25 June 2007 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 1 - AISC Faculty Fellowship Cross-section Stability of Structural Steel
Tom, Please find enclosed the first progress report for the AISC Faculty Fellowship. The report summarizes research efforts to study the cross-section stability of structural steel, and to extend the Direct Strength Method to hot-rolled steel sections. The focus of the work in this initial period has been on graduate student training, and performing preliminary parametric studies. The parametric studies reported herein focus on local buckling of W-sections including web-flange interaction, and comparisons of the AISC, AISI – Effective Width, and AISI – Direct Strength design methods for columns with slender cross-sections. In addition, so that students and practitioners can become more familiar with tools for predicting cross-section stability, a series of educational tutorials have been created that explore the finite strip method. Sincerely,
Mina Seif ([email protected]) Graduate Research Assistant
Ben Schafer ([email protected]) Associate Professor
2
Summary of Progress
The primary goal of this AISC funded research is to study and assess the cross-
section stability of structural steel. A timeline and brief synopsis follows.
Research begins March 2006
(Note, Mina Seif joined project in October 2006)
Progress Report #1 June 2007
Completed work:
• Performed axial and major axis bending elastic cross-section stability analysis on the W- sections in the AISC (v3) shapes database using the finite strip elastic buckling analysis software CUFSM.
• Evaluated and found simple design formulas for plate buckling coefficients of W-sections in local buckling that include web-flange interaction.
• Reformulated the AISC, AISI, and DSM column design equations into a single notation so that the methods can be readily compared to one another, and so that the centrality of elastic buckling predictions for all the methods could be readily observed.
• Performed a parametric study on AISC, AISI, and DSM column design equations for W-sections to compare and contrast the design methods.
• Created educational tutorials to explore elastic cross-section stability of structural steel with the finite strip method, tutorials include clear learning objectives, step-by-step instructions, and complementary homework problems for students.
3
Table of Contents
1 Introduction..............................................................................................................................5
1.1 Cross-section stability and the finite strip method.......................................................... 5 1.2 Impact of high yield strength steel on cross-section stability ......................................... 6 1.3 Direct Strength Method................................................................................................... 8 1.4 Challenges....................................................................................................................... 9
2 Elastic buckling finite strip analysis of the AISC sections database ....................................11
2.1 Objectives and methodology......................................................................................... 11 2.2 Results for W-sections .................................................................................................. 11 2.3 Comparison of k values ................................................................................................ 16 2.4 Development of approximate design expressions for k of W-sections......................... 17 2.5 Overall summary of web-flange interaction ................................................................. 21 2.6 Ongoing / future work................................................................................................... 25
3 Comparing the AISC, AISI, and DSM design methods ........................................................26
3.1 AISC ............................................................................................................................. 28 3.2 AISI (AISI – Effective Width Method) ........................................................................ 31 3.3 DSM (AISI – Direct Strength Method) ........................................................................ 31 3.4 Direct comparison of design expressions ..................................................................... 31 3.5 Stub column comparison............................................................................................... 34 3.6 Long column comparisons............................................................................................ 38 3.7 Ongoing / future work................................................................................................... 42
4 Educational materials.............................................................................................................44
4.1 Objective ....................................................................................................................... 44 4.2 Work Products .............................................................................................................. 44
4.2.1 Tutorial 1: Cross-section stability of a W36x150 using the finite strip method... 45 4.2.2 Tutorial 2: Cross-section stability of a W36x150 exploring higher modes and the interaction of modes.............................................................................................................. 46 4.2.3 Tutorial 3: Exploring how cross-section changes influence cross-section stability – an extension to Tutorial #1 ................................................................................................ 46 4.2.4 Exercises: Homework exercises related to Tutorials 1 and 3 on cross-section stability (doc) ........................................................................................................................ 47
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5 Conclusions............................................................................................................................50
6 References..............................................................................................................................52
A Appendix: Further finite strip analysis of structural steel cross-sections ..............................53
B Appendix: Explicit parametric study of W-section columns comparing AISC, AISI, and
DSM design methods.....................................................................................................................56
Sample long column analysis.................................................................................................... 56 W14 stub columns by explicit parametric study....................................................................... 57
W14 section with varied flange thickness ............................................................................ 58 W14 section with varied web thickness................................................................................ 61
W36 Stub column results .......................................................................................................... 65 W36 section with varied flange thickness ............................................................................ 65 W36 section with varied web thickness................................................................................ 69
C Appendix: Educational materials (PowerPoint slides) ..........................................................74
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1 Introduction Cross-section stability refers to those aspects of member stability that may be
evaluated in isolation of the entire structure. For an individual structural steel member
cross-section stability typically refers to those instabilities that are driven by plate
bending within the cross-section; commonly known as flange local buckling or web
local buckling, or more generically, just local buckling1. In structural steel design the
primary method for dealing with local buckling has been (a) to employ plate buckling
solutions to predict when such buckling modes occur and (b) to try to avoid their
occurrence in common sections through the application of slenderness limits.
1.1 Cross-section stability and the finite strip method
The application of isolated plate buckling solutions in design ignores significant
advances that have been made in cross-section stability analysis. For example, the
analysis of Figure 1.1 was performed using an open source finite strip analysis program
developed by the senior author, and provides the local buckling modes of a common
AISC W-section where web-flange interaction has been properly included. A cross-
section stability analysis can provide a more accurate prediction of elastic (and even
inelastic) cross-section stability and provides a more direct way to understand the
stability behavior of the cross-section as a whole instead of attempting to make
idealizations about the flange or web in isolation of the member.
1 More recently in the scientific/academic community attention has been paid to instabilities that appear to combine plate bending instability with global member instability; the so-called distortional buckling modes. Whether these modes are truly distinct cross-section stability modes, or a combination of modes is a matter of some debate at this time. The author would argue that in W-sections distortional buckling as it is most commonly referred to in the literature is just a combination of local and global buckling, see e.g. Schafer and Adany (2005).
6
Figure 1.1 Finite strip analysis of a W14x109 showing local buckling modes relevant to this cross-section
1.2 Impact of high yield strength steel on cross-section stability
Good reasons exist to try to avoid local buckling in common sections, maximum
strength and ductility can typically be achieved when local buckling modes do not
occur. However, completely avoiding local buckling ignores the beneficial post-
buckling reserve that can exist in this mode, and may limit the yield strength that a
particular cross-section is used for. As yield stress increases the potential for cross-
section stability to control the strength increases in kind. Metallurgists have not been
able to appreciably change the modulus of steel, but the yield stress has certainly seen
significant changes over time from mild steel, 36 ksi, to high-strength steel of 50, 65 and
even 70 ksi (e.g., A913) to high performance steel of 70 and 100 ksi, and today the slow
but steady emergence of ultra high-strength steel with yield greater than 100 ksi.
An illustration of the impact of higher yield stress steels on cross-section stability
is provided in Figure 1.2. Consider the flange slenderness limits of the AISC Specification
7
(AISC 2005) as shown in Figure 1.2a. As the yield stress increases the flange slenderness
limits decrease. The histogram on the right of Figure 1.2a provides the flange
slenderness of all W-shapes currently listed in the AISC Manual. The strictest limit is
the flange slenderness limit for fully compact beams (λp bending). How many W-shapes
become noncompact as the yield stress increases?
0 20 40 60 80 100 1200
2
4
6
8
10
12
14
16
18
20
yield stress (ksi)
flang
e sl
ende
rnes
s (b
f/2t f)
flange slenderness limits
λr bendingλr compressionλp bending
0 500
2
4
6
8
10
12
14
16
18
20
W-shapes
flange slenderness
histogram offlange slender-ness for AISCmanual W-shapes
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
100
yield stress (ksi)
web
sle
nder
ness
(h/t w
)
web slenderness limits
λr bendingλp bendingλr compression
0 500
10
20
30
40
50
60
70
80
90
100
W-shapes
web slenderness
histogram of web slendernessfor AISC manualW-shapes
(a) flange slenderness (b) web slenderness Figure 1.2 Impact of yield stress on slenderness limits compared with current slenderness of W-shapes
Based on flange slenderness (Figure 1.2a) At 36 ksi, only 1 of the 267 standard W-
sections is noncompact, at 50 ksi 11 W-sections, at 65 ksi 27 W-sections, at 70 ksi 39, at
100 ksi 94, at 120 ksi 119 W-sections. As steel yield stress approaches ultra high strength
steels nearly ½ of the standard W-sections become noncompact. Web slenderness is
portrayed in Figure 1.2b, as the figure illustrates, for columns the number of W-sections
that have slender webs increases dramatically. While not all W-sections would be used
as columns, many of the W12 and W14’s which are compact at 36 and 50 ksi reach the
slender regime as yield stresses push up to 100 ksi.
8
Full cross-section stability analysis (e.g., using the finite strip method) can provide
a somewhat more nuanced picture than Figure 1.2. Ignoring web-flange interaction in
the development of the λ limits employed in Figure 1.2 may be misleading. Consider, for web
slenderness (Figure 1.2b) for 36 ksi and even 50 ksi yield stress steel the flange is generally
compact and stable, thus it is possible to approximately consider the web in isolation of the
flange. However, for higher strength steels web-flange interaction from the noncompact
flanges becomes a greater issue, and past assumptions may need to be reinvestigated.
Further, issues which are readily apparent in the cross-section analysis such as multiple
local buckling modes (Figure 1.1) are not reflected in current design.
1.3 Direct Strength Method
Given that cross-section stability can be predicted with increasing ease, and given
that the future use of higher yield strength steel implies an increased reliance on
noncompact sections in structural steel, then design methods for structural steel that
can easily and readily handle locally unstable cross-sections are needed. One candidate
for such a method is the Direct Strength Method (DSM) recently developed and
adopted for cold-formed steel sections (AISI-S100 2007, Appendix 1).
DSM does not require the calculation of effective properties, nor consideration of
slenderness parameters (bf/2tf, Q, etc.). Instead, DSM relies on an accurate cross-section
stability analysis as the primary input to prediction of member capacity. Simple
strength curves are used for each cross-section stability limit state, in the case of cold-
formed steel sections this includes: local, distortional, and global buckling limit states.
9
DSM’s advantages include: simplifying the design procedure for slender cross-sections,
properly accounting for interaction between elements (e.g. web-flange) in local
buckling, and treating distortional buckling explicitly in design. In addition, as the AISC
Specification moves towards advanced analysis methods in general, DSM potentially
couples well with these methods by providing a tool for cross-section analysis that can
readily incorporate local buckling effects2. Significant work remains to extend DSM to
hot-rolled steel structural shapes.
1.4 Challenges
Although DSM for cold-formed steel provides a solid basis for further study, a
number of challenges exist before DSM style calculations can be verified for structural
steel shapes. Compared with cold-formed steel sections, hot-rolled sections have large
thickness variations in the cross-section which lead them to have unique cross-section
stability modes (Figure 1.1). It is also known that inelastic buckling is more important in
structural steel shapes than in cold-formed steel, thus the influence of residual stresses
and strain hardening must be explicitly considered – perhaps in quite different ways
than for cold-formed steel. Further, investigation of high and ultra-high strength steel
suffers from a lack of test data in many common situations. In addition, if more
structural steel shapes use high strength steel (and therefore become noncompact) the
impact on ductility, particularly in cases where it is inherently assumed to exist but not
explicitly checked, needs to be carefully considered.
2 To date a significant complication in most beam element formulations being suggested for advanced frame analysis (i.e., fiber element methods) is their inability to account for local buckling.
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However, with current desktop computing power, availability of open source
analysis packages (i.e., CUFSM: Schafer and Adany 2006) new analysis methods that
allow the user to isolate individual cross-section buckling modes (i.e., cFSM: Adany and
Schafer 2007), and recent progress in the design methods for locally unstable sections
(i.e., DSM: AISI-S100 2007, Appendix 1), it seems that now is a good time to take a fresh
look at cross-section stability of structural steel shapes.
