April 2010 AISC - Department of Civil Engineering Faculty Fellowship Progress... · April 2010 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 4 ‐ AISC

April 2010 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 4 ‐ AISC Faculty Fellowship Cross‐section Stability of Structural Steel

Tom, Please find enclosed the fourth 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 finite element parametric analysis reported herein focuses on local‐global interaction of buckling modes, and comparisons of the AISC, AISI – Effective Width, and AISI – Direct Strength design methods for locally slender columns and beams with variable lengths at preselected slenderness ratios. Sincerely,

Mina Seif ([email protected]) Graduate Research Assistant

Ben Schafer ([email protected]) Associate Professor

mailto:[email protected]

mailto:[email protected]

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 finite strip elastic buckling analysis 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

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learning objectives, step‐by‐step instructions, and complementary homework problems for students.

Publications:

Schafer, B.W., Seif, M., “Comparison of Design Methods for Locally Slender Steel Columns” SSRC Annual Stability Conference, Nashville, TN, April 2008.

Progress Report #2 April 2008

Completed work:

• Performed axial, positive and negative major axis bending, and positive and negative minor axis bending finite strip elastic cross‐section buckling stability analysis on all the sections in the AISC (v3) shapes database using the finite strip elastic buckling analysis software CUFSM.

• Evaluated and determined simple design formulas that include web‐flange interaction for local plate buckling coefficients of all structural steel section types.

• Performed ABAQUS finite element elastic buckling analyses on W‐sections, comparing and assessing a variety of element types and mesh densities.

• Initiated an ABAQUS nonlinear finite element analysis parameter study on W‐section stub columns, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations.

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Publications:

Seif, M., Schafer, B.W., “Elastic Buckling Finite Strip Analysis of the AISC Sections Database and Proposed Local Plate Buckling Coefficients” Structures Congress, Austin, TX, April 2009.


Completed work:

• Studied the influence of the variation of some design parameters on the ultimate strength of W‐section steel stub columns; further understanding, highlighting, and quantifying the uncertainties of parameters that lead to the divergence of the columns strength than what one might typically expect.

• Performed an ABAQUS nonlinear finite element analysis parameter study on W‐section stub columns, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations.

• Performed a similar nonlinear finite element analysis parameter study on W‐section short beams, assessing and comparing results to the strengths predicted by AISC, AISI, and DSM beam equations.

• Initiated a nonlinear finite element analysis parameter study for columns with variable lengths at preselected slenderness ratios, as a step towards the completion of a database that will allow extension of the Direct Strength Method to hot‐rolled steel sections.

Publications:

Seif, M., Schafer, B.W., “Finite element comparison of design methods for locally slender steel beams and columns” SSRC Annual Stability Conference, Phoenix, AZ, April 2009.

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Completed work:

• Studied the influence of the variation of some design parameters on the ultimate strength of W‐section steel long columns and beams where local and global buckling modes interact; further understanding, highlighting, and quantifying the uncertainties of parameters that lead to the divergence of the columns strength than what one might typically expect.

• Performed an ABAQUS nonlinear finite element analysis parameter study on W‐section columns with variable lengths at preselected slenderness ratios, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations.

• Performed a similar nonlinear finite element analysis parameter study on W‐section beams with variable lengths at preselected slenderness ratios, assessing and comparing results to the strengths predicted by AISC, AISI, and DSM beam equations.

• Started studying the stress and strain distributions in all the sections analyzed in the nonlinear finite element analysis parameter study database as a final step towards the extension of the Direct Strength Method to hot‐rolled steel sections.

Publications:

Seif, M., Schafer, B.W. “Elastic Local Buckling of Structural Steel Shapes.” Journal of Constructional Steel Research (JCSR), doi:10.1016/j.jcsr.2010.03.015. Seif, M., Schafer, B.W., “Design methods for local-global interaction of locally slender steel members” SSRC Annual Stability Conference, Orlando, FL, May 2010. Seif, M., Schafer, B.W., “Cross-sectional Stability of Structural Steel.” International Conference of Stability and Ductility of Steel Structures (SDSS) Proceedings, Rio de Janeiro, Brazil, September 2010, In Press.

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Table of Contents

Summary of Progress .......................................................................................................2

1 Introduction..................................................................................................................8

2 Finite Element Comparison of Design Methods for Local-Global Interaction of

Locally Slender Steel Beams and Columns .......................................................................12

2.1 Introduction and Motivation ..............................................................................12 2.2 Design Methods and Equations .........................................................................13 2.3 Parameter Study and Modeling..........................................................................15 2.3.1 Approach..................................................................................................15 2.3.2 Geometric Variation: Element Local Slenderness.....................................16 2.3.3 Geometric Variation: Member Length.......................................................19 2.3.4 Finite Element Modeling ...........................................................................20

2.3.4.1 Mesh and element selection .................................................................. 20 2.3.4.2 Material modeling................................................................................. 21 2.3.4.3 Residual stresses ................................................................................... 22 2.3.4.4 Geometric imperfections....................................................................... 23

2.4 Results................................................................................................................26 2.4.1 Columns ...................................................................................................27 2.4.2 Beams ........................................................................................................32

2.5 Discussion..........................................................................................................37 2.5.1 Columns ...................................................................................................37 2.5.2 Beams ........................................................................................................38 2.5.3 Overall ......................................................................................................42

