005a) Specification. questions may arise about the appropriate calculation of these loads, one can always use the more fundamental 2.2. Elastic analysis and design methods in AISC out-of-plumb geometry. For example, the total base shear (2005a) due to any out-of-plumbness is always zero, and thus the The AISC (2005a) Specification defines three specific total base shear due to the notional loads also must be elastic analysis and design methods. These are: zero. As shown in Fig. 5, the notional loads arise from the 1.The direct analysis method, detailed in Appendix 7, sum of the P∆ shear forces above and below each level. o 2.The effective length method, detailed in Section The P∆ shear at the base of the structure offsets the sum o C2.2a and of all the horizontal notional loads in the analysis model. 3.The first-order analysis method, detailed in Section Explicit modeling of the out-of-plumbness also removes C2.2b. the need to calculate different notional loads for different Table 1, summarizes the key attributes of each of these load combinations. methods. Within the restrictions specified on their usage, The direct analysis method provides an improved and provided that effects such as connection rotations or representation of the structure’s distributed plasticity member axial and shear deformations are properly forces and moments at the strength limit of the most considered in the analysis when these attributes are critical member or members. Due to this improvement in important, each of the above methods is intended to the calculation of the internal forces and moments, AISC comprehensively address all of the effects listed at the (2005a) bases its calculation of P , the column nominal ni beginning of Section 1. The following subsections provide strength for checking the in-plane resistance in Eqs. (5), an overview of these AISC (2005a) procedures. The on the actual unsupported length in the plane of bending. reader is referred to Deierlein (2004), Nair (2005a), Nair In short, the need to calculate in-plane effective length (2005b) and White and Kim (2006) for additional (K) factors is eliminated. discussions. Interestingly, the use of P =P for members with ni y compact cross-section elements was considered in the 2.2.1. Direct analysis method development of the direct analysis approach (Maleck, The direct analysis method is the only one of the above 2001). Although this is a viable option, it requires the three procedures that is generally applicable to all types modeling of out-of-straightness in the analysis for members of frames. This method involves two simple modifications subjected to large axial compression (to properly capture to the second-order elastic analysis: (1) the use of a in-plane limit states dominated by non-sway column nominal reduced elastic stiffness and (2) the inclusion of flexural buckling). The modeling of member out-of- a nominal initial out-of-plumbness. These two devices are straightness adds an additional level of complexity to the adjustments to the analysis that approximate the internal analysis, and in many steel structures, P based on the ni forces and moments from the type of distributed plasticity actual unsupported length is only slightly smaller than P. y analysis explained in the previous section. The reduced Therefore, AISC (2005a) uses P based on the actual ni elastic stiffness is taken as 0.8 of the nominal elastic unsupported length (K=1) to capture the influence of stiffness of the structure, except in members subjected to potential in-plane non-sway column flexural buckling. large axial loads of αP >0.5P, where the member For certain member types, such as tapered members, r y flexural rigidity is taken as 0.8 times the column inelastic there are significant advantages to using the cross-section stiffness reduction factor τ (see Table 1). The base axial strength rather than the nominal buckling strength b nominal out-of-plumbness is taken as ∆ =L/500, the P as the axial strength term in the beam-column o ni same value as discussed previously for distributed plasticity interaction check (White and Kim, 2006). In cases where analysis. However, the direct analysis provisions permit the member axial loads are small and the cross-section is the use of a smaller nominal out-of-plumbness where compact, the column and beam-column resistances are justified. For instance, when the sidesway amplification represented accurately using P =P, without the inclusion ni y ∆ /∆ is smaller than 1.5, AISC (2005a) permits the of any member out-of-straightness in the analysis. This 2nd 1st out-of-plumbness effect to be neglected when the simplification is appropriate whenever αP <0.1P . For r eL associated notional load (see the discussion below) is members with cross-section elements that are slender smaller than the corresponding applied lateral load. under axial compression, White and Kim (2006) show Many engineers may prefer to model the above out-of- that P may be taken as QP when the above limit is ni y plumbness effects by using notional lateral loads. As satisfied, where Q is the AISC form factor accounting for shown in Fig. 5, if the framing above and below a given local buckling effects with the web edge stress f taken as vertical load elevation has the same ∆ /L, the out-of- F. o y plumbness effect can be represented accurately by AISC (2005a) introduces a plethora of rules intended to applying a notional lateral load of N =Y∆ /L at the level allow (and provide limits on) the use of various idealizations i