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, USA2Civil and Environmental Engineering, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA

3RAM International, 2744 Loker Avenue West, Carlsbad, CA, 92010, USA4Structural 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. Thenew Specification defines the general requirements for stability analysis and design and gives engineers the freedom to selector devise their own methods within these constraints. It also provides several specific procedures. This paper first gives anoverview of the elastic analysis and design procedures in AISC (2005) as well as specific second-order distributed plasticitymethods upon which, in part, these procedures are based. The relationship between the AISC elastic provisions and the refinedinelastic methods is explained. Secondly, the paper highlights one interpretation of the AISC inelastic analysis and designprovisions that greatly facilitates the application of elastic-plastic hinge methods of analysis. The paper closes by presentingfour 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

Chapter C of the AISC (2005a) Specification, Stability

Analysis and Design, states that any analysis and design

method that addresses the following effects on the overall

stability of the structure and its elements is permitted:

1. Flexural, axial and shear deformations (in members,

connections and other components),

2. Reduction in stiffness (and corresponding increases

in deformations) due to residual stresses and material

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-∆o and P-δ

o effects, which are caused by the

member axial loads acting through unavoidable

initial ∆o and δ

o geometric imperfections (within

fabrication and erection tolerances) relative to the

ideal configuration of the structure.

The above Chapter C statement gives engineers the

freedom to select or devise methods that are best suited

for the various structure types encountered in practice. It

allows for innovation as long as there is proper

consideration of the physical effects that influence the

structural response.

Since the 1961 AISC Specification, when the column

effective length concept was first introduced by AISC,

*Corresponding authorTel: +404-894-5839, Fax: +404-894-2278E-mail: dwhite@ce.gatech.edu

Figure 1. Second-order P-∆ and P-δ effects (from Whiteand Kim (2006)).

72 Donald W. White et al.

American analysis and design methods have addressed all

of the above effects in some fashion whenever they are

deemed to have an important influence on the structural

response. Also, there has always been implicit recognition

that engineers can use their professional judgment to

disregard specific effects (e.g., member shear deformations,

connection deformations, etc.) whenever they are considered

negligible. Member yielding, residual stress effects, and

geometric imperfection effects traditionally have been

addressed in the formulation of member design resistances,

and, with minor exceptions, have not been considered in

the analysis. In some cases, engineers have included a

nominal out-of-plumbness effect in the analysis of gravity

load combinations, particularly if the geometry and loading

are symmetric. Strictly speaking, this is not necessary for

the in-plane strength assessment of beam-columns in the

prior AISC Specifications. However, this practice is

necessary to determine P-∆o effects on bracing forces,

beam moments, connection moments, and in-plane moments

for checking the out-of-plane resistance of beam-columns.

Also, plastic design procedures incorporate material

yielding into the analysis, typically by the use of idealized

plastic hinge models in adequately braced compact-

section members. However, geometric imperfection and

residual stress effects traditionally are not included in this

type of analysis.

The 1961 AISC Specification, and other AISC

Specifications up until 1986, relied strictly on the

structural analysis only for calculation of geometrically

linear (first-order) forces and moments in the idealized

geometrically-perfect, nominally-elastic (or elastic-plastic)

structure. For elastic analysis and design, second-order

(P-∆ and P-δ) effects were addressed very discreetly via

the following beam-column strength interaction equation:

(1)

The expression

(2)

in Eq. (1) amplifies the member flexural stresses fb to

account for the P-∆ and P-δ effects.

Generally, Eq. (2) gives only a coarse approximation of

the true second-order effects. LeMessurier (1977) and

others later addressed better ways to determine the

amplified bending stresses in general rectangular frames,

all within the context of Allowable Stress Design (ASD).

However, even today (2005), engineers using the AISC

(1989) ASD provisions can apply Eq. (2) in ways that

significantly under-estimate the physical second-order

effects in certain cases. The accuracy hinges largely on

the proper determination of Fe, the member axial stress at

incipient elastic buckling (considering the interaction of

the member with the rest of the structure). The parameter

Fe' is calculated by dividing this buckling stress by the

factor of safety for column elastic buckling, 23/12 = 1.92.

The axial stress Fe' is determined typically using the

equation

(3)

where KLb/rb is the column effective slenderness ratio

in the plane of bending, and K is the effective length

factor associated with the above buckling solution. Also,

this K factor is used typically in calculating the column

axial resistance Fa. If desired, Fe' = Fe/1.92 and Fa can be

determined directly from the buckling analysis model.

The term Cm in Eqs. (1) and (2) is discussed subsequently.

AISC (1986) LRFD was the first American Specification

to refer explicitly to the calculation of second-order

moments from a structural analysis. This specification

introduced the following two-equation format for the

beam-column resistance:

for (4a)

for (4b)

where Mu is defined as the maximum second-order

elastic moment along the member length. AISC (1986)

states that Mu may be determined from a second-order

elastic analysis using factored loads. However, it also

provides a procedure that uses amplification factors to

calculate the second-order elastic moments from a first-

order elastic analysis. This procedure is in fact an

approximate second-order analysis. The moments Mu in

Eqs. (4) are the second-order elastic moments in an

idealized model of the structure assuming initially perfect

geometry and completely linear elastic material response.

In all the AISC Specifications from AISC (1961)

through AISC (1989) ASD and AISC (1999) LRFD, the

influence of geometric imperfections and residual stresses

was addressed solely within the calculation of the

member resistances (Fa and Fb in ASD and Pn and Mn in

LRFD). The new direct analysis provisions in AISC

(2005a) recognize that specific advantages can be

realized by moving an appropriate nominal consideration

of these effects out of the resistance side and into the

structural analysis side of the design equations. By

incorporating an appropriate nominal consideration of

these effects in the analysis, the resistance side of the

design equations is greatly simplified, and the accuracy of

the design checks is generally improved. These attributes

are discussed further in the subsequent sections.

All of the above analysis and design procedures are

based inherently on the use of second-order analysis

fa

Fa

-----Cmfb

1fa

Fe'

------–⎝ ⎠⎛ ⎞F

b

------------------------+ 1.0≤

Cm

1fa

Fe'

------–⎝ ⎠⎛ ⎞------------------ AF=

Fe'

12π2E

23 KLbrb

⁄( )2----------------------------=

Pu

2φcPn

--------------M

u

φbM

n

------------+ 1.0≤Pu

φcPn

---------- 0.2<

Pu

φcPn

----------8

9---M

u

φbM

n

------------+ 1.0≤Pu

φcPn

---------- 0.2≥

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 73

(first-order elastic analysis with amplifiers being one type

of second-order elastic analysis). Elastic analysis does not

include the consideration of the member resistances in

itself. Therefore, all of the above elastic methods must

include member resistance equations. However, the

method of analysis and the equations for checking the

member resistances are inextricably linked. Changes in

the analysis calculation of the required strengths (e.g., faand fbCm/(1 − fa/Fe') in Eq. (1) or Pu and Mu in Eqs. (4))

can lead to simplifications in the calculation of the

member resistances (typically Fa in Eq. (1) or Pn in Eqs.

(4)). Specifically, if the structural analysis is configured

to provide an appropriate representation of the internal

member forces, the in-plane resistance of the structure

can be checked entirely on a cross-section by cross-

section basis.

AISC (2005a) adopts the equations from AISC (1999)

LRFD as a base representation of the beam-column

resistances for all of its analysis and design procedures.

However, the notation is generalized such that the

equations apply to both ASD and LRFD:

for (5b, AISC H1-1a)1

for (5a, AISC H1-1b)

The terms in these equations are defined as follows:

Pr = the required axial compressive strength, determined

in ASD by analyzing the structure under 1.6 times the

ASD load combinations and then dividing the results by

1.6, or determined in LRFD by analyzing the structure

under the LRFD load combinations.

Mr = the required flexural strength, determined in ASD

by analyzing the structure under 1.6 times the ASD load

combinations and then dividing the results by 1.6, or

determined in LRFD by analyzing the structure under the

LRFD load combinations.

Pc = the allowable or design axial compressive strength,

given by Pn/Ωc in ASD or by φcPn in LRFD, where Pn is

the nominal compressive resistance determined in accordance

with Chapter E.

Mc = the allowable or design flexural strength, given by

Mn/Ωb in ASD or by φbMn in LRFD, where Mn is the

corresponding nominal resistance determined in accordance

with Chapter F.

φc and φb = resistance factors for axial compression and

bending, both equal to 0.9.

Ωc and Ωb = factors of safety for axial compression and

bending, both equal to 1.67.

The above 1.6 factor for ASD is smaller than the

column safety factor of 1.92 in the AISC ASD (1989)

amplification of the flexural stresses (see Eqs. (1) through

(3)). However, ASD-H1 also states that Cm shall be taken

as 0.85 in Eqs. (1) and (2) for frames subject to joint

translation. This Cm value typically underestimates the

sidesway amplification effects (Salmon and Johnson, 1996).

