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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≤⎝ ⎠⎛ ⎞
1032 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 2016
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
Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1033
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
--------------------= =
1034 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 2016
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.
Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1035
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).
1036 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 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,
Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1037
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)
1038 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 2016
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.
Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1039
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.
1040 Donald W. White et al. / International Journal of Steel Structures, 16(4), 1029-1042, 2016
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
Comprehensive Stability Design of Planar Steel Members and Framing Systems via Inelastic Buckling Analysis 1041
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
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