DESIGN METHODS FOR LOCAL-GLOBAL
INTERACTION OF LOCALLY SLENDER
STEEL MEMBERS
M. Seif1 and B.W. Schafer2
ABSTRACT
The work presented herein is part of a continuing effort towards fully
understanding, and taking advantage of the use of, locally slender
cross-sections in structural steel. Previous efforts compared three
design methods for locally slender steel short beams and stub columns;
(i) AISC, and two methods from cold-formed steel specifications that
focus on locally slender cross-sections: (ii) AISI-Effective Width, and
(iii) AISI-Direct Strength Method (DSM). Parametric studies
employing material and geometric nonlinear, shell-element based FE
analysis were used to understand and highlight the parameters that lead
to the divergence between the capacity predictions of the different
design methods. This work presents an extension of the previous
parametric studies to long columns and beams, where the locally
slender cross-sections may interact with global (flexural, lateraltorsional, etc.) buckling modes. Of particular interest are; the
divergence of strength predictions between the three design methods,
the accuracy of the AISC methodology which handles local-global
interaction quite differently for beams than for columns, and comparing
the more elaborate AISC methods for beams to those utilized in AISI.
The final goal of this research is to propose improvements to DSM so
that it may be readily applied to structural steel with locally slender
cross-sections, and in doing so provide a means to simplify and
improve the design of locally slender structural steel sections.
1
2
PhD Candidate, Johns Hopkins University, mina.seif@jhu.edu
Associate Prof. and Chair, Johns Hopkins University, schafer@jhu.edu
INTRODUCTION
With the advent of high and ultra-high yield strength steels, the
increased yield stress drives even standard hot-rolled steel shapes from
locally compact to locally slender (noncompact or slender), making it
inefficient to avoid such cross-sections in the design of hot-rolled steel
structural members (see Seif and Schafer 2009a,b for details). Efficient
and reliable strength predictions are needed for locally slender hotrolled steel cross-sections. Analysis of existing AISC (2005) provisions
for locally slender stub columns and short beams (Seif and Schafer
2009a) indicated geometric regions where AISC design may be
excessively conservative, and other regions where it may be moderately
unconservative. The work herein represents a direct extension of
previous studies on stub columns and short beams (Seif and Schafer
2009a) now to include long columns and long beams, where the locally
slender cross-sections may interact with global (flexural, lateraltorsional, etc.) buckling modes.
DESIGN METHODS
The design of locally slender steel cross-sections may be completed by
a variety of methods, three of which are examined in this study: (1) The
AISC method, as embodied in the 2005 AISC Specification, labeled
AISC herein, (2) The AISI Effective Width Method from the main
body of the 2007 AISI Specification for cold-formed steel, labeled
AISI herein, and, (3) The Direct Strength Method as given in Appendix
1 of the 2007 AISI Specification, labeled DSM herein.
For each of these three design methods the expressions for strength
prediction of locally slender braced columns and beams have been
provided in a common notation in Seif and Schafer (2009a). In those
equations the centrality of elastic local buckling is made clear. For long
(unbraced) columns and beams global buckling must be considered as
well as local-global interaction.
In AISC, AISI, and DSM global column buckling is predicted using the
same (single) expression. However, local-global interaction is handled
by the Q-factor method in AISC, the unified method in AISI, and a
variation of the unified method in DSM. In all cases the global strength
is reduced due to local cross-section slenderness. The Q-factor
approach reduces the strength and increases the long-column
slenderness to arrive at its reduction. The unified method uses the
effective area of the column at the long column buckling stress. DSM
uses a similar approach, but the effective area calculation is replaced by
a reduction of the full cross-section (at the long column strength).