11
2 Elastic buckling finite strip analysis of the AISC sections database
2.1 Objectives and methodology
Finite strip analysis was performed on the W-sections in the shape database (v3)
from the AISC (2005) Manual of Steel Construction. The analysis was completed using
CUFSM version 3.12 (Schafer and Adany 2006). Sections were simplified to their centerline
geometry (the increased width in the k-zone was thus ignored). The analysis was used
to investigate the elastic local buckling behavior of the section (thus including web-
flange interaction) so that the exact elastic local buckling values of the plate buckling
coefficients, ck ’s, could be compared to those used within the AISC Specification.
Further, based on the exact values for elastic local buckling, approximate design
expressions that include web-flange interaction for kc are developed for the W-sections.
Finally, the exact ck values are also used to help assess and compare the available AISC,
AISI, and DSM design methods.
2.2 Results for W-sections
The finite strip analysis results are converted into local plate buckling coefficients
for comparison to existing design provisions and for the development of new
approximate design expressions. For example, for a W-section in pure compression the
finite strip analysis will provide the elastic local buckling stress, fcrl. The plate buckling
solution for the elastic buckling of the flange outstand (width = bf/2) is:
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( )2
2
2 2112 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
ν−π
=f
ffcrb b
tEkf 2.1
where:
fk : Flange local plate buckling coefficient.
bf : Full Flange width.
ft : Flange thickness.
E: Modulus of elasticity
ν: Poisson’s ratio
Setting fcrb = fcrl and solving for kf:
( ) 2
2
2
2112
⎟⎟⎠
⎞⎜⎜⎝
⎛
πν−
=f
fcrf t
bE
fk l 2.2
Thus, using Eq. 2.2 the flange plate buckling coefficient, kf, including web-flange
interaction can be calculated from each finite strip analysis of a W-section. For the AISC
W-sections in pure compression, the resulting kf’s are provided Figure 2.1(a) and (b).
Figure 2.1(a) highlights that the flange plate buckling coefficient is not independent of
the web slenderness h/tw, i.e., web-flange interaction is real and unavoidable. Figure
2.1(b) shows that if both web and flange slenderness are considered that relatively
simple functional relationships may exist for predicting when local buckling occurs.
Similar analysis for the web may be completed, whereby:
( )2
2
2
112⎟⎠⎞
⎜⎝⎛
ν−π
=htEkf w
wcrh 2.3
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and after setting fcrh = fcrl, and solving for kw:
( ) 2
2
2112⎟⎟⎠
⎞⎜⎜⎝
⎛π
ν−=
wcrw t
hE
fk l 2.4
where:
wk : Web local plate buckling coefficient.
h : Clear distance between flanges less the fillet (see AISC 2005).
wt : Web thickness.
The web plate buckling coefficients are provided for the W-sections in pure
compression Figure 2.1(c) and (d). The web plate buckling coefficient is dependent on
the flange slenderness, but again a simple combination of slenderness may adequately
describe the plate buckling coefficient, as shown in Figure 2.1(d).
Since local buckling is calculated for the cross-section, not the plates, the flange
and web plate buckling coefficients are related. Recognizing that fcrb=fcrl and fcrh=fcrl then
Eq. 2.1 must be equal to Eq. 2.3, resulting in the desired kf-kw relations:
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2 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
f
fwwf t
bhtkk or
222
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
wf
ffw t
hbt
kk 2.5
The plate buckling coefficients for the W-sections in major-axis bending are
provided in Figure 2.2 in a similar format to Figure 2.1. The basic conclusions for
bending are similar to compression – web-flange interaction strongly influences kf and
kw results, but simple functional relations for predicting the k’s appear possible.
14
a)
(b)
(c)
(d)
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
web slenderness h/tw
flang
e, k
f
W-sections
2 3 4 5 6 7 8 9 10 11 120
2
4
6
flange slenderness bf/(2tf)
web
, kw
1 2 3 4 5 6 7 8 9 10 110
0.2
0.4
0.6
0.8
(h/tw )(2tf/bf)
k f
1 2 3 4 5 6 7 8 9 10 110
2
4
6
(h/tw )(2tf/bf)
k w
Figure 2.1 Flange and web local buckling coefficients for the W-sections under axial loading.
15
a)
(b)
(c)
(d)
0 10 20 30 40 50 600.2
0.4
0.6
0.8
1
web slenderness h/tw
flang
e, k
f
W-sections
0 2 4 6 8 10 120
10
20
30
40
flange slenderness bf/(2tf)
web
, kw
1 2 3 4 5 6 7 8 9 10 110.2
0.4
0.6
0.8
1
(h/tw )(2tf/bf)
k f
1 2 3 4 5 6 7 8 9 10 110
10
20
30
40
(h/tw )(2tf/bf)
k w
Figure 2.2 Flange and web local buckling coefficients for the W-sections under bending loading.
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2.3 Comparison of k values
The typically cited theoretical limits3 for the local plate bucking coefficient, kf, of an
isolated flange (an unstiffened element) vary from the 0.43 (simply supported on one
longitudinal edge free on the other longitudinal edge) to 1.3 (fixed on one longitudinal
edge free on the other longitudinal edge). The AISC Specification assumes a kf value of
0.7 for determining slenderness limits and effective width calculations. The finite strip
results of the AISC W-sections show that in compression kf varies from 0.04 to 0.62 and
in major-axis bending (even though the flange is still in compression) kf varies from 0.25
to 0.74. Though many sections have results in the neighborhood of 0.7, the large
variation from this constant value is clear in Figure 2.1 and Figure 2.2.
The theoretical limits for the local plate buckling coefficient, kw, of an isolated web
(a stiffened element) vary from 4 to 7 (simply supported to fixed edges) in pure
compression and from 24 to 42 in pure bending. The AISC Specification assumes a kw of 5
in pure compression and 32 in pure bending (based on the λp limit). The exact elastic
local buckling wk values vary from 1.9 to 5.7 in pure compression and 2.2 to 30.5 in
major-axis bending. As with the flange values, it is clear that web-flange interaction
plays a significant role for the webs.
For both the web and flange results, not only is their a large difference between
the assumed AISC Specification k values and those calculated from the finite strip
analysis, but also the finite strip values are outside the bounds of the theoretical isolated
3 the theoretical limits provided here are the limits for an isolated plate which has simple supports at the loaded edges and varying support along the longitudinal edges, see Galambos (1998) or Salmon and Johnson (1996) .
17
plate solutions. For example, for the flange in pure compression (Figure 2.1) a number
of the kf values are below the 0.43 value for an isolated unstiffened element simply
supported on one side and free on the other side. In these cases the web local buckling
is detrimental and actually driving the flange local buckling to values much lower than
isolated plate buckling solutions. In essence the situation for the flange at the web-
flange juncture is worse than simply supported as the flange must provide rotational
stiffness to the web for the section to remain stable. Traditionally, it has been assumed
that plate buckling coefficients between simply supported and fixed values provide
reasonable bounds (e.g., see Salmon and Johnson 1996), but if local buckling of the
entire cross-section is considered, then a much wider range of k values are possible.
2.4 Development of approximate design expressions for k of W-sections
While it may be preferable to always perform a unique cross-section stability
analysis, it may not be necessary in all cases. Figure 2.1 and Figure 2.2 show that kf and
kw for W-sections are, within good approximation, a function of the flange and web
slenderness. (This is not a unique observation and has been examined with respect to
ductility limits and in other situations in the past).
A series of simple empirical expressions were developed to provide an
approximate means of predicting the flange and web local plate buckling coefficients.
For the flange of a W-section in pure compression, see Figure 2.3, and:
( )( )( ) 31261 .ffwf b/tt/h/.k̂ = 2.6
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for the web of a W-section in pure compression, see Figure 2.4:
( )( )( ) 1802511 52 .b/tt/h/.k̂/ .ffww += 2.7
for the flange of a W-section in major-axis bending, see Figure 2.5
( )( )( ) 451201901 52 .b/tt/h.k̂/ .ffwf += 2.8
for the web of a W-section in major-axis bending, see Figure 2.6.
( )( )( ) 01502511 2 .b/tt/h/.k̂/ ffww += 2.9
Where the “^” in the above expressions denotes that these are estimates of the
actual kf or kw values. In addition, per Eq. 2.5, separate expressions for kf and kw are not
strictly necessary. If fk̂ is adequate then it may be used to directly provide an estimate
of kw via Eq. 2.5. The expressions above, as well as the potential simplifications using
Eq. 2.5 will be further analyzed and examined for their accuracy as part of the future
work of this project. For now, Eq. 2.6 – 2.9 and the accompanying figures demonstrate
the efficacy of this potential empirical approach for generating more accurate k values.
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1 2 3 4 5 6 7 8 9 10 110
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(h/tw )(2tf/bf)
k f
kf from CUFSM
kf=1.6/[(h/tw )(2tf/bf)]1.3
Figure 2.3 Flange local buckling coefficient, fk , obtained from both the CUFSM finite strip analysis and the
proposed equation, versus the web to flange ratio of slenderness, ( )( )ffw btth /2/ , for the W-sections under axial loading.
1 2 3 4 5 6 7 8 9 10 11
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
(h/tw )(2tf/bf)
1/k w
1/kw from CUFSM
1/kw =1.5/[(h/tw )(2tf/bf)]2.5+0.18
Figure 2.4 Inverse of the web local buckling coefficient, wk , obtained from both the CUFSM finite strip
analysis and the proposed equation, versus the web to flange ratio of slenderness, ( )( )ffw btth /2/ , for the W-sections under axial loading.
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.71
1.5
2
2.5
3
3.5
4
4.5
1/(h/tw )(2tf/bf)
1/k f
1/kf from CUFSM
1/kf=0.019[(h/tw )(2tf/bf)]2.5+1.45
Figure 2.5 Inverse of the Flange local buckling coefficient, fk , obtained from both the CUFSM finite strip analysis and the proposed equation, versus the inverse of the web to flange ratio of
slenderness, ( )( )ffw btth /2/ , for the W-sections under bending loading.
1 2 3 4 5 6 7 8 9 10 110
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
(h/tw )(2tf/bf)
1/k w
1/kw from CUFSM
1/kw =1.5/[(h/tw )(2tf/bf)]2+0.015
Figure 2.6 Inverse of the web local buckling coefficient, wk , obtained from both the CUFSM finite strip
analysis and the proposed equation, versus the web to flange ratio of slenderness, ( )( )ffw btth /2/ , for the W-sections under bending loading.
21
2.5 Overall summary of web-flange interaction
Web-flange interaction for a W-section is a function of four geometric variables h,
tw, bf, and tf as well as loading (compression, bending, etc) and material parameters.
With respect to the geometric variables, two non-dimensional pairs are in common use:
h/tw and bf/2tf; however given 4 free geometric variables a third non-dimensional pair
must also influence the solution, with h/bf or tf/tw being the obvious candidates. In this
section the elastic local buckling of W-sections is again examined, but with particular
attention paid to (a) how these non-dimensional parameters may predict the elastic
local buckling and (b) where typical series of W-sections fall with respect to these non-
dimensional parameters. The greatest attention is paid to the W14 and W36 series of
sections since they are used in Section 3 (and Appendix B) of this report to examine a
variety of different strength prediction methodologies.
A series of contours for the flange plate buckling coefficient (kf), including web-
flange interaction, are produced for the W-sections in pure compression in Figure 2.7
and Figure 2.8. The flange benefits from web-flange interaction the greatest when the
flange itself is slender (bf/2tf is high) the web is stocky (h/tw is low) the section is square,
not tall and narrow (h/b near 1) and the flange thickness is near the web thickness (tf/tw
approaching 1). Those trends remain the same for a W-section in major-axis bending
(Figure 2.9 and Figure 2.10), but in general web-flange interaction is not as pronounced
in bending as the tensile stress on the web stabilize the generally slender web and
increase its local buckling stress to that more similar to the flange, thus decreasing
interaction in local buckling substantially for most common sections.
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h/tw
b f/2t f
10 20 30 40 50
3
4
5
6
7
8
9
10
11 Kf contour
allW14W36
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2.7 Contour plot of the flange local buckling coefficient, fk , with reference to the normalized web
slenderness, ( )wth / , and the normalized flange slenderness, ( )ff tb 2/ , for the W-sections under axial loading.
h/b
t f/t w
1 1.5 2 2.5 3
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9Kf contour
allW14W36
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2.8 . Contour plot of the flange local buckling coefficient, fk , with reference to the normalized section
lengths, ( )bh / , and the normalized section thicknesses, ( )wf tt / , for the W-sections under axial loading.
