comprehensive stability design of planar steel members and

www.springer.com/journal/13296

International Journal of Steel Structures 16(4): 1029-1042 (2016)

DOI 10.1007/s13296-016-0070-3

ISSN 1598-2351 (Print)

ISSN 2093-6311 (Online)

Comprehensive Stability Design of Planar Steel Members and

Framing Systems via Inelastic Buckling Analysis

Donald W. White*, Woo Yong Jeong, and Oğuzhan Toğay

School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA

Abstract

This paper presents a comprehensive approach for the design of planar structural steel members and framing systems usinga direct computational buckling analysis configured with appropriate column, beam and beam-column inelastic stiffnessreduction factors. The stiffness reduction factors are derived from the ANSI/AISC 360-16 Specification column, beam andbeam-column strength provisions. The resulting procedure provides a rigorous check of all member in-plane and out-of-planedesign resistances accounting for continuity effects across braced points as well as lateral and/or rotational restraint from otherframing. The method allows for the consideration of any type and configuration of stability bracing. With this approach, nomember effective length (K) or moment gradient and/or load height (Cb) factors are required. The buckling analysis rigorouslycaptures the stability behavior commonly approximated by these factors. A pre-buckling analysis is conducted using the AISCDirect Analysis Method (the DM) to account for second-order effects on the in-plane internal forces. The buckling analysis iscombined with cross-section strength checks based on the AISC Specification resistance equations to fully capture all themember strength limit states. This approach provides a particularly powerful mechanism for the design of frames utilizinggeneral stepped and/or tapered I-section members.

Keywords: Buckling Analysis, Inelastic Stiffness Reduction Factors, Stability Design

1. Introduction

Within the context of the Effective Length Method of

design (the ELM), engineers have often calculated inelastic

buckling effective length (K) factors to achieve a more

accurate and economical design of columns. This process

involves the determination of a stiffness reduction factor,

τ, which captures the loss of rigidity of the column due to

the spread of yielding, including initial residual stress

effects, as a function of the magnitude of the column axial

force. Various tau factor equations have been used in practice,

but there is only one that fully captures the implicit inelastic

stiffness reduction associated with the column strength

curve of the ANSI/AISC 360-16 Specification (AISC 2016)

and the prior AISC unified LRFD/ASD and LRFD

Specifications. This tau factor typically is referred to as

τa. What many engineers do not realize is that the ELM

does not actually require the calculation of K factors at all.

The column theoretical buckling load can be computed

directly and used in determining the design resistance

rather than being obtained implicitly via the use of K.

Furthermore, if the stiffness reduction 0.9×0.877×τa is

incorporated within a buckling analysis of the member or

structural system, the calculations can be set up such that,

if the member or structure buckles at a given multiple of

the required design load, ΓPu in Load and Resistance

Factor Design (LRFD), ΓPu is equal to the design axial

resistance φcPn. If the load multiplier required to reach

buckling is greater than 1.0, given the column stiffnesses

calculated based on 0.9×0.877×τa, the member or

structure satisfies the AISC Specification column strength

requirements without the need for further checking. The

column strength requirements are inherently included in

the buckling calculations.

The above approach can be applied not only to account

for column end rotational restraint from supports or other

structural framing. In addition, it can be employed to

rigorously evaluate the column strength given the stiffness

of various types and combinations of stability bracing.

Furthermore, since the bracing stiffness requirements of

the AISC (2016) and the prior 2005 and 2010 AISC

Appendix 6 are based on multiplying the ideal bracing

stiffness, which is the bracing stiffness necessary to achieve

a column buckling strength equal to the required column

Received January 31, 2016; accepted May 8, 2016;published online December 31, 2016© KSSC and Springer 2016

*Corresponding authorTel: +1-404-894-5839, Fax: +1-404-894-2278E-mail: [email protected]

1030 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 2016

axial load, by 2/φ=2/0.75 (using LRFD), a buckling analysis

that incorporates the column τa factor(s) can be used as a

rigorous stiffness check of the AISC column stability

bracing requirements. Even more powerful is that the

above approach can be extended to the bracing and

member design of beams and beam-columns. Similar

methods have been developed in the prior research by

Trahair and Hancock (2004) and Trahair (2009 and 2010)

and have been shown to provide accurate estimates of

column, beam and beam-column resistances for various

types of geometries, loadings and member restraints. More

recently, Kucukler et al. (2015a & b) have developed

comparable procedures in the context of design to

Eurocode 3.

This paper first reviews the development and proper

use of the column stiffness reduction factor, τa. It then

focuses on the extension of column buckling analysis

procedures based on τa to the rigorous assessment of

general I-section beam and beam-column strength limit

states via a buckling analysis with appropriate inelastic

stiffness reduction factors.

2. Column Buckling Analysis using the AISC Inelastic Stiffness Reduction Factor τa

The column inelastic stiffness reduction factor τa is the

most appropriate stiffness reduction estimate corresponding

to the AISC design assessment of steel columns via buckling

analysis. This is because τa is derived inherently from the

AISC column strength curve. Therefore, when τa is

configured properly with a buckling analysis, the internal

axial force in the column(s) is equal to φcPn at incipient

buckling of the analysis model. The τa factor accounts

implicitly for residual stress and initial geometric imper-

fection effects, as well as the traditional higher margin of

safety specified by AISC for slender columns. This factor

is not the most appropriate inelastic stiffness reduction for

a second-order load-deflection analysis, such as an analysis

conducted to satisfy the requirements of the Direct

Analysis Method of design (the DM). The separate τb

factor has been adopted by AISC for use with the Direct

Analysis Method. The τb factor is intended to account

predominantly just for nominal residual stress effects. If

used with a second-order load-deflection analysis per the

DM, the τa factor gives falsely high internal forces and

correspondingly low strength predictions. This is because

the engineer would effectively be double-counting geometric

imperfection effects in the DM if τa were used, since

geometric imperfections are modeled explicitly as part of

the DM calculation of the internal forces. It is recommended

that τa can be used, along with an eigenvalue buckling

analysis, to provide a rigorous assessment of the member

resistances, given the internal forces obtained from the

above DM second-order load-deflection analysis. The use

of τa in a buckling analysis allows the engineer to obtain

a rigorous prediction of column strengths per the AISC

column strength equations, accounting for continuity effects

across braced points and lateral and/or rotational restraint

from other framing, including general stability bracing. With

this approach, no separate checking of the corresponding

underlying Specification member resistance equations is

needed. The mechanical responses associated with effective

length (K) factors are captured rigorously without the

difficulty of determining these factors. (The subsequent

extension of this approach to beam members also eliminates

the need for moment gradient and load height (Cb)

factors.) Refined estimates of the member strengths are

obtained without the modeling of detailed member out-

of-straightness imperfections.