2.6 Summary and Conclusion ..................................................................................43

3 Strain Distribution in Locally Slender Structural Steel Cross-Sections ....................45

3.1 Introduction........................................................................................................45

4 References..................................................................................................................51

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Appendix A : NRC Research Proposal..............................................................................53

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1 Introduction The research work presented in this progress report represents a continuing

effort towards a fuller understanding of hot‐rolled steel cross‐sectional local

stability. Typically, locally slender cross‐sections are avoided in the design of

hot‐rolled steel structural elements, but completely avoiding local buckling

ignores the beneficial post‐buckling reserve that exists in this mode. With the

appearance of high and ultra‐high yield strength steels this practice may become

uneconomical, as the local slenderness limits for a section to remain compact are

function of the yield stress. Currently, the AISC employs the Q‐factor approach

when slender elements exist in the cross‐section, but analysis in Progress Report

#1 indicates geometric regions where the Q‐factor approach may be overly

conservative, and other regions where it may be moderately unconservative as

well. It is postulated that a more accurate accounting of web‐flange interaction

will create a more robust method for the design of high yield stress structural

steel cross‐sections that are locally slender.

Progress Report #1 summarized how the locally slender W‐section column

design equations from the AISC Q‐factor approach, AISI Effective Width

Method, and AISI Direct Strength Method (DSM) can be reformulated and

arranged into a common set of notation. This common notation highlights the

central role of cross‐section stability in predicting member strength.

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Progress Report #2, provided results of finite strip elastic cross‐section

buckling analysis performed on all the sections in the AISC (v3) shapes database

(2005) under: axial, positive and negative major‐axis bending, and positive and

negative minor‐axis bending. The results were used to evaluate the plate local

buckling coefficients underlying the AISC cross‐section compactness limits (e.g.,

bf/2tf and h/tw limits). In addition, the finite strip results provided the basis for the

creation of simple design formulas for local plate buckling that include web‐

flange interaction, and better represent the elastic stability behavior of structural

steel sections, for all different loading types. Those design formulas are

essentially a proposed replacement for the AISC’s Table B4.1 which defines the

slenderness limits.

Progress Report #2 also provided a comparison and assessment of the

different two‐dimensional shell elements which are commonly used in modeling

structural steel. The assessment is completed through finite element elastic

buckling analysis performed on W‐sections using a variety of element types and

mesh densities in the program ABAQUS. The concluding section of that report

discussed the initiation of a finite element parameter study (performed in

ABAQUS) on W‐section stub columns.

Progress Report #3, provided a finite element reliability analysis study on

hot rolled W‐sectioned structural steel columns. The study aimed to assess the

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influence of the variation of some design parameters on the ultimate strength of

such type of members; further understanding, highlighting, and quantifying the

uncertainties of parameters that lead to the divergence of the columns strength

than what one might typically expect.

Progress Report #3 also presented and discussed a nonlinear finite element

analysis parameter study (performed in ABAQUS) on W‐section stub columns

and short beams. The study aimed to highlight the parameters that lead to the

divergence of the section strength capacity predictions, provided by the different

design methods: AISC, AISI, and DSM design equations.

The first part of this document, Progress Report #4, discusses the extension

of the parameter study presented in Progress report #3 to include longer columns

and beams, thus including global buckling modes and the effect of local‐global

mode interactions. The columns and beams in this study have variable lengths at

preselected slenderness ratios. This extension leads to a further completion of a

database of failure mechanisms of W-sections at different element slenderness

ratios.

The second part of this report describes the current ongoing work where

the strain distributions observed in the failure mechanisms of the FE parameter

study database are closely examined, and compared to finite strip analysis

results using CUFSM, as well as to theoretical distributions. This will allow us to

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utilize the elastic buckling information, for cross‐sections with large variations in

element slenderness, and ultimately propose improvements to DSM so it may be

applied to hot‐rolled structural steel with locally slender cross‐sections.

The Appendix of this report shows a copy of a research proposal titled

“Multi‐scale Structural Stability under Realistic Fire Loading”. The proposal was

submitted to the National Research Council (NRC), as part of a post‐doctoral

fellowship application, and it aims to extend this research where the effect of

realistic fire loading scenarios on locally slender structural steel members will be

studied.

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2 Finite Element Comparison of Design Methods for Local-Global Interaction of Locally Slender Steel Beams and Columns

2.1 Introduction and Motivation

With the advent of high and ultra‐high yield strength steels, the increased

yield stress drives even standard hot‐rolled steel shapes from locally compact to

locally slender (noncompact or slender), making it inefficient to avoid such cross‐

sections in the design of hot‐rolled steel structural members (see Seif and Schafer

2009a and 2009b for details). Efficient and reliable strength predictions are

needed for locally slender hot‐rolled steel cross‐sections. Analysis of existing

AISC (2005) provisions for locally slender stub columns and short beams (Seif

and Schafer 2009a) indicated geometric regions where AISC design may be

excessively conservative, and other regions where it may be moderately

unconservative. The work on the stub columns and short beams isolated and

studied the effect of local buckling modes on the predicted strength. However,

most failures occur do to combinations and interactions between local and global

buckling modes (see Figure 2-15 and Figure 2-22). The work herein represents a

direct extension of previous studies on stub columns and short beams (Seif and

Schafer 2009b and 2009c) now to include long columns and long beams, where

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the locally slender cross‐sections may interact with global (flexural, lateral‐

torsional, etc.) buckling modes.