23
h/tw
b f/2t f
10 20 30 40 502
3
4
5
6
7
8
9
10
11 Kf contour
allW14W36
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2.9 Contour plot of the flange local buckling coefficient, fk , with reference to the normalized web
slenderness, ( )wth / , and the normalized flange slenderness, ( )ff tb 2/ , for the W-sections under bending.
h/b
t f/t w
1 1.5 2 2.5 3
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9Kf contour
allW14W36
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2.10 Contour plot of the flange local buckling coefficient, fk , with reference to the normalized
section lengths, ( )bh / , and the normalized section thicknesses, ( )wf tt / , for the W-sections under bending.
24
Turning now to the W14 and W36 sections, which are studied in further detail in
Section 3. Web-flange interaction exists for the W14 and W36 sections, but largely is
constant within a series. Most of the W14 sections are within a single kf contour and
most of the W36 sections are within one of two kf contours (exceptions certainly exist,
but the basic trends are clear). The plots also highlight the actual geometry of the W14
and W36 sections, which is useful for later parameter studies.
The W14, most typically used as a column, has a specific geometry. For a W14,
h/bf ~ 0.6, tf/tw ~ 1.6, h ~ 14 in. deep, as a result bf is known through the h/bf to be
approximately 23 in. and tf remains the primary variable to vary (with bf/2tf known to
range from approximately 2 to 10) therefore tf ranges from 1.2 to 5.8 in.. For W36, which
is most typically used as a girder, two ranges exist in the first, h/bf ~ 1.9, tf/tw ~ 1.8, h ~
36 in., bf is therefore 19 in., and bf/2tf varies from 2 to 7, therefore tf ranges from 1.4 to
4.8 in., in the second, h/bf ~ 2.7, tf/tw ~ 1.6, h ~ 36 in., bf is therefore 13 in., and bf/2tf
varies from 2 to 7, therefore tf ranges from 0.9 to 3.2 in. These ranges provide practical
geometric limits to the parameter studies conducted in Section 3 and Appendix B.
25
2.6 Ongoing / future work
• Provide a comprehensive literature review of cross-section stability solution
methods including elastic and inelastic stability solutions.
• Extend the finite strip analyses for local buckling and the prediction of the
local plate buckling coefficients for W-sections to minor axis bending and
extend the analysis to cover the rest of the shapes in the AISC (v3) shapes
database (at least WT, C, L, and HSS). See Appendix A for the beginning of
this work.
• Propose simplified formulas for estimating the local buckling coefficients
for all types of sections similar to those proposed for the W-sections.
• Verify the accuracy of proposed formulas against the exact elastic local
buckling values from finite strip analyses.
26
3 Comparing the AISC, AISI, and DSM design methods A number of different methods exist for the design of steel columns with slender
cross-sections, three of which are detailed here: AISC, AISI, and DSM. The AISC
method, as embodied in the 2005 AISC Specification, uses the Q-factor approach to adjust
the global slenderness in the inelastic regime of the column curve to account for local-
global interaction, and further uses a mixture of effective width (for stiffened elements)
and average stress (for unstiffened elements) to determine the final reduced strength.
The AISI method, from the main body of the 2007 AISI Specification for cold-formed
steel, uses the effective width approach. In the AISI method the global column curve is
unmodified but the column area is reduced to account for local buckling in both
stiffened and unstiffened elements via the same effective width equation. Finally, the
DSM or Direct Strength Method, as given in Appendix 1 of the 2007 AISI Specification
for cold-formed steel, uses a new approach where the global column strength is
determined and then reduced to account for local buckling based on the local buckling
cross-section slenderness.
To provide more definitive comparisons between these three methods the
formulas are detailed in the subsequent sections for a centerline model of a W-section in
compression. The formulas are presented in a common set of notation so that they may
be more directly compared. Intermediate steps are shown only for the AISC formulas.
In addition, the format of presentation is modified from that used directly in the
respective Specifications so that (1) the methods may be most readily compared to one
another and (2) the key input parameters are brought to light.
27
Basic definitions:
nP : Nominal section compressive strength.
gA : Gross area of the section.
b : Half of the flange width (bf = 2b).
ft : Flange thickness.
h : Height of section, between the two flange centerlines.
wt : Web thickness.
ef : Elastic global critical buckling stress, e.g., ( )2
2
rKLEπ .
L : Laterally unbraced length of the member.
r : Governing radius of gyration.
K : Effective length factor.
yf : Yield stress.
crbf : Flange elastic critical local buckling stress = ( )2
2
2
112 ⎟⎟⎠
⎞⎜⎜⎝
⎛ν−
πbtEk f
f .
crhf : Web elastic critical buckling stress = ( )2
2
2
112⎟⎠⎞
⎜⎝⎛
ν−π
htEk w
w .
fk : Flange local buckling coefficient.
wk : Web local buckling coefficient.
E : Young’s modulus of elasticity.
v : Poisson’s ratio.
lcrf : Section local buckling stress, e.g., determined by finite strip analysis
28
3.1 AISC
The AISC design procedure for a column with slender elements is summarized in
Section E7 of the 2005 AISC Specification. Focusing on a centerline model of a W-section,
the relevant sub-sections and equations are Section E7, Eq.’s E7-1 through E7-3, Section
E7.1, Eq.’s E7.4 – E7.6, and Section E7.2, Eq.’s E7-16 and E7-17. Specifically, the
compressive strength is found via:
⎪⎩
⎪⎨⎧
=e
y)f/f(Q
gn f.f).(QAP
ye
87706580 for:
ye
ye
Qf.fQf.f
440440
<≥
3.1
where:
asQQQ = 3.2
and sQ is a flange reduction factor for unstiffened elements that depends on the
flange slenderness as follows:
yf fEtb /56.0/ ≤ 0.1=sQ 3.3
yfy fEtbfE /03.1//56.0 << Ef
tbQ y
fs ⎟
⎟⎠
⎞⎜⎜⎝
⎛−= 74.0415.1 3.4
yf fEtb /03.1/ ≥ 2
69.0
⎟⎟⎠
⎞⎜⎜⎝
⎛=
fy
s
tbf
EQ 3.5
aQ is a web reduction factor, defined as the ratio between the effective area of the cross
section, using an effective height, eh , as shown below, to the total cross sectional area:
29
gwegeffa /AthA/AQ == , 3.6
where he is defined as
fEth w /49.1/ ≤ hhe = 3.7
fEth w /49.1/ ≤ hfE
thfEth
wwe ≤⎥
⎦
⎤⎢⎣
⎡−=
)/(34.0192.1 3.8
and where
effn /APf = 3.9
In this form determination of f, and thus he and Qa requires iteration. The AISC
Specification notes that f may be conservatively set to fy. More practically, a reasonable
estimate of the f from the iteration may be had without iteration – simply by using the
stress from the global buckling column curve with Q = 1, i.e.,
estimated ⎪⎩
⎪⎨⎧
=e
yff
fff
ye
877.0)658.0( )/(
for: ye
ye
ffff
44.044.0
<≥
3.10
This approximation to f is conservative since the f from Eq. 3.10 will always be
greater than the f resulting from Eq. 3.1 (because Q is strictly less than 1), but Eq. 3.10’s
approximation for f is also always less than or equal to fy.
The AISC expressions may be rewritten slightly to better contrast them with their
AISI counterparts and highlight the role of cross-section stability:
ngn f̂AP = 3.11
30
⎪⎩
⎪⎨⎧
<≥
= 440 if 8770
440 if 6580
yasee
yasey)f/f(QQ
asn fQQ.ff.
fQQ.ff).(QQf̂
yeas
3.12
The Q factors may be written directly in terms of the flange and web critical
buckling stresses as shown in Eq.’s 3.13 through 3.17. sQ , the flange reduction factor
depends on crbf as follows:
:ff ycrb 2≥ 0.1=sQ 3.13
:fff ycrby 253
<< crb
ys f
f..Q 5904151 −= 3.14
:ff ycrb 53
≤ y
crbs f
f.Q 11= 3.15
while aQ , the web reduction factor depends on crhf as follows:
:ffcrh 2> 0.1=aQ 3.16
:ffcrh 2≤ g
wcrhcrha A
htf
f.f
f.Q ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−−= 16019011 3.17
Note, that the ratio of the web area to the gross area appears due to the AISC
methodology where only stiffened elements are treated as being reduced to effective
width, and hence effective area.
31
3.2 AISI (AISI – Effective Width Method)
The AISI effective width method is detailed in the 2007 AISI Specification (AISI-
S100 2007). The long column (global buckling) design expressions are provided in
Section C4.1 of AISI-S100, the effective width reductions follow Section B2.1 for the web
(stiffened element) and B3.1 for the flange (unstiffened element). The expressions
provided in Table 3.1 and Table 3.2 are not in the same format as AISI-S100 but have
been derived here for the purposes of comparison to the AISC expressions.
3.3 DSM (AISI – Direct Strength Method)
The AISI Direct Strength Method (DSM) is detailed in Appendix 1 of the 2007 AISI
Specification. The long column (global buckling) design expression is identical to that in
C4.1 of the main AISI Specification. The local buckling strength uses the long column
strength as its maximum capacity. The DSM expressions provided in Table 3.1 and
Table 3.2 have been formulated for comparison to the AISC and AISI effective width
expressions, and are not in the same form as shown in DSM Appendix 1.
3.4 Direct comparison of design expressions
The design expressions for all three methods, in a common notation system, are
provided in Table 3.1 for the general case of a W-section column and Table 3.2 for a W-
section stub column assuming cross-section local buckling (fcrl) is used in place of
isolated plate buckling solutions (fcrb and fcrh). Although the expressions appear quite
different in the format of their original Specification’s – in this format (Table 3.1) they
can be seen to have many similarities.
32
Table 3.1 Comparison of column design equations for a slender W-section in a common notation* AISC
inputs to find Pn Ag = gross area fe = global buckling stress fy = yield stress fcrb = flange local buckling fcrh = web local buckling htw/Ag = web/gross area Comments: shifts the slenderness in the global column curve in the inelastic range only, assumes that unstiffened elements (flange) should be referenced to fy, only applies an effective width style reduction to stiffened elements (the web), includes an awkward iteration for web stress f.
1 with determined
2 if 16019011
2 if 01
53 if 11
253 if 5904151
2 if 01
440 if 8770440 if 6580
===
⎪⎩
⎪⎨
⎧
≤⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−−
>
=
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≤
<<−
≥
=
⎪⎩
⎪⎨⎧
<≥
=
=
asnga
n
crhg
wcrhcrh
crh
a
ycrby
crb
ycrbycrb
y
ycrb
s
yasee
yasey)f/f(QQ
asn
ngn
QQf̂~AQ
Pf
ffAht
ff.
ff.
ff.Q
ffff.
fffff
..
ff.
Q
fQQ.ff.fQQ.ff).(QQ
f̂
f̂APyeas
AISI-Eff. Width inputs to find Pn Ag = gross area fe = global buckling stress fy = yield stress fcrb = flange local buckling fcrh = web local buckling btf = flange area htw = web area Comments: no shift in global column curve, effective width used for stiffened and unstiffened ele-ments.
⎪⎩
⎪⎨
⎧
<⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥=ρρ=
⎪⎩
⎪⎨
⎧
<⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥=ρρ=
ρ+ρ=
⎪⎩
⎪⎨⎧
<≥
=
=
ncrhn
crh
n
crh
ncrh
hhe
ncrbn
crb
n
crb
ncrb
bbe
whfbeff
yee
yey)f/f(
n
neffn
f.fff
ff.
f.fhh
f.fff
ff.
f.fbb
htbtA
f.ff.f.ff).(
f
fAPye
22 if 2201
22 if 1 where
22 if 2201
22 if 1 where
4
440 if 8770440 if 6580
AISI-DSM inputs to find Pn Ag = gross area fe = global buckling stress fy = yield stress fcrl = local buckling stress Comments: similar to AISI but reductions on whole section and “effective width” equation modified.