The τa factor has been used extensively for the

calculation of column inelastic effective lengths for use in

the AISC Effective Length Method of design (the ELM).

However, as has been well documented, one does not

have to actually calculate column effective lengths to use

the ELM. The ELM can be employed with an explicit

buckling analysis to determine the column internal axial

force at theoretical buckling, Pe, or the corresponding

column axial stress Fe (in essence a “direct” buckling

analysis, but the word “direct” is being avoided here to

avoid confusion with the DM). This type of application of

the ELM, including the application of τa, is documented

in detail in (ASCE 1997).

An expression for τa can be derived as follows. The

derivation is shown only in the context of AISC LRFD to

keep the development succinct and clear.

Generally, one can write the AISC factored column

design resistance as

φcPn=0.9(0.877) Peτ=0.9(0.877)τaPe (1)

where 0.9 is the resistance factor used by the AISC

Specification for column axial compression, 0.877 is a

factor applied to the elastic column buckling resistance in

the AISC Specification to obtain the nominal column elastic

buckling resistance (accounting for geometric imperfection

and partial yielding effects for columns that fail by theoretical

elastic buckling, as well as an implicit increased margin

of safety for slender columns), and τa is the column

inelastic stiffness reduction factor. The column inelastic

buckling load, considering just τa and not including the

additional 0.9 and 0.877 factors, may be written as

Peτ=τaPe (2)

where Pe is the theoretical column elastic buckling load.

For nonslender element columns with , or

, the AISC column inelastic strength

equation may be written as

(3)

Peτ

Py

-------4

9--->

φcPn

φcPy

---------- 0.390>⎝ ⎠⎛ ⎞

φcPn

φcPy

---------- 0.658

τaPy

Peτ

-----------

=

Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1031

Upon recognizing, from Eq. (1) that

(4)

this relationship may be substituted into Eq. (3), giving

(5)

Using this form of the column inelastic strength equation,

one can take the natural logarithm of both of its sides and

then solve for τa as follows:

(6)

(7)

(8)

(The φc factor is included in both the numerator and

denominator of the fractions in Eq. (8) to facilitate the

next step of the development shown below.) The final Eq.

(8) has been used widely for column inelastic buckling

calculations in the context of the AISC Specification.

This equation can be applied most clearly by substituting

an internal axial force ΓPu for φcPn, such that τa can be

thought of conceptually as an effective reduction on the

member flexural rigidity (EI) at a given level of axial

load. As such, the τa equation becomes, for :

(9a)

This equation is valid only for column buckling within

the inelastic range. For elastic buckling

controls and

τa=1 (9b)

The above τa expressions can be employed with

buckling analysis capabilities, such as those provided by

the programs Mastan (Ziemian, 2016) and SABRE2

(White et al., 2016), to explicitly (or “directly”) calculate

the maximum column strength for any axially loaded

problem.

The most streamlined application of τa with a buckling

analysis to determine column strengths is as follows:

(1) Construct an overall buckling analysis model for the

problem at hand. (This can be done easily for basic

structural problems using programs such as Mastan and

SABRE2.)

(2) Apply the desired factored loads from a given

LRFD load combination to the above model. These

applied loads produce the column internal axial forces Pu.

(3) Reduce the elastic modulus of the structural

members, E, by 0.9×0.877=0.7893.

(4) Reduce the member moments of inertia by τa, based

on Pu, using Eqs. (9) with Γ=1. (Alternately, this step and

step 3 may be replaced by a single step where either the

elastic modulus E or the moment of inertia I is reduced by

the net stiffness reduction factor, SRF=0.9×0.877×τa.)

(5) Solve for the buckling load of the above inelastic

model. Initially, do this with the above Γ=1 and ΓPu=Pu.

The buckling analysis returns the eigenvalue, γ, which is

the multiple of the applied loads at incipient buckling.

Subsequently, scale the applied loads by the common

applied load scale factor Γ, calculate the τa values using

the corresponding internal forces ΓPu, and again solve for

the multiple of the current loading, γ, at which the system

buckles. Iterate these calculations until γ=1, indicating

that the system buckles at the specified load level Γ. The

corresponding internal axial forces ΓPu at incipient buckling

are then “directly” equal to the column axial capacities

φcPn.

This is a very powerful approach to obtain the most

accurate column design axial strengths, accounting for

the inelastic characteristics of the different members at

the strength limit state, continuity effects across braced

points with adjacent member lengths and any other member

end and intermediate restraints.

Traditionally, the ELM has been used with a basic

version of this approach to determine the influence of end

rotational restraints on columns. The member restraints

need not be limited to just column end rotational restraints

though. When applied to column buckling problems, the

above procedure gives an accurate estimate of the column

strength accounting for the restraint offered by any

general bracing system. The engineer simply needs to

include the lateral and/or torsional stiffnesses provided by

the bracing system in the buckling analysis.

In fact, if desired, rather than solving for the column

buckling load for a given set of bracing stiffnesses, one

can consider a given LRFD applied factored loading Pu

(with Γ=1) and then solve for the required bracing

stiffnesses necessary to develop the critical buckling

strength equal to this factored load level. These bracing

stiffnesses are commonly referred to as the ideal bracing

stiffness values, βi, corresponding to a given desired load

level Pu. The required bracing stiffness is then taken as

2βi/φ, with φ=0.75, per the AISC Appendix 6.

Some engineers have suggested that 0.8τb, the general

stiffness reduction used in the AISC Direct Analysis Method,

should be used for all problems, including calculation of

column inelastic buckling loads. This is certainly possible,

but such an approach misses the clear advantage of having

a buckling analysis procedure that rigorously determines

the value of φcPn accounting for all end and intermediate

Peτ

Pn

0.877-------------=

φcPn

φcPy

---------- 0.658

0.877τaPy

Pn

-------------------------

=

lnφcPn

φcPy

----------⎝ ⎠⎛ ⎞ ln 0.658

0.877τaPy

Pn

-------------------------

⎝ ⎠⎜ ⎟⎛ ⎞

=

lnφcPn

φcPy

----------⎝ ⎠⎛ ⎞ 0.877τa

Py

Pn

-----ln 0.658( )=

τa 2.724φcPn

φcPy

----------lnφcPn

φcPy

----------⎝ ⎠⎛ ⎞–=

ΓPu

φcPy

---------- 0.390>⎝ ⎠⎛ ⎞

τa 2.724ΓPu

φcPy

----------lnΓPu

φcPy

----------⎝ ⎠⎛ ⎞–=

ΓPu

φcPy

---------- 0.390≤⎝ ⎠⎛ ⎞


restraint effects. This issue can be understood by comparing

the net stiffness reduction factors (SRFs) 0.9×0.877×τa

and 0.8τb as shown in Fig. 1. The SRF 0.8τb generally

does not give an accurate estimate of the column strength

φcPn when used in a buckling analysis calculation. It does

perform reasonably well at giving an appropriate but less

rigorous estimate of the column strengths if used as part

of a second-order analysis in which appropriate geometric

imperfections are included per the requirements of the

DM.