AISC and AISI/DSM use different formats for the global (lateraltorsional buckling) provisions of beams. However, for no moment
gradient (Cb = 1) the resulting expressions are actually quite similar
with the exception that AISI only provides capacities up to first yield
(My) for sections subject to lateral-torsional buckling. For AISI/DSM
local-global interaction in beams is treated in the same conceptual
manner as for columns; not so for AISC, which uses nothing like the Qfactor approach, and instead provides direct reductions based on the
flange and web plate slenderness (also see White 2008). A result of
AISC’s approach in not adopting one consistent philosophy for localglobal interaction in beams is some unusual changes in strength as local
slenderness is varied.
FE PARAMETRIC STUDY
A nonlinear finite element (FE) analysis parameter study was carried
out for the purpose of understanding and highlighting the parameters
that lead to the divergence between the capacity predictions of the
different design methods under axial and bending loads. The FE
analysis is extended herein to long members, where the locally slender
cross-sections may interact with global (flexural, lateral-torsional, etc.)
buckling modes.
Sections and boundary conditions
Based on the authors’ judgment, AISC W14 and W36 sections were
selected for the study as representing “common” sections for columns
and beams in high-rise buildings. The W14x233 section is
approximately the average dimensions for the W14 group and the
W36x330 for the W36 group. All sections are modeled with globally
pinned, warping fixed boundary conditions, and loaded via incremental
displacement for the columns and rotation for the beams.
Geometric variation – cross sectional
To examine the impact of slenderness in the local buckling mode, and
the impact of web-flange interaction in I-sections, four series of
parametric studies are performed under axial and bending loading, as
further described in Table 1: W14FI: a W14x233 section with a
modified Flange thickness, that varies Independently from all other
dimensions, W14FR: a W14x233 section with variable Flange
thickness, but the web thickness set so that the Ratio of the flange-toweb thickness remains the same as the original W14x233, W36FR: a
W36x330 section with variable Web thickness, but the flange thickness
set so that the Ratio of the flange-to-web thickness remains the same as
the original W36x330, and W36WI: a W36x330 section with a
variable Web thickness, that varies Independently from all other
dimensions.
Table 1 Parametric study of W-sections
W14x233
W14FI
W14FR
W36x330
W36FR
W36WI
bf/2tf
4.62
varied
varied
4.54
varied
fixed
h/tw
13.35
fixed
varied
35.15
varied
varied
h/bf
0.90
fixed
fixed
2.13
fixed
fixed
tf/tw
1.61
varied
fixed
1.81
fixed
varied
For the purpose of this study, element thicknesses were varied between
0.05 in. (1.27 mm) and 3.0 in. (76.2 mm). While not strictly realistic,
the values chosen here are for the purposes of comparing and
exercising the design methods up to and through their extreme limits.
Local slenderness may be understood as the square root of the ratio of
the yield stress to the local buckling stress (i.e., √fy/fcr). The element
local buckling stress is proportional to the square of the element
thickness, thus the local slenderness is proportional to 1/t. Here element
thickness is varied and used as the primary proxy for investigating local
slenderness, in the future, material property variations are also needed.
Geometric variation – lengths
The initial FE analysis (Seif and Schafer 2009a) was conducted on stub
(short) members, avoiding global (i.e., flexural, or lateral-torsional)
buckling modes, and focusing on local buckling modes alone. The
length of the studied members was determined according to the stub
column definitions of SSRC (i.e., Galambos 1998), and fixed at that
length.
To examine the impact of local-global buckling modes interaction on
the strength of locally slender members, longer members are included
in the FE parameter study taking the member’s length as a variable in
the parameter study. Each member’s length is determined so to achieve
certain preset slenderness parameter values, where the slenderness
parameter, λ, is defined in terms of the member’s length and crosssection dimensions.
For columns, two groups of analysis were chosen to be performed at
axial slenderness parameter, λc, values fixed at 0.90 and 1.50, where λc
is defined as:
c 
fy
fcr
euler

fy
KL
r  2E
(1)
Note that varying the thicknesses (flange, web, or both at constant
ratio) will vary the moment of inertia, I, and the cross-sectional area, A,
and accordingly the radius of gyration, r. The member’s length, L, is
then back-calculated to maintain the specified λc values.