661 if 1501
661 if 1
440 if 8770440 if 6580
4040
⎪⎩
⎪⎨
⎧
<⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥
=ρ
ρ=
⎪⎩
⎪⎨⎧
<≥
=
=
ncr
.
n
cr
.
n
cr
ncr
geff
yee
yey)f/f(
n
neffn
f.fff
ff.
f.f
AA
f.ff.f.ff).(
f
fAPye
lll
l
* centerline model of W-section, in practice AISC and AISI use slightly different k values for fcrb and fcrh.
33
Table 3.2 Comparison of stub column design equations for a slender W-section when cross-section elastic local buckling replaces isolated plate buckling solutions, i.e., fcrl = fcrb = fcrh
and when global buckling is assumed to be fully braced.
AISC inputs to find Pn Ag = gross area fy = yield stress fcrl = local buckling stress htw/Ag = web/gross area Comments: adoption of fcrl for fcrb and fcrh does not simplify the AISC methodology significantly. Unstiffened and stiffened elements are treated inherently differently in the AISC methodology.
⎪⎩
⎪⎨
⎧
≤⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−
>
=
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≤
<<−
≥
=
=
ycrg
w
y
cr
y
cr
ycr
a
ycry
cr
ycrycr
y
ycr
s
ygasn
ffAht
ff.
ff.
ff.
Q
ffff.
fffff
..
ff.
Q
fAQQP
2 if 16019011
2 if 01
53 if 11
253 if 5904151
2 if 01
lll
l
ll
l
l
l
AISI-Eff. Width inputs to find Pn Ag = gross area fy = yield stress fcrl = local buckling stress Comments: when fcrl is used for fcrb and fcrh the methodology becomes the same as DSM, but with a more conservative local buckling predictor equation.
⎪⎩
⎪⎨
⎧
<⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥
=ρ
ρ=
=
ycry
cr
y
cr
ycr
geff
yeffn
f.fff
ff.
f.f
AA
fAP
22 if 2201
22 if 1
lll
l
AISI-DSM inputs to find Pn Ag = gross area fy = yield stress fcrl = local buckling stress Comments: no change from general case
661 if 1501
661 if 1
4040
⎪⎪⎩
⎪⎪⎨
⎧
<⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−
≥
=ρ
ρ=
=
ycr
.
y
cr
.
y
cr
ycr
geff
yeffn
f.fff
ff.
f.f
AA
fAP
lll
l
It is hoped that in the provided format, Table 3.1, it is made clear that the number of
free parameters in slender column design is actually significantly less than one might
typically think. Based on Table 3.1, and performing a simple non-dimensional analysis,
the parameters for determining the column strength of an idealized W-section are:
34
AISC: Pn/Py = f (fe/fy, fcrb/fy, fcrh/fy, htw/Ag)
AISI: Pn/Py = f (fe/fy, fcrb/fy, fcrh/fy, htw/Ag or 2bftf/Ag)
DSM: Pn/Py = f (fe/fy, fcrl/fy,)
The central role of elastic buckling prediction both globally (fe) and locally (fcrb, fcrh or fcrl)
in determining the strength of the column is clear given the parameters above. Further,
the “direct” nature of the DSM approach is highlighted as DSM only uses ratios of
critical buckling values to determine the strength; where AISC and AISI still involve
cross-section parameters beyond determination of gross area and critical stress.
3.5 Stub column comparison
Since all three methods use the same global buckling column curve (though AISC
uses the Q-factor approach which adjusts the slenderness used within the curve) the
initial focus of the comparison is on a stub column – and thus local buckling only.
Predicted stub column capacities via the three design methods are provided in Figure
3.1. Since the results are dependent on the cross-section geometry (namely, the htw/Ag
ratio) some care must be taken when comparing the methods.
Figure 3.1(a) provides the stub column comparison for the range of geometry
typical of the heavier W14 columns. For W14 columns all three methods yield nearly the
same strength even for cross-sections reduced as much as 40% from the squash load
due to local buckling (i.e. Pn/Py = 0.6). For more slender cross-sections (i.e., crby f/f >
1.2) the AISC method becomes more conservative than AISI and DSM; which essentially
provide the same solution for this column.
35
For a W36 column fcrb and fcrh are very different (as opposed to a W14 when they
are nearly the same), with the web local buckling stress, fcrh, being significantly lower
than the flange local buckling stress, fcrb. In addition, W36 columns have a greater
percentage of total material in the web (higher htw/Ag than a W14). For the W36’s AISC
and AISI provide essentially the same solution over the anticipated flange slenderness
range. However, DSM which accounts for the web-flange interaction in a very different
manner from the other two methods assumes the W36 remains compact up to higher
flange slenderness, but provides a more severe reduction as the flange slenderness
increases.
Since the W36 provides a definite contrast between DSM, and AISI and AISC the
analysis is extended over a wider slenderness range in Figure 3.2. (Note, flange
slenderness crby f/f greater than 2 is unlikely for these sections even at yield stress
approaching 100 ksi). For the W36 geometry AISI and AISC provide the same solution
even as reductions move from just the web, to include the flange. Only when the flange
reduction reaches the final branch of the AISC Qs curve (fcrb<3/5fy) and the design stress
is reduced essentially to its elastic value of 1.1fcrb does the AISC method diverge from
AISI, and in assuming essentially no post-buckling reserve for the unstiffened element
flange, provide a more conservative solution. In contrast, the DSM solution provides a
continuous reduction and at high slenderness predicts strength between AISI and AISC.
36
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
local flange slenderness (fy/fcrb)0.5
Pn/P
y
Stub ColumnTypical heavy W14 dimensions:htw /Ag=0.2 2bftf/Ag=0.8
fcrb/fcrh=0.8, fcrb/fcrlocal=1.3
AISCAISI - Eff. WidthDSM (AISI App. 1)
(a) W14 stub column (flange slenderness varies within W14 series and due to change in fy)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
local web slenderness (fy/fcrh)0.5
Pn/P
y
Stub ColumnTypical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
AISCAISI - Eff. WidthDSM (AISI App. 1)
(b) W36 stub column (web slenderness varies within W36 series and due to change in fy)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
local slenderness (fy/fcrlocal)0.5
Pn/P
y
local slenderness of W14'sat fy=36 ksi from ~ 0.1 to 0.8at fy=100 ksi from ~ 0.1 to 1.3
AISC (max htw /Ag)
AISC (min htw /Ag)
AISI - Eff. WidthDSM (AISI App. 1)
(c) Any W-section stub column, but cross-section local bucking fcrl has replaced plate buckling fcrb, fcrh in
the design expressions per Table 3.2 Figure 3.1 Predicted stub column capacities by the AISC, AISI, and DSM methods
37
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
local web slenderness (fy/fcrh)0.5
Pn/P
y
Stub Column
Typical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
flange becomes partially effective
transitioning through AISC Qs equations
AISCAISI - Eff. WidthDSM (AISI App. 1)
Figure 3.2 Predicted stub column capacity of a W36 section, same analysis as Figure 3.1(b) but examined over
a wider slenderness range to highlight the differences in the methods
The stub column strength for the case where cross-section elastic local buckling
analysis (fcrl) is used instead of the isolated plate solutions (fcrb and fcrh) is provided in
Figure 3.1(c), while the actual design expressions for this case are provided in Table 3.2.
Since one of the hypotheses of this work is that the use of cross-section stability analysis
may prove useful, this plot provides an interesting contrast to the previous two plots of
Figure 3.1, as it shows that directly introducing fcrl into existing AISC or AISI methods
may be overly conservative. The DSM solution provides strictly greater predictions of
columns strength than both AISI and AISC for a stub column capacity calculated in this
manner. The development of the DSM to an expression different than AISI is exactly
because comparisons to cold-formed steel columns showed that when cross-section
local buckling was used as the parameter stub column strength follows the DSM curve,
not the AISI curve. It is postulated that similar conclusions will be reached for AISC
sections, though the exact change to a similar DSM curve is not yet known.
38
3.6 Long column comparisons
The column design expressions for AISC, AISI, and DSM, as summarized in Table
3.1, are examined for the same three cases as the stub columns in the previous section:
W14 columns (Figure 3.3), W36 columns (Figure 3.4), and general W-sections where fcrl
is substituted for fcrb and fcrh (Figure 3.5). For each case all three methods are examined
as the global slenderness ( ey f/f ) is varied from 0 to 2, and for four different cross-
section slenderness values (subfigures (a) – (d)). The cross-section slenderness is
systematically increased in the subfigures: (a) provides the results for a fully compact
section, (b) for a local slenderness of 0.8, which corresponds approximately to the most
slender W14 at fy=36 ksi, (c) for a local slenderness of 1.3, a locally slender W14 at fy=100
ksi, and (d) for a local slenderness of 2 which corresponds to a section with high local
slenderness – fcr=¼fy.
The results for the W14 long columns are provided in Figure 3.4, and the basic
conclusions are similar in many respects to the stub column results of Figure 3.1(a):
AISC, AISI, and DSM provide similar capacities except at high local slenderness where
AISC provides a much more conservative prediction than AISI or DSM. AISC’s Q-factor
approach changes the shape of the column curve (i.e., 0.658Q(fe/fy) instead of 0.658(fe/fy))
and the asymptote (Qfy) for a stub column. Figure 3.2 shows that the change in shape is
not significant as neither AISI nor DSM make this change and the basic results are
similar as long as the stub column asymptote is similar. Thus, for the AISC curve the
stub column asymptote (Qfy) is the only change of practical significance. This is not
39
particularly surprising since prior to the adoption of the unified method in AISI, the
cold-formed steel specification also used the Q-factor approach. Part of the justification
for moving to a unified effective width approach was that the most significant change to
the column curve results was the asymptote (stub column value) not the global
slenderness change.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Flange local slenderness (fy/fcrb)0.5 = 0.1
Typical heavy W14 dimensions:htw /Ag=0.2 2bftf/Ag=0.8
fcrb/fcrh=0.8, fcrb/fcrlocal=1.3
AISCAISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Flange local slenderness (fy/fcrb)0.5 = 0.8
Typical heavy W14 dimensions:htw /Ag=0.2 2bftf/Ag=0.8
fcrb/fcrh=0.8, fcrb/fcrlocal=1.3
AISCAISI - Eff. WidthDSM (AISI App. 1)
(a) compact: flange slenderness of 0.1, i.e. fcrb = 100fy (b) local flange slenderness of 0.8, i.e. fcrb = 1.56fy
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Flange local slenderness (fy/fcrb)0.5 = 1.3
Typical heavy W14 dimensions:htw /Ag=0.2 2bftf/Ag=0.8
fcrb/fcrh=0.8, fcrb/fcrlocal=1.3
AISCAISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Flange local slenderness (fy/fcrb)0.5 = 2
Typical heavy W14 dimensions:htw /Ag=0.2 2bftf/Ag=0.8
fcrb/fcrh=0.8, fcrb/fcrlocal=1.3
AISCAISI - Eff. WidthDSM (AISI App. 1)
(c) local flange slenderness of 1.3, i.e. fcrb = 0.59fy (b) local flange slenderness of 2, i.e. fcrb = 0.25fy Figure 3.3 Predicted long column capacities of typical W14 columns by the AISC, AISI, and DSM methods
for varying flange local slenderness following the formulas of Table 3.1
40
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Web local slenderness (fy/fcrh)0.5 = 0.1
Typical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
AISCAISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Web local slenderness (fy/fcrh)0.5 = 0.8
Typical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
AISCAISI - Eff. WidthDSM (AISI App. 1)
(a) compact: web slenderness of 0.1, i.e. fcrh = 100fy (b) local web slenderness of 0.8, i.e. fcrh = 1.56fy
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Web local slenderness (fy/fcrh)0.5 = 1.3
Typical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
AISCAISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Web local slenderness (fy/fcrh)0.5 = 2
Typical heavy W36 dimensions:htw /Ag=0.4 2bftf/Ag=0.6
fcrb/fcrh=8, fcrb/fcrlocal=6
AISCAISI - Eff. WidthDSM (AISI App. 1)
(c) local web slenderness of 1.3, i.e. fcrh = 0.59fy (b) local web slenderness of 2, i.e. fcrh = 0.25fy Figure 3.4 Predicted long column capacities of typical W36 (as a column) by the AISC, AISI, and DSM
methods for varying web local slenderness following the formulas of Table 3.1
Comparison of the W36 columns is provided in Figure 3.4. The most interesting
results occur for the most slender cross-section, Figure 3.4(d), which shows that AISC
provides the most liberal prediction of the column strength (though still similar to
AISI), which is the opposite of the W14’s where AISC provided the most conservative
prediction. In practice this implies that AISC penalizes slender unstiffened elements
(the flange) more than AISI and rewards slender stiffened elements (the web) more than
AISI, thus the ratio of the area of stiffened elements to the area of unstiffened elements
or the web-to-flange area ratios influence the AISC predictions relative to AISI or DSM a
41
great deal. The behavior of DSM is similar to what was observed in the stub column
predictions of Figure 3.1(b): DSM provides a higher capacity at low web (local)
slenderness, but as the web slenderness increases the predicted overall decrease in the
capacity is greater than AISC or AISI. Thus, DSM assumes a greater reduction in the
slender column strength due to local buckling driven by the web than AISC or AISI.