Nevertheless, the ASD moment amplifier summarized in

Eq. (2) is still conservative in many practical cases. This

is because the predominant second-order effects are often

associated solely with the structure sidesway. Equation

(1) applies a single amplifier indiscriminately to the total

flexural stresses from both sidesway and non-sway

displacements. The amplification factor procedure in

AISC (1999) LRFD and AISC (2005a) is more accurate,

but involves a cumbersome subdivision of the analysis

into separate no-translation (nt) and lateral-translation (lt)

parts. Kuchenbecker et al. (2004) and White et al. (2005a

& b) outline an amplified first-order elastic analysis

approach that provides good accuracy for rectangular

framing. This approach avoids the above cumbersome

attributes of the AISC (1999 & 2005a) amplification factor

procedure.

In many cases, Eqs. (5) provide a more liberal

characterization of the beam-column resistances than the

multiple beam-column strength curves in AISC ASD

(1989). Also, AISC (2005a) provides the following

alternative interaction equation to characterize the out-of-

plane flexural-torsional buckling resistance of doubly-

symmetric I-section members subjected to axial compression

and major-axis bending moment:

(6, AISC H1-2)

where Pco is the out-of-plane column strength and Mcx

is the flexural strength with respect to lateral-torsional

buckling. White and Kim (2006) provide a detailed

discussion of the background to Eq. (6) and explain that

this equation should be applied only for doubly-symmetric

compact I-section members.

In the subsequent developments, it is useful to consider

the characterization of separate in-plane and out-of-plane

beam-column resistances. This can be accomplished by

using Eqs. (5) with different definitions of Pc and Mc, or

by using Eqs. (5) to characterize the in-plane strength and

Eq. (6), where it is applicable, to characterize the out-of-

plane strength. The in-plane resistance is estimated with

Eqs. (5) by neglecting: (1) out-of-plane flexural, torsional

or flexural-torsional buckling in the calculation of Pc and

(2) lateral-torsional buckling in the calculation of Mc. The

out-of-plane resistance is estimated with Eqs. (5) by using

Pc = Pco and using the governing resistance from Chapter

F for Mc (White and Kim, 2006).

For checking the in-plane and out-of-plane strength of

general I-section members, Eqs. (5) must be used. However,

for doubly-symmetric compact I-section members subjected

to in-plane major-axis bending and large axial loads, Eq.

(6) provides a more liberal assessment of the out-of-plane

Pr

Pc

-----8

9---M

r

Mc

------+ 1.0≤P

r

Pc

----- 0.2≥

Pr

2Pc

--------8

9---M

r

Mc

------+ 1.0≤P

r

Pc

----- 0.2<

Pr

Pco

-------M

r

Mcx

--------⎝ ⎠⎛ ⎞

2

+ 1.0≤

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

Mry/Mcy is smaller than 0.05 in the out-of-plane direction,

where Mry and Mcy are the required moment and the

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

paper as a distributed plasticity analysis.

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 δo = L/1000, where L is the

unsupported length in the plane of bending.

• An out-of-plumbness of ∆o = L/500, the maximum

tolerance specified in the AISC (2005b) Code of Standard

Practice.

• The Lehigh (Galambos and Ketter, 1959) residual

stress pattern shown in Fig. 2.

• An elastic-perfectly plastic material stress-strain

response.

These results are not surprising, since Eqs. (5) were

originally developed in part based on calibration to results

from this type of analysis (ASCE, 1997; Surovek-Maleck

and White, 2004). Since the AISC (2005a) analysis and

design procedures do not make any distinction between

hot-rolled and general built-up members, the above

nominal geometric imperfections and residual stresses are

sufficient to capture the Specification requirements for all

compact I-section member types. Figure 3 defines two

simple cases that illustrate the above findings. These are

W10 × 60 cantilever beam-columns with L/r = 40 subjected

to axial and transverse loads at their free ends. In the first

case, the member is subjected to major-axis bending,

while in the second case it is subjected to minor-axis

bending. Figure 4 compares the nominal strength interaction

curves obtained from the above type of distributed

plasticity analysis to the corresponding AISC (2005a)

beam-column strength curve. The AISC curve is obtained

using the AISC (2005a) effective length method with K =

2, and is the same for both the minor and major-axis

bending examples, since their L/r values are the same in

the plane of bending. Martinez-Garcia (2002), Surovek-

Maleck and White (2003), and Deierlein (2003) summarize

the results from other more comprehensive studies.

If the above distributed plasticity analysis is to be used

in an AISC (2005a) LRFD context, the resistance factors

φc = φb = 0.9 must be included. One way of doing this is

to determine the nominal beam-column strength curves as

shown in Fig. 4, and then to multiply both the abscissa

and the ordinate by φc = φb = 0.9 to obtain the final

member design resistance. However, identical results are

obtained if both the yield strength Fy and the elastic

modulus E are factored by 0.9. If only the yield strength

Fy is factored by 0.9, the design strengths are overestimated

for highly slender columns that fail by elastic buckling.

The factoring of both E and Fy by 0.9 up front is

preferred, since this approach facilitates the general

inelastic analysis and design of structural systems. There

is no straightforward way of applying distributed plasticity

analysis, or any other form of inelastic analysis, in the

context of ASD. In as such, AISC (2005a) disallows the

use of inelastic analysis for this approach. Most of the

Figure 2. Lehigh (Galambos and Ketter 1959) residualstress pattern.

Figure 3. Example cantilever beam-columns.

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 75

subsequent discussions in this paper are phrased in the

context of LRFD.

The distributed plasticity analysis solution captures the

in-plane design resistances of compact I-section members

completely when it includes the above attributes. Therefore,

the in-plane beam, column and beam-column resistance

checks are automatically satisfied for these member types

if the distributed plasticity analysis shows that the

structure supports the design loadings. The engineer does

not need to perform any separate evaluation of the in-

plane member resistances. The Australian Standard

AS4100 (SAA, 1990) was the first to explicitly permit

this type of analysis and design. This Standard coined the

term advanced analysis, in a very specific context, to

denote an analysis that supersedes the Specification member

strength checks. Subsequently, various SSRC publications

(e.g., White and Chen (1993)) as well as other research

papers and reports have adopted this terminology.

In many practical steel building structures, member

out-of-straightness has little to no effect on the frame

resistance. White and Nukala (1997) explain that when

αPr is less than PeL/7 ≅ 0.15PeL, where

α = 1.6 for ASD or 1.0 for LFRD and

PeL = π2EI/L2

using the moment of inertia I in the plane

of bending,

out-of-straightness effects can be neglected generally in

the distributed plasticity analysis (i.e., δo can be taken equal

to zero). Otherwise, one must determine the appropriate

direction for the member out-of-straightness. Usually, the

strength is reduced the most due to member out-of-

straightness if δo is specified in the direction of the

member deformations (relative to its chord) due to the

applied loads. However, in some cases, it is advisable to

specify the various member δo values in the pattern of the

fundamental buckling mode obtained from an eigenvalue

buckling analysis. In these cases, the direction of the δo

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 αPr <

0.15PeL rule allows the engineer to disregard these

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

loadings.

At the present time (2005), distributed plasticity analysis

has been applied most commonly in research studies

investigating the resistance of frames composed of

adequately braced compact-section members. However,

in design practice, the member strengths can be governed

by out-of-plane buckling, flange or web local buckling, or

combinations of these strength limits. These limit states

cannot be captured by a planar distributed plasticity

analysis. Although some progress has been made on 3D

distributed plasticity methods (e.g., see Pi and Trahair

(1994), Izzudin and Smith (1996), Teh and Clarke (1998),

Battini and Pacoste (2002) and Nukala and White (2004)

among others), the complexity and cost of the analysis is

significantly greater. Furthermore, the appropriate handling

of residual stresses, geometric imperfections, warping

continuity at beam-to-column joints, local-overall member

buckling interactions, restraint from and interaction with

floor slabs, and other important attributes that can influence

the 3D response have not been studied thoroughly at this

time. In fact, many of these effects are considered in

rather simplistic ways in ordinary design practices, e.g.,

out-of-plane and spatial beam-column resistances are

based typically on the assumption of unrestrained warping

at the member ends. Nevertheless, in frames subjected

predominantly to in-plane loading, the authors assert that

distributed plasticity analysis can be utilized to determine

an accurate estimate of the internal forces in the structure.

These forces then can be checked against Specification

member resistance equations corresponding to any of the

above 3D limit states not included in the analysis.

Various analysis refinements are possible relative to the

above procedures. For instance, a number of approaches

have been suggested for reducing the nominal out-of-

plumbness relative to base values, e.g., see the Commentary

of AISC (2005), White et al. (2003) and CEN (2003).

Also, other nominal residual stress distributions as well as

more complete stress-strain models (e.g., models including

strain-hardening) can be incorporated within the distributed

plasticity analysis. However, these considerations introduce

Figure 4. Nominal strength curves by distributed plasticityanalysis versus the AISC (2005a) effective length methodfor the example W10 × 60 cantilever beam-columns.

