M. Seif and B.W. Schafer

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Nov 15, 2013 (3 years and 10 months ago)

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DESIGN METHODS FOR LOCAL
-
GLOBAL
INTERACTION OF

LOCALLY SLENDER
STEEL MEMBERS



M. Seif
1

and B.W. Schafer
2





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)
A
ISI
-
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 ca
pacity 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, lateral
-
torsional, 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 el
aborate 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 simpli
fy and
improve the design of locally slender structural steel sections.




1

PhD Candidate, Johns Hopkins University, mina.seif@jhu.edu

2

Associate Prof. and Chair, Johns Hopkins University, schafer@jhu.edu

INTRODUCTION

With the advent of high and ul
tra
-
high yield strength steels,
the
increased
yield stress drives even standard
hot
-
rolled steel
shapes from
loc
ally compact to locally slend
er
(noncompact

or slender
)
, making it
inefficient to avoid such
cross
-
sections in
the
design of hot
-
rolled steel
structural members

(
see
Seif and Schafer 2009
a,
b for detail
s
).

Efficient
and reliable strength predictions are needed for locally slender hot
-
r
olled steel cross
-
sections. Analysis of existing AISC (2005) provisions
for locally slender stub columns and short beams (
Seif and Schafer
2009
a) 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, lateral
-
torsional, 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 Specifica
tion, 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 e
lastic 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 s
trength 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 redu
ction of the full cross
-
section (at the long column strength).

AISC and AISI/DSM use different formats for the global (lateral
-
torsional buckling) provisions of beams. However, for no moment
gradient (
C
b

= 1) the resulting expressions are actually quite si
milar
with the exception that AISI only provides capacities up to first yield
(
M
y
) 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 Q
-
factor 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 local
-
global 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 diverg
ence between the capacity predictions of the
different design methods

under axial and bending loads. T
he
FE
analysis is
extended herein to long

members
,
where the locally slender
cross
-
sections may interact with global (flexural, lateral
-
torsional, etc.)
b
uckling modes.

Sections and boundary conditions

Based on the authors’ judgment, AISC
W14 and W36 sections
were

selected
for the study

as
represent
ing

“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 variatio
n


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
W14
x233 section with a
modified
F
lan
ge thickness,

that varies
I
ndependently from
all other
dimensions
,
W14FR
: a

W14
x233 section with variable
F
lange
thickness, but the web thickness set so that the
R
atio of the flange
-
to
-
web thickness remains the same as the original W14x233,
W36FR
: a

W36
x330 section with variable
W
eb

thickness, but the flange thickness
set so that the
R
atio of the flange
-
to
-
web thickness remains the same as
the original W36x330, and
W36WI
: a

W36
x330 secti
on with a
variable
W
eb thickness,
that varies
I
ndependently from all other
dimensions.

Table 1 Parametric study of W
-
sections


b
f
/2
t
f

h
/
t
w

h
/
b
f

t
f
/
t
w

W14x233

4.62

13.35

0.90

1.61

W14FI

varied

fixed

fixed

varied

W14FR

varied

varied

fixed

fixed

W36x330

4
.54

35.15

2.13

1.81

W36FR

varied

varied

fixed

fixed

W36WI

fixed

varied

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 ar
e 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., √
f
y
/
f
cr
). The element
loca
l 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 propert
y variations are also needed.

Geometric variation


lengths

T
he
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 definition
s 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 m
embers, 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 t
he slenderness
parameter,
λ
, is defined in terms of the member’s length and cross
-
section dimensions
.

For columns, two groups of analysis were chosen to be performed at
axial
slenderness parameter,
λ
c
,

values fixed at 0.90 and 1.50, where
λ
c

is defined as:

2
euler
f f
KL
y y
c
f r
cr
E


 


(1)

Note that v
arying 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 gyrati
on,
r
. The
member’s
length,
L
, is
then back
-
calculated to maintain
the specified
λ
c

values
.