Similarly for beams, two groups of analysis were chosen to be
performed at flexural slenderness parameter, λe, values fixed at 0.60
and 1.34, which are the AISI values defining the non-compact from the
compact and slender members respectively (see, e.g. Shifferaw and
Schafer 2008). The λe is defined as:
e 
My
M cre
(2)
Fixing λe, the critical buckling moment, Mcre, is calculated for each
section. Mcre is also defined as follows:
M cre  Cb
 2 EI y GJ
L2

 2 EI y 2Cw
L4
(3)
Again, note that varying the thicknesses (flange, web, or both at
constant ratio) will vary all the parameters on the right hand side of Eq.
(3). Accordingly, the member’s length, L, is then back-calculated to
maintain the specified λe values.
Mesh and element selection
ABAQUS was used to perform the analysis. Members were modeled
using S4 shell elements. The S4 element has six degrees of freedom per
node, adopts bilinear interpolation for the displacement and rotation
fields, incorporates finite membrane strains, and its shear stiffness is
yielded by “full” integration. Considering computational speed and
accuracy it was decided that a mesh density of five elements across
each flange outstand, ten across the web, and an aspect ratio of 1 was
adequate for this study. A typical model is shown in Figure 2. The
choice of element type and density are based on comparisons with
three-dimensional solid elements as reported in Seif and Schafer
(2008,2009a). It is noted that some debate exists in the literature
regarding the selection of the S4 vs. S4R element (see, e.g. Dinis and
Camotim 2006, and Earls 2001).
Material modeling
The material model follows classical metal plasticity: Von Mises yield
criteria, associated flow, and isotropic hardening. The uniaxial -
diagram is provided in Figure 1a, and is similar to that employed by
Engineering Stress (ksi)
Barth et al. (2005). The curve is converted to a true - curve for the
ABAQUS analysis.
fu= 65
fy= 50
Slope,
E’=145
-
Slope,
Est =720
 st =0.011
-
 c  0 .3 f y


bf t f

+  t   c 

b
t

t
(
d

2
t
)
f
f
w
f


Slope,
E =29000
y
-
-
-
Engineering Strain
(a) uniaxial - relation
(b) residual stress distribution
Figure 1 Idealized material model (a) - and (b) residual stresses
Residual stresses
For this work, the classic and commonly used distribution of Galambos
and Ketter (1959), as shown in Figure 1b, is employed. Similar to other
researchers (e.g., Jung and White 2006) the residual stresses are defined
in the finite element analysis as initial longitudinal stresses, and given
as the average value across the element at its center. (See Seif and
Schafer 2009a for further discussion).
Geometric imperfections
Geometric imperfections have an important role to play in any collapse
analysis involving stability. For the previous work on short (stub)
members, the imperfections were defined by scaling the local buckling
eigenmode from elastic buckling analysis. Since the focus at this point
is on longer members, global buckling modes are also included. Initial
geometric imperfections are added through linearly superposing a
scaled local and a scaled global eigenmode solution from a finite strip
analysis performed on each section, using CUFSM (Schafer, B.W.,
Ádány, S. 2006). The local buckling mode is scaled so that the
maximum nodal displacement is equal to the greater of bf /150 or d/150
which is a commonly employed magnitude (see, e.g Kian and Lee
2002), while the global buckling mode is scaled so that the maximum
nodal displacement is equal to L/1000, as shown in Figure 2
Figure 2 Typical column buckling modes (left: local, right: global) and
initial geometric imperfections for the analysis (a) ABAQUS isometric, (b)
ABAQUS front view, and (c) CUFSM front view, with scaling factors.