i o under consideration, where Y is the total vertical load and approximations (see Table 1). This characteristic is i Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 77 Table 1. Summary of specific AISC (2005a) elastic analysis-design procedures adapted from White and Kim (2006) Direct analysis (Appendix 7) Effective length (Section C2.2a) First-order analysis (Section C2.2b) Limitations on the ∆ /∆ ≤ 1.5, αP ≤ 0.5P None ∆ /∆ ≤ 1.5 2nd 1st r y use of the method 2nd 1st (See Note 5) First-order, B1 is applied to the Type of analysis Second-order (See Note 1) Second-order (See Note 1) member total moment Structure geometry Nominal (See Note 2) Nominal Nominal used in the analysis 0.002Y Minimum if ∆ /∆ Notional load to be i 2nd 1st ≤ 1.5 Additive if ∆ /∆ > 1.5 0.002Y minimum 2.1(∆/L)Yi ≥ 0.0042Y additive applied in the analysis 2nd 1st i i (See Note 2) 0.8 * Nominal, except EI = eff 0.8τEI when αP > 0.5P Nominal Nominal b r y (See Note 3) Effective stiffness τ = 4[αP/P(1−αP/P)] b r y r y used in the analysis Use of τ =1 is permitted in all b cases if additional notional loads of 0.001Y are applied, additive i to other lateral loads P in moment-frame columns is ni P is based on the unsupported based on a buckling analysis or ni length in the plane of bending, the corresponding effective L (i.e., K=1) length KL; P in all other cases i ni is based on KL = L (i.e., K = 1). In-plane flexural i i P is based on the unsupported buckling strength P If αP < 0.15P , or if a member lennigth in the plane of bending, L ni out-ofr-straightneLess of 0.001L or i the equivalent notional loading If ∆ /∆ < 1.1, K may be taken 2nd 1st is included in the analysis, P equal to one in all cases. ni may be taken equal to P y (See Note 4) P is based on the unsupported length in the out-of-plane direction, L Out-of-plane flexural no o buckling strength P Alternatively, P may be based on an out-of-plane buckling analysis or the corresponding effective no length KL (see nNoote 4) o General Note. ∆ /∆ is the ratio of the 2nd-order drift to the 1st-order drift (for rectangular frames, ∆ /∆ may be taken as B calculated by Section C2.1b2n)d. ∆/1Lstt is the largest 1st-order drift from all the stories in the structure. In structures that2 nhdave1s tflexible diaphragms2, the ∆/L in each story is taken as the average drift weighted in proportion to the vertical load, or alternatively, the maximum drift. All ∆ /∆ and ∆/L ratios shall be calculated using the LRFD load combinations or using a factor of α=1.6 applied to the gravity loads in ASD.2 Tndhe1 fstactor a is 1.0 for LRFD and 1.6 for ASD. The term Y is the total gravity load applied at a given level of the structure. P is the member elastic buckling resistance based on the actual unsupiported length in the plane of bending, π2EI/L2 for prismatic members.e L Note 1. Any legitimate method of second-order analysis that includes both P∆ and Pδ effects is permitted, including 1st-order analysis with amplifiers. For αP <0.15P , a P-large delta (P-∆) analysis using one element per member generally provides an accurate solution for the sidesway displacemr ents andeL the corresponding internal second-order forces and moments. However, for members with αP >0.05P , either multiple elements must be used per member to obtain accurate second-order internal moments (unconservative error lress than eoLr equal to 5%) in general from a P-large delta analysis, or a P-small delta amplifier must be applied to the element internal moments. Accurate general P-∆ analysis solutions may be obtained by maintaining αP <0.05P , where P =π2EI/l2 is the Euler buckling load in the plane of bending based on the element length l. Second-order analysis methrods that edlirectly in cellude both P-∆ and P-δ effects at the element level generally provide better accuracy than P-large delta analysis procedures. The target of 5% maximum unconservative error is based on the original development of the AISC LRFD beam-column strength equations (ASCE 1997; Surovek-Maleck and White 2004a). Note 2. A nominal initial out-of-plumbness of ∆/L= 0.002 may be used directly in lieu of applying 0.002Y minimum or additive notional loads. o i Note 3. The nominal stiffness and geometry should be employed for checking serviceability limit states. The reduced effective stiffness and the notional loads or nominal initial out-of-plumbness are required only in considering strength limit states. Note 4. AISC (2005) does not explicitly state this provision in the context of the direct analysis method. This provision is encompassed within the Chapter C requirements for general stability analysis and design, which allow any method of analysis and design that addresses the effects listed at the beginning of Section 1. Note 5. The largest unconservative error associated with the limit αP <0.1P is approximately 5% and occurs for a simply-supported, concentrically loaded column with zero moment and αP =0.1P =φPr . The teaLrget of 5% maximum unconservative error is based on the original development of the AISC LRFD beam-column srtrength eeLquaticonys (ASCE 1997; Surovek-Maleck and White 2004a). Note 6. The 1st-order analysis method does not account for the influence of significant axial compression in the rafters or beams of portal frames. Therefore, this method strictly should not be applied for the analysis and design of the primary moment frames in these types of structures. 