3. Net Column Stiffness Reduction Factor (SRF) for Columns with Slender Cross-Section Elements

The ANSI/AISC 360-16 Specification (AISC 2016) has

adopted a unified effective width approach to characterize

the axial resistance of members having slender cross-

section elements under uniform axial compression. Using

this approach, the member axial resistance is expressed

simply as

φcPn=0.9Fcr Ae (10)

where Fcr is the column critical stress determined using

the member gross cross-section properties and Ae is the

cross-section effective area obtained by summing the

effective widths times the thicknesses for all the cross-

section elements. In many practical situations, the most

economical welded I-section members have slender webs.

Beam-type rolled wide flange sections, i.e., sections that

have a depth-to-flange width d/bf greater than about 1.7,

also often have slender webs. Therefore, it is important to

define how the column inelastic SRFs should be deter-

mined for these cases. Stated succinctly, the column net

SRF is given by

(11)

for these cross-section types, where τa is calculated from

Eqs. (9), but using the ratio instead of , and

φcPye=0.9Fy Ae (12)

In addition, the plate effective widths should be calculated

based on the axial stress f =ΓPu/Ae. As such, since Ae is

dependent on f, while f is also dependent on Ae, Ae and f

generally must be solved for iteratively. These iterations

are reasonably fast and are simple to handle numerically.

For manual calculation, f may be taken conservatively as

Fy . For columns with simply-supported end conditions,

the above calculations produce the same result as the

streamlined unified effective width equations in the AISC

(2016) Specification. For columns with general end and

intermediate restraints, the above calculations account for

the idealized relative stiffnesses in the structural system

with the same rigor as the more basic method for

nonslender element members explained in Section 2.

4. Basic Column Inelastic Buckling Example

Figure 2 shows a basic L-shaped frame subjected to a

vertical load at its knee. The frame’s in-plane inelastic

buckling mode is shown in the figure along with the

column τa and the load at incipient buckling φcPn=ΓP.

The column inelastic K factor, back-calculated by setting

SRF times the column elastic flexural buckling load Pe

equal to φcPn=ΓP from the inelastic buckling solution, is

shown as well. This result matches with a traditional

iterative calculation (Yura 1971) using the AISC τa values.

The inelastic K of 0.861 is smaller than the K factor of

0.894 back-calculated from an elastic buckling analysis

of the frame. Because of the column inelastic stiffness

reduction, the rotational restraint provided by the elastic

beam at the top of the column is more effective, resulting

SRF 0.9 0.877× τa

Ae

Ag

-----×=

ΓPu

φcPye

------------ΓPu

φcPy

----------

Figure 2. Column inelastic buckling analysis example.

Figure 1. Comparison of the net column stiffness reductionfactors (SRFs) 0.9×0.877×τa and 0.8τb


in a 2.6% increase in the column axial resistance for this

problem (a small but measureable increase). A buckling

analysis with the inelastic SRF incorporated integrally

within the calculations captures these attributes of the

response directly and explicitly.

5. Inelastic Lateral Torsional Buckling (LTB) Analysis using the Stiffness Reduction Obtained from the AISC LTB Strength Curves, τltb

Generally, one can write the factored AISC LRFD

beam LTB design resistance as

φbMn=φbRbMeτ=0.9RbτltbMe (13)

where Me represents the theoretical beam elastic LTB

resistance, Meτ represents the beam inelastic LTB strength,

and Rb is the web bend buckling strength reduction factor,

equal to 1.0 for a compact or noncompact web I-section.

The term τltb is the SRF corresponding to the nominal

AISC LTB strength curves. The derivation of this factor

parallels the derivation of the basic column SRF, τa,

presented in Section 2. To keep the presentation succinct,

the derivation of τltb is not provided here. Rather, just the

resulting equations for τltb are summarized. These equations

are as follows. For all types of I-section members, when

where

τltb=1 (14a)

However, for compact and noncompact web I-section

members with ,

τltb= (14b)

and the corresponding net stiffness reduction factor is

SRF=0.9τltb (14c)

where:

(15)

is the so-called LTB “plateau strength,” equal to 0.9Mp

for a compact-web cross-section,

(16)

and

(17)

(The additional common variables in these equations are

defined in the glossary at the end of the paper.)

Conversely, for slender-web I-sections with

, the following simpler form is obtained

in comparison to Eq. (14b):

(18a)

and the corresponding net stiffness reduction factor is

SRF=0.9Rbτltb (18b)

where Rh is the hybrid cross-section factor, which is not

considered in the ANSI/AISC 360 Specification, but is

addressed by similar strength equations in the AASHTO

LRFD Specifications (AASHTO 2015). Furthermore, for

compact- and noncompact-web sections, one can write

(19)

where φbMmax is the general “plateau strength” taken as

the minimum of the independent flexural strengths

calculated from the “flexural yielding” maximum limit

for LTB (using the AISC terminology), flange local buckling

(FLB), and tension flange yielding (TFY) as applicable:

φbMmax=min(φbMmax.LTB, φbMn.FLB, φbMn.TFY) (20)

The advantage of writing m as shown in Eq. (19) is that

the ratio can vary from zero to a maximum value

of 1.0, as can . These ratios facilitate the description

of an interpolated beam-column net SRF discussed

subsequently. For slender-web I-sections, one can write

(21)

where

Mmax.LTB=RbRhMyc (22)

Figures 3 and 4 illustrate how τltb varies relative to the

well-known column stiffness reduction factor τa for

representative beam- and column-type wide flange sections

respectively. The behavior of τltb for slender-web I-

sections is similar to that shown for the beam-type

mRbFL

Fyc

-----------≤ mΓMu

φbMyc

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

FL

Fyc

------- mφbM

max.LTB

φbMyc

-------------------------< <

Y4X

2

6.76X2 Fyc

E-------

⎝ ⎠⎛ ⎞

2

m2

2Y2

+

---------------------------------------------------------

φbMmax.LTB 0.9RpcMyc=

Y m

1m

Rpc

-------–⎝ ⎠⎛ ⎞

1FL

RpcFyc

---------------–⎝ ⎠⎛ ⎞----------------------------

Lr

rt

----Lp

rt

-----–⎝ ⎠⎛ ⎞ Lp

rt

-----+Fyc

E-------⎝ ⎠

⎛ ⎞ 1

1.95----------⎝ ⎠

⎛ ⎞=

X2 Sxcho

J------------=

RbFL

Fyc

----------- mφbMmax

φbMyc

-----------------< <

τltb

m

Rb

-----

Rh

m

Rb

-----–⎝ ⎠⎛ ⎞

Rh

FL

Fyc

-------–⎝ ⎠⎛ ⎞-----------------------

Fyc

FL

-------1.1

π-------–⎝ ⎠

⎛ ⎞ 1.1

π-------+

2

=

mΓMu

φbMyc

--------------ΓMu

φbMmax.LTB Rpc⁄

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

Rpc

ΓMu

φbMmax.LTB

------------------------- Rpc

ΓMu

φbMmax

-----------------Mmax

Mmax.LTB

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

ΓMu

φbMmax

-----------------

ΓPu

φcPye

------------

mΓMu

φbMyc

--------------ΓMu

φbMmax.LTB RbRh⁄

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

RbRh

ΓMu

φbMmax.LTB

------------------------- RbRh

ΓMu

φbMmax

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

max

Mmax.LTB

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


W21×44 section.