RESULTS
As discussed previously (see Table 1), the parametric study is broken
into 4 groups: W14FI, W14FR, W36FR, and W36WI analyzed at
different preset slenderness limits. Here the results of the parametric
study are presented for each group, including comparisons to the AISC,
AISI, and DSM design methods. Analysis results are provided first for
the columns, then the beams. Due to limited space the results are
condensed, see Seif and Schafer (2010) for full results and discussion.
Columns: ABAQUS results for the parametric study of locally slender
long columns (denoted with “·” and given for the 4 parametric studies)
are reported as a function of long column slenderness ( c~0.25, 0.9, and
1.5) in Figure 3. In Figure 3 the standard (compact) W14 and W36
cross-sections have been denoted with a “*”. If the long column curve
is exact, the “*” would be in perfect agreement with the upper curve
shown. As can be observed, as the local slenderness is increased the
strength predictions fall further and further below the global column
(upper) curve, which for compact/fully-effective sections is identical in
AISC, AISI, and DSM. Also highlighted in Figure 3, so that a locally
slender section may be observed, is the cross-sections with a backcalculated Q or Aeff/Ag ≈ 0.7, denoted with a “o”, and the AISC and
AISI (both effective width and DSM) strength curves for Q or Aeff/Ag =
0.7. Figure 3 does not allow for a complete study of the impact of local
slenderness as a full family of strength curves would need to be
generated and each point compared to a different curve. Rather than do
this, to compare all the sections in a given study the results are
expressed as a function of local slenderness (at a given global
slenderness c).
W14FI
W14FR
1
0.5
0.5
n
P /P
y
1
0
0
0.5
1
1.5
0
0
0.5
W36FI
1
1.5
W36WI
1
0.5
0.5
n
P /P
y
1
0
0
0.5
1
Lambdac
1.5
0
0
0.5
1
Lambdac
1.5
Figure 3 ABAQUS results for the parametric study reported as a function
of long column slenderness
W14FI
W14FR
1
0.5
0.5
n
P /P
y
1
0
1
2
3
W36FR
0
AISC
1
AISI
DSM
ABAQUS
1
0.5
0.5
3
W36WI
n
P /P
y
1
2
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 4 Results of column parametric study for 4 study groups (stub)
1
0.5
0.5
n
P /P
y
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
P /P
y
AISC
1
0.5 AISI
0
DSM
123
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 5 Results of column parametric study for 3 design methods (stub)
W14FI
W14FR
1
0.5
0.5
n
P /P
y
1
0
1
2
3
W36FR
0
AISC
AISI
DSM
ABAQUS
1
0.5
0.5
2
3
W36WI
n
P /P
y
1
1
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 6 Results of column parametric study for 4 study groups (λc=0.9)
1
0.5
0.5
n
P /P
y
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
P /P
y
AISC
1
0.5
AISI
0
123
DSM
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 7 Results of column parametric study for 3 design methods (λc=0.9)
W14FI
W14FR
1
0.5
0.5
n
P /P
y
1
0
1
2
3
W36FR
AISC0
AISI
DSM
ABAQUS
1
0.5
0.5
2
3
W36WI
n
P /P
y
1
1
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 8 Results of column parametric study for 4 study groups (λc=1.5)
1
0.5
0.5
n
P /P
y
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
P /P
y
1
0.5 AISC
0 AISI
123
DSM
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 9 Results of column parametric study for 3 design methods (λc=1.5)
Figure 10 Deformed shapes for a W14FI section (a) λc=0.9, (b) λc=1.5
Complete comparisons of the studied columns with the AISC, AISI,
and DSM methods are provided in Figures 4 through 9. Figures 4 and 5
provide the summary of results for the stub column study of Seif and
Schafer 2009a. In a similar manner, Figure 6 and Figure 8 present the
results for each of the 4 parameter studies at λc=0.9 and λc=1.5
respectively. Figure 7 and Figure 9 present all 4 studies directly
compared against each of the design methods, for λc=0.9 and λc=1.5
respectively. All results are plotted as a function of elastic local
slenderness of the cross-section: √fy/fcrl, determined by finite strip
analysis. Finally, Figure 10 provides the deformed shapes for a W14
section at λc=0.9 and λc=1.5. The figure shows the interaction between
the local and global (about the minor axis) buckling modes.