78 Donald W. White et al. length method when the second-order amplification of the sidesway displacements is larger than 1.5, i.e., ∆ /∆ > 2nd 1st 1.5 (based on the nominal elastic stiffness of the structure). This is because the effective length method significantly underestimates the internal forces and moments in certain cases when this limit is exceeded (Deierlein, 2003 & 2004; Kuchenbecker et al., 2004; White and Kim, 2006). For structures with ∆ /∆ >1.5, 2nd 1st AISC (2005a) requires the use of the direct analysis method. Correspondingly, when using the direct analysis approach with structures having ∆ /∆ <1.5, AISC 2nd 1st (2005a) allows the engineer to apply the notional lateral loads (or the corresponding nominal out-of-plumbness) as minimum values solely in the gravity-only load combinations. For column and beam-column in-plane strength assessment in moment frames, the effective length approach focuses on the calculation of the member axial stresses F at ei incipient buckling of an appropriately selected model (the subscript “i” is used to denote in-plane flexural buckling). This buckling model is usually some type of subassembly that is isolated from the rest of the structural system (ASCE, 1997). Engineers often handle the elastic buckling stresses (F ) implicitly, via the corresponding column ei effective lengths KL. The effective length is related to the i underlying elastic buckling stress via the relationship Figure 5. Relationship between notional lateral loads and nominal out-of-plumbness. π2E F =-------------------- (7a) ei (KL⁄r)2 i i typical of design Specifications and is not new to these or AISC provisions. One can avoid the need to consider all of these caveats by using the direct analysis method, (π2E)⁄(L⁄r)2 K = ---------------------i------i---- (7b) calculating P as the applicable column strength based i F on the actuanl unsupported length, in-plane or out-of- ei plane, including an elastic stiffness reduction factor of 0.8 In the effective length method, the effects of residual or 0.8τ in the analysis as applicable, and including a stresses, P-∆ effects and P-δ effects are addressed b o o uniform initial out-of-plumbness of 0.002L in the analysis implicitly by the calculation of P from the column ni relative to the perfect geometry of the structure. strength equations. These equations can be written in terms of either KL or F (AISC 2005a). Unfortunately, i ei 2.2.2. Effective length method the selection of an appropriate subassembly buckling The AISC (2005a) effective length method is the same model generally requires considerable skill and judgment. as the traditional AISC method of analysis and design, As a result, there is a wide range of different buckling but with the addition of a notional minimum lateral load models and K factor equations. In certain cases, subtle for gravity-only load combinations. This minimum lateral differences in the models can produce radically different load accounts for the influence of nominal geometric results (ASCE, 1997). imperfections on the brace forces, beam moments, In particular, one should note that a rigorous buckling connection moments and in-plane moments used for out- analysis of the entire structure does not necessarily of-plane strength design of beam-columns. In actuality, provide an appropriate F or K (ASCE, 1997). Members ei i the effects of any physical out-of-plumbness are present that have small axial stress F at the buckling limit ei for all load combinations. However, these effects are (relative to π2E/(L/r)2) tend to have large values for K i i i overwhelmed by the effects of the primary lateral loads in from Eq. (7b). In some cases, these large K values are i all the ASCE 7 (ASCE, 2005) lateral load combinations, justified while in other cases they are not. If the member as long as the structure’s sidesway amplification is not is indeed participating in the governing buckling mode, a excessive. Therefore, in the AISC (2005a) effective length large K is justified. If the member is largely undergoing i method, the notional lateral loads are specified solely as rigid-body motion in the governing buckling mode, or if minimum lateral loads in gravity-only load combinations. it has a relatively light axial load and is predominantly AISC (2005a) does not allow the use of the effective serving to restrain the buckling of other members, a large Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 79 K value often is not justified. The distinction between often significantly more accurate. Therefore, the first- i these two situations requires engineering judgment. Some order analysis method is not considered further in this of the situations requiring the greatest exercise of paper. judgment to avoid excessively large K values include: (1) columns in the upper stories of tall buildings, (2) columns 2.2.4. General comments with highly flexible and/or weak connections and (3) Both the direct analysis and effective length methods beams or rafters in portal frames, which may have require a second-order elastic analysis. However, any significant axial compression due to the horizontal thrust second-order elastic analysis procedure is sufficient, from the base of the frame. including first-order analysis with amplifiers, assuming There is no simple way of quantifying the relative that the procedure is sufficiently accurate or conservative. participation of a given member in the overall buckling The above methods differ in the way that they handle of the structure or subassembly under consideration. geometric imperfection and distributed yielding effects in Quantifying the participation