Finally, as is true for all of the long column methods, since the same global buckling
column curve is used, at high global slenderness all of the methods eventually
converge.
A general comparison of the AISC, AISI, and DSM design methods for W-sections
is possible if the local cross-section stability solution (fcrl) is used in place of the isolated
plate buckling solutions (fcrb and fcrh), such a comparison is provided in Figure 3.5.
Comparisons between the design methods remain similar to the stub column
comparisons of Figure 3.1(c): DSM predicts a consistently greater strength than AISC or
AISI, and AISC is most conservative when the flange (unstiffened element) contributes
more to the strength. The DSM column curve is known to fit available cold-formed steel
column data better than the AISI effective width method, when the plate buckling
solutions (fcrb and fcrh) are replaced by the cross-section local buckling (fcrl) solution. The
difference in strength predictions at high local slenderness is quite large – and suggests
that the AISC design philosophy may be overly conservative if cross-section stability
solutions are adopted with no other change. Further, the importance of this
conservatism is increasing as higher yield stress cross-sections are considered.
42
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Local slenderness (fy/fcrlocal)
0.5 = 0.1 AISC (max htw /Ag)
AISC (min htw /Ag)
AISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Local slenderness (fy/fcrlocal)
0.5 = 0.8 AISC (max htw /Ag)
AISC (min htw /Ag)
AISI - Eff. WidthDSM (AISI App. 1)
(a) compact: local slenderness of 0.1, i.e. fcrl = 100fy (b) local slenderness of 0.8, i.e. fcrl = 1.56fy
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Local slenderness (fy/fcrlocal)
0.5 = 1.3 AISC (max htw /Ag)
AISC (min htw /Ag)
AISI - Eff. WidthDSM (AISI App. 1)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
0.6
0.8
1
global slenderness (fy/fe)0.5
Pn/P
y
Local slenderness (fy/fcrlocal)
0.5 = 2 AISC (max htw /Ag)
AISC (min htw /Ag)
AISI - Eff. WidthDSM (AISI App. 1)
(c) local slenderness of 1.3, i.e. fcrl = 0.59fy (b) local slenderness of 2, i.e. fcrl = 0.25fy
Figure 3.5 Predicted long column capacities of columns with slender cross-sections by the AISC, AISI, and DSM methods following the formulas of Table 3.1, but where, cross-section elastic local buckling replaces
isolated plate buckling solutions, i.e., fcrb = fcrh = fcrl and AISC htw/Ag range reflects range of W14’s
3.7 Ongoing / future work
Significant future work remains related to the comparison of these design
methods including:
• Provide a comprehensive literature review of the development of the ASIC
Q-factor, AISI unified effective width, and AISI - DSM design methods.
43
• Extend the work initiated herein on the W14 and W36 sections under pure
axial loading, to include a wider range of W-sections, other section types,
and loading cases including major and minor axis bending.
• Gather available test data for hot-rolled steel (recent work of Don White et
al. will be extremely helpful in this regard) and cold-formed steel to make a
statistical comparison of the predictive methods against tested cross-
sections.
• Initiate a parametric study, using ABAQUS, to extend the test database for
comparison to the design methods. Particular attention will be placed on
understanding the regimes where the AISC and DSM methods give
divergent results. The role of cross-section details (k-zone, etc.)
imperfections, residual stresses, and material yield stress and parameters
(strain hardening, etc.) on the results and comparisons will be completed.
• Propose design improvements to DSM for its application to structural steel.
• Explore the possibility of modifications to a small group of standard cross-
sections that may be appropriate for higher yield stress applications where
serviceability has to be balanced against strength.
• Examine the impact of using cross-section local buckling values on the
currently used slenderness limits in the AISC Specification.
44
4 Educational materials
4.1 Objective
The objective of the educational materials developed in support of this project are
to provide tools, tutorials, and educational aids related to cross-section stability of
structural steel shapes so that educators, students, and engineers may explore these
concepts more readily. Further, the developed educational aids are intended to be
appropriate for courses in steel design using structural steel at the undergraduate and
graduate levels.
Figure 4.1 Project web site for dissemination of educational materials (www.ce.jhu.edu/bschafer/aisc)
4.2 Work Products
A series of tutorials and accompanying software files and homework problems
have been created in support of the educational objectives. All of the materials are
disseminated from the project website, Figure 4.1, www.ce.jhu.edu/bschafer/aisc. The
tutorials provide detailed instructions and background and discussion in the use of
45
CUFSM (www.ce.jhu.edu/bschafer/cufsm) an open source software tool for cross-
section stability analysis developed by the senior author. In addition, a series of CUFSM
examples files have been created for the following structural steel shapes: W36x150,
W14x120, C5x9, L4x4x1/2, WT 18x150, and an HSS 4x4x1/2. The files are used and
references in the tutorials and in the homework problems.
4.2.1 Tutorial 1: Cross-section stability of a W36x150 using the finite strip method
Tutorial 1 covers cross-section stability analysis of a W36x150 in compression and
major-axis bending using CUFSM. Step by step instructions are provided for the novice
user of CUFSM. The tutorial is prepared as a series of PowerPoint slides that could be
guided by an instructor or used for independent learning. The slides are available
online, but are also included here (6 to a page) in Appendix C. The learning objectives
of tutorial 1 are:
1) Identify all the buckling modes in a W-section,
i. for columns explore flexural (Euler) buckling and local buckling,
ii. for beams explore lateral-torsional buckling and local buckling;
2) Predict the buckling stress (load or moment) for identified buckling modes,
3) Learn the interface of a simple program for exploring cross-section stability of
any AISC section and learn finite strip method concepts such as half-wavelength
of the bucking mode, and buckling load factor associated with the applied
stresses.
46
4.2.2 Tutorial 2: Cross-section stability of a W36x150 exploring higher modes and the interaction of modes
Tutorial 2 follows on directly from the first tutorial on the W36x150, but here the
target audience is graduate students, advanced undergraduates, or practitioners /
designers comfortable with basic stability concepts. The learning objectives of tutorial 2
are as follows:
1) Understand the role of “higher” buckling modes in the analysis of a W-
section, including
a. how higher buckling modes relate to strong-axis, weak-axis, and
torsional buckling in columns,
b. what higher buckling modes mean for local buckling, and
c. when knowledge of higher buckling modes may be useful in design;
2) Understand how interaction of modes may be identified and quantified
using CUFSM for a W-section.
4.2.3 Tutorial 3: Exploring how cross-section changes influence cross-section stability – an extension to Tutorial #1
Tutorial 3 leads directly from the first tutorial on the W36x150 but helps the
analyst further develops their skills in cross-section stability analysis using the finite
strip method. The target audience for this tutorial is undergraduate and graduate
students (though the intent is that all the material be clear and understandable to an
undergraduate). The focus of the tutorial is how to modify the cross-section and then
examine the changing in the local and global buckling results. The intent of this tutorial
47
is to develop the skills necessary so that active (not canned) homework can be assigned
where students explore cross-section stability for themselves. The specific learning
objectives for the tutorial are:
1) Study the impact of flange width, web thickness, and flange-to-web fillet
size on a W36x150 section,
2) Learn how to change the cross-section in CUFSM,
3) Learn how to compare analysis results to study the impact of changing the
cross-section.
4.2.4 Exercises: Homework exercises related to Tutorials 1 and 3 on cross-section stability (doc)
The developed exercises include simple homework problems that cover tutorials 1
and 3, as well as homework problems that require the student to apply the knowledge
from tutorials 1 and 3, in addition homework exercises related to the other W, C, L, WT,
and HSS are provided and some simple group problems that require a fuller
exploration of cross-section stability of structural shapes. The developed exercises
follow:
48
Cross-section stability of structural steel
Exercises targeted at undergraduate level
Software www.ce.jhu.edu/bschafer/cufsm Tutorials www.ce.jhu.edu/bschafer/aisc
[Checkup on Tutorial #1]
1) Using finite strip analysis and the program CUFSM, what is the elastic local buckling stress of a W36x150 section in pure compression?
2) Again for this W36x150, at what moment does elastic local buckling occur if bending is about the strong axis?
[Applying Tutorial #1]
1) For a W36x150 what is the elastic weak-axis flexural buckling stress for a pin-ended member which is 30 ft. long?
2) Again for the same W36x150, at what moment does elastic local buckling occur if bending is about the weak axis?
[Checkup on Tutorial #3]
1) How does the elastic local buckling stress change if the web thickness is increased in a W36x150 to be the same as the flange thickness for the section in pure compression?
2) How does the elastic local buckling stress change if the flange thickness is decreased by 2 in. in a W36x150 subject to pure compression?
[Applying Tutorial #3]
1) Load the W14x120 cross-section and make the same changes as Tutorial 3 on this cross-section and observe the impact, i.e.,
a. determine the elastic local buckling stress and global buckling stress at 40 ft. for the W14x120 in pure compression
b. make the web thickness the same as the flange thickness and examine the impact on the local buckling stress for the section in pure compression, report the change in the local buckling stress global buckling stress, and observed behavior.
c. make the flange 2 in. narrower and examine the impact on the local buckling stress for the section in pure compression, report the change in local buckling stress, global buckling stress, and observed behavior.
d. how is the W14x120 different in its response than the W36x150? [Exploration of other cross-sections]
1) For each of the following cross-sections find the elastic local buckling stress in (i) pure compression and (ii) for restrained bending about the global x-x axis using CUFSM (a) W14x120 (b) C5x9 (c) L4x4x1/2 (d) WT 18x150 (e) HSS 4x4x1/2
49
2) Compare your W36x150 results with your WT18x150, what is the impact of slicing the W36x150 in half on local buckling? on global buckling?
3) Compare your L4x4x1/2 with the HSS4x4x1/2, what is the impact of having all four side connected as in the HSS section as opposed to the L-section on local buckling? on global buckling?
[Small project / Group project]
1) Consider pure compression load on the W36x150, keeping the total area constant modify the cross-section such that the local buckling stress is increased by at least 50%. (You are asked to use creativity and trial and error to find this solution, note you can see your current “area” in the properties page it should be nearly the same as the original W36x150 area – you may change thickness, dimension, whatever you choose in this part)
2) Can you modify the cross-section in such a way that the local buckling stress increases AND the global buckling stress at 20 ft. also increases AND the section remains its centerline depth of 35 in. all while keeping the area constant? (To achieve this you may have to be quite creative in how you use the material and the improvements may be modest, see how much better you can make this section..)
50
5 Conclusions Local buckling solutions that include the full cross-section can provide
significantly different predictions of the elastic buckling stress when compared to
isolated plate buckling solutions typically employed in design specifications. In
particular, the inclusion of web-flange interaction, inherent in any proper cross-section
stability analysis may yield significantly different predictions of compactness limits and
reductions (Qs, Qa) in cross-section strength. Simplified empirical expressions that
approximate cross-section stability analysis are possible for AISC sections, and sample
expressions are provided for W-sections. For certain classes of W-sections, for example
W14’s, the consistent variation in geometry (as W14’s step up in weight) results in little
change in web-flange interaction with respect to elastic local buckling within a class.