Beams: For the beams the predicted capacities from the nonlinear
collapse analysis in ABAQUS are shown for each of the 4 parameter
groups in Figures 11, 13, and 15; for the short specimens, intermediate
length specimens at λe=0.6, and long specimens at λe=1.34 respectively.
Results are also compared against the design methods directly in
Figures 12, 14, and 16 for the same three lengths (short, intermediate,
long). In all the preceding plots the local slenderness √fy/fcr (or
equivalently √My/Mcr) is plotted against the capacity, normalized to the
plastic moment, Mp. Finally, Figure 17 provides the deformed shapes
for a W36 section with a slender web at λe=0.6 and λe=1.34
(intermediate and long lengths); indicating the interaction between the
local and lateral-torsional buckling mode at failure.
W14FI
W14FR
1
0.5
0.5
n
M /M
p
1
0
1
2
3
W36FR
0
AISC
1
AISI
DSM
ABAQUS
1
0.5
0.5
3
W36WI
n
M /M
p
1
2
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 11 Results of beam parametric study for 4 study groups (short)
1
0.5
0.5
n
M /M
p
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
M /M
p
AISC
1
AISI
0.5
0
DSM
123
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 12 Results of beam parametric study for 3 design methods (short)
W14FI
W14FR
1
0.5
0.5
n
M /M
p
1
0
1
2
3
W36FR
0
AISC
AISI
DSM
ABAQUS
1
0.5
0.5
2
3
W36WI
n
M /M
p
1
1
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 13 Results of beam parametric study for 4 study groups (λe=0.6)
1
0.5
0.5
n
M /M
p
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
M /M
p
AISC
1
0.5
0AISI
123
DSM
0.5
ABAQUS
0
1
2
3
(fy /fcrl)0.5
Figure 14 Results of beam parametric study for 3 design methods (λe=0.6)
W14FI
W14FR
1
0.5
0.5
n
M /M
p
1
0
1
2
3
W36FR
0
AISC
AISI
DSM
ABAQUS
1
0.5
0.5
2
3
W36WI
n
M /M
p
1
1
0
1
2
3
0
1
0.5
2
3
(fy /fcrl)0.5
(fy /fcrl)
Figure 15 Results of beam parametric study for 4 study groups (λe=1.34)
1
0.5
0.5
n
M /M
p
1
0
1
2
3
0
1
2
3
(fy /fcrl)0.5
1
n
M /M
p
AISC
1
0.5 AISI
0
123
DSM
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 15 Results of beam parametric study for 3 design methods (λe=1.34)
Figure 17 Deformed shapes for a W36WI section (a) λe=0.6, (b) λe=1.34
DISCUSSION OF DESIGN METHOD PERFORMANCE
Columns: Unlike the case of stub columns, where the AISI’s
implementation of the Effective Width Method provided, by far, the
best prediction of the column capacity, there isn’t a specific design
method that outperforms the others when it comes to predicting the
capacity of longer columns. (Recall all methods use the same global
column curve, but reduce the strength in different manners to account
for local-global interaction.) For longer columns, similar to stub
columns, AISC provides reliable predictions when the flange is nonslender; however AISC is unduly conservative whenever the flanges
become slender (regardless of the web). The level of conservatism is
large enough to make AISC design with slender flanges completely
uneconomical. AISI works well in nearly all cases; however, when the
flange is specifically varied the unified method for reducing the column
capacity does not properly capture the reduction in global capacity
(through loss of I, note Figure 7 and 9 W14FI study). DSM’s accuracy
is excellent when the flange and web vary at fixed ratios, and
conservative (sometimes significantly) when one element is markedly
more slender than its neighbor.