2.2 Design Methods and Equations

The design of locally slender steel cross‐sections may be completed by a

variety of methods, three of which are examined here: (1) The hot‐rolled steel

AISC method, as embodied in the 2005 AISC Specification, labeled AISC herein,

(2) The AISI Effective Width Method from the main body of the 2007 AISI

Specification for cold‐formed steel, labeled AISI herein, and, (3) The Direct

Strength Method as given in Appendix 1 of the 2007 AISI Specification, labeled

DSM herein.

For each of these three design methods the expressions for strength

prediction of locally slender braced columns and beams have been provided in a

common notation in Seif and Schafer (2008, 2009b, and 2009c). In those equations

the centrality of elastic local buckling is made clear. For long (unbraced) columns

and beams global buckling must be considered as well as local‐global interaction.

In AISC, AISI, and DSM global column buckling is predicted using the

same (single) expression. However, local‐global interaction is handled by the Q‐

factor method in AISC, the unified method in AISI, and a variation of the unified

method in DSM. In all cases the global strength is reduced due to local cross‐

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section slenderness. The Q‐factor approach reduces the strength and increases

the long‐column slenderness to arrive at its reduction. The unified method uses

the effective area of the column at the long column buckling stress. DSM uses a

similar approach, but the effective area calculation is replaced by a reduction of

the full cross‐section (at the long column strength). Figure 2-1 shows the effect of

how the AISC and AISI differently handle the predicted strength reduction due

to global slenderness. Also it shows the column global slenderness regions that

were covered in Progress Report #3 and Progress Report #4.

cλ

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

stub

cλ

/n yP P

Progress Report #3Local modes

Progress Report #4Local-Global modes interaction

cλ

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

stub

cλ

/n yP P

Progress Report #3Local modes

Progress Report #4Local-Global modes interaction

Figure 2-1 Effect of global column slenderness on predicted strength, and regions covered in the parametric studies of progress reports #3 and #4

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AISC and AISI/DSM use different formats for the global (lateral-torsional

buckling) provisions of beams. However, for no moment gradient (Cb = 1) the

resulting expressions are actually quite similar with the exception that AISI only

provides capacities up to first yield (My) for sections subject to lateral-torsional

buckling. For AISI/DSM local-global interaction in beams is treated in the same

conceptual manner as for columns; not so for AISC, which uses nothing like the

Q-factor approach, and instead provides direct reductions based on the flange

and web plate slenderness (also see White 2008). A result of AISC’s approach in

not adopting one consistent philosophy for local-global interaction in beams is

some unusual changes in strength as local slenderness is varied.

2.3 Parameter Study and Modeling

2.3.1 Approach The purpose of the nonlinear finite element (FE) analysis parameter study

studied herein is the understanding and highlighting of the parameters that lead

to the divergence between the capacity predictions of the different design

methods under axial and bending loads.

Previous FE analysis (Seif and Schafer 2009b and 2009c) was conducted on

stub (short) members, avoiding global (i.e., flexural, or lateral-torsional) buckling

modes, and focusing on local buckling modes alone. The length of the studied

members was determined according to the stub column definitions of SSRC (i.e.,

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Galambos 1998). The FE analysis herein is extended to longer members, thus

including global buckling modes where the interaction between local and global

modes is allowed and examined. The columns and beams in this study have

variable lengths at preselected slenderness ratios. Based on the authors

judgment, AISC W14 and W36 sections are selected for the study as representing

“common” sections for columns and beams in high-rise buildings. The W14x233

section is approximately the average dimensions for the W14 group and the

W36x330 for the W36 group. All sections are modeled with globally pinned,

warping fixed boundary conditions, and loaded via incremental displacement or

rotation for the columns and beams respectively.

2.3.2 Geometric Variation: Element Local Slenderness To examine the impact of slenderness in the local-global buckling mode

interaction, and the impact of web-flange interaction in I-sections, four series of

parametric studies are performed under axial and bending loading at preselected

slenderness ratios:

• W14FI: a W14x233 section with a modified Flange thickness, that

varies Independently from all other dimensions,

• W14FR: a W14x233 section with variable Flange thickness, but the

web thickness set so that the Ratio of the flange-to-web thickness

remains the same as the original W14x233,

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• W36FR: a W36x330 section with variable Web thickness, but the

flange thickness set so that the Ratio of the flange-to-web thickness

remains the same as the original W36x330, and

• W36WI: a W36x330 section with a variable Web thickness, that

varies Independently from all other dimensions,

as summarized in Table 2-1 and Figure 2-2. Figure 2-2 indicates that for the

W14FI group, the web slenderness is held constant (compact), while the flange

slenderness varies from compact to noncompact and slender. Similarly, for the

W36WI group, the flange slenderness is held constant (compact) while the web

slenderness is varied for compact to noncompact and slender. Finally the W14FR

and W36WR groups range a whole range of slenderness combinations.