The LTB inelastic stiffness reduction factor, τltb, is

generally somewhat larger (i.e., reduces the capacity less)

than the corresponding column inelastic stiffness reduction

factor, τa, for a given normalized load ratio or

It should be noted that based on the AISC LTB

strength curves, I-section beams still have significant effective

inelastic stiffness when reaches the plateau resistance

φbMmax. For the above W21×44 and W14×257 examples,

τltb=0.223 and 0.180 respectively when this level of

loading is reached.

In addition to the above, if the internal moment at

incipient buckling, ΓMu, is larger than the corresponding

φbMmax at the most critically loaded cross-section, based

on an analysis in which the τltb values are calculated using

the corresponding internal moments throughout the

length of the members, the “plateau strength” has been

reached at the critical cross-section; hence, the design

strength is the applied load level at which the internal

moment at the critical cross-section is equal to φbMmax.

Since both of the above example cross-sections are

doubly symmetric and have compact flanges, φbMmax=

φbMmax.LTB. For doubly-symmetric sections having a non-

compact or slender compression flange, or singly-symmetric

sections with a noncompact or slender compression

flange and hc>h (usually associated with the compression

flange being smaller than the tension flange), flange local

buckling (FLB) governs the plateau resistance and φbMmax

=φbMmax.FLB. For singly-symmetric sections with hc<h,

either FLB or tension flange yielding (TFY) can govern

for the plateau resistance. For doubly- or singly-

symmetric sections with compact webs and compact

flanges, φbMmax = φbMmax .LTB =φbMmax .FLB =φbMmax .TFY=

φbMp, or as stated in the AISC Specification, the limit

state is “yielding” and the other limit states do not apply.

For proper calculation of the LTB resistance from a

buckling analysis, several requirements must be satisfied:

(1) In the context of doubly-symmetric I-section members,

the buckling analysis software must rigorously include

the contributions from the warping rigidity ECw , the St.

Venant torsional rigidity GJ, and the lateral bending rigidity

EIy .

(2) In addition, for singly-symmetric I-section members,

the buckling analysis must account rigorously for the

behavior associated with the shear center differing from

the cross-section centroidal axis, which relates to the

monosymmetry factor, βx, in analytical equations for the

LTB resistance of these types of beams.

(3) The SRF=0.9 Rb τltb should be applied equally to

each of the elastic stiffness contributions GJ, ECw and EIy,

at a given cross-section, for the execution of the buckling

analysis. Physically, it can be argued that the effective

reduction in the St. Venant torsional rigidity of an inelastic

beam typically is not as large as the reduction in the

effective EIy and ECw values. However, the use of an

equal reduction on all three rigidities (at a given cross-

section) is simple and sufficient. Furthermore, equal

reduction on all three cross-section rigidities reproduces

the beam LTB resistance from the AISC Specification

equations exactly for cases involving uniform bending

and simply-supported end conditions.

(4) A separate SRF of 0.9×0.877×τa should be applied

to the elastic stiffness contributions EA, EIx and EQx

(where Qx is the first moment of the area about the

reference axis of the cross-section, equal to zero when the

reference axis is the cross-section centroidal axis, but

non-zero in cases such as singly-symmetric section

members, where it is common for the shear center to be

taken as the reference axis in the structural analysis). For

beam members subjected to zero axial load, τa=1. Beam-

column members are addressed subsequently. Furthermore,

it should be noted that for singly-symmetric sections, the

moment about the centroidal axis is to be used in the

calculation of τltb.

(5) The internal force state upon which the buckling

ΓPu

φcPye

------------

ΓMu

φbMmax

-----------------

ΓMu

Figure 3. Column and beam τ factors for a W21×44representative beam-type wide flange section.

Figure 4. Column and beam τ factors for a W14×257representative column-type wide flange section.


analysis is based is to be determined using the elastic

properties of the structure, using the Direct Analysis

Method (the DM) rules specified in Chapter C of the

ANSI/AISC 360-16 Specification. Per the Direct Analysis

Method of design, all the cross-section elastic stiffnesses are

reduced generally by 0.8τb in a load-deflection analysis

employed to determine the internal forces. In addition,

Chapter C specifies nominal initial imperfections of the

points of intersection of the members in the structure (i.e.,

“system imperfections”, or equivalent notional loads,

corresponding to these imperfections). In cases involving

large axial loads and potential member non-sway failure

in the plane of bending, or for the assessment of structures

such as arches, it is also appropriate to consider comparable

in-plane member out-of-straightness values.

In general, the Chapter C requirements entail that an

inelastic nonlinear buckling analysis must be used to assess

the member resistances. This is a buckling analysis in which

the pre-buckling displacement effects are considered, via

the use of a second-order load-deflection analysis to

determine the internal forces. That is, Chapter C requires

that the in-plane internal forces must be calculated by a

second-order load-deflection analysis including stiffness

reduction and geometric imperfection effects. Given a

selected level of applied load and the corresponding internal

forces, the SRFs are calculated. Then the buckling analysis

is performed to evaluate the member resistances. If the

buckling eigenvalue γ is greater than 1.0, the buckling

resistance is greater than the current load level. The AISC

member out-of-plane buckling resistance equations are

never directly employed in the evaluation of the member

resistances. Instead, a buckling analysis with SRFs based

on the AISC member resistance equations is employed in

combination with a check of the cross-section strength

limits associated with the AISC member resistance equations.

In some cases, the member failure mode determined

from the buckling analysis may involve an overall buckling

of the entire structural system; however, in many situations,

the member failure will involve a localized member buckling

involving several unbraced lengths in the vicinity of a

critical region, or for beams, a failure by reaching the

FLB, the TFY limit state, or the “plateau resistance” of

the LTB curve (referred to by AISC as the flexural

yielding limit state). Related cross-section strength limits

for beam-columns are addressed subsequently.