76 Donald W. White et al.

additional complexities that must be addressed. The

above procedures fully satisfy the base requirements of

the AISC (2005a) Specification.

2.2. Elastic analysis and design methods in AISC

(2005a)

The AISC (2005a) Specification defines three specific

elastic analysis and design methods. These are:

1. The direct analysis method, detailed in Appendix 7,

2. The effective length method, detailed in Section

C2.2a and

3. The first-order analysis method, detailed in Section

C2.2b.

Table 1, summarizes the key attributes of each of these

methods. Within the restrictions specified on their usage,

and provided that effects such as connection rotations or

member axial and shear deformations are properly

considered in the analysis when these attributes are

important, each of the above methods is intended to

comprehensively address all of the effects listed at the

beginning of Section 1. The following subsections provide

an overview of these AISC (2005a) procedures. The

reader is referred to Deierlein (2004), Nair (2005a), Nair

(2005b) and White and Kim (2006) for additional

discussions.

2.2.1. Direct analysis method

The direct analysis method is the only one of the above

three procedures that is generally applicable to all types

of frames. This method involves two simple modifications

to the second-order elastic analysis: (1) the use of a

nominal reduced elastic stiffness and (2) the inclusion of

a nominal initial out-of-plumbness. These two devices are

adjustments to the analysis that approximate the internal

forces and moments from the type of distributed plasticity

analysis explained in the previous section. The reduced

elastic stiffness is taken as 0.8 of the nominal elastic

stiffness of the structure, except in members subjected to

large axial loads of αPr > 0.5Py, where the member

flexural rigidity is taken as 0.8 times the column inelastic

stiffness reduction factor τb (see Table 1). The base

nominal out-of-plumbness is taken as ∆o = L/500, the

same value as discussed previously for distributed plasticity

analysis. However, the direct analysis provisions permit

the use of a smaller nominal out-of-plumbness where

justified. For instance, when the sidesway amplification

∆2nd/∆1st is smaller than 1.5, AISC (2005a) permits the

out-of-plumbness effect to be neglected when the

associated notional load (see the discussion below) is

smaller than the corresponding applied lateral load.

Many engineers may prefer to model the above out-of-

plumbness effects by using notional lateral loads. As

shown in Fig. 5, if the framing above and below a given

vertical load elevation has the same ∆o/L, the out-of-

plumbness effect can be represented accurately by

applying a notional lateral load of Ni = Yi∆o/L at the level

under consideration, where Yi is the total vertical load

applied at this level. Table 1 emphasizes the use of

notional lateral loads. However, in general cases where

questions may arise about the appropriate calculation of

these loads, one can always use the more fundamental

out-of-plumb geometry. For example, the total base shear

due to any out-of-plumbness is always zero, and thus the

total base shear due to the notional loads also must be

zero. As shown in Fig. 5, the notional loads arise from the

sum of the P∆o shear forces above and below each level.

The P∆o shear at the base of the structure offsets the sum

of all the horizontal notional loads in the analysis model.

Explicit modeling of the out-of-plumbness also removes

the need to calculate different notional loads for different

load combinations.

The direct analysis method provides an improved

representation of the structure’s distributed plasticity

forces and moments at the strength limit of the most

critical member or members. Due to this improvement in

the calculation of the internal forces and moments, AISC

(2005a) bases its calculation of Pni, the column nominal

strength for checking the in-plane resistance in Eqs. (5),

on the actual unsupported length in the plane of bending.

In short, the need to calculate in-plane effective length

(K) factors is eliminated.

Interestingly, the use of Pni = Py for members with

compact cross-section elements was considered in the

development of the direct analysis approach (Maleck,

2001). Although this is a viable option, it requires the

modeling of out-of-straightness in the analysis for members

subjected to large axial compression (to properly capture

in-plane limit states dominated by non-sway column

flexural buckling). The modeling of member out-of-

straightness adds an additional level of complexity to the

analysis, and in many steel structures, Pni based on the

actual unsupported length is only slightly smaller than Py.

Therefore, AISC (2005a) uses Pni based on the actual

unsupported length (K = 1) to capture the influence of

potential in-plane non-sway column flexural buckling.

For certain member types, such as tapered members,

there are significant advantages to using the cross-section

axial strength rather than the nominal buckling strength

Pni as the axial strength term in the beam-column

interaction check (White and Kim, 2006). In cases where

the member axial loads are small and the cross-section is

compact, the column and beam-column resistances are

represented accurately using Pni = Py, without the inclusion

of any member out-of-straightness in the analysis. This

simplification is appropriate whenever αPr < 0.1PeL. For

members with cross-section elements that are slender

under axial compression, White and Kim (2006) show

that Pni may be taken as QPy when the above limit is

satisfied, where Q is the AISC form factor accounting for

local buckling effects with the web edge stress f taken as

Fy.

AISC (2005a) introduces a plethora of rules intended to

allow (and provide limits on) the use of various idealizations

and approximations (see Table 1). This characteristic is

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 theuse of the method

None ∆2nd/∆1st ≤ 1.5∆2nd/∆1st ≤ 1.5, αPr ≤ 0.5Py

(See Note 5)

Type of analysis Second-order (See Note 1) Second-order (See Note 1)First-order, B1 is applied to the member total moment

Structure geometryused in the analysis

Nominal (See Note 2) Nominal Nominal

Notional load to beapplied in the analysis

0.002Yi Minimum if ∆2nd/∆1st

≤ 1.5 Additive if ∆2nd/∆1st > 1.5 (See Note 2)

0.002Yi minimum 2.1(∆/L)Yi ≥ 0.0042Yi additive

Effective stiffnessused in the analysis

0.8 * Nominal, except EIeff = 0.8τbEI when αPr > 0.5Py

(See Note 3)Nominal Nominal

τb = 4[αPr/Py(1 − αPr/Py)]

Use of τb = 1 is permitted in all cases if additional notional loads of 0.001Yi are applied, additive to other lateral loads

In-plane flexuralbuckling strength Pni

Pni is based on the unsupported length in the plane of bending,Li (i.e., K = 1)

Pni in moment-frame columns is based on a buckling analysis or the corresponding effective length KL; Pni in all other cases is based on KLi = Li (i.e., K = 1).

Pni is based on the unsupported length in the plane of bending, Li

If αPr < 0.15PeL, or if a member out-of-straightness of 0.001L or the equivalent notional loading is included in the analysis, Pni

may be taken equal to Py

(See Note 4)

If ∆2nd/∆1st < 1.1, K may be taken equal to one in all cases.

Out-of-plane flexuralbuckling strength Pno

Pno is based on the unsupported length in the out-of-plane direction, Lo

Alternatively, Pno may be based on an out-of-plane buckling analysis or the corresponding effective length KLo (see Note 4)

General Note. ∆2nd/∆1stt is the ratio of the 2nd-order drift to the 1st-order drift (for rectangular frames, ∆2nd/∆1st may be taken as B2 calculatedby Section C2.1b). ∆/L is the largest 1st-order drift from all the stories in the structure. In structures that have flexible diaphragms, the ∆/L ineach story is taken as the average drift weighted in proportion to the vertical load, or alternatively, the maximum drift. All ∆2nd/∆1st and ∆/Lratios shall be calculated using the LRFD load combinations or using a factor of α = 1.6 applied to the gravity loads in ASD. The factor a is1.0 for LRFD and 1.6 for ASD. The term Yi is the total gravity load applied at a given level of the structure. PeL is the member elasticbuckling resistance based on the actual unsupported length in the plane of bending, π2EI/L2 for prismatic members. Note 1. Any legitimate method of second-order analysis that includes both P∆ and Pδ effects is permitted, including 1st-order analysis withamplifiers. For αPr < 0.15PeL, a P-large delta (P-∆) analysis using one element per member generally provides an accurate solution for thesidesway displacements and the corresponding internal second-order forces and moments. However, for members with αPr > 0.05PeL,either multiple elements must be used per member to obtain accurate second-order internal moments (unconservative error less than orequal to 5%) in general from a P-large delta analysis, or a P-small delta amplifier must be applied to the element internal moments. Accurategeneral P-∆ analysis solutions may be obtained by maintaining αPr < 0.05Pel, where P el = π2EI/l2 is the Euler buckling load in the plane ofbending based on the element length l. Second-order analysis methods that directly include both P-∆ and P-δ effects at the element levelgenerally provide better accuracy than P-large delta analysis procedures. The target of 5% maximum unconservative error is based on theoriginal 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 ∆o/L = 0.002 may be used directly in lieu of applying 0.002Yi minimum or additive notionalloads. Note 3. The nominal stiffness and geometry should be employed for checking serviceability limit states. The reduced effective stiffness andthe 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 encompassedwithin the Chapter C requirements for general stability analysis and design, which allow any method of analysis and design that addressesthe effects listed at the beginning of Section 1. Note 5. The largest unconservative error associated with the limit αPr < 0.1PeL is approximately 5% and occurs for a simply-supported,concentrically loaded column with zero moment and αPr = 0.1PeL = φcPy. The target of 5% maximum unconservative error is based on theoriginal development of the AISC LRFD beam-column strength equations (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 portalframes. Therefore, this method strictly should not be applied for the analysis and design of the primary moment frames in these types ofstructures.