Table 2-1 Parametric study of W-sections

bf/2tf h/tw h/bf tf/tw W14x233 4.62 13.35 0.90 1.61 W14FI varied fixed fixed varied W14FR varied varied fixed fixed W36x330 4.54 35.15 2.13 1.81 W36FR varied varied fixed fixed W36WI Fixed varied fixed varied

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0 50 100 1500

5

10

15

20

25

30

35

h/tw

b f/2t f

kf=0.1

kw

=36k

f=0.05

kw

=36

kf=0.5

kw

=29

kf=0.5

kw

=29

kf=0.5

kw

=27

kf=0.6

kw

=6.0k

f=1.2

kw

=0.5

kf=0.6

kw=5.6

kf=0.6

kw

=5.0

kf=0.9

kw

=2.1

λpw λrw

λpf

λrf

W36WI

W36FR

W14FR

W14FI

fy = 50 ksi (345 MPa)

Figure 2-2 Variation of parameters as a function of h/tw and bf/2tf with back-calculated elastic

buckling k values, and AISC λ limts for beams shown.

For the purpose of this study, element thicknesses were varied between 0.05

in. (1.27 mm) and 3.0 in. (76.2 mm). While not strictly realistic, the values chosen

here are for the purposes of comparing and exercising the design methods up to

and through their extreme limits. Local slenderness may be understood as the

square root of the ratio of the yield stress to the local buckling stress (i.e., √fy/fcr).

The element local buckling stress is proportional to the square of the element

thickness, thus the local slenderness is proportional to 1/t. Here element

thickness is varied and used as a proxy for investigating local slenderness, in the

future, material property variations are also needed.

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2.3.3 Geometric Variation: Member Length The initial FE analysis (Seif and Schafer 2009b and 2009c) was conducted on

stub (short) members, avoiding global (i.e., flexural, or lateral-torsional) buckling

modes, and focusing on local buckling modes alone. The length of the studied

members was determined according to the stub column definitions of SSRC (i.e.,

Galambos 1998), and fixed at that length.

To examine the impact of local-global buckling modes interaction on the

strength of locally slender members, longer members are included in the FE

parameter study taking the member’s length as a variable in the parameter

study. Each member’s length is determined so to achieve certain preset

slenderness parameter values, where the slenderness parameter, λ, is defined in

terms of the member’s length and cross-section dimensions.

For columns, two groups of analysis were chosen to be performed at axial

slenderness parameter, λc, values fixed at 0.90 and 1.50, where λc is defined as:

2euler

y yc

cr

f fKLf r

λπ

= =E (2-1)

Note that varying the thicknesses (flange, web, or both at constant ratio)

will vary the moment of inertia, I, and the cross-sectional area, A, and

accordingly the radius of gyration, r. The member’s length, L, is then back-

calculated to maintain the specified λc values.

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Similarly for beams, two groups of analysis were chosen to be performed at

flexural slenderness parameter, λe, values fixed at 0.60 and 1.34, which are the

AISI values defining the non-compact from the compact and slender members

respectively (see, e.g. Shifferaw and Schafer 2008). The λe is defined as:

ye

cre

MM

λ = (2-2)

Fixing λe, the critical buckling moment, Mcre, is calculated for each section.

Mcre is also defined as follows:

2 2

2 4y y

cre b

EI GJ EI CM C

L Lπ π π

= +2

w (2-3)

Again, note that varying the thicknesses (flange, web, or both at constant

ratio) will vary all the parameters on the right hand side of Eq. (3). Accordingly,

the member’s length, L, is then back-calculated to maintain the specified λe

values.

2.3.4 Finite Element Modeling

2.3.4.1 Mesh and element selection

ABAQUS was used to perform the analysis. Members were modeled using

S4 shell elements. The S4 element has six degrees of freedom per node, adopts

bilinear interpolation for the displacement and rotation fields, incorporates finite

membrane strains, and its shear stiffness is yielded by “full” integration.

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Considering computational speed and accuracy it was decided that a mesh

density of five elements across each flange outstand, ten across the web, and an

aspect ratio of 1 was adequate for this study. The choice of element type and

density are based on comparisons with three-dimensional solid elements as

reported in Seif and Schafer (2008 and 2009b). It is noted that some debate exists

in the literature regarding the selection of the S4 vs. S4R element (see, e.g. Dinis

and Camotim 2006, and Earls 2001).

2.3.4.2 Material modeling

The material model used is similar to that of Barth, K.E. et al. (2005). It

follows classical metal plasticity: Von Mises yield criteria, associated flow, and

isotropic hardening. The uniaxial σ-ε diagram is provided in Figure 2-3 is

defined for the finite element analysis as a multi-linear stress-strain response,

consisting of an elastic region, a yield plateau, and a strain hardening region. The

elastic region is defined by the modulus of elasticity, E, and the yield stress, fy.

The yield plateau is defined by a small slope of E’ ~ E/200, to help in avoiding

numerical instabilities during analysis. A strain hardening modulus Est = 145 ksi

which initiates at a strain of 0.011 was chosen. The curve shown in Figure 2-3 is

converted to a true stress-strain curve for the analysis.

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fu= 65

fy= 50

Eng

inee

ring

Str

ess (

ksi)

Engineering Strain

yε stε

Slope, E =29000

Slope, Est=720Slope, Est=720

Slope, E’=145

=0.011

fu= 65

fy= 50

Eng

inee

ring

Str

ess (

ksi)

Engineering Strain

yε stε

Slope, E =29000

Slope, Est=720Slope, Est=720

Slope, E’=145

=0.011

Figure 2-3 Idealized engineering stress‐strain curve used for analysis.