For problems involving only beam members or involving

braced concentrically loaded columns with negligible pre-

buckling displacements, an inelastic linear buckling

analysis provides an acceptable solution. This type of

bucking analysis entails the use of a first-order elastic

analysis to determine the system internal forces, followed

by the calculation of the corresponding SRFs and the

execution of the buckling analysis. Although individual

beams and braced concentrically loaded columns can be

properly evaluated using a first-order analysis to determine

the internal forces, it should be noted that for general

beam-column members, the AISC Effective Length Method

(the ELM) is generally not sufficient for design using the

buckling analysis based procedures discussed here. This

is because the influence of geometric imperfections and

stiffness reductions on the second-order internal forces

must be considered for general beam-column members

when the in-plane strengths are based on cross-section

strength limits. The beam-column stiffness reduction factor

applied to ECw, GJ and EIy to capture the out-of-plane

resistance (discussed subsequently) is based on an estimate

of the “true” second-order member internal forces.

Furthermore, the in-plane beam-column stability behavior

is handled directly as a second-order load-deflection problem

via the DM in this work, using cross-section strength

limits based on the AISC Specification provisions.

The software SABRE2 (White et al., 2016) automates

and satisfies all the above requirements for general doubly-

or singly-symmetric I-section members with prismatic or

non-prismatic stepped and/or tapered geometries, as well

as frames composed of these types of members. Figure 5

shows a snapshot of the main window of SABRE2. The

SABRE2 software provides streamlined capabilities for

defining all attributes of the above types of problems, as

well as the definition of out-of-plane point and panel

bracing along the lengths of the members.

A sufficient number of elements per member must be

employed in the above LTB solutions. For frame elements

based on thin-walled open-section beam theory and cubic

Hermitian interpolation of the transverse displacements

and twists along the element length (the type of frame

element employed by SABRE2), four elements within

each unbraced length tend to be sufficient. In addition, for

inelastic buckling cases involving a moment gradient, the

variation of the inelastic stiffness along the member

length must be captured. Eight elements within each span

between the major-axis bending support locations tends

to be sufficient to capture the variation in the SRFs for

problems that do not have any reversal of the sign of the

moment within the span. Sixteen elements within each

span between major-axis bending support locations tends

Figure 5. Screen shot of a clear-span frame analysis beingconducted using the SABRE2 software (White et al.,2016).


to be sufficient for problems involving fully-reversed

curvature bending. The frame elements in SABRE2 use a

five-point Gauss-Lobatto numerical integration along their

length to capture the variations in the SRFs along the

length of each element.

Obviously, the above inelastic LTB solutions are not

manual engineering solutions. However, for that matter,

neither is the general second-order elastic analysis of an

indeterminate frame. Although engineers can conduct

approximate analysis to perform initial sizing of the

members in an indeterminate frame structure, commonly

they do not rely on these analyses, manual moment distri-

bution calculations, etc. for final design at this day and

time. With the appropriate software implementation of

the above τa and τltb calculations using a frame element

based on thin-walled open-section beam theory, the above

procedure is quite easy to apply. The software performs

the appropriate elastic matrix structural analysis to determine

the required member internal forces. Then it performs an

inelastic eigenvalue buckling analysis based on these

forces to evaluate the design. If the software automatically

handles the internal inelastic stiffness reductions based on

the magnitude of the internal forces, as is performed in

SABRE2, the inelastic buckling analysis is relatively

straightforward to apply.

This approach can be quite powerful to provide highly

accurate consideration of end restraints, continuity across

braced points, general moment gradient and finite bracing

stiffness effects on the LTB resistance of beam and frame

members. One key attribute of the power of this approach

is that, similar to the τa approach for column buckling,

once one has determined the load level corresponding to

incipient inelastic buckling using the τltb factor, the internal

forces in the model at the buckling load correspond

precisely to the design moment resistances φbMn. This

approach allows the consideration of any and all restraints

from bracing and member end conditions to be directly

and automatically considered in the design assessment, by

including them in the structural analysis model. Regarding

the assessment of the required stiffnesses for stability

bracing, this assessment is accomplished as a direct and

integral part of the calculation of the member LTB

resistances. If the buckling eigenvalue γ is greater than

1.0 from the buckling analysis, with the internal element

stiffnesses calculated based on the τltb equations given the

internal forces at a load level Γ, then the beam has

sufficient design strength for LTB at that load level. In

addition, SABRE2 calculates the φbMmax values associated

with flange local buckling and tension flange yielding, as

applicable, and checks these. If the critical beam cross-

section reaches φbMmax based on Eq. (20), with Γ≥1 and

with the reference load taken as the required loading from

a given LRFD load combination, the system maximum

load is governed by reaching this “plateau resistance”

prior to the occurrence of LTB. (Note that other limit

states such as web crippling, connection limit states, etc.

must be checked separately, just as they would be in

ordinary design.)

If desired, rather than solving for the beam LTB load

given bracing stiffnesses, one can consider a given LRFD

applied factored loading Mu (with Γ=1) and then solve

for the ideal bracing stiffnesses necessary to develop this

factored load level at buckling. The ideal bracing stiffnesses

are then multiplied by 2/φ to obtain the required bracing

stiffnesses for design.

6. Validation and Demonstration of the LTB Stiffness Reduction Equations

Consider the LTB resistance of a suite of W21×44

beams having torsionally and flexurally simply-supported

end conditions and unbraced lengths ranging from zero to

6.096 m. Figure 6 shows the results for the uniform bending

case as well as a basic moment gradient case involving an

applied moment at one end and zero moment at the other

end of the beams. All of the buckling analysis calculations

in this example, and in the subsequent examples are

performed using the SABRE2 software (White et al., 2016).

The following observations can be made from this LTB

study:

(1) The buckling analysis results for the LTB resistance

under uniform bending match exactly with the calculations

from the AISC (2016) Section F2 equations. Therefore,

only one curve is shown for the uniform bending case in

Fig. 6.

(2) The buckling analysis results for the LTB resistance

under the moment gradient fit closely with the calculations

from AISC Section F2 using a moment gradient factor

Cb=1.75. However, this rigorous LTB curve is slightly

different from the one obtained using the Section F2 LTB

equations. The differences between these curves are important,

and may be explained as follows:

(a) For longer unbraced lengths, where the beam is

elastic and τltb=1, the buckling load determined from

SABRE2 is approximately 6 % larger than the capacity

determined from the AISC Section F2 equations with Cb

taken as 1.75. The SABRE2 solution is a more accurate

assessment in this case. The 1.75 value for Cb is a lower-

bound approximation developed by Salvadori (1955).