78 Donald W. White et al.

typical of design Specifications and is not new to these

AISC provisions. One can avoid the need to consider all

of these caveats by using the direct analysis method,

calculating Pn as the applicable column strength based

on the actual unsupported length, in-plane or out-of-

plane, including an elastic stiffness reduction factor of 0.8

or 0.8τb in the analysis as applicable, and including a

uniform initial out-of-plumbness of 0.002L in the analysis

relative to the perfect geometry of the structure.

2.2.2. Effective length method

The AISC (2005a) effective length method is the same

as the traditional AISC method of analysis and design,

but with the addition of a notional minimum lateral load

for gravity-only load combinations. This minimum lateral

load accounts for the influence of nominal geometric

imperfections on the brace forces, beam moments,

connection moments and in-plane moments used for out-

of-plane strength design of beam-columns. In actuality,

the effects of any physical out-of-plumbness are present

for all load combinations. However, these effects are

overwhelmed by the effects of the primary lateral loads in

all the ASCE 7 (ASCE, 2005) lateral load combinations,

as long as the structure’s sidesway amplification is not

excessive. Therefore, in the AISC (2005a) effective length

method, the notional lateral loads are specified solely as

minimum lateral loads in gravity-only load combinations.

AISC (2005a) does not allow the use of the effective

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 ∆2nd/∆1st > 1.5,

AISC (2005a) requires the use of the direct analysis

method. Correspondingly, when using the direct analysis

approach with structures having ∆2nd/∆1st < 1.5, AISC

(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 Fei at

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 (Fei) implicitly, via the corresponding column

effective lengths KLi. The effective length is related to the

underlying elastic buckling stress via the relationship

(7a)

or

(7b)

In the effective length method, the effects of residual

stresses, P-∆o effects and P-δo effects are addressed

implicitly by the calculation of Pni from the column

strength equations. These equations can be written in

terms of either KLi or Fei (AISC 2005a). Unfortunately,

the selection of an appropriate subassembly buckling

model generally requires considerable skill and judgment.

As a result, there is a wide range of different buckling

models and K factor equations. In certain cases, subtle

differences in the models can produce radically different

results (ASCE, 1997).

In particular, one should note that a rigorous buckling

analysis of the entire structure does not necessarily

provide an appropriate Fei or Ki (ASCE, 1997). Members

that have small axial stress Fei at the buckling limit

(relative to π2E/(Li/ri)2) tend to have large values for Ki

from Eq. (7b). In some cases, these large Ki values are

justified while in other cases they are not. If the member

is indeed participating in the governing buckling mode, a

large Ki is justified. If the member is largely undergoing

rigid-body motion in the governing buckling mode, or if

it has a relatively light axial load and is predominantly

serving to restrain the buckling of other members, a large

Fei

π2E

KLiri

⁄( )2

--------------------=

Ki

π2E( ) L

iri

⁄( )2

⁄

Fei

-------------------------------=

Figure 5. Relationship between notional lateral loads andnominal out-of-plumbness.

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 79

Ki value often is not justified. The distinction between

these two situations requires engineering judgment. Some

of the situations requiring the greatest exercise of

judgment to avoid excessively large K values include: (1)

columns in the upper stories of tall buildings, (2) columns

with highly flexible and/or weak connections and (3)

beams or rafters in portal frames, which may have

significant axial compression due to the horizontal thrust

from the base of the frame.

There is no simple way of quantifying the relative

participation of a given member in the overall buckling

of the structure or subassembly under consideration.

Quantifying the participation requires an analysis of the

sensitivity of the buckling load to variations in the

member sizes. Even if one conducted such an analysis,

there is no established metric for judging when Eq. (7b)

should or should not be used. Engineers typically base

their effective length calculations on story-by-story models

to avoid the first of the above situations. They idealize

columns with weak and/or flexible connections as pin-

ended leaner columns with K = 1 to avoid the second of

the above situations. Lastly, many engineers utilize K = 1

for design of the beams or rafters in portal frames for the

axial compression effects, although Eq. (7b) may suggest

K > 1 based on the Fei from an eigenvalue buckling

analysis.

The direct analysis method provides a more straightforward

and accurate way of addressing frame in-plane stability

considerations. By including an appropriately reduced

nominal elastic stiffness, an appropriate nominal out-of-

plumbness of the structure, and an appropriate out-of-

straightness (for members subjected to high axial loads)

in the analysis, the member in-plane length effects can be

removed entirely from the resistance side of the design

equations. The member in-plane column strength Pni may

be taken simply as Py for members that satisfy the

previously discussed caveats. In-plane stability is addressed

by estimating the actual required internal cross-section

strengths Pr and Mr directly from the analysis, and by

comparing these required strengths against appropriate

cross-section based resistances. Alternatively, to avoid the

need to include out-of-straightness effects in members

with large axial loads, Pni may be calculated using the

actual in-plane unsupported length Li (K = 1).

2.2.3. First-order analysis method

The first-order analysis method, summarized in Table

1, is implicitly a simplified conservative application of

the direct analysis approach, targeted at rectangular or

tiered building frames. White and Kim (2006) detail the

conservative assumptions invoked in the development of

this procedure. Although the first-order analysis method

can be useful for simplified analysis and design of some

types of frames, this method is really just a direct analysis

with a number of simplifying assumptions. There are

numerous other ways to apply direct analysis using an

approximate second-order analysis, many of which are

often significantly more accurate. Therefore, the first-

order analysis method is not considered further in this

paper.

2.2.4. General comments

Both the direct analysis and effective length methods

require a second-order elastic analysis. However, any

second-order elastic analysis procedure is sufficient,

including first-order analysis with amplifiers, assuming

that the procedure is sufficiently accurate or conservative.

The above methods differ in the way that they handle

geometric imperfection and distributed yielding effects in

the second-order analysis model and in the member

resistance equations.

The beam-column out-of-plane resistance check is the

same in both of the above methods, albeit with different

values of Pr and Mr. In AISC (2005a), the simplest out-

of-plane beam-column resistance check is given by Eqs.

(5) but with Pn = Pno, where Pno is the out-of-plane

flexural, torsional or flexural-torsional buckling strength

of the member as a concentrically-loaded column. Eq. (6)

generally provides a more liberal estimate of the out-of-

plane flexural-torsional resistance of doubly-symmetric

compact I-section members.

2.3. Design by direct elastic-plastic hinge analysis

Since 1961, the AISC Specifications have permitted the

use of plastic analysis and design in cases where members

subjected to plastic hinging satisfy requirements that

ensure their ductility. However, the AISC Specifications

from 1969 through 1999 have also generally required the

engineer to supplement the plastic analysis by beam-

column strength interaction checks in which the axial

resistance term is based on a member effective length.

This practice adds significant complexity to the AISC

plastic analysis and design procedures. Furthermore, at

best, the resulting beam-column interaction equations

provide only an approximate assessment of the frame

stability behavior under progressive plastic hinge formation.

In many cases, they overly restrict the forces and moments

in sway frame columns (Ziemian et al., 1992; McGuire,

1995).

The AISC (2005a) direct analysis method can be

extended to provide an attractive alternative to the above

procedures. The extension is very simple - for members

that satisfy separate requirements to ensure sufficiently

ductile response (i.e., sufficient rotation capacity), moment

redistribution is allowed based on the assumption of

elastic-perfectly plastic hinge behavior at the limit of the

member resistance from Eqs. (5). This extension satisfies

the AISC (2005a) Appendix I provisions for inelastic

analysis and design. The separate ductility requirements

in AISC (2005a) Appendix 1 include restrictions on:

• The material yield strength Fy,

• The flange and web slenderness values bf/2tf and hp/tw,

• The member out-of-plane slenderness Lb/ry,

• The magnitude of the axial force αPr (α = 1.0 for

80 Donald W. White et al.

LRFD loadings) and

• The connection details.

These restrictions tend to preclude the formation of

plastic hinges in beams and beam-columns with noncompact

flanges or webs, in beams where the resistance is governed

by out-of-plane lateral-torsional buckling, and in beam-

columns with out-of-plane unbraced lengths large enough

such that the resistance is governed by Eq. (6). Therefore,

for members subjected to in-plane loading and where

inelastic redistribution is allowed, the strength either is

given or is accurately approximated by Eqs. (5) with Mn

= Mp and with Pn = Pni calculated using the actual member

unsupported length in the plane of bending. Furthermore,

when αPr is less than 0.1PeL, Pni = Py is an acceptable

approximation. The approximation Pni = Py also can be

used for general I-section members as long as an appropriate

out-of-straightness is included in the second-order

analysis. Members that do not satisfy all the requirements

necessary to ensure ductile response must be designed

elastically in the manner discussed in Section 2.2.1.