2.3.4.3 Residual stresses

For this work, the classic and commonly used distribution of Galambos and

Ketter (1959), as shown in Figure 2-4, is employed. Similar to other researchers

(e.g., Jung and White 2006) the residual stresses are defined in the finite element

analysis as initial longitudinal stresses, and given as the average value across the

element at its center. (See Seif and Schafer 2009b for further discussion).

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---

--

+

yc f3.0=σ

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

)2( fwff

ffct tdttb

tbσσ

----

---

--

+

yc f3.0=σ

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+=

)2( fwff

ffct tdttb

tbσσ

----

Figure 2-4 Residual stress distribution used for analysis as given by Galambos and Ketter (1959).

2.3.4.4 Geometric imperfections

Geometric imperfections have an important role to play in any collapse

analysis involving stability. For the previous work on short (stub) members, the

imperfections were defined by scaling the local buckling eigenmode from elastic

buckling analysis. Since the focus at this point is on longer members, global

buckling modes are also included. Initial geometric imperfections are added

through linearly superposing a scaled local and a scaled global eigenmode

solution from a finite strip analysis performed on each section, using CUFSM

(Schafer, B.W., Ádány, S. 2006). Figure 2-5 shows a typical CUFSM curve, where

the local and global buckling modes are determined. It is noted that the global

buckling mode is that at a half wave-length equal to the member’s unbraced

length, L, while the local buckling mode is at the minima of the curve. For the

purposes of this study, the local buckling is chosen as the mode closest to the

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minima that can fit as a whole number of half waves within the member’s

unbraced length. The local buckling mode as shown in Figure 2-6, is scaled so

that the maximum nodal displacement is equal to the greater of bf/150 or d/150

which is a commonly employed magnitude (see, e.g Kian and Lee 2002), while

the global buckling mode is scaled so that the maximum nodal displacement is

equal to L/1000, as shown in Figure 2-7.

100 101 102 1030

100

200

300

400

500

600

700

800

900

1000

Half wave length

Load

x

x

L

Local

Global

Figure 2-5 Typical CUFSM curve where local and global buckling modes are determined.

24

d

bf

bf /150

d /150

d /150

bf /150

(a)

(b)

(c)

d

bf

bf /150

d /150

d /150

bf /150

(a)

(b)

(c)

Figure 2-6 Typical local buckling mode and initial geometrical imperfections for the analysis (a) ABAQUS 3D view, (b) ABAQUS front view, and (c) CUFSM front view, with typical

scaling factors.

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L /1000

L /1000

(b)

(c)

(a) (a)

Figure 2-7 Typical global (flexural) buckling mode and initial geometrical imperfections for the analysis (a) ABAQUS 3D view, (b) ABAQUS front view, and (c) CUFSM front view, with

typical scaling factors.

2.4 Results

As discussed previously (see table 2-1 and Figure 2-2), the parametric study

is broken into 4 groups: W14FI, W14FR, W36FR, and W36WI analyzed at

different preset slenderness limits. Here the results of the parametric study are

presented for each group, including comparisons to the AISC, AISI, and DSM

26

design methods. Analysis results are provided first for the columns, then the

beams.

2.4.1 Columns ABAQUS results for the parametric study of locally slender long columns

(denoted with “· ” and given for the 4 parametric studies) are reported as a

function of long column slenderness (λc~0.25, 0.9, and 1.5) in Figure 2-8. In Figure

2-8 the standard (compact) W14 and W36 cross-sections have been denoted with

a “*”. If the long column curve is exact, the “*” would be in perfect agreement

with the upper curve shown. As can be observed, as the local slenderness is

increased the strength predictions fall further and further below the global

column (upper) curve, which for compact/fully-effective sections is identical in

AISC, AISI, and DSM. Also highlighted in Figure 2-8, so that a locally slender

section may be observed, is the cross-sections with a back-calculated Q or Aeff/Ag

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

strength curves for Q or Aeff/Ag = 0.7. Figure 2-8 does not allow for a complete

study of the impact of local slenderness as a full family of strength curves would

need to be generated and each point compared to a different curve. Rather than

do this, to compare all the sections in a given study the results are expressed as a

function of local slenderness (at a given global slenderness, λc).

27

0 0.5 1 1.50

0.5

1

W14FI

Pn/P

y

0 0.5 1 1.50

0.5

1

W14FR

0 0.5 1 1.50

0.5

1

W36FI

Pn/P

y

Lambdac

0 0.5 1 1.50

0.5

1

W36WI

Lambdac

Figure 2-8 ABAQUS results for the parametric study reported as a function of long column slenderness

Complete comparisons of the studied columns with the AISC, AISI, and

DSM methods are provided in Figure 2-9 through Figure 2-14. Figure 2-9 and

Figure 2-10 provide the summary of results for the stub column study of Seif and

Schafer (2009b and 2009c). In a similar manner, Figure 2-11 and Figure 2-13

present the results for each of the 4 parameter studies at λc=0.9 and λc=1.5

respectively. Figure 2-12 and Figure 2-14 present all 4 studies directly compared

against each of the design methods, for λc=0.9 and λc=1.5 respectively. All results

are plotted as a function of elastic local slenderness of the cross-section: √fy/fcrl,

28

determined by finite strip analysis. Finally, Figure 2-15 provides the deformed

shapes for a W14 section at λc=0.9 and λc=1.5. The figure shows the interaction

between the local and global (about the minor axis) buckling modes.