Although the design of columns with slender cross-sections can appear
complicated given the current expressions and their format – the actual number of
parameters used by design specifications is quite small. The elastic buckling stress and
the yield stress are by far the most important variables for determining the capacity. In
particular, for AISC: Pn/Py = f (fe/fy, fcrb/fy, fcrh/fy, htw/Ag) and for the Direct Strength
Method (DSM): Pn/Py = f (fe/fy, fcrl/fy,) where Pn is the nominal column capacity, Py the
squash load, fe the global elastic buckling stress, fcrb the flange local buckling stress, fcrh
the web local buckling stress, fcrl the local buckling stress, Ag the gross area, and htw the
web area. The premise of the DSM method is that the improved prediction that comes
51
with using cross-section fcrl instead of isolated plate solutions fcrb and fcrh leads to the
ability to use simpler design procedures, a premise proven for cold-formed steel.
Three design methods are compared for W-section columns: AISC (2005
Specification), AISI (AISI-S100-07 main body effective width method) and DSM (AISI-
S100-07 Appendix 1). For common sections, at common slenderness values, the three
design methods yield generally similar results, but differences do exist. The AISC
treatment of unstiffened elements is generally found to be more conservative than AISI,
particularly as the unstiffened element (i.e., the flange of a W-section) becomes more
slender. The AISC treatment of stiffened elements is generally found to be similar, but
slightly less conservative than AISI. The predictions of DSM may follow different trends
than AISI, or AISC, as shown with respect to a W36 column, where DSM predicts the
section is fully compact at a much higher slenderness than AISC or AISI, but that the
strength reductions occur much more sharply once slenderness increases. The
parameters that lead to significant differences between the design methods will be the
focus of a planned parametric study using nonlinear finite element analysis.
A series of educational tutorials related to cross-section stability have been
created. These tutorials provide detailed step-by-step instructions for performing cross-
section stability analysis for the engineer or student that has never performed such
analyses, as well as more advanced tutorials appropriate for graduate level study.
Example files, specific learning objectives, and complementary homework exercises are
all available on the project web site (www.ce.jhu.edu/bschafer/aisc).
52
6 References Ádány, S., Schafer, B.W. (2007) “A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method.” Elsevier, Journal of Constructional Steel Research (In Press 2007). AISC (2005). Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, IL. ANSI/ASIC 360-05 AISI (2007). North American Specification for the Design of Cold-Formed Steel Structures. American Iron and Steel Institute, Washington, D.C., AISI-S100. Schafer, B.W., Ádány, S. (2005). “Understanding and classifying local, distortional and global buckling in open thin-walled members.” Proceedings of the Structural Stability Research Council Annual Stability Conference, May, 2005. Montreal, Quebec, Canada. 27-46. Schafer, B.W., Ádány, S. (2006). “Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods.” Proceedings of the Eighteenth International Specialty Conference on Cold-Formed Steel Structures, Orlando, FL. 39-54. Galambos, T. (1998) “Guide to Stability Design Criteria for Metal Structures”. 5th ed., Wiley, New York, NY, 815-822. Salmon, C.G., Johnson, J.S. (1996) “Steel structures: design and behavior: emphasizing load and resistance factor design” HarperCollins College Publishers, New York, NY.
53
A Appendix: Further finite strip analysis of structural steel cross-sections
Finite strip analysis of all the sections in the AISC shape database has been
initiated. Although this report focuses on the analysis of W-sections this appendix
reports on some of the additional work that has recently been completed on WT-
sections; as summarized in Figure A.1 and Figure A.2. The presented analysis indicates
that simplified expressions approximating the cross-section stability analysis of the
finite strip method are possible for the WT-sections as well.
54
5 10 15 20 25 30 350
0.1
0.2
0.3
0.4
web slenderness h/tw
flang
e, k
f
WT-sections
2 3 4 5 6 7 8 91
1.2
1.4
1.6
1.8
flange slenderness bf/(2tf)
web
, kw
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50
0.1
0.2
0.3
0.4
(h/tw )(2tf/bf)
k f
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.51
1.2
1.4
1.6
1.8
(h/tw )(2tf/bf)
k w
Figure A.1 Flange and web local buckling coefficients for the WT-sections under axial loading.
55
0 5 10 15 20 25 30 350
0.5
1
web slenderness h/tw
flang
e, k
f
WT-sections
0 2 4 6 8 10 121
1.5
2
2.5
flange slenderness bf/(2tf)
web
, kw
1 2 3 4 5 6 70
0.5
1
(h/tw )(2tf/bf)
k f
1 2 3 4 5 6 71
1.5
2
2.5
(h/tw )(2tf/bf)
k w
Figure A.2 Flange and web local buckling coefficients for the WT-sections under bending loading.
56
B Appendix: Explicit parametric study of W-section columns comparing AISC, AISI, and DSM design methods
Preliminary to the study of W-sections presented in Section 3 an explicit
parametric study using actual dimensions (h, b, tf, tw etc.), not non-dimensional
variables (fcrb/fy, htw/Ag etc.), was completed. This study, presented in this appendix, is
provided to show a complete picture of the work completed during the first year of
research. Further, some observations with respect to the behavior of the design methods
presented herein are useful in planning for the anticipated nonlinear finite element
analysis which cannot use non-dimensional variables, but must instead use explicit
variables as was completed here.
Sample long column analysis
Figure B.1 shows a plot of the normalized nominal strengths, yn PP / , obtained by
both the AISC and AISI design methods versus the normalized length, rL / , for a W18
section loaded in compression for a case with a thick flange ( 8.0=ft ”) and another with
a thin flange ( 5.0=ft ”). Figure B.2 shows a similar plot but with a case of a thick web
( 6.0=wt ”) and another with a thin web ( 2.0=wt ”).
For compact sections, all of the design methods give the same strengths; however
as any element of the section becomes slender (Q<1, or partially effective), the different
methods diverge. The differences typically become greater for members with higher
yield stress, since this is equivalent to making the cross-section more slender.
57
0 50 100 150 200 250 3000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L/r
Pn/P
y
AISC with large tf (0.8")
AISC with small tf (0.5")
AISI with large tf (0.8")
AISI with small tf (0.5")
Figure B.1 Normalized nominal strengths, yn PP / , obtained by both the AISC and AISI design methods
versus the normalized length, rL / , for a W18 section loaded in compression with different flange thicknesses.
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L/r
Pn/P
y
AISC with large tw (0.6")
AISC with small tw (0.2")
AISI with large tw (0.6")
AISI with small tw (0.2")
Figure B.2 Normalized nominal strengths, yn PP / , obtained by both the AISC and AISI design methods
versus the normalized length, rL / , for a W18 section loaded in compression with different web thicknesses.
W14 stub columns by explicit parametric study
In order to further investigate the effect of element thicknesses on section nominal
strength, W14 and W36 sections were chosen to represent most commonly used
columns and beams respectively. To begin, the members were considered to be fully
braced and their stub column capacity was examined.
58
W14 section with varied flange thickness For the W14 sections, the web height, h, and the flange width, b, were fixed to
13.5” each, which is the average value for all the W14 sections. First the web thickness,
wt , was fixed to 1.0” (which also is the average value of all the W14 sections), while the
flange thickness, ft , was varied over a range between 0.3” to 5.0” which represents the
range in which all W14 sections flanges fall within. The nominal strength was calculated
for all the sections using:
• AISC design procedure with fk =0.7 and wk =5.0.
• AISI design procedure with fk =0.7 and wk =5.0 (this is different than the
AISI code assigned values of fk =0.43 and wk =4.0, but is completed here so
that the comparison between the methods can be as similar as possible).
• DSM design procedure with crf as an output from the finite strip analysis.
• AISC design procedure with fk and wk values back-calculated from the
finite strip analysis.
• AISI design procedure with fk and wk values back-calculated from the
finite strip analysis.
Figure B.3 shows a plot of the normalized nominal strengths, yn PP / , obtained by
all different methods described above versus the inverse of the flange reduction factor,
59
sQ , for the chosen W14 sections loaded in compression, with yield strength, 50=yf ksi.
Figure B.4 shows similar plots but with 100=yf ksi.
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40
0.2
0.4
0.6
0.8
1
1.2
1/Qs
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.3 Normalized nominal strengths, yn PP / , versus the inverse of the flange reduction factor, sQ , for
the chosen W14 sections, with a variable flange thickness loaded in compression, 50=yf ksi.
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60
0.2
0.4
0.6
0.8
1
1.2
1/Qs
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.4 Normalized nominal strengths, yn PP / , versus the inverse of the flange reduction factor, sQ , for
the chosen W14 sections, with a variable flange thickness, loaded in compression, 100=yf ksi.
60
It is clear from the figures that the different design methods give significantly
different results, and that the differences become more significant for higher yield
strengths.
Figs. B.5, B.6, and B.7 show plots of the normalized nominal strengths, yn PP / ,
obtained by the different design methods described above versus a local slenderness
cry ff / for the chosen W14 sections loaded in compression for yield strengths of 50,
70, and 100 ksi, respectively. Again, It is clear from the figures that the different design
methods give significantly different results, and that the differences become more
significant for higher yield strengths.
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.5 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable flange thickness, loaded in compression, 50=yf ksi
61
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.6 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable flange thickness, loaded in compression, 70=yf ksi
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
0.2
0.4
0.6
0.8
1
1.2
(fy./fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.7 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable flange thickness, loaded in compression, 100=yf ksi
W14 section with varied web thickness To check the effect of the web thickness, similar calculations were made on the
W14 sections where, as used for the flange effect calculations, the web height, h, and the
flange width, b, were fixed to 13.5” each, which is the average value for all the W14
62
sections. Now, flange thickness, ft , was set to 1.69” which also is the average value of
all the W14 sections, while the web thickness, wt , was varied over a range between 0.2”
to 3.1” which represents the range in which all W14 sections web thicknesses fall within.
The nominal strength was calculated for all the sections using the same five design
procedures.
Figure B.8 shows a plot of the normalized nominal strengths, yn PP / , obtained by
the different design methods versus the inverse of the web reduction factor, aQ , for the
chosen W14 sections loaded in compression, with yield strength, 50=yf ksi. Figure B.9
shows similar plots but with 100=yf ksi.
1 1.005 1.01 1.015 1.02 1.0250.8
0.85
0.9
0.95
1
1.05
1.1
1/Qa
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.8 Normalized nominal strengths, yn PP / , versus the inverse of the web reduction factor, aQ , for
the chosen W14 sections, with a variable web thickness, loaded in compression, 50=yf ksi.
63
1 1.005 1.01 1.015 1.02 1.025 1.03 1.0350.8
0.85
0.9
0.95
1
1.05
1.1
1/Qa
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.9 Normalized nominal strengths, yn PP / , versus the inverse of the web reduction factor, aQ , for
the chosen W14 sections, with a variable web thickness, loaded in compression, 100=yf ksi.
For the case of web thickness variation for the same aQ value, the AISC and AISI
methods give nearly identical results, but when web-flange interaction is included the
methods diverge (a) from one another and (b) from the previous solutions. The
influence of web-flange appears greater in W14 sections as web thickness is varied than
flange thickness.
Figures B.10 through B.12 are similar to Figures B.5 through B.7, and show plots of
the normalized nominal strengths, yn PP / , obtained by the design methods but here
plotted with respect to the local slenderness cry ff / for the chosen W14 sections
loaded in compression for yield strengths of 50, 70, and 100 ksi’s, respectively, again for
the case of varying the web thickness.
64
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.10 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable web thickness, loaded in compression, 50=yf ksi.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.11 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable web thickness, loaded in compression, 70=yf ksi.
65
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.12 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W14 sections, with a variable web thickness, loaded in compression, 100=yf ksi.
W36 Stub column results
W36 section with varied flange thickness Similar calculations and analysis were performed on the W36 sections. For the
W36 sections, the web height, h, was fixed to 33.0”and the flange width, b, was fixed to
15.0”, which are the average values for all the W36 sections. First the web thickness, wt ,
was fixed to 1.0” which also is the average value of all the W36 sections, while the
flange thickness, ft , was varied over a range between 0.7” to 4.3” which represents the
range in which all W36 sections flanges fall within. The nominal strength was calculated
for all the sections using the same five design procedures used for the W14 sections.