The SABRE2 solution is approximately 11 % larger than

the solution with Cb=1.67 obtained from AISC Eq. (F1-

1) for this problem. AISC Eq. (F1-1), originally developed

by Kirby and Nethercot (1979), gives a “lower” lower-

bound solution than Professor Salvadori’s equation for

this problem.

(b) For intermediate unbraced lengths at which the

maximum moment at incipient buckling is larger than

φbFLSxc=0.9(0.7FySxc) (equal to 290 kN-m for the W21×44), the inelastic buckling analysis solution is again fully

consistent with the AISC Section F2 equations, but is a

more accurate assessment of the LTB resistance than the

direct use of the AISC Section F2 equations. In this case,


as the buckling resistance increases above φbFLSxc, some

reduction in the LTB resistance occurs due to the onset of

yielding at the locations where the internal moment is

largest. The approach taken in AISC Chapter F is to

simply scale the uniform bending LTB resistance by Cb,

but with a cap of φbMmax on the flexural resistance. As

discussed by Yura et al. (1978), this approach tends to

over-predict the true response to some extent in the

vicinity of the point where the elastic or inelastic LTB

design strength curve intersects φbMmax, although the

approximation is considered to be acceptable. The LTB

resistances obtained from the buckling analysis are slightly

smaller than those obtained directly from the AISC

Chapter F2 equations in the vicinity of the location where

the LTB resistance reaches the plateau resistance φbMmax,

reflecting the more rigorous accounting for inelastic

stiffness reduction effects on the LTB resistance in the

buckling analysis based solution.

7. Roof Girder Design Example

Figure 7 shows a roof girder design example adapted

from a suite of example problems developed by the AISC

Ad hoc Committee on Stability Bracing (AISC 2002).

The girder has a 21.3 m span and is subjected to gravity

loading applied from outset roof purlins connected to its

top flange and spaced at 1.52 m. The girder ends are

assumed to be flexurally and torsionally simply supported.

The top flange in this problem is braced at the purlin

locations by light-weight roof deck panels, having a shear

panel stiffness of 0.876 kN/mm. Flange diagonal braces

are provided from the purlins to the bottom flange at the

mid-span of the girder plus at two additional locations on

each side of the mid-span with a spacing of 3.05 m

between each of these positions. These diagonal braces

restrain the lateral movement of the bottom flange

relative to the top flange, and therefore they are classified

as torsional braces. The provided elastic torsional bracing

stiffness is modeled as βT=723 kN-m/rad. These torsional

braces combine with the panel lateral bracing from the

roof deck to provide out-of-plane stability to the roof

girder. The above bracing stiffnesses are divided by 2/φand the corresponding reduced stiffnesses are employed

for inelastic buckling analysis, per the AISC (2016)

Appendix 6 requirements.

Figure 8 shows the governing overall lateral-torsional

buckling mode for this roof girder, determined using

SABRE2. The lines shown with a diamond symbol in the

horizontal plane at the top flange level represent the shear

panel bracing from the roof deck, and the circular lines at

the mid-span and at two locations on each side of the

mid-span represent the torsional bracing from the roof

purlins and the framing of a flange diagonal to the bottom

flange of the girder. The arrow symbols indicate zero

displacement constraints. Figure 9 shows the variation in

the net SRF along the length of the girder obtained from

the SABRE2 solution.

The applied load scale factor on the required vertical

gravity load at incipient inelastic buckling of the girder is

Γ=1.010. Therefore, the girder and its bracing system are

sufficient to support the required LRFD loading. One can

observe a noticeable lateral deformation within the adjacent

shear panels on each side of the mid-span torsional brace

in Fig. 8. This indicates that the light roof panel bracing

is providing slightly less than full bracing at the first

braced point on each side of the mid-span. Nevertheless,

the overall design strength is slightly larger than the

required strength from the LRFD loading. Figure 9 shows

that significant yielding is developed both at the mid-span

and at the girder ends when the girder reaches its maximum

design resistance.

It is important to note that the accurate assessment of

the combined bracing stiffnesses is somewhat challenging

for this problem using any method other than the SABRE2

Figure 7. Roof girder example, adapted from (AISC2002).

Figure 6. Lateral torsional buckling design resistances for W21×44 beams (Fy=345 MPa), calculated using the AISC Specification equations and using buckling analysis with

the corresponding stiffness reduction factor 0.9τltb (Rb=1.0)


buckling analysis. The basic requirements specified in

AISC Appendix 6 do not address combined lateral and

torsional partial bracing. The AISC (2016) Appendix 1

provisions provide guidance for the use of advanced load-

deflection analysis methods for the general stability design.

However, the application of these methods necessitates

the modeling of an appropriate initial out-of-alignment of

the girder braced points (in the out-of-plane direction) as

well as out-of-straightness of the girder flanges between

the braced points. The geometric imperfections needed to

evaluate the different bracing components are in general

different for each of the bracing components, and the

geometric imperfections necessary to evaluate the maximum

strength of the girder are in general different from those

necessary to evaluate the bracing components. One can

rule out the need to perform many of these analyses by

identifying the girder critical unbraced lengths as well as

the critical bracing components. However, short of the

type of buckling analysis provided by SABRE2, it can be

difficult to assess which unbraced lengths and which bracing

components are indeed the critical ones. SABRE2 provides

not only an assessment of the adequacy of the bracing

system stiffnesses, but it also provides a “direct” check of

the member design resistance given the member’s bracing

restraints and end boundary conditions.

The only shortcoming of the above buckling analysis

approach, in the context of the above type of design

problem, is that this approach does not provide any direct

estimate of the bracing strength requirements. However,

based on numerous results from experimental testing and

from refined FEA simulation of experimental tests, it is

recommended that the simple member force percentage

rules of Appendix 6 can be used to specify the minimum

required strengths for the different bracing components

(AISC 2016).

8. Stiffness Reduction Factors for Beam-Columns

Traditional beam-column strength interaction equations

utilize a simple interpolation between the member axial

strength in the absence of bending, φcPn, and the member

flexural strength in the absence of axial loading, φbMn.

Given the stiffness reduction factors SRF=0.9×0.877×τa

for axial load only and SRF=0.9 Rb τltb for bending

only, one might expect that a simple interpolation between

these stiffness reduction factors would provide an accurate

representation of the net SRF for beam-column members.

The authors have found that the following interpolation

between the cross-section column and beam net SRF

values provides an accurate characterization of I-section

beam-column strengths:

(1) The unity check value with respect to the cross-

section maximum strength is obtained using the equations

for (23a)

and

for (23b)

(2) The above UC value is employed in the τa and τltb

equations instead of the ratios and

(3) The angle

(24)

is calculated. This angle is the position of the current

force point within a normalized x-y interaction plot of the

axial and moment strength ratios for a given cross-

section.