Beams: The AISC predictions are overall best characterized as
conservative, often excessively so when compared with the FE
predictions. The strength prediction as the web and flange move from
compact, to non-compact, to slender often have abrupt transitions as the
related design methods use different formulae in these different local
slenderness ranges. For example, see the W36WI study at e=0.6 of
Figure 13. In general the expressions related to local flange slenderness
provide smooth but quite conservative design predictions, while those
related to local web slenderness suffer from the abrupt transitions. The
study shows that the AISC expressions are essentially intended for
compact, and semi-compact sections; but for locally slender sections
the results are safe, but unduly conservative. An important proviso to
this conclusion, particularly for long beams, is that users must take care
when utilizing the approximations provided in AISC as in some cases
the conservatism is derived from these approximations as opposed to
the fundamentals of the design approach itself. For example, Figure 18
provides the change in AISC’s results for the W14FI (e=1.34)
depending on whether or not the User Note approximation suggested
for the lateral-torsional buckling stress (Eq. F2-4) is utilized – it is clear
the use of this approximation must be done with care.
AISC
AISI
DSM
ABAQUS
(a)
1
0.8
0.6
0.6
M /M
p
0.8
(b)
n
n
M /M
p
1
0.4
0.4
0.2
0.2
0
0.5
1
1.5
(fy/fcrl)0.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
(fy/fcrl)0.5
Figure 18 Beam results of W14FI study group at λe=1.34; (a) AISC without
Eq. F2-4’s approximation, (b) AISC with Eq. F2-4’s approximation
AISI’s Effective Width Method is overall the best performer in
comparison with the FE results. However, the method is unconservative
for long beams with locally slender webs (see the W36WI study at
e=1.34 of Figure 15). Note, as per AISI for any section which is
subject to lateral-torsional buckling (such as those studied here) the
capacity is limited to My as shown. The DSM results for beams are in
excellent agreement at all lengths when the flange and web slenderness
vary at a fixed ratio (the W14FR and W36FR studies). The method has
smooth transitions in all ranges of local slenderness. However, when
one of the elements becomes significantly more slender than its
neighbor DSM assumes the entire cross-section capacity degrades and
this assumption becomes excessively conservative particularly for the
W36WI cases, though less so than AISC. Note, multiple curves are
presented for DSM in Figures 12, 14, and 16 because of the
normalization to Mp (as opposed to My) and further the inelastic
bending provisions allowing strengths up to Mp, as proposed for DSM
and currently under ballot at AISI, are utilized here.
CONCLUSIONS
Three methods for the design of locally slender steel cross-sections are
studied herein: the AISC Specification, the AISI Specification
(effective width method) and DSM (the Direct Strength Method as
adopted in Appendix 1 of the AISI Specification). A parametric study
utilizing nonlinear shell element-based finite element analysis focusing
on the collapse capacity of W14 and W36 sections, where both the
member length and the flange slenderness, and/or web slenderness are
systematically varied (from compact, to noncompact, to slender in the
parlance of AISC) was completed. AISC’s solutions are accurate in the
range where they see their greatest current use, but excessively
conservative for columns with locally slender sections, particularly for
flanges (unstiffened elements) and for beams with slender webs. AISI’s
effective width method is a reliable predictor; only for the beam studies
does AISI provide unconservative solutions when the web is slender
and the beam long. DSM provides a consistently conservative, and
conceptually simple prediction method and is highly accurate when
both the flange and web slenderness vary together, but the elastic webflange interaction assumed in the method is not always realized and the
method is overly conservative when one element is significantly more
slender than another. Work is now underway to address this limitation.
ACKNOWLEDGMENTS
The authors of this paper gratefully acknowledge the support of the
AISC, and the AISC Faculty Fellowship program in this research. Any
views or opinions expressed in this paper are those of the authors.
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