Similarly for beams, two groups of analysis were chosen to be
performed at flexural
slenderness parameter,
λ
e
,

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
λ
e

is defined as:

M
y
e
M
cre




(2)

Fixing
λ
e
, the critical buckling moment,
M
cre
, is calculated for each
s
ection.
M
cre

is also defined as follows:

2 2 2
2 4
EI GJ EI C
y y w
M C
cre b
L L
  
 

(3)

Again, note that v
arying the thicknesses (flange, web, or both at
constant ratio) will vary
all
the
parameters on the right hand si
de of Eq.
(3)
.
Accordingly, t
he
member’s
length,
L
, is then back
-
calculated to
maintain
the specified
λ
e

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 ro
tation
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 tha
t 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 fl
ow, and isotropic hardening. The uniaxial

-


diagram is provided in
Fi
gure
1
a, and
is similar to that
employed by
Barth et al. (2005)
. The curve is converted to a
true

-


curve for the
ABAQUS
analysis.


f
u
=
65
f
y
=
50
Engineering Stress (
ksi
)
Engineering Strain
y

st

Slope,
E =29000
Slope,
E
st
=720
Slope,
E
st
=720
Slope,
E

=145
=0.011
f
u
=
65
f
y
=
50
Engineering Stress (
ksi
)
Engineering Strain
y

st

Slope,
E =29000
Slope,
E
st
=720
Slope,
E
st
=720
Slope,
E

=145
=0.011

-
-
-
-
-
+
y
c
f
3
.
0













)
2
(
f
w
f
f
f
f
c
t
t
d
t
t
b
t
b


-
-
-
-
-
-
-
-
-
+
y
c
f
3
.
0













)
2
(
f
w
f
f
f
f
c
t
t
d
t
t
b
t
b


-
-
-
-

(a) uniaxial

-


relation

(b) residual stress distribution

Fi
gure
1

Idealized
material model (a)

-


and⡢⤠re獩sua氠獴ses獥s


Residual stresses

For this work
,

the classic and commonly used distribution of
Galambos
and Ketter (1959)
, as shown in
Fi
gure
1
b,
is emplo
yed.
Similar to other
researchers (e.g., Jung and White 2006) t
he 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 fo
r 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
eigen
mode 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

solut
ion

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
b
f
/150

or
d/150

which is a commonly employed magni
tude (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 mode
s (left: local, right: global)

and
i
nitial geometric imperfections for the analysis (a) ABAQUS
isometric
, (b)
ABAQUS front view, and (c) CUFSM front view, with scaling factor
s
.


RESULTS

As discussed previously (see
Table
1), the parametric study is broken
into 4 gro
ups: 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 “*” woul
d 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 back
-
calculated
Q

or
A
eff
/
A
g

≈ 0.7, denoted with a “o”, and the AISC and
AISI (both effective width and DSM) strength curves for
Q

or
A
eff
/
A
g

=
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
gen
erated 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
).

0
0.5
1
1.5
0
0.5
1
W14FI
P
n
/P
y
0
0.5
1
1.5
0
0.5
1
W14FR
0
0.5
1
1.5
0
0.5
1
W36FI
P
n
/P
y
Lambda
c
0
0.5
1
1.5
0
0.5
1
W36WI
Lambda
c

Figure
3

ABAQ
US results for the parametric study reported as a function
of long column slenderness

1
2
3
0
0.5
1
P
n
/P
y
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
W36FR
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure
4

Results of column parametric study for 4 study groups (stub)

1
2
3
0
0.5
1
P
n
/P
y
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
5

Results of column parametric study fo
r 3 design methods (stub)

1
2
3
0
0.5
1
P
n
/P
y
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
W36FR
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure
6

Results of column parametric study for 4 study groups (
λ
c
=0.9)

1
2
3
0
0.5
1
P
n
/P
y
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
7

Results of column parametric study for 3 design methods (
λ
c
=0.9)

1
2
3
0
0.5
1
P
n
/P
y
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
W36FR
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure
8

Results of column parametric study for 4 study groups (
λ
c
=1.5)

1
2
3
0
0.5
1
P
n
/P
y
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
P
n
/P
y
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
9

