Schafer AISC Fac Fellow Progress Report 1 - June 2007 AISC Fac Fellow...25 June 2007 To: Tom...

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

4

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.

10

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:

12

( )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

13

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:

22

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


flang

e, k

f

W-sections

0 2 4 6 8 10 120

10

20

30

40


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.

16

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

18

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.

20

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.

22

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


(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


flange becomes partially effective

transitioning through AISC Qs equations


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




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


Pn/P

y





(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


Pn/P

y





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


Pn/P

y

Flange local slenderness (fy/fcrb)0.5 = 2




(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


Pn/P

y

Web local slenderness (fy/fcrh)0.5 = 0.1




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


Pn/P

y





(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


Pn/P

y





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


Pn/P

y

Web local slenderness (fy/fcrh)0.5 = 2




(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


Pn/P

y

Local slenderness (fy/fcrlocal)

0.5 = 0.1 AISC (max htw /Ag)

AISC (min htw /Ag)


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


Pn/P

y



AISC (min htw /Ag)


(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


Pn/P

y



AISC (min htw /Ag)


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


Pn/P

y


0.5 = 2 AISC (max htw /Ag)

AISC (min htw /Ag)


(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


flang

e, k

f

WT-sections

2 3 4 5 6 7 8 91

1.2

1.4

1.6

1.8


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


flang

e, k

f

WT-sections

0 2 4 6 8 10 121

1.5

2

2.5


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.


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)


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)


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)


AISI (kf FSM)



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)


AISI (kf FSM)



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.


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)


AISI (kw FSM)



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)


AISI (kw FSM)


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)


AISI (kf FSM)



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)


AISI (kw FSM)



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)


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)


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)


AISI (kf FSM)



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)


AISI (kf FSM)



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)


AISI (kf FSM)



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)


AISI (kw FSM)



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)


AISI (kw FSM)



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)


AISI (kw FSM)



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)


AISI (kw FSM)



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)


AISI (kw FSM)



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.


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


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)



• (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


Acknowledgments



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





Pcr,local = 47.12 x 42.6= 2007 k

or






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)




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.


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)



• (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


Acknowledgments



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





Pcr,local = 47.12 x 42.6= 2007 k

or






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)




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.





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





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

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