(4) The net SRF representing the beam-column response

is determined using the interpolation equation

(25)

Ae

Ag

-----

UCΓPu

φcPye

------------8

9---

ΓMu

φbMmax

-----------------+=ΓPu

φcPye

------------ 0.2≥

UCΓPu

2φcPye

---------------ΓMu

φbMmax

-----------------+=ΓPu

φcPye

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

ΓPu

φcPye

------------ΓMu

φbMmax

-----------------

ζ atanΓPu φcPye⁄

ΓMu φbMmax

⁄------------------------------⎝ ⎠

⎛ ⎞=

SRFζ

90o

--------⎝ ⎠⎛ ⎞0.9 0.877× τa

Ae

Ag

-----× 1ζ

90o

--------–⎝ ⎠⎛ ⎞0.9Rbτltb+=

Figure 8. Governing overall lateral-torsional bucklingmode for the roof girder.

Figure 9. Variation of the net stiffness reduction factor(SRF) along the length of the roof girder at its maximumdesign resistance corresponding to Γ=1.01.


where ζ is expressed in degrees. This SRF value is applied

to ECw, EIy and GJ. The separate column SRF=0.9×0.877

×τa , with τa obtained directly from the column strength

ratio is applied to EIx, EQx and EA, as discussed

previously in Section 5. In addition, it should be noted

that the area ratio in the above expressions is

determined directly from the axial force ΓMu as discussed

previously in Section 3. The following section demonstrates

the quality of the beam-column strength estimates obtained

from the above approach for several basic validation cases.

9. Beam-Column Examples

Figures 10 and 11 show the beam-column strength curves

obtained from SABRE2 for several suites of flexurally

and torsionally simply-supported beam-columns subjected

to uniform primary bending moment and a linear primary

moment gradient loading respectively. It should be noted

that ΓMu in these figures is the maximum calculated

second-order moment along the member lengths. The

following observations may be made from these plots:

(1) For the shorter members, the strength envelopes are

essentially equal to the fully-effective cross-section plastic

strength curves (i.e., Eqs. 23 with Pye=Py=Fy Ag) at

smaller axial load values. This result corresponds to the

mechanics of the beam-column strength problem reported

by Cuk et al. (1986) and approximated by a combination

of Eqs. (H1-1) and (H1-2) in the AISC (2010 and 2016)

Specifications.

(2) For larger axial load values, the strength envelopes

“peel away” from the above in-plane strengths at a

particular axial load level and approach the out-of-plane

column strength of these simply-supported members in

the limit of zero bending and pure axial compression. The

strength envelopes in these regions are slightly convex

due to the loss of effectiveness of the W21×44 webs with

increasing axial force. That is, the W21×44 webs are

slender under uniform axial compression. The members

with longer lengths are not able to develop φbMn=φbMp

for the case of zero axial load; rather, their bending resistance

is governed by out-of-plane lateral torsional buckling.

The moment gradient loadings allow the development of

φbMn=φbMp for larger member unbraced lengths in the

case of zero axial load.

(3) For the longest W21x44 members considered (i.e.,

L=4.57 m), subjected to uniform primary bending moment,

the failure is entirely due to elastic beam and beam-column

lateral torsional buckling. In this case, the resulting strength

envelope ranges between 0.9×0.877 Pe for pure axial

compression and zero bending and 0.9 RbMe for pure

bending and zero axial compression, where Pe is the

theoretical out-of-plane flexural buckling load, Me is the

theoretical lateral torsional buckling moment for pure

bending, and Rb=1 for this problem. The shape of the

strength curve in this case matches exactly with the

theoretical beam-column elastic LTB resistance, scaled

by an interpolated reduction factor ranging from 0.9×0.877 to 0.9 Rb.

It is not possible to perform beam-column strength

calculations with this level of rigor by a second-order

load-deflection analysis combined with the traditional

application of separate manual strength interaction equations.

SABRE2 incorporates the AISC member strength equations

ubiquitously within a buckling analysis, via calculated net

stiffness reduction factors (SRFs), to provide a more

rigorous characterization of the member resistances.

10. Conclusions

Traditional design of structural steel members and

frames involves the use of a second-order elastic load-

deflection analysis to estimate member internal forces,

followed by the application of separate “manual” member

Ae

Ag

-----

ΓPu

φcPye

------------

Ae

Ag

-----

Figure 10. Beam-column strength curves obtained fromSABRE2 for several suites of flexurally and torsionallysimply-supported W21x44 members subjected to uniformprimary bending moment and uniform axial compression.

Figure 11. Beam-column strength curves obtained fromSABRE2 for several suites of flexurally and torsionallysimply-supported W21x44 members subjected to uniformaxial compression and moment gradient loading from anapplied bending moment at one end, zero bendingmoment at the opposite end.


resistance equations. These “manual” equations commonly

require the calculation of various design strength factors,

such as different member effective length factors (K)

corresponding to ideal flexural and torsional column

buckling as well as beam lateral torsional buckling (often

estimated based on ideal elastic buckling, but sometimes

determined using more tedious inelastic buckling estimates),

and moment gradient and load height modifiers, which

are represented by the term Cb in the AISC Specification

provisions. These factors become increasingly tedious to

calculate and increasingly tenuous in terms of the resulting

accuracy for problems involving general loadings and

general member intermediate (flexible bracing) and end

(flexible rotational and/or translational) restraints. In addition,

the assessment of the adequacy of flexible intermediate

lateral and/or torsional bracing generally requires the

consideration of the member inelastic stiffness properties

at the strength limit. Emerging advanced design procedures,

such as those identified in the AISC (2016) Appendix 1,

focus on an explicit analysis of the second-order load-

deflection response of geometrically imperfect members

and systems; however, these procedures are relatively

expensive to conduct and can require the consideration of

a tremendous number of load combinations times geometric

imperfection patterns to perform a complete assessment

of an overall structural system.

This paper presents a comprehensive method for the

design of planar structural steel members and systems via

the use of buckling analysis combined with appropriate

column, beam and beam-column strength reduction factors

(SRFs). The net SRFs are derived from the current ANSI/

AISC 360-16 Specification (AISC 2016) column, beam

and beam-column strength provisions. The buckling analysis

is combined with a check of member cross-section strength

limits based on the AISC Specification provisions. The

resulting procedure provides a rigorous check of member

design resistances, accounting for continuity effects across

braced points, as well as lateral and/or rotational restraint

from other framing including any type and configuration

of stability bracing. Although the emphasis of the examples

presented in this paper is on basic demonstration and

validation problems involving prismatic members, the

recommended procedure is particularly powerful for the

design assessment of frames utilizing general stepped

and/or tapered I-section members.