The above type of analysis and design is referred to in

this paper as the direct elastic-plastic hinge method. In

this method, the elastic stiffness of the structure is reduced

to account for distributed yielding effects neglected in the

elastic-plastic hinge idealization. Also, a nominal initial

out-of-plumbness is included to account for geometric

imperfection effects. These devices eliminate the need to

calculate and apply column effective lengths in the

context of inelastic design, as long as the second-order

effects are captured in the elastic-plastic hinge analysis.

Furthermore, these devices allow the engineer to take

advantage of second-order elastic-plastic hinge analysis

software. This kind of software is becoming more and

more available in engineering practice.

Other limits are possible within which the engineer may

be permitted to perform a classical rigid-plastic analysis

and design, e.g., see King (2001) and Davies and Brown

(1996). These limits are not addressed in this study.

3. Illustrative Examples

This section provides a number of basic examples

aimed at illustrating the relative merits of the various

analysis and design procedures outlined in Section 2. The

first example addresses the strength predictions for one of

the major-axis bending cases of the Fig. 3 cantilever

beam-columns. Since this is a statically determinate

structure, inelastic analysis and design does not provide

any advantage. This example is representative of numerous

nonredundant stability critical benchmark problems

considered in the development of the direct analysis

method (Deierlein 2003; Surovek-Maleck and White

2003 & 2004). The second example is a nonredundant

portal frame previously posed by LeMessurier (1977).

This frame exhibits significant sway under gravity loadings,

due to lack of symmetry of its geometry. Also, the beam

governs its maximum resistance rather than the columns.

The third example is a hypothetical industrial building

frame that meets representative service drift requirements.

However, the behavior of the lateral load resisting system

is sensitive to stability considerations to the extent that

the structure’s limit load is reached when the first column

plastic hinge forms in the direct elastic-plastic hinge

analysis. Lastly, a fourth example is provided in which the

lateral load resisting columns are subjected to substantial

non-sway gravity load moments. This frame exhibits

some reserve strength beyond the first plastic hinge for its

critical load combination. All of these examples are

individual members or single-story rectangular structures

with either fully-restrained (FR) or simple connections.

The reader is referred to Maleck (2001), Martinez-Garcia

(2002), Deierlein (2003), Surovek et al. (2005), White et

al. (2005a & b) and White and Kim (2006) for consideration

of a broader range of structure types including multi-story

frames, braced and combined framing systems, trussed

framing, PR frames, gabled frames, and frames with slender

element section members, singly-symmetric members,

nonprismatic members, and/or members with non-

constant axial load along their lengths.

In all the following examples, the appropriate stiffness

reduction factor in the direct analysis is 0.8. That is, P/Py

is always less than 0.5, and therefore τb = 1.0. The

distributed plasticity analysis is conducted using a factor

of 0.9 on the elastic stiffness E and the strength Fy in all

cases. The axial force P is close to or smaller than 0.1PeL

in all the above problems. Therefore, the column out-of-

straightness is neglected and, to illustrate the validity of

this option, Pni is taken equal to Py in the direct analysis

solutions. A nominal column out-of-straightness is included

in the direction of the member curvature due to the applied

loadings in all of the distributed plasticity solutions, to

demonstrate that the direct analysis solutions give accurate

predictions for these cases with out-of-straightness

neglected. All of the direct and distributed plasticity

analyses are conducted using an out-of-plumbness of

∆o = 0.002L, specified in the direction of the drift under

the applied loads. For the effective length method, a

second-order elastic analysis is conducted using the nominal

elastic stiffness and the initially perfect geometry.

The second-order elastic and distributed plasticity

solutions are conducted using the GT-Sabre software system

(Chang 2005). A flexibility-based element (Alemdar and

White 2005), which is based on an equilibrium-based

distribution of moments and a correspondingly accurate

representation of inelastic curvatures (including the influence

of transverse distributed loads), is used for the beams.

The mixed element developed by Alemdar and White

(2005), which is capable of accurate modeling of the

second-order moments and inelastic curvatures, is used

for the beam-columns. A small modulus of 0.001E is

assumed for the yielded material. The direct elastic-

plastic hinge solutions are conducted using the Mastan2

software system (McGuire et al., 2000). For major-axis

bending, the Mastan2 yield surface, with axial force and

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 81

bending moment anchor points of 0.9Py and 0.9Mp, is

slightly more liberal than the yield surface corresponding

to Eqs. (5) with Pc = 0.9Py and Mc = 0.9Mp.

At the strength design loads, ∆2nd/∆1st, which is

equivalent to the sidesway amplifier B2 in AISC (2005a),

is larger than 1.5 in all of these examples. Therefore,

AISC (2005a) disallows the use of the effective length

method for all of these structures. The effective length

method is applied in the examples to illustrate the

implications of its use for these types of frames.

3.1. Cantilever column

Figure 6 compares the results predicted by the direct

analysis and the effective length methods to the results

from distributed plasticity analysis for the Fig. 3 beam-

column subjected to major-axis bending and the specific

loading of H = 0.01P. The direct (elastic) and direct

elastic-plastic hinge analysis methods are one and the

same up to the maximum strength limit for this example,

since the structure is statically determinate. Figure 6

shows the force-point traces at the column base (i.e., the

variation of the member base moment and axial force for

increasing levels of the applied loads) determined by the

direct analysis, effective length and distributed plasticity

analysis solutions.

Figure 6 also shows the factored beam-column strength

curves for the effective length and the direct analysis

methods (with φc = φb = 0.9). In this problem, the member

in-plane strength governs in both the direct analysis and

the effective length solutions if the member is braced at

its top and bottom in the out-of-plane direction, K = 0.7 is

used for the calculation of Pno, and Eq. (6) is used for the

out-of-plane strength assessment. Therefore, Fig. 6 focuses

only on the in-plane resistance. The direct analysis in-

plane strength curve is based on φcPni = 0.9Py (although P

is slightly larger than 0.1PeL). The corresponding axial

strength for the effective length method is φcPni = 0.9

(0.626Py) based on K = 2, or KL/rx = 80. The anchor point

on the horizontal axis is φbMn = 0.9Mp for both of the

above strength curves.

The direct analysis provides a reasonably accurate

estimate of the distributed plasticity internal force and

moment up to the predicted maximum resistance. The

force point trace from the distributed plasticity analysis

indicates a larger moment than determined by the second-

order elastic analysis of the nominally-elastic initially-

perfect structure by the effective length method. This is

due to the stiffness reduction factor of φc = 0.9 in the

distributed plasticity analysis as well as the effects of

distributed yielding as the maximum strength limit is

approached. Figure 7 shows the ratio of the effective

moment of inertia Ie for the beam-column (i.e., the

moment of inertia of its elastic core) to the elastic cross-

section moment of inertia at the distributed plasticity

analysis limit load. The member experiences significant

yielding at the strength limit, but a full plastic hinge has

not yet formed at its base. The maximum value of P on

the force-point trace from the distributed plasticity

analysis corresponds to the limit load in the refined

inelastic solution. Correspondingly, the design strength

for the effective length and the direct analysis methods is

defined by the intersection of their force-point traces and

the corresponding beam-column strength curves. Both the

effective length and direct analysis provisions are calibrated

such that these intersection points give an accurate to

conservative estimate of the “actual” maximum strength

represented by the distributed plasticity solution.

The direct analysis method accounts for all the key

attributes that influence the in-plane system stability effects

directly within the analysis. Hence, its force point trace

may be compared against the cross-section strength limit

given by Eqs. (5) with Pni = Py and Mn = Mp. Conversely,

the effective length method accounts for the in-plane

system stability effects by reducing the member axial

strength Pni via the effective length KL, or the elastic

buckling stress Fei, obtained implicitly or explicitly from

an appropriately configured buckling analysis.

Table 2 summarizes the results at the calculated strength

Figure 6. Comparison of effective length and direct analysismethod beam-column strength interaction calculations todistributed plasticity analysis, example cantilever beam-column subjected to major-axis bending, adapted fromWhite and Kim (2006).

Figure 7. Ratio of the effective moment of inertia Ie at

the distributed plasticity limit load to the elastic cross-section moment of inertia I, example cantilever beam-column subjected to major-axis bending.

82 Donald W. White et al.

limits by each of the above methods. The ratios of the

base moments Mmax = HL + P(∆ + ∆o) to the primary

moment HL indicate the magnitude of the second-order

effects. The axial load at the direct analysis strength limit,

which is representative of the strength in terms of the

total applied load, is 5% higher than that obtained from

the distributed plasticity analysis. Conversely, the axial

load at the effective length method beam-column strength

limit is 4% smaller than that obtained from the distributed

plasticity solution. Both of these estimates are within the

targeted upper bound of 5% unconservative error relative

to distributed plasticity analysis established in the original

development of the AISC LRFD beam-column strength

equations (ASCE 1997; Surovek-Maleck and White 2004).