requires an analysis of the the second-order analysis model and in the member sensitivity of the buckling load to variations in the resistance equations. member sizes. Even if one conducted such an analysis, The beam-column out-of-plane resistance check is the there is no established metric for judging when Eq. (7b) same in both of the above methods, albeit with different should or should not be used. Engineers typically base values of P and M. In AISC (2005a), the simplest out- r r their effective length calculations on story-by-story models of-plane beam-column resistance check is given by Eqs. to avoid the first of the above situations. They idealize (5) but with P =P , where P is the out-of-plane n no no columns with weak and/or flexible connections as pin- flexural, torsional or flexural-torsional buckling strength ended leaner columns with K=1 to avoid the second of of the member as a concentrically-loaded column. Eq. (6) the above situations. Lastly, many engineers utilize K=1 generally provides a more liberal estimate of the out-of- for design of the beams or rafters in portal frames for the plane flexural-torsional resistance of doubly-symmetric axial compression effects, although Eq. (7b) may suggest compact I-section members. K>1 based on the F from an eigenvalue buckling ei analysis. 2.3. Design by direct elastic-plastic hinge analysis The direct analysis method provides a more straightforward Since 1961, the AISC Specifications have permitted the and accurate way of addressing frame in-plane stability use of plastic analysis and design in cases where members considerations. By including an appropriately reduced subjected to plastic hinging satisfy requirements that nominal elastic stiffness, an appropriate nominal out-of- ensure their ductility. However, the AISC Specifications plumbness of the structure, and an appropriate out-of- from 1969 through 1999 have also generally required the straightness (for members subjected to high axial loads) engineer to supplement the plastic analysis by beam- in the analysis, the member in-plane length effects can be column strength interaction checks in which the axial removed entirely from the resistance side of the design resistance term is based on a member effective length. equations. The member in-plane column strength P may This practice adds significant complexity to the AISC ni be taken simply as P for members that satisfy the plastic analysis and design procedures. Furthermore, at y previously discussed caveats. In-plane stability is addressed best, the resulting beam-column interaction equations by estimating the actual required internal cross-section provide only an approximate assessment of the frame strengths P and M directly from the analysis, and by stability behavior under progressive plastic hinge formation. r r comparing these required strengths against appropriate In many cases, they overly restrict the forces and moments cross-section based resistances. Alternatively, to avoid the in sway frame columns (Ziemian et al., 1992; McGuire, need to include out-of-straightness effects in members 1995). with large axial loads, P may be calculated using the The AISC (2005a) direct analysis method can be ni actual in-plane unsupported length L (K=1). extended to provide an attractive alternative to the above i procedures. The extension is very simple-for members 2.2.3. First-order analysis method that satisfy separate requirements to ensure sufficiently The first-order analysis method, summarized in Table ductile response (i.e., sufficient rotation capacity), moment 1, is implicitly a simplified conservative application of redistribution is allowed based on the assumption of the direct analysis approach, targeted at rectangular or elastic-perfectly plastic hinge behavior at the limit of the tiered building frames. White and Kim (2006) detail the member resistance from Eqs. (5). This extension satisfies conservative assumptions invoked in the development of the AISC (2005a) Appendix I provisions for inelastic this procedure. Although the first-order analysis method analysis and design. The separate ductility requirements can be useful for simplified analysis and design of some in AISC (2005a) Appendix 1 include restrictions on: types of frames, this method is really just a direct analysis •The material yield strength F, y with a number of simplifying assumptions. There are •The flange and web slenderness values b/2t and h /t , f f p w numerous other ways to apply direct analysis using an •The member out-of-plane slenderness L /r, b y approximate second-order analysis, many of which are •The magnitude of the axial force αP (α=1.0 for r 80 Donald W. White et al. LRFD loadings) and The third example is a hypothetical industrial building •The connection details. frame that meets representative service drift requirements. These restrictions tend to preclude the formation of However, the behavior of the lateral load resisting system plastic hinges in beams and beam-columns with noncompact is sensitive to stability considerations to the extent that flanges or webs, in beams where the resistance is governed the structure’s limit load is reached when the first column by out-of-plane lateral-torsional buckling, and in beam- plastic hinge forms in the direct elastic-plastic hinge columns with out-of-plane unbraced lengths large enough analysis. Lastly, a fourth example is provided in which the such that the resistance is governed by Eq. (6). Therefore, lateral load resisting columns are subjected