1 2 30

0.5

1

Pn/P

y

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

(fy/fcrl)0.5

Pn/P

y

W36FR

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS

Figure 2-9 Results of column parametric study for 4 study groups (stub)

29

1 2 30

0.5

1

Pn/P

y

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Pn/P

y

12300.51

AISCAISIDSMABAQUS

Figure 2-10 Results of column parametric study for 3 design methods (stub)

1 2 30

0.5

1

Pn/P

y

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Pn/P

y

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS

Figure 2-11 Results of column parametric study for 4 study groups (λc=0.9)

30

1 2 30

0.5

1

Pn/P

y

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Pn/P

y

1230

0.51

AISCAISIDSMABAQUS

Figure 2-12 Results of column parametric study for 3 design methods (λc=0.9)

1 2 30

0.5

1

Pn/P

y

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Pn/P

y

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS

Figure 2-13 Results of column parametric study for 4 study groups (λc=1.5)

31

1 2 30

0.5

1

Pn/P

y

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Pn/P

y

12300.51

AISCAISIDSMABAQUS

Figure 2-14 Results of column parametric study for 3 design methods (λc=1.5)

Figure 2-15 Deformed shapes for a W14FI section (a) λc=0.9, (b) λc=1.5

2.4.2 Beams For the beams the predicted capacities from the nonlinear collapse analysis

in ABAQUS are shown for each of the 4 parameter groups in Figure 2-16, Figure

2-18, and Figure 2-20; for the short specimens, intermediate length specimens at

λe=0.6, and long specimens at λe=1.34 respectively. Results are also compared

32

against the design methods directly in Figure 2-17, Figure 2-19, and Figure 2-21

for the same three lengths (short, intermediate, long). In all the preceding plots

the local slenderness √fy/fcr (or equivalently √My/Mcr) is plotted against the

capacity, normalized to the plastic moment, Mp. Finally, Figure 2-22 provides the

deformed shapes for a W36 section with a slender web at λe=0.6 and λe=1.34

(intermediate and long lengths); indicating the interaction between the local and

lateral-torsional buckling mode at failure.

33

1 2 30

0.5

1

Mn/M

p

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS

Figure 2-16 Results of beam parametric study for 4 study groups (short)

1 2 30

0.5

1

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Mn/M

p

12300.51

AISCAISIDSMABAQUS

Figure 2-17 Results of beam parametric study for 3 design methods (short)

34

1 2 30

0.5

1

Mn/M

p

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS

Figure 2-18 Results of beam parametric study for 4 study groups (λe=0.6)

1 2 30

0.5

1

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Mn/M

p

1 2 300.51

AISCAISIDSMABAQUS

Figure 2-19 Results of beam parametric study for 3 design methods (λe=0.6)

35

1 2 30

0.5

1

Mn/M

p

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS


1 2 30

0.5

1

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Mn/M

p

12300.51

AISCAISIDSMABAQUS


36

Figure 2-22 Deformed shapes for a W36WI section (a) λe=0.6, (b) λe=1.34

2.5 Discussion

The focus of the following discussion is the performance of the design

methods in comparison with the capacities predicted by the nonlinear finite

element analysis.

2.5.1 Columns Unlike the case of stub columns, where the AISI’s implementation of the

Effective Width Method provided, by far, the best prediction of the column

capacity, there isn’t a specific design method that outperforms the others when it

comes to predicting the capacity of longer columns. (Recall all methods use the

same global column curve, but reduce the strength in different manners to

account for local-global interaction.) For longer columns, similar to stub columns,

AISC provides reliable predictions when the flange is non-slender; however

AISC is unduly conservative whenever the flanges become slender (regardless of

the web). The level of conservatism is large enough to make AISC design with

37

slender flanges completely uneconomical. AISI works well in nearly all cases;

however, when the flange is specifically varied the unified method for reducing

the column capacity does not properly capture the reduction in global capacity

(through loss of I). DSM’s accuracy is excellent when the flange and web vary at

fixed ratios, and conservative (sometimes significantly) when one element is

markedly more slender than its neighbor.

2.5.2 Beams The AISC predictions are overall best characterized as conservative, often

excessively so when compared with the FE predictions. The strength prediction

as the web and flange move from compact, to non-compact, to slender often have

abrupt transitions as the related design methods use different formulae in these

different local slenderness ranges. For example, see the W36WI study at λe=0.6 of

Figure 2-18. In general the expressions related to local flange slenderness provide

smooth but quite conservative design predictions, while those related to local

web slenderness suffer from the abrupt transitions. The study shows that the

AISC expressions are essentially intended for compact, and semi-compact

sections; but for locally slender sections the results are safe, but unduly

conservative. An important proviso to this conclusion, particularly for long

beams, is that users must take care when utilizing the approximations provided

in AISC as in some cases the conservatism is derived from these approximations

as opposed to the fundamentals of the design approach itself. For example, the

AISC’s Equation F2-4 for lateral-torsional buckling stress is:

38

22

2 1 0.078bcr

x o tsb

ts

C E LJcFS h rL

r

π ⎛ ⎞⎛ ⎞= + ⎜ ⎟⎜ ⎟⎛ ⎞ ⎝ ⎠⎝ ⎠⎜ ⎟⎝ ⎠

b (2-4)

AISC allows the approximation of the term under the square root to be

taken equal to 1.0. that approximation is reasonable for compact sections.