The difference in the calculated internal moments is

much larger. This difference is expected since the effective

length approach compensates for the underestimation of

the actual moments by reducing the value of the axial

resistance term Pni, whereas the direct analysis method

imposes additional requirements on the analysis to obtain

an improved estimate of the actual internal moments.

This more accurate calculation of the internal moments

also influences the design of the restraining members and

their connections. For instance, in this example, the direct

analysis base moments are more representative of the

actual moments required at this position to support the

applied loads at the limit of the member resistance. In

other words, the direct analysis method provides a direct

calculation of the required strengths for all of the structural

components. Conversely, the effective length approach

generally needs supplementary rules for calculation of the

required component strengths. In AISC (2005a), these

supplementary rules are: (1) A minimum notional lateral

load is applied in gravity-only load combinations, and (2)

The effective length method is limited to frames having

∆2nd/∆1st < 1.5 (see Table 1).

3.2. LeMessurier’s (1977) Example 3

The structure shown in Fig. 8 is a market shed deliberately

designed to have slender columns. LeMessurier (1977)

presents this frame as his Example 3. He explains that

column CD is on an open side and is slender for architectural

reasons. Column AB is selected to limit the first-order

drift under nominal wind to a maximum of 25.4 mm (1

in). The discussions below focus on checking the maximum

resistance of this frame by LRFD using the various

analysis and design procedures. The girder is assumed to

be adequately braced in the out-of-plane direction such

that Mn = Mp. Column AB is braced at its ends in the out-

of-plane direction. This bracing is sufficient such that its

resistance is governed by in-plane limit states. A live-to-

dead load ratio of 3.0 is assumed for the LRFD

calculations. The loading 1.2∆ + 1.6Lr + 0.8W, with the

wind applied in the direction of the sway under the

gravity load, is the most critical of the ASCE 7-05 load

combinations for this structure. The following discussions

focus on the behavior of the frame under this load

combination.

Figure 9 shows the fraction of the design load versus

the drift ∆/L from the different analysis methods, Fig. 10

shows the moment diagrams and the diagrams of the

effective moment of inertia Ie at the distributed plasticity

analysis limit load, Fig. 11 shows the deflected shape, the

location of the single plastic hinge and the design load

fraction at the formation of the plastic hinge from the

direct elastic-plastic hinge analysis, Fig. 12 shows the

applied fraction of the design load versus the girder

maximum bending moment from the different analysis

methods, and Fig. 13 shows the force-point trace at the

Table 2. Summary of calculated design strengths, cantilever beam-column example

Pmax

(kN) Mmax

(kN-m) Mmax

/HL Pmax/P

max (Distributed Plasticity)

Traditional effective length 1590 119 1.68 0.96

Direct analysis 1730 217 2.79 1.05

Distributed plasticity analysis 1650 198 2.68

Figure 8. LeMessurier’s (1977) Example 3.

Figure 9. Design load fraction versus the story drift forLeMessurier’s Example 3.

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 83

top of column AB from the different analysis methods.

One can observe that the direct elastic-plastic hinge

analysis provides a reasonable estimate of the distributed

plasticity solution. Due to the use of the smaller stiffness

reduction factor in the plastic hinge analysis (0.8 versus

0.9), the elastic deflections are slightly larger than those

predicted by the distributed plasticity analysis. However,

the sway deflection at the limit load is slightly smaller in

the plastic hinge analysis. The maximum resistances in all

of the analysis solutions correspond to the formation of a

plastic hinge in the girder. The second major slope

discontinuity in the direct analysis load-deflection curve

(Fig. 9) occurs when a plastic hinge forms at the top of

column AB. The resulting overall shape of the direct

analysis load-deflection curve matches quite well with the

distributed plasticity analysis results.

Due to the smaller drift predicted in the second-order

elastic analysis by the effective length method, the

maximum girder moments are slightly smaller for a given

design load fraction. As a result, the effective length

method predicts a maximum strength of the structure (at

the formation of a plastic hinge with a resistance of 0.9Mp

in the girder) at 0.987 of the loading 1.2∆ + 1.6Lr + 0.8W.

The plastic hinge and distributed plasticity analysis methods

predict corresponding strength limits of 0.975 and 0.959

of the design loading. Although there is substantial

distributed yielding along the length of the girder at the

distributed plasticity strength limit (see Fig. 10), this

yielding has a small impact on the girder maximum

second-order moment.

Figure 10. Diagrams of member moments and effective moment of inertia Ie at the distributed plasticity analysis limit

load, LeMessurier’s Example 3.

Figure 11. Deflected shape, location of plastic hinge, anddesign load fraction at the formation of the plastic hingefrom direct elastic-plastic hinge analysis, LeMessurier’sExample 3.

Figure 12. Applied fraction of the design load versus thegirder maximum bending moment obtained from the differentanalysis-design methods, LeMessurier’s Example 3.

Figure 13. Force-point trace at the top of column ABobtained from the different analysis-design methods,LeMessurier’s Example 3.

84 Donald W. White et al.

3.3. Maleck’s (2001) Industrial Building Frame

Figure 14 shows an example 11-bay industrial building

frame developed by Maleck (2001) and studied further in

Martinez-Garcia (2002), Deierlein (2003) and Surovek-

Maleck and White (2004). Five gravity columns are

located on each side of the interior moment frame. In

practice, the exterior bays of this type of frame might be

designed with simply-supported joist girders, hence the

large number of gravity columns. Large gravity loads are

specified to simulate conditions in some single-story industrial

buildings, such as automobile plants (Springfield, 1991),

in which large equipment loads are located on the roof.

Significant P-∆ effects are transmitted to the moment

frame from the large number of gravity (leaner) columns.

However, the design lateral loadings from wind, etc. are

relatively small.

The same girder size (W27x84) is used throughout the

structure in Fig. 14. This size is selected to resist the

moments in the simply-supported exterior bays of the

frame. Although a W24 × 84 has sufficient strength to

withstand the design gravity loads, the deeper W27 × 84

profile is selected to reduce the lateral drift. The tops of

the two W10 × 49 interior columns are rigidly connected

to the middle W27 × 84 girder. The middle girder is

continuous over the tops of these columns, and the

columns are attached to the bottom of the girder by end-

plate connections. Base restraint is provided for the

lateral-load resisting columns to limit the drift under a

service loading of 1.0∆ + 0.5Lr+ 0.7W to L/400. The service

drift is calculated on the nominally-elastic geometrically-

perfect frame, that is, no stiffness reduction or geometric

imperfections are applied for the service load analysis.

The strength design of the girders and the columns in

the above structure is governed by the LRFD gravity load

combination 1.2∆ + 1.6Lr. The ASCE7-05 required inclusion

of 0.8W with this combination is neglected to accentuate

the frame stability effects. The following discussions

focus on the frame behavior under this load combination.

The columns are again governed by the frame in-plane

stability limit states when Eq. (6) is used for the out-of-

plane strength checks. The reader is referred to Surovek-

Maleck and White (2004) and to Kuchenbecker et al.

(2004) for further details of the strength and service load

calculations.

Figures 15 through 19 present the same results as Figs.

9 through 13 for LeMessurier’s (1977) Example 3

structure, but illustrate the responses for Maleck’s (2001)

frame. Figure 19 shows the force-point trace at the top of

the left-hand lateral load resisting column. This location

has the most critical combination of column axial force

and bending moment. Figure 18 shows the girder negative

bending moment versus the design load fraction just to

the left of the left-hand column. This is the location of the

largest girder bending moment.

Again, the direct elastic-plastic hinge analysis provides

a reasonable estimate of the distributed plasticity analysis

solution. Due to the smaller stiffness reduction factor in

the plastic hinge analysis (0.8 versus 0.9), the elastic

deflections are slightly larger than those predicted by the

distributed plasticity analysis. Also, the sway deflection

at the limit load is slightly larger in the plastic hinge

analysis. The maximum resistance in the plastic hinge

analysis (1.155 of the design load level) corresponds to

the formation of a plastic hinge at the top of the left-hand

column. Subsequently, plastic hinges form at the other

ends of the lateral load resisting columns within the post-

peak range of the response at 1.115, 1.075 and 0.930 of

the design load level (see Fig. 17). If the example frame

were not so sensitive to stability effects, there would be

Figure 14. Maleck’s (2001) 11-bay industrial building frame.

Figure 15. Design load fraction versus the story drift forMaleck’s frame.

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 85

some reserve strength associated with the redistribution

of the left-hand column moments from its top to its base,

as well as the redistribution of moments from the left-

hand column to the right-hand column. Of course, elastic

analysis and design discounts this reserve system strength.