Glossary of Terms

A Gross cross-sectional area

Ae Effective cross-section area corresponding to a

given axial compression force, accounting for

plate local buckling effects

Ag Gross cross-sectional area

Cb Beam moment gradient and load height factor

Cw Warping constant

E Modulus of elasticity of steel, taken equal to

200 GPa

G Shear modulus of elasticity of steel, taken

equal to 77.2 GPa

FL Magnitude of flexural stress in the compression

flange at which flange local buckling or lateral

torsional buckling is taken to be influenced by

yielding in the AISC Specification

Fcr Column critical buckling stress

Fe Member internal axial compression stress at

incipient elastic buckling of the member or

structural system

Fy Specified minimum yield stress

Fyc Specified minimum yield stress of the compression

flange

I Moment of inertia in the plane of bending

Ix Moment of inertia about the major principal

axis

Iy Moment of inertia about the minor principal

axis

Iyc Moment of inertia of the compression flange

about the axis of the web

J St. Venant torsional constant

K Column or beam effective length factor

L Member length

Lbr Member unbraced length between the braced

points

Lp Prismatic beam unbraced length limit within

which the AISC nominal lateral torsional buckling

(LTB) resistance under uniform bending is

equal to the plateau resistance Mmax.LTB

Lr Prismatic beam unbraced length limit beyond

which the nominal AISC lateral torsional buckling

(LTB) resistance is taken as the theoretical

elastic LTB resistance

Me Elastic lateral torsional buckling moment of

the member for the case of zero axial

compressive force

Meτ Member maximum buckling moment equal to

τltb Meτ

Mmax Largest potential moment that can be developed

in the beam cross-section considering the three

potential governing limit states of compression

flange yielding, compression flange local buckling,

and tension flange yielding

Mmax.FLB Largest potential moment that can be developed

in the beam cross-section considering only the

compression flange local buckling limit state

Mmax.LTB Largest potential moment that can be developed

prior to lateral torsional buckling for sufficiently

short member lengths, considering only the

lateral torsional buckling limit state; commonly

referred to as the LTB “plateau resistance,”

and referred to in the AISC Specification as

the limit state of yielding or compression flange

yielding

Mmax.TFY Largest potential moment that can be developed

in the beam cross-section considering only the


tension flange yielding limit state

Mn Member nominal flexural strength

Mp Cross-section plastic bending moment

Mu Cross-section internal moment corresponding

to a given ASCE LRFD load combination

Myc Moment at nominal yielding of the extreme

fiber of the compression flange

Pe Member internal axial compression force at

incipient elastic buckling of the member or

structural system

Peτ Member axial force equal to τa Pe

Pn Member nominal axial resistance

Pu Cross-section internal axial load corresponding

to a given ASCE LRFD load combination

Py Cross-section yield axial load, equal to AgFy

Pye Yield axial load based on the effective cross-

section area, taken equal AeFy

Qx First moment of the gross cross-sectional area

about the reference axis of the cross-section

Rb Cross-section bend buckling strength reduction

factor, AASHTO (2015) notation, represented

by the symbol Rpg in the AISC Specification

Rh Hybrid girder factor from AASHTO (2015)

Rpc AISC LRFD web plastification factor, or effective

plastic section modulus considering web

slenderness effects

Sx Elastic section modulus

Sxc Elastic section modulus to the extreme fiber of

the compression flange for major-axis bending

SRF Stiffness reduction factor employed for the

buckling analysis

UC Unity check value with respect to the cross-

section maximum strength, given by Eqs. (23)

X Cross-section major-axis flexural to St. Venant

torsional property ratio, taken as

Y Intermediate factor used in calculating τltb for

compact and noncompact web I-sections,

given by Eq. (16)

bf Width of flange

d Full nominal depth of the cross-section

f Member axial stress equal to ΓPu/Ae

h Clear distance between the flanges less the

fillet radius for rolled I-sections; clear distance

between the flanges for welded I-sections

hc Twice the distance from the centroid of the

cross-section to the inside face of the compression

flange less the fillet radius for rolled I-sections,

and to the inside face of the compression

flange for welded I-sections

ho Distance between the flange centroids

m normalized cross-section moment ΓMu/φbMyc

rt radius of gyration of the compression flange

plus one-third of the web area in compression

due to the application of major-axis bending

moment alone

Γ Applied design load scale factor

ΓMu Cross-section internal moment corresponding

to a given value of the load scale factor ΓΓPu Cross-section axial force corresponding to a

given value of the load scale factor ΓβT Provided torsional bracing stiffness

βbr Provided lateral bracing stiffness

βx Cross-section monosymmetry factor

βi Ideal bracing stiffness, defined as the bracing

stiffness at which the member or structure and

its bracing system buckle at the required

design load

φ AISC LRFD resistance factor on bracing stiffness,

equal to 0.75

φb AISC LRFD resistance factor for flexure

φc AISC LRFD resistance factor for axial compression

γ Eigenvalue obtained from the buckling analysis,

equal to the multiple of the current loading

corresponding to buckling, given the stiffness

properties associated with the current loading

state

τa Column inelastic stiffness reduction factor not

including the additional factors 0.9×0.877×Ae/

Ag

τb Stiffness reduction factor applied to the member

flexural rigidity for a second-order load-deflection

analysis per the AISC Direct Analysis Method

τltb Beam lateral torsional buckling stiffness reduction

factor not including the additional factors

0.9Rb

ζ Angle of the force point within the normalized

x-y interaction plot of the cross-section axial

and moment strength ratios, given by Eq. (24)

References

AASHTO (2015). AASHTO LRFD Bridge Design

Specifications. 7th Edition with 2015 Interim Revisions,

American Association of State Highway and

Transportation Officials, Washington, DC.

AISC (2016). Specification for Structural Steel Buildings.

ANSI/AISC 360-16, American Institute of Steel

Construction, Chicago, IL.

AISC (2010). Specification for Structural Steel Buildings.

ANSI/AISC 360-10, American Institute of Steel

Construction, Chicago, IL.

AISC (2002). “Example Problems Illustrating the Use of the

New Bracing Provisions-Section C3, Spec and

Commentary”, Ad hoc Committee on Stability Bracing,

November.

ASCE (1997). Effective Length and Notional Load

Approaches for Assessing Frame Stability: Implications

for American Steel Design. American Society of Civil

Engineers, Reston, VA.

Cuk, P.E., Rogers, D.F. and Trahair, N.S. (1986). “Inelastic

Buckling of Continuous Steel Beam-Columns.” Journal

of Constructional Steel Research, 6(1), pp. 21-52.

Kirby, P.A. and Nethercot, D.A. (1979). Design for

Sxcho J⁄


Structural Stability. Wiley, New York.

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Torsional Buckling Assessment of Steel Beams Through

a Stiffness Reduction Method,” Journal of Constructional

Steel Research, 109, pp. 87-100.

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