However for very slender sections it blows up (~5.0 for sections in this study).

Figure 2-23 provides the change in AISC’s results for the W14FI (λe=1.34)

depending on whether or not the approximation suggested for the lateral-

torsional buckling stress (Eq. F2-4) is utilized – it is clear the use of this

approximation must be done with care.

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(fy/fcrl)0.5

Mn/M

p

AISCAISIDSMABAQUS

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(fy/fcrl)0.5

Mn/M

p

(a) (b)

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(fy/fcrl)0.5

Mn/M

p

AISCAISIDSMABAQUS

0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

(fy/fcrl)0.5

Mn/M

p

(a) (b)

Figure 2-23 Beam results of W14FI study group at λe=1.34; (a) AISC without Eq. F2-4’s approximation, (b) AISC with Eq. F2-4’s approximation

Figure 2-24 through Figure 2-27 are re-presentations of Figure 2-18 through

Figure 2-21, but with the exact computation of the lateral-torsional buckling

stress (Eq. F2-4) utilized.

39

1 2 30

0.5

1

Mn/M

p

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS


1 2 30

0.5

1

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Mn/M

p

12300.51

AISCAISIDSMABAQUS


40

1 2 30

0.5

1

Mn/M

p

W14FI

1 2 30

0.5

1

W14FR

1 2 30

0.5

1

W36FR

(fy/fcrl)0.5

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

W36WI

AISCAISIDSMABAQUS


1 2 30

0.5

1

Mn/M

p

1 2 30

0.5

1

(fy/fcrl)0.5

1 2 30

0.5

1

(fy/fcrl)0.5

Mn/M

p

12300.51

AISCAISIDSMABAQUS


41

AISI’s Effective Width Method is overall the best performer in comparison

with the FE results. However, the method is unconservative for long beams with

locally slender webs (see the W36WI study at λe=1.34 of Figure 2-18). Note, as per

AISI for any section which is subject to lateral-torsional buckling (such as those

studied here) the capacity is limited to My as shown. The DSM results for beams

are in excellent agreement at all lengths when the flange and web slenderness

vary at a fixed ratio (the W14FR and W36FR studies). The method has smooth

transitions in all ranges of local slenderness. However, when one of the elements

becomes significantly more slender than its neighbor DSM assumes the entire

cross-section capacity degrades and this assumption becomes excessively

conservative particularly for the W36WI cases, though less so than AISC. Note,

multiple curves are presented for DSM in Figures 2-15, 2-17, and 2-19 because of

the normalization to Mp (as opposed to My) and further the inelastic bending

provisions allowing strengths up to Mp, as proposed for DSM and currently

under ballot at AISI, are utilized here.

2.5.3 Overall AISC’s solutions are overly approximate for locally slender sections and

deserve improvement, particularly for flanges (unstiffened elements). AISI’s

effective width, while the most complicated of the methods, appears to provide

the most accurate solution, particularly for braced (stub) columns. The simplicity

of DSM is obvious in the expressions and the curves, but the elastic web-flange

interaction assumed in the method is not always realized. DSM provides a

42

consistently conservative, and conceptually simple prediction method that is

worthy of further study.

2.6 Summary and Conclusion

The design of locally slender steel cross-sections may be completed by a

variety of methods. For braced (short) columns and beams, design expressions in

common notation are provided for the AISC Specification, the AISI Specification

(effective width method) and DSM the Direct Strength Method (as adopted in

Appendix 1 of the AISI Specification as an alternative design procedure). The key

parameters, found throughout all 3 design methods, are the elastic local

(element, or member) buckling stress and the material yield stress. The design

expressions indicate significantly different solution methodologies to this

common problem, particularly for beams.

A parametric study of braced (short) columns and beams is conducted with

nonlinear finite element models in ABAQUS, deformed to collapse, and

compared with the AISC, AISI, and DSM design predictions. The parametric

study focuses on W14 and W36 sections, where through modification of element

thicknesses, the flange slenderness, and/or web slenderness are systematically

varied (from compact, to noncompact, to slender in the parlance of AISC).

The results indicate that AISC is overly conservative when the flange is

slender, AISC’s assumption of little to no post-buckling reserve in unstiffened

elements is not borne out by the analysis. AISI’s effective width method is a

43

reliable predictor, only for the beam studies does AISI provide overly

conservative solutions when the web is compact but the flange slender. DSM

provides reliable predictions when both the flange and web slenderness vary

together, but is overly conservative when one element is significantly more

slender than another. Additional work on long beams and columns with local-

global interaction is underway.

44

3 Strain Distribution in Locally Slender Structural Steel Cross-Sections

3.1 Introduction

The work presented herein is part of the continuing effort towards fully

understanding the local stability, including the beneficial web-flange interaction,

of structural steel. Through describing and analyzing a series of finite element

(FE) analysis, efforts in Progress Report #3 and this Progress Report #4 showed

comparisons of three design methods for locally slender steel short beams and

stub columns; (i) AISC, and two methods from cold-formed steel specifications

which focus on locally slender cross-sections: (ii) AISI-Effective Width, and (iii)

AISI-Direct Strength Method (DSM). It was shown that in AISC, AISI, and DSM

global column buckling is predicted using the same (single) expression.