The distributed plasticity analysis predicts a similar

response; however, at the maximum load level (at 1.158

of the design loading), there are no fully-formed plastic

hinges in the structure (see Fig. 16). The limit load is

reached due to a combination of P-∆ effects along with

the progressive softening of the columns and girders due

to the spread of plasticity through their cross-sections and

along their lengths. At the distributed plasticity analysis

limit load, the girders are significantly yielded on the left-

hand side of each of the columns (see Fig. 16). However,

this yielding is highly localized due to the moment

gradient in the girders at these positions.

Again, the second-order elastic analysis of the perfect

nominally-elastic structure by the effective length method

results in significantly smaller overall drift at the predicted

strength limit (see Fig. 15). The corresponding force-

point trace for the columns intersects the column in-plane

strength curve at a design load fraction of 0.946. The

effective length method beam-column strength for this

frame is based on a column effective length factor of K =

2.12 using Eq. (C-C2-5) of the AISC (2005a) Commentary.

The conservatism of this strength check is due to two

causes: (1) a substantial fraction of the column internal

moments are due to non-sway gravity loading and (2) the

AISC (2005a) effective length method requires the inclusion

of a minimum notional lateral load with gravity-only load

combinations. The original calibration of the effective

length method in AISC LRFD (1986) did not include any

sway frames that were subjected to gravity loads causing

non-sway member moments. The loadings in the original

benchmark solutions were all applied directly at the

beam-column joints such that the non-sway moments

were zero (ASCE 1997; Surovek-Maleck and White

2003; Deierlein 2003). The above conservatism of the

AISC (2005a) effective length method strength checks,

Figure 16. Diagrams of member moments M and effective moment of inertia Ie at the distributed plasticity analysis limit

load, Maleck’s frame, 1.2∆ + 1.6Lr.

Figure 17. Deflected shape, plastic hinge locations, anddesign load fraction at the formation of the plastic hingespredicted by direct elastic-plastic hinge analysis, Maleck’sframe, 1.2∆ + 1.6L

r.

Figure 18. Applied fraction of the design load versus themaximum girder bending moment obtained from thedifferent analysis-design methods, Maleck’s frame, 1.2∆ +1.6L

r.

86 Donald W. White et al.

due to the significant non-sway column moments, is

much larger in the next example (see Section 3.4). Also,

the original AISC (2005a) beam-column strengths were

calibrated to give accurate to conservative predictions of

the results from distributed plasticity analysis without the

inclusion of any notional lateral loads. However, even if

the beam-column strength checks in the example frame

are conducted with zero notional lateral load, a conservative

estimate of 1.119 of the design loading still is obtained

for the maximum frame resistance.

Although the above effective length method checks are

conservative for the in-plane strength of the columns of

Maleck’s frame, the moments for checking the out-of-

plane strength of these members as well as the design of

the end-plate connections and column bases are dramatically

underestimated. Figure 19 shows that the maximum

moment at the top of the left-hand column is only 54.6

kN-m when the beam-column interaction equation is

intersected by the force-point trace of the effective length

method using zero notional load. However, this moment

is 186.9 kN-m at the distributed plasticity analysis limit

load (242% larger). The minimum lateral load requirement

in the AISC (2005a) effective length method increases the

predicted value to 100 kN-m when the beam-column

strength condition is reached. This moment is also

significantly smaller than the moment predicted by the

distributed plasticity analysis. However, in this case, the

conservatism of the in-plane strength check prevents the

beam-columns from reaching a state in which their in-

plane bending moments increase substantially with only a

minor increase in the applied load. The force-point trace

for the AISC (2005a) effective length method is a reasonable

approximation of the result predicted by the distributed

plasticity analysis up to the point at which its in-plane

beam-column strength condition is reached. The direct

elastic-plastic hinge analysis gives a conservative estimate

of the connection moment at the top of the left-hand

column of 231.7 kN-m at its limit load (24% larger than

the value from the distributed plasticity analysis). However,

this method does a good job of estimating the maximum

overall capacity of the frame. This is evidenced by the

similarity of the peak ordinate values for the direct and

distributed plasticity analysis methods in Figs. 15 and 19.

3.4. AISC Specification Committee (2004) Design

Problem 13 (DP-13)

Figure 20 shows a single story rectangular frame posed

as one of many design problems during the final Specification

Committee validation and checking of the AISC (2005a)

provisions. The interior columns in this structure are all

leaner columns. All the beam-to-column connections are

simple except for the connections to the exterior columns,

which are fully-restrained. The exterior columns are

subjected to relatively light axial loads, whereas they

experience substantial gravity load moments in addition

to the wind load moments. Also, the frame has significant

second-order effects, i.e., amplification of the member

sidesway internal bending moments. The columns are

braced in the out-of-plane direction at their base and at

the roof height. Simple base conditions are assumed in

both the in-plane and out-of-plane directions, and simple

connections are assumed in the out-of-plane direction at

the column tops. This problem is considered using the

following LRFD load combinations:

Load Case 1 (LC1): 1.2D + 1.6S

Load Case 2 (LC2): 1.2D + 0.5S + 1.6W

Figures 21 and 22 show the applied fraction of the

design loads versus the story drift by the distributed

plasticity analysis, direct elastic-plastic hinge analysis and

effective length methods for these load combinations. The

distributed plasticity analysis gives a maximum in-plane

capacity of 1.13 times the factored design load level for

LC1 and 1.06 of the factored design load level for LC2.

Each of the figures shows two curves determined from a

direct elastic-plastic hinge analysis, one in which a

stiffness reduction factor (SRF) of only 0.9 (the same as

the value used in the distributed plasticity analysis) is

employed, and the other in which a SRF of 0.8 (derived

from the AISC (2005a) direct analysis provisions) is

employed. For LC2, the direct analysis matches the

Figure 19. Force-point trace at the top of left-hand lateral-load resisting column obtained from the different analysis-design methods, Maleck’s frame, 1.2∆ + 1.6L

r.

Figure 20. AISC Specification Committee Design Problem13 (DP-13).

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 87

distributed plasticity solution closely when the larger SRF

of 0.9 is employed. However, for LC1, the direct analysis

over-predicts the maximum load capacity by 12% when

the SRF is taken as 0.9. Use of the recommended SRF of

0.8 results in a predicted maximum load capacity of 1.19

of the design load level for LC1 (5% larger than the

distributed plasticity analysis solution, which is equal to

the acceptable tolerance on the unconservative error

established in the original development of the LRFD

beam-column strength interaction equations (ASCE 1997;

Surovek-Maleck and White 2004)).

The load-drift curves for the effective length method

analysis indicate a stiffer load deflection response. These

curves end at the load level at which the right-hand

exterior beam-column force-point trace intersects the

effective length method strength envelope. As indicated

in the previous example, the original AISC effective length

method (no notional minimum lateral load) tends to be

conservative for beam-columns in which a large fraction

of the bending moments are associated with non-sway

gravity loading. To isolate the conservatism associated with

the original effective length calculations, the effective

length solution for LC1 is conducted without including

the minimum notional load required by AISC (2005a).

Since LC2 includes a nonzero lateral load, there is no

additional notional lateral load requirement for this load

case in AISC (2005a). The original effective length

procedure, which does not include any notional lateral

load, suggests that only 0.78 and 0.82 of the design loads

can be applied to the DP-13 frame for LC1 and LC2

respectively. That is, the effective length method gives a

capacity of only 0.78/1.13 = 0.69 and 0.82/1.06 = 0.77 of

the in-plane capacity from the distributed plasticity

analyses for LC1 and LC2. If the AISC (2005a) minimum

lateral load is included for LC1, the predicted capacity is

reduced only a slight additional amount to 0.76 of the

design load level.

Table 3 summarizes the fractions of the design loads

giving a value of 1.0 for the in-plane strength check from

Eqs. (5) for the AISC (2005a) direct (elastic) and effective

length methods, as well as the maximum capacities

predicted by the direct elastic-plastic hinge analysis using

Mastan2, and the distributed plasticity analysis using GT-

Sabre. The direct (elastic) analysis method gives a slightly

smaller maximum strength than the direct elastic-plastic

hinge analysis using Mastan2 for LC1 (1.17 versus 1.19).

This is due to the slightly more liberal cross-section plastic

hinge strength assumed in Mastan2 compared to Eqs. (5).

For LC2, the direct (elastic) analysis force-point trace

reaches Eqs. (5) in the right-hand exterior column at a

design load fraction of 0.95. This value is comparable to

the first plastic hinge formation at the top of the right-

hand column at 0.96 of the design loading in Mastan2.

However, as indicated in Table 3 and illustrated by the

Figure 21. Design load fraction versus the story drift forDP-13 load case 1 (LC1).

Figure 22. Design load fraction versus the story drift forDP-13 load case 2 (LC2).

Table 3. Applied fraction of the design loads giving a value of 1.0 for the in-plane strength check from Eqs. (5) for theAISC (2005a) direct (elastic) and effective length methods with and without minimum lateral loads, maximum capacitiespredicted by direct elastic-plastic hinge analysis using Mastan2, and maximum capacities predicted by distributed plasticityanalysis using GT-Sabre, DP-13 LC1

LC1 LC2

Direct (elastic) analysis 1.17 0.95

AISC (2005a) effective length 0.76 0.82

Effective length with zero Ni 0.78 0.82

Direct elastic-plastic hinge analysis first-hinge strength, Mastan2 1.19 0.96

Direct elastic-plastic hinge analysis limit load, Mastan2 1.19 1.02

Distributed plasticity analysis 1.13 1.06

88 Donald W. White et al.

direct elastic-plastic hinge load-drift curve in Fig. 22, the

maximum capacity of the frame is not reached under the

LC2 loading until a design load fraction of 1.02 by the

direct elastic-plastic hinge analysis (versus 1.06 by the

distributed plasticity solution).

Figure 23 illustrates the behavior at the strength limit

predicted by the direct elastic-plastic hinge and distributed

plasticity solutions for this load combination. The effective

moment of inertia Ie is reduced to zero at the top of the

right-hand column at the limit load in the distributed

plasticity analysis. Also, the bending rigidities are

significantly reduced in the positive bending region of the

left-most girder and at the top of the left-hand column at

this load level. However, significant portions of these

cross-sections are still elastic. The direct-elastic plastic

hinge analysis predicts the formation of a sway mechanism,

by plastic hinging at the tops of each of the exterior

columns, at its predicted capacity of 1.02 of LC2.

Figure 24 shows the corresponding predictions for

LC1. In this case, no plastic hinges have fully formed at

any location in the frame (i.e, Ie/I > 0 everywhere) at the

predicted capacity of 1.13 of the design loading in the

distributed plasticity analysis. The direct elastic-plastic

hinge solution reaches its limit load (at 1.19 of the design

loading) when a hinge forms at the top of the right-hand

column.

The key reason for the underestimation of the load

capacities by the effective length method is illustrated in

Fig. 25. This figure compares the force-point trace at the

top of the right-hand column predicted by the original

effective length method (no notional minimum lateral

load) and the AISC (2005a) direct (elastic) analysis

method to the corresponding force-point trace from the

distributed plasticity analysis for LC1. The AISC (2005a)

strength envelopes (Eqs. (5)) are shown as dashed curves

in the figure. The significant in-plane stability effects in

Figure 23. Plastic hinges and deflected shape from direct elastic-plastic hinge analysis and effective moment of inertia Ie

at the distributed plasticity analysis limit load, DP-13 LC2.

Figure 24. Plastic hinges and deflected shape from direct elastic-plastic hinge analysis and effective moment of inertia Ieat the distributed plasticity analysis limit load, DP-13 LC1.

Figure 25. Force-point traces for the right-hand beam-column by the original effective length method, the directanalysis method and the distributed plasticity analysismethod versus the effective length and direct analysisstrength curves, DP-13 LC1.

Stability Analysis and Design of Steel Building Frames: The AISC (2005) Specification and Beyond 89

this example result in a Pn(KL) of only 0.098Py for LC1.

The corresponding effective length factor for the right-

hand column is 5.02 using a rigorous sidesway buckling

analysis (comparable values are obtained using the

equations in the AISC (2005a) Commentary, as long as

the leaner column effects and the effect of the simple

connections at the inside ends of the exterior girders are

correctly incorporated).

The direct analysis method gives a substantially larger

estimate of the in-plane resistance because it focuses on

a more realistic estimate of the internal moments and the

corresponding member resistances. In determining the

anchor point Pn(KL) for its beam-column strength envelope,

the effective length method overemphasizes the response

of the structure to idealized loads causing uniform axial

compression in all of the columns. The physical strength

of the DP-13 frame is dominated by its amplified internal

moments, a large fraction of which is not related to the

sidesway of the structure. The limit of the DP-13 frame

resistance is tied essentially to the flexural resistance of

the right-hand beam-column and the amplified internal

moments in this member, not by a column failure under

concentrically-applied axial loads.

When the effective length method is applied to the DP-

13 frame, the design resistance is governed by the in-

plane strength of the leeward beam-column (Eqs. (5)) for

both LC1 and LC2 (rather than the out-of-plane strength

as represented by Eq. (6)). However, the out-of-plane

strength based on Eq. (6) governs slightly relative to the

in-plane strength in both the distributed plasticity analysis

as well as the direct analysis methods. Eq. (6) is intersected

by the direct (elastic) analysis force-point trace for the

right-hand column at 1.16 and 0.94 of LC1 and LC2

respectively. The right-hand column does not satisfy the

AISC (2005a) unbraced length requirements for inelastic

design using braces only at its ends. Additional out-of-plane

bracing would be required to satisfy these requirements

such that the inelastic reserve strength illustrated by Fig.

22 can be utilized based on the AISC requirements.

4. Conclusions

This paper explains the AISC (2005a) stability analysis

and design provisions with an emphasis on their relationship

to refined second-order inelastic analysis methods and on

how they can facilitate the use of second-order inelastic

analysis in practical design.

First, an appropriate application of second-order distributed

plasticity analysis is discussed for refined assessment of

adequately braced compact I-section members and structural

systems. The base AISC (2005a) beam-column strength

interaction equations have been developed in part by

calibration against the results from refined inelastic

analyses of this type. The paper recommends that for

distributed plasticity analysis, both the material elastic

stiffnesses (E) and strengths (Fy) should be reduced up

front by the factors φb = φc = 0.9. This is necessary to

obtain the same result as determined by analyzing a

member or structure with nominal E and Fy and

subsequently factoring the abscissa and ordinate of the

resulting strength curves by φb and φc. The appropriate

nominal residual stress, geometric imperfection, and

material stress-strain idealizations for use of distributed

plasticity analysis in a design context are discussed. Also,

a simple limit on the magnitude of the axial force is

provided within which member out-of-straightness may

be neglected in a distributed plasticity analysis.

Secondly, three specific elastic methods of analysis and

design detailed in AISC (2005a) are discussed in the

context of the above refined inelastic analysis and design

procedure. These are the direct analysis, the effective

length and the first-order analysis methods. The new

AISC (2005a) direct analysis method provides a better

approximation of the results from refined distributed

plasticity analysis. By improving the accuracy of the

analysis, the direct analysis method allows for greater

simplicity in the design checks by eliminating the need

for effective length factors. Also, a more accurate

characterization of the overall strength is attained in

general.

Thirdly, the paper presents a basic extension of the

direct (elastic) analysis method that satisfies the intent of

the Specification and captures the beneficial effects of

inelastic redistribution from compact adequately-braced

members after the individual member resistances are

reached. This extension, termed the direct elastic-plastic

hinge analysis method, is obtained simply by approximating

the response of these types of members by an elastic-

perfectly plastic hinge model at the Specification limit of

the member resistance.

Lastly, four examples are provided that illustrate key

characteristics of each of the above analysis and design

methods. The direct elastic and elastic-plastic hinge analysis

methods provide accurate elastic and inelastic approximations

of the refined distributed plasticity solutions. The effective

length analysis and design method is shown to give

accurate to conservative solutions for the in-plane beam-

column resistances in all of the example cases. However,

the effective length method significantly underestimates

the internal moments that the beam, connection, and out-

of-plane beam-column resistances must accommodate in

certain cases. Furthermore, the effective length method is

shown to give conservative solutions for beam-columns

in sway frames where a large percentage of the moments

are due to non-sway gravity loading. AISC (2005a)

requires the use of a notional minimum lateral load in

gravity-only load combinations, and limits the application

of the effective length method to frames having ∆2nd/∆1st

< 1.5 to control these errors. The AISC (2005a) direct

elastic analysis method is generally applicable to all types

of frame structures. The direct elastic-plastic hinge method

is a useful extension of this method that facilitates

simplified inelastic analysis and design.

90 Donald W. White et al.

Acknowledgments

Many of the concepts discussed in this paper have

benefited greatly from discussions with the members of

the AISC Technical Committee 10 (TC10), the Structural

Stability Research Council (SSRC) Task Group 4 on

Frames, and the former SSRC Task Group 29 on Inelastic

Analysis for Frame Design. Professor J. Yura of the

University of Texas at Austin, Dr. S. Nair of Teng and

Associates, Inc. and Prof. G.G. Deierlien of Stanford

University are thanked for their guidance of the TC10

efforts toward the implementation of the AISC (2005a)

provisions. Various portions of this work were funded by

the National Science Foundation, the Georgia Institute of

Technology, the American Society of Civil Engineers,

and the Metal Building Manufacturers Association. The

funding by these organizations is gratefully acknowledged.

The opinions, findings and conclusions expressed in this

paper are the authors’ and do not necessarily reflect the

views of the above individuals, groups and organizations.

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