to substantial for members subjected to in-plane loading and where non-sway gravity load moments. This frame exhibits inelastic redistribution is allowed, the strength either is some reserve strength beyond the first plastic hinge for its given or is accurately approximated by Eqs. (5) with M critical load combination. All of these examples are n =M and with P =P calculated using the actual member individual members or single-story rectangular structures p n ni unsupported length in the plane of bending. Furthermore, with either fully-restrained (FR) or simple connections. when αP is less than 0.1P , P =P is an acceptable The reader is referred to Maleck (2001), Martinez-Garcia r eL ni y approximation. The approximation P =P also can be (2002), Deierlein (2003), Surovek et al. (2005), White et ni y used for general I-section members as long as an appropriate al. (2005a & b) and White and Kim (2006) for consideration out-of-straightness is included in the second-order of a broader range of structure types including multi-story analysis. Members that do not satisfy all the requirements frames, braced and combined framing systems, trussed necessary to ensure ductile response must be designed framing, PR frames, gabled frames, and frames with slender elastically in the manner discussed in Section 2.2.1. element section members, singly-symmetric members, The above type of analysis and design is referred to in nonprismatic members, and/or members with non- this paper as the direct elastic-plastic hinge method. In constant axial load along their lengths. this method, the elastic stiffness of the structure is reduced In all the following examples, the appropriate stiffness to account for distributed yielding effects neglected in the reduction factor in the direct analysis is 0.8. That is, P/P y elastic-plastic hinge idealization. Also, a nominal initial is always less than 0.5, and therefore τ =1.0. The b out-of-plumbness is included to account for geometric distributed plasticity analysis is conducted using a factor imperfection effects. These devices eliminate the need to of 0.9 on the elastic stiffness E and the strength F in all y calculate and apply column effective lengths in the cases. The axial force P is close to or smaller than 0.1P eL context of inelastic design, as long as the second-order in all the above problems. Therefore, the column out-of- effects are captured in the elastic-plastic hinge analysis. straightness is neglected and, to illustrate the validity of Furthermore, these devices allow the engineer to take this option, P is taken equal to P in the direct analysis ni y advantage of second-order elastic-plastic hinge analysis solutions. A nominal column out-of-straightness is included software. This kind of software is becoming more and in the direction of the member curvature due to the applied more available in engineering practice. loadings in all of the distributed plasticity solutions, to Other limits are possible within which the engineer may demonstrate that the direct analysis solutions give accurate be permitted to perform a classical rigid-plastic analysis predictions for these cases with out-of-straightness and design, e.g., see King (2001) and Davies and Brown neglected. All of the direct and distributed plasticity (1996). These limits are not addressed in this study. analyses are conducted using an out-of-plumbness of ∆ =0.002L, specified in the direction of the drift under 3. Illustrative Examples thoe applied loads. For the effective length method, a second-order elastic analysis is conducted using the nominal This section provides a number of basic examples elastic stiffness and the initially perfect geometry. aimed at illustrating the relative merits of the various The second-order elastic and distributed plasticity analysis and design procedures outlined in Section 2. The solutions are conducted using the GT-Sabre software system first example addresses the strength predictions for one of (Chang 2005). A flexibility-based element (Alemdar and the major-axis bending cases of the Fig. 3 cantilever White 2005), which is based on an equilibrium-based beam-columns. Since this is a statically determinate distribution of moments and a correspondingly accurate structure, inelastic analysis and design does not provide representation of inelastic curvatures (including the influence any advantage. This example is representative of numerous of transverse distributed loads), is used for the beams. nonredundant stability critical benchmark problems The mixed element developed by Alemdar and White considered in the development of the direct analysis (2005), which is capable of accurate modeling of the method (Deierlein 2003; Surovek-Maleck and White second-order moments and inelastic curvatures, is used 2003 & 2004). The second example is a nonredundant for the beam-columns. A small modulus of 0.001E is portal frame previously posed by LeMessurier (1977). assumed for the yielded material. The direct elastic- This frame exhibits significant sway under gravity loadings, plastic hinge solutions are conducted using the Mastan2 due to lack of symmetry of its geometry. Also, the beam software system (McGuire et al., 2000). For major-axis governs its maximum resistance rather than the columns. bending, the Mastan2 yield surface, with axial force and

Stability Analysis and Design of Steel Building Frames Using the 2005 AISC Specification

Similar Stability Analysis and Design of Steel Building Frames Using the 2005 AISC Specification