However, local-global interaction is handled by the Q-factor method in AISC, the

unified method in AISI, and a variation of the unified method in DSM. In all

cases the global strength is reduced due to local cross-section slenderness. The Q-

factor approach reduces the strength and increases the long-column slenderness

to arrive at its reduction. The unified method uses the effective area of the

column at the long column buckling stress. DSM uses a similar approach.

The underlying mechanics of a locally unstable cross-section at failure

involves a complex nonlinear stress-strain state in the cross-section. Next

generation design methods should at least in part reflect this stress-strain state in

45

their predictions; Q-factor approach does not, while the unified method

simplifies the distributions based on elements of the cross-section. We seek here

a simple means to utilize knowledge of the complete cross sectional stability

(including element interactions) to predict this fundamental underlying stress-

strain distribution. To develop what at its heart is essentially a semi-empirical

method, we have used nonlinear FE collapse analysis to generate information on

the stress-strain state at collapse for locally slender steel cross sections. The

nonlinear FE analysis parameter study, using ABAQUS, that was used in

Progress Report #3 and this Progress Report #4 for the purpose of understanding

and highlighting the parameters that lead to the divergence between the capacity

predictions of the different design methods, lead to the establishment of a

database of failure mechanisms of W-sections at different element slenderness

ratios. The main objective of this current work is to closely examine the strain

distributions observed in the failure mechanisms of the FE parameter study

database, and compare them to finite strip analysis results using CUFSM, as well

as to theoretical distributions. The final goal of this research is to propose

improvements to DSM so it may be applied to hot-rolled structural steel with

locally slender cross-sections.

The stress distributions and the strain distributions of all sections are

examined along the length of the member as well as through the thickness of the

elements. Distributions throught the thickness of the elements are examined at

different levels; at the top, mid-thickness, and bottom, as well as the average

46

through the through the thickness. Figure 3-1 shows the average stress

distribution through the thickness for the four study groups at different

thicknesses.

-50

0Avg str at different sections for W14arstress34

-40 -20 0

-50

0

thinoriginalthick

-50

0Avg str at different sections for W14afstress34

-40 -20 0

-50

0

thinoriginalthick

-50

0Avg str at different sections for W36awstress34

-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-40 -20 0

-50

0

thinoriginalthick

-50


-40 -20 0

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

(a) (b)

(c) (d)

Figure 3-1 Average stress distributions through the thickness for the four study groups at different thicknesses: (a) W14FR, (b) W14FI, (c) W36FR, and (d) W36WI.

47

Figure 3-2 shows the average strain distribution through the thickness for

the four study groups at different thicknesses.

-5

0x 10-3 Avg str at different sections for W36arstrain34

-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

-5

0x 10-3 Avg str at different sections for W36awstrain34

-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

-5


-0.01 0 0.01

-5

0x 10-3

thinoriginalthick

-5

0x 10-3 Avg str at different sections for W14afstrain34

-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

-5


-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

-5

0x 10-3 Avg str at different sections for W36awstrain34

-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

-5


-0.01 0 0.01

-5

0x 10-3

thinoriginalthick

-5

0x 10-3 Avg str at different sections for W14afstrain34

-5 0 5

x 10-3

-5

0x 10-3

thinoriginalthick

(a) (b)

(c) (d)

Figure 3-2 Average strain distributions through the thickness for the four study groups at different thicknesses: (a) W14FR, (b) W14FI, (c) W36FR, and (d) W36WI.

48

Figure 3-3 shows stress distribution at the mid-thickness for the four study

groups at different thicknesses.

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

-50


-50 0 50

-50

0

thinoriginalthick

(a) (b)

(c) (d)

Figure 3-3 Stress distributions at mid-thickness for the four study groups at different thicknesses: (a) W14FR, (b) W14FI, (c) W36FR, and (d) W36WI.

49

Figure 3-4 shows strain distribution at the mid-thickness for the four study

groups at different thicknesses.

-50

x 10-3 Avg str at different sections for W36arstrain34

-0.01 0 0.01

-50

x 10-3

thinoriginalthick

-50

x 10-3 Avg str at different sections for W36awstrain34

-5 0 5

x 10-3-5

0x 10-3

thinoriginalthick

-50

x 10-3 Avg str at different sections for W14afstrain34

-0.01 0 0.01

-50

x 10-3

thinoriginalthick

-0.01-0.005

0Avg str at different sections for W14arstrain34

-0.02 0 0.02

-0.01-0.005

0

thinoriginalthick

-50

x 10-3 Avg str at different sections for W36arstrain34

-0.01 0 0.01

-50

x 10-3

thinoriginalthick

-50

x 10-3 Avg str at different sections for W36awstrain34

-5 0 5

x 10-3-5

0x 10-3

thinoriginalthick

-50

x 10-3 Avg str at different sections for W14afstrain34

-0.01 0 0.01

-50

x 10-3

thinoriginalthick

-0.01-0.005

0Avg str at different sections for W14arstrain34

-0.02 0 0.02

-0.01-0.005

0

thinoriginalthick

(a) (b)

(c) (d)

Figure 3-4 Strain distributions at mid-thickness for the four study groups at different thicknesses: (a) W14FR, (b) W14FI, (c) W36FR, and (d