Results of c
olumn parametric study for 3 design methods (
λ
c
=
1
.
5
)


Figure
10

Deformed shapes for a W14FI section (a)
λ
c
=
0.9, (b)

λ
c
=
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
λ
c
=0.9

and
λ
c
=
1.5
respectively
. Figure 7 an
d Figure 9 present all 4 studies directly
compared against each of the design methods, for
λ
c
=0.9

and
λ
c
=
1.5
respectively
. All results are plotted as a function of elastic local
slenderness of the cross
-
section: √
f
y
/
f
cr
l

, determined by finite strip
analys
is. Finally, Figure 10 provides the deformed shapes for a W14
section at
λ
c
=0.9

and
λ
c
=
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 nonlin
ear
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
λ
e
=0.6, and long specimens at
λ
e
=1.34 respectively.
Results are also compared against the desi
gn methods directly in
Figures 12, 14, and 16 for the same three lengths (short, intermediate,
long). In all the preceding plots the local slenderness √
f
y
/
f
cr


(or
equivalently √
M
y
/
M
cr

) is plotted against the capacity, normalized to the
plastic moment,
M
p
. Finally, Figure 17 provides the deformed shapes
for a W36 section with a slender web at
λ
e
=0.
6 and
λ
e
=
1.34
(intermediate and long lengths); indicating the interaction between the
local and lateral
-
torsional buckling mode at failure.

1
2
3
0
0.5
1
M
n
/M
p
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
W36FR
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure
11

Results of beam parametric study for 4 study groups (short)

1
2
3
0
0.5
1
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
12

Results of beam parametric study for 3 design methods (short)


1
2
3
0
0.5
1
M
n
/M
p
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
W36FR
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure
13

Results of beam parametric study for 4 st
udy groups (
λ
e
=0.6)


1
2
3
0
0.5
1
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
14

Results of beam parametric study for 3 design methods (
λ
e
=0.6)


1
2
3
0
0.5
1
M
n
/M
p
W14FI
1
2
3
0
0.5
1
W14FR
1
2
3
0
0.5
1
W36FR
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
W36WI


AISC
AISI
DSM
ABAQUS

Figure 15 Results of beam parametric study for 4 study groups (
λ
e
=1.34)


1
2
3
0
0.5
1
M
n
/M
p
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
1
2
3
0
0.5
1
(f
y
/f
c
r
l
)
0
.
5
M
n
/M
p
1
2
3
0
0.5
1


AISC
AISI
DSM
ABAQUS

Figure
15

Results of beam parametric study f
or 3 design methods (
λ
e
=1.34)


Figure 17
Deformed shapes for a W36WI section (a)
λ
e
=
0.6, (b)

λ
e
=
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 i
n different manners to account
for local
-
global interaction.) For longer columns, similar to stub
columns, AISC provides reliable predictions when the flange is non
-
slender; however AISC is unduly conservative whenever the flanges
become slender (regardles
s 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
capac
ity 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 ma
rkedly
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 slenderne
ss
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 t
he lateral
-
torsional buckling stress (Eq. F2
-
4) is utilized


it is clear
the use of this approximation must be done with care.

0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
(fy/fcrl)
0
.
5
M
n
/M
p


AISC
AISI
DSM
ABAQUS
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
(fy/fcrl)
0
.
5
M
n
/M
p


(a)
(b)
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
(fy/fcrl)
0
.
5
M
n
/M
p


AISC
AISI
DSM
ABAQUS
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
(fy/fcrl)
0
.
5
M
n
/M
p


(a)
(b)

Figure 18 Beam results of W14FI study group at
λ
e
=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 slen
der 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 M
y

as shown. The DSM results for beams are in
excellent agreeme
nt 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
n
eighbor 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
M
p

(as opposed to
M
y
) and further the inelastic
bending provisions allowing strengths up to
M
p
, as proposed for DSM
and currently under ballot at AISI, are utilized here.

CONCLUSIONS

Three methods for the design of locally slender ste
el 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
-
base
d 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. A
ISI’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 web
-
flange 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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