Figures B.13 though B.17 show similar plots to those shown in figures B.3 through B.7.
Figure B.13 shows a plot of the normalized nominal strengths, yn PP / , obtained by all of
66
the design methods versus the inverse of the flange reduction factor, sQ , for the chosen
W36 sections loaded in compression, with yield strength, 50=yf ksi. Figure B.14 shows
similar plots but with 100=yf ksi.
Figures B.15 through B.17 are similar to figures B.5 through B.7, showing plots of
the normalized nominal strengths, yn PP / , obtained by all different methods described
above versus the local slenderness cry ff / for the chosen W36 sections loaded in
compression for yield strengths of 50, 70, and 100 ksi’s, respectively, for the case of
varying the flange thickness.
1 1.002 1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 1.020
0.2
0.4
0.6
0.8
1
1.2
1/Qs
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.13 Normalized nominal strengths, yn PP / , versus the inverse of the flange reduction factor, sQ ,
for the chosen W36 sections, with a variable flange thickness, loaded in compression, 50=yf ksi.
67
1 1.05 1.1 1.15 1.2 1.250
0.2
0.4
0.6
0.8
1
1.2
1/Qs
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.14 Normalized nominal strengths, yn PP / , versus the inverse of the flange reduction factor, sQ ,
for the chosen W36 sections, with a variable flange thickness, loaded in compression, 100=yf ksi.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.15 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable flange thickness, loaded in compression, 50=yf ksi
68
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.16 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable flange thickness, loaded in compression, 70=yf ksi
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
Figure B.17 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable flange thickness, loaded in compression, 100=yf ksi
Interesting, for this parameter stuffy, as shown in Figure B.17, the AISC design
procedure starts “looping back” when using the fk values back-calculated from the
finite strip analysis. For the same critical stress, cry ff / , gives multiple strength
69
values for the nominal strength (due to the methods dependence on the htw/Ag ratio in
addition to local slenderness). This “looping” results from the aQ term in the AISC
design procedure calculations. Figure B.18 shows the effect of the aQ term in a plot
between the reduction factors, aQ and sQ , versus the local slenderness, cry ff / .
0.8 0.85 0.9 0.95 1 1.050.65
0.7
0.75
0.8
0.85
0.9
0.95
1
(fy/fcr)0.5
Q
Qa
Qs
Q
Figure B.18 Reduction factor, Q, versus a local slenderness, cry ff / , for the chosen W36 sections, with a
variable flange thickness, loaded in compression, 100=yf ksi.
W36 section with varied web thickness To check the effect of the web thickness, similar calculations were made on the
W36 sections where, as used for the flange effect calculations, the web height, h, was
fixed to 33.0” and the flange width, b, was fixed to 13.5”, which are the average values
for all the W36 sections. Now, flange thickness, ft , was set to 1.83” which also is the
average value of all the W36 sections, while the web thickness, wt , was varied over a
range between 0.6” to 2.4” which represents the range in which all W36 sections web
70
thicknesses fall within. The nominal strength was also calculated for all the sections
using the same five design procedures mentioned before.
Figures B.20 though B.24 show similar plots to those shown in figures B.8 through
B.12. Figure B.20 shows a plot of the normalized nominal strengths, yn PP / , obtained by
all different methods described above versus the inverse of the web reduction factor,
aQ , for the chosen W36 sections loaded in compression, with yield strength, 50=yf ksi.
Figure B.21 shows similar plots but with 100=yf ksi.
Figures 3.22 through 3.24 are similar to figures. B.10, through B.12, showing plots
of the normalized nominal strengths, yn PP / , obtained by all different methods
described above versus a local slenderness, cry ff / , for the chosen W36 sections
loaded in compression for yield strengths of 50, 70, and 100 ksi’s, respectively, for the
case of varying the flange thickness. Again, it is clear from the figures that the different
design methods give significantly different results, and that the differences become
more significant for higher yield strengths.
71
1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10.8
0.85
0.9
0.95
1
1.05
1.1
1/Qa
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.19 Normalized nominal strengths, yn PP / , versus the inverse of the web reduction factor, aQ , for
the chosen W36 sections, with a variable web thickness, loaded in compression, 50=yf ksi.
1 1.05 1.1 1.150.8
0.85
0.9
0.95
1
1.05
1.1
1/Qa
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.20 Normalized nominal strengths, yn PP / , versus the inverse of the web reduction factor, aQ , for
the chosen W36 sections, with a variable web thickness, loaded in compression, 100=yf ksi.
For the case of web thickness variation of the W36 sections, it is clear from the
figures that the different design methods give significantly different results. It is also
noticed that, for the same aQ value, the AISC and AISI methods gave results that are
72
close in value to each other for the case of using the 0.5=wk and also for the case of
using the wk values obtained from the finite strip analysis but the two cases gave
significantly different results, which keeps diverging as aQ decreases. It is believed that
the real behavior follows some intermediate trend, which in this case, is closer to the
DSM results. It is also noticeable from the figures that unlike with the flange reduction
factor, sQ , for the case of the web reduction factor, aQ , the normalized strength, yn PP / ,
sharply drops as the aQ value decreases.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.20
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.21 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable web thickness, loaded in compression, 50=yf ksi.
73
0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.22 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable web thickness, loaded in compression, 70=yf ksi.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
0.2
0.4
0.6
0.8
1
1.2
(fy/fcr)0.5
Pn/P
y
AISC (kw =5.0)
AISI (kw =5.0)
DSM (FSM)AISC (kw FSM)
AISI (kw FSM)
Figure B.23 Normalized nominal strengths, yn PP / , versus a local slenderness, cry ff / , for the chosen
W36 sections, with a variable web thickness, loaded in compression, 100=yf ksi.
74
C Appendix: Educational materials (PowerPoint slides)
Tutorial 1:Cross-section stability of a W36x150
Learning how to use and interpret finite strip method results for cross-section stability of hot-rolled steel members
prepared by Ben Schafer, Johns Hopkins University, version 1.0
Acknowledgments
• Preparation of this tutorial was funded in part through the AISC faculty fellowship program.
• Views and opinions expressed herein are those of the author, not AISC.
Learning objectives• Identify all the buckling modes in a W-section
– For columns explore flexural (Euler) buckling and local buckling– For beams explore lateral-torsional buckling and local buckling
• Predict the buckling stress (load or moment) for identified buckling modes
• Learn the interface of a simple program for exploring cross-section stability of any AISC section and learn finite strip method concepts such as
– half-wavelength of the bucking mode– buckling load factor associated with the applied stresses
Going further with other tutorials...• Show how changes in the cross-section
– change the buckling modes– change the buckling stress (load or moment)
• Explore the provided WT, C, L, HSS sections..• Exploring higher modes, and the interaction of buckling modes• Understand how the results relate to the AISC Specification
Start CUFSM
• The program may be downloaded from www.ce.jhu.edu/bschafer/cufsm
• Instructions for initializing the program are available online
select the input page
load a file
select W36x150(these files are available online where you down-loaded this tutorial)
questionmarks givemore info...
nodeelement
The geometry is defined by nodesand elements, youcan change these as you like, here aW36x150 is shown
Each element hasmaterial propertiesassociated with itin this example E is29000 ksi, and is0.3. (Each element also has a thickness)
the model is evaluatedfor many different“lengths” this allowsus to explore all the buckling modes, moreon this soon.
select propertiesbasic properties of the cross-section,you can compare themwith the AISC manualthey will be close, buthere we use a straightline model – so theywon’t be identical.
advanced note:these properties are
provided for convenience,but the program does notactually use them to calculate the buckling behavior of the section,instead plate theory is used throughout tomodel the section.
Let’s explore one of the ways we can apply loads
enter 1 here
uncheck this box
press this button to generate stress
max referencestress
appliedreferencemoment
Generated stressdistribution
when done, goback to the inputpage
this last column ofthe node entriesreflects the appliedreference stress.
now, go backto the propertiespage
put compression of1 ksi on ths section,enter 1, uncheck Mxxgenerate stress –should get thisdistribution...
go back to the inputpage when you are done.
notestresses areall 1.0 now(+ = comp.)
analyze thesection
Finite strip analysis results – lots to takein here!
buckled shape, herewe can figure out whattype of buckling modewe are looking at, is itlocal? global? etc.
half-wave vs.load factor plothere we find thebuckling load andwe find the critical buckling lengths...
undeformed shape
buckled shape
the little red dottells you where you are
at half-wavelength = 22.6and load factor = 48.7
explaining load factor and half-wavelength
move the little red dotto the minimum on the curve with thesecontrols, then selectplot shape and youwill get this bucklingmode shape result.
Local buckling
How do you know this is local buckling?Where is flange local buckling?Where is web local buckling?
In the beginning, looking at the buckledshape in 3D can help a lot...
select (and be patient)
web and flange local buckling is shown
remember, appliedload is a uniform compressive stressof 1.0 ksi
let’s rotate this section so we cansee the bucklingfrom the end onview.
buckled shape at“midspan” of the half-wavelength, thisis the 2D buckledshape
Go back to 2D nowand see if the shapemakes more sense...
we call this local bucklingbecause the elements whichmake up the section aredistorted/bent in-plane.
Also, the half-wavelength ismuch shorter than typicalphysical member length, infact the half-wavelength is less than the largest dimensionof the section (this is typical).
At what stress orload is this elastic local buckling predicted to occur at?
our referenceload of 42.6 k
or, equivalentlyour referencestress of 1.0ksi every-where..
you also can get a quick check on the applied stress by selectingthis plot within the post-processor.
Pref = 42.6 korfref = 1.0 ksi
load factor for localbuckling = 47.12
Pcr,local = 47.12 x 42.6= 2007 k
or
fcr,local = 47.12 x 1.0 ksi= 47.12 ksi
now let’s take a look at long half-wavelengths
change half-wavelengthto ~480” = 40ft and plotthe shape to get the result shown here. try out the
3D shape tobetter see thismode...
this is weak axis flexuralbuckling...
note that for flexuralbuckling the cross-section elements donot distort/bend, thefull cross-sectiontranslates/rotatesrigidly in-plane.
Pref = 42.6 korfref = 1.0 ksi
load factor for globalflexural buckling = 7.6at 40 ft. length
Pcr = 7.6 x 42.6 k= 324 k
or
fcr = 7.6 x 1.0 ksi= 7.6 ksi
Column summary• A W36x150 under pure compression (a column)
has two important cross-section stability elastic buckling modes
• (1) Local buckling which occurs at a stress of 47 ksi and may repeat along the length of a member every 27 in. (it’s half-wavelength)
• (2) Global flexural buckling, which for a 40 ft. long member occurs at a stress of 7.6 ksi (other member lengths may be selected from the curve provided from the analysis results)
A W36x150 is really intended for beam applications more than columns, let’ssee how it behaves as a beam...
go back to thepropertiespage
enter a referencestress of 1.0 ksi
calculate
uncheck P
reference momentis 500.5 kip-in.
generate stress
check everythingout on the input page, you can even look at thestress dist.to double check..
then analyze
Results page...
move to the firstminimum to explorelocal buckling of thisbeam further
Local buckling..
Mcr,local = 231 x 500 kip-in.= 115,500 kip-in.= 9,625 kip-ft
fcr,local = 231 x 1.0 ksi= 231 ksi
compression
tension
tension helps stiffenthe bottom of the weband elevates local buckling a great deal.
local buckling half-wavelength is 25.6 in.,as shown here in the 3D plot of thebucklingmode
what about long half-wavelengths, say 40’?
Lateral-torsional buckling..
In-plane the cross-sectionremains rigid and onlyundergoes lateral translationand twist (torsion), as shownin this buckling mode shape
Lateral-torsional buckling..
Mcr = 15.8 x 500 kip-in.= 7,900 kip-in.= 660 kip-ft
fcr = 15.8 x 1.0 ksi= 15.8 ksi
also predicted by this classical formula:
Beam summary• A W36x150 under major-axis bending (a beam)
has two important cross-section stability elastic buckling modes
• (1) Local buckling which occurs at a stress of 231 ksi and may repeat along the length of a member every 26 in. (it’s half-wavelength)
• (2) Global lateral-torsional buckling, which for a 40 ft. long member occurs at a stress of 15.8 ksi(other member lengths may be selected from the curve provided from the analysis results)
Learning objectives• Identify all the buckling modes in a W-section
– For columns explore flexural (Euler) buckling and local buckling– For beams explore lateral-torsional buckling and local buckling
• Predict the buckling stress (load or moment) for identified buckling modes
• Learn the interface of a simple program for exploring cross-section stability of any AISC section and learn finite strip method concepts such as
– half-wavelength of the bucking mode– buckling load factor associated with the applied stresses
Going further with other tutorials...• Show how changes in the cross-section
– change the buckling modes– change the buckling stress (load or moment)
• Explore the provided WT, C, L, HSS sections..• Exploring higher modes, and the interaction of buckling modes• Understand how the results relate to the AISC Specification
Tutorial 2:Cross-section stability of a W36x150
Exploring higher modes, and the interaction of buckling modes
prepared by Ben Schafer, Johns Hopkins University, version 1.0
Acknowledgments
• Preparation of this tutorial was funded in part through the AISC faculty fellowship program.
• Views and opinions expressed herein are those of the author, not AISC.
Target audience
• This tutorial is targeted at the advanced undergraduate/beginning graduate level. Some familiarity with structural stability is assumed in the provided discussion.
• It is also assumed that Tutorial #1 has been completed and thus some familiarity with the use of CUFSM is assumed.
Learning objectives• Understand the role of “higher” buckling modes
in the analysis of a W-section, including– how higher buckling modes relate to strong-axis,
weak-axis, and torsional buckling in columns– what higher buckling modes mean for local buckling– when knowledge of higher buckling modes may be
useful in design
• Understand how interaction of modes may be identified and quantified using CUFSM for a W-section
Summary of Tutorial #1• A W36x150 beam was analyzed using the
finite strip method available in CUFSM for pure compression and major axis bending.
• For pure compression local buckling and flexural buckling were identified as the critical buckling modes.
• For major axis bending local buckling and lateral-torsional buckling were identifies as the critical bucklig modes.
W36x150 column – review of Tutorial 1
web and flange local buckling is shown
remember, appliedload is a uniform compressive stressof 1.0 ksi
Pref = 42.6 korfref = 1.0 ksi
load factor for localbuckling = 47.12
Pcr,local = 47.12 x 42.6= 2007 k
or
fcr,local = 47.12 x 1.0 ksi= 47.12 ksi
this is weak axis flexuralbuckling...
note that for flexuralbuckling the cross-section elements donot distort/bend, thefull cross-sectiontranslates/rotatesrigidly in-plane.
Pref = 42.6 korfref = 1.0 ksi
load factor for globalflexural buckling = 7.6at 40 ft. length
Pcr = 7.6 x 42.6 k= 324 k
or
fcr = 7.6 x 1.0 ksi= 7.6 ksi
Tutorial #1: Column summary• A W36x150 under pure compression (a column)
has two important cross-section stability elastic buckling modes
• (1) Local buckling which occurs at a stress of 47 ksi and may repeat along the length of a member every 27 in. (it’s half-wavelength)
• (2) Global flexural buckling, which for a 40 ft. long member occurs at a stress of 7.6 ksi (other member lengths may be selected from the curve provided from the analysis results)
Higher modes
• We know global buckling of a column has more than one mode.. for instance the buckling can occur about the strong or weak axis:
• How is this reflected in CUFSM?
• Let’s investigate higher modes...
torsional buckling,mode 2 at 40 ft. istorsional buckling ofthe W36x150 at 21ksi.
a mid-height brace wouldremove weak-axis flexuralbuckling, but may still allowthis torsional buckling mode,so in some very specificsituations the higher moderesults could be quite useful.
strong axis flexural buckling, mode 3 at 40 ft. is strong axis flexural buckling ofthe W36x150 at 244ksi.
strong axis flexural buckling, mode 3 at 40 ft. is strong axis flexural buckling ofthe W36x150 at 244ksi.
what are these?
local buckling,mode 2 has a half-wavelength of 15.5 in.and a buckling stressof 157 ksi.
a mid-height brace wouldremove mode 1 local buckling, but may still allowthis mode 2 local mode,so in some very specificsituations the higher moderesults could be quite useful.
Higher modes summary• At every half-wavelength investigated many buckling
modes are revealed – the lowest of which is the 1st
mode.• Higher modes are those buckling modes at a given half-
wavelength that have higher buckling stresses (load or moment) than the 1st mode.
• Higher buckling modes become more important as bracing and different boundary conditions are considered.
• In this W36x150 example it may be of surprise to some that torsional buckling may be the limiting global buckling mode when weak-axis flexural buckling is restricted
W36x150 beam – review of Tutorial 1
Local buckling..
Mcr,local = 231 x 500 kip-in.= 115,500 kip-in.= 9,625 kip-ft
fcr,local = 231 x 1.0 ksi= 231 ksi
compression
tension
tension helps stiffenthe bottom of the weband elevates local buckling a great deal.
Lateral-torsional buckling..
Mcr = 15.8 x 500 kip-in.= 7,900 kip-in.= 660 kip-ft
fcr = 15.8 x 1.0 ksi= 15.8 ksi
also predicted by this classical formula:
Tutorial #1: Beam summary• A W36x150 under major-axis bending (a beam)
has two important cross-section stability elastic buckling modes
• (1) Local buckling which occurs at a stress of 231 ksi and may repeat along the length of a member every 26 in. (it’s half-wavelength)
• (2) Global lateral-torsional buckling, which for a 40 ft. long member occurs at a stress of 15.8 ksi(other member lengths may be selected from the curve provided from the analysis results)
Interaction of modes• In the analysis so far we have looked at
two half-wavelengths:– 26 in.: which is the first minimum in the curve
and exhibits local buckling– 480in. or 40 ft.: which is the longest length
investigated & exhibits lateral-torsional buckling
• What happens at other half-wavelengths? How about at 100in.?
local and lateral-torsional bucklinginteracting... (this result is at a lowerstress/moment thanjust lateral-torsionalbuckling)
explore lengthsin here you willsee localbuckling
explore out here youwill see lateral-torsionalhere we
see a mix
constrained Finite Strip Method
• Using the constrained Finite Strip Method (cFSM) we can formalize our predictions of modal interactions.
• The cFSM was developed and implemented by Schafer and Adany and can be explored in CUFSM
• Let’s run an analysis with cFSM on and examine the modal interactions..
go back tothe inputpage
turn on cFSM
analyze
select classify...
cFSM modal classificationresults at half-wavelengthof 100 in. As given,70% lateral-torsional21% local9% other
(other buckling modesprimarily include shear effects)
Local
Global
click here next
cFSM decomposition
• One of the useful features of cFSM is the ability to focus on only one type of buckling mode at a time, for example, local buckling..
uncheck Globaland Other andanalyze the section.
Local only,result aftercFSM analysisand classification
some more detailed thoughts about when these interactions matter
conclusion? interaction of the buckling modes is only of seriousconcern from an FSM analysis when it is identified on the downwardslope of a traditional (m=1, single half-wave) finite strip result.
It should also be noted that interactions such as those identified herein these elastic buckling results are not considered in the AISC Spec.
Tutorial 3:Exploring how cross-section changes
influence cross-section stability
an extension to Tutorial 1
prepared by Ben Schafer, Johns Hopkins University, version 1.0
Acknowledgments
• Preparation of this tutorial was funded in part through the AISC faculty fellowship program.
• Views and opinions expressed herein are those of the author, not AISC.
Target audience
• This tutorial is targeted at the under-graduate level.
• It is also assumed that Tutorial #1 has been completed and thus some familiarity with the use of CUFSM is assumed.
Learning objectives
• Study the impact of flange width, web thickness, and flange-to-web fillet size on a W-section
• Learn how to change the cross-section in CUFSM
• Learn how to compare analysis results to study the impact of changing the cross-section
Summary of Tutorial #1• A W36x150 beam was analyzed using the
finite strip method available in CUFSM for pure compression and major axis bending.
• For pure compression local buckling and flexural buckling were identified as the critical buckling modes.
• For major axis bending local buckling and lateral-torsional buckling were identifies as the critical buckling modes.
W36x150 column – review of Tutorial 1
web and flange local buckling is shown
remember, appliedload is a uniform compressive stressof 1.0 ksi
Pref = 42.6 korfref = 1.0 ksi
load factor for localbuckling = 47.12
Pcr,local = 47.12 x 42.6= 2007 k
or
fcr,local = 47.12 x 1.0 ksi= 47.12 ksi
this is weak axis flexuralbuckling...
note that for flexuralbuckling the cross-section elements donot distort/bend, thefull cross-sectiontranslates/rotatesrigidly in-plane.
Pref = 42.6 korfref = 1.0 ksi
load factor for globalflexural buckling = 7.6at 40 ft. length
Pcr = 7.6 x 42.6 k= 324 k
or
fcr = 7.6 x 1.0 ksi= 7.6 ksi
Tutorial #1: Column summary• A W36x150 under pure compression (a column)
has two important cross-section stability elastic buckling modes
• (1) Local buckling which occurs at a stress of 47 ksi and may repeat along the length of a member every 27 in. (it’s half-wavelength)
• (2) Global flexural buckling, which for a 40 ft. long member occurs at a stress of 7.6 ksi (other member lengths may be selected from the curve provided from the analysis results)
Modifying the cross-section• Once we start changing the depth, width,
thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability.
• Let’s– see what happens when the web thickness is set
equal to the flange thickness– see what happens when the flange width is reduced
by 2 inches.
Modifying the cross-section• Once we start changing the depth, width,
thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability.
• Let’s– see what happens when the web thickness is set
equal to the flange thickness– see what happens when the flange width is reduced
by 2 inches.
load up the defaultW36x150
change the webthickness to 0.9 in
the model shouldlook like this now.
default post-processorresults, change thehalf-wavelength to thelocal buckling minimum
local buckling at astress of 84.6 ksi
let’s save this fileand load up the originalfile, so we can compare.
load the actualW36x150
now we can readily seethat the local bucklingstress increases from47 ksi to 85 ksi.
(Advanced note: if one was usingplate theory the prediction wouldbe that the buckling stress should increase by (new thickness/old thickness)2
but the increase is slightly less here becausethe web and flange interact – somethingthat finite strip modeling includes.)
At longer length the section with the thickerweb buckles at slightlylower stress, this reflectsthe increased area, withlittle increas in momentof inertia that results withthis modification.
W36x150 @ 40’fcr= 7.6 ksiPcr= 324 k“W36x150” w/ tw=tffcr=6.2 ksiPcr=328 k
Modifying the cross-section• Once we start changing the depth, width,
thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability.
• Let’s– see what happens when the web thickness is set
equal to the flange thickness– see what happens when the flange width is reduced
by 2 inches.
Modifying the cross-section...
The W36x150 we have been studying in local buckling is largely dominated bythe web. Do the fillets at the ends of the web help things at all?
Let’s make an approximate model to look into this effect.
Load up the W36x150 modeland go to theinput page.
Let’s divide upthese elementsso that we can increase the thickness of theweb, near the flange to approx-imate the role ofthe fillet.
now divide element 5 at0.2 of its length..
the model shouldlook this this now,let’s change thethickness ofelements 5 andelements 10 to 2tw=2x0.6=1.2in.
save this result, so that wecan load up earlier resultsand compare them. After hitting save above I namedmy file “W36x150 withapprox fillet” this now showsup to the left and in the plotbelow.
next, let’s load theoriginal centerline model W36x150...
After loading “W36x150”now I have two files ofresults and I can seeboth buckling curves andmay select either buckingmode shape.
Let’s change the axis limitsbelow to focus more on local buckling..
the reference stress is 1.0 ksi, thefillet increases local buckling from47 ksi to 54 ksi, a real change in this case.
of course global flexuralbuckling out in this rangechanges very little sincethe moment of inertiachanges only a smallamount when the filletis modeled
Other modifications...• Change the web depth and explore the
change in the buckling properties• Add a longitudinal stiffener at mid-depth of
the web and explore• Modify the material properties to see what
happens if your W-section is made of plastic or aluminium, etc.
• Add a spring (to model a brace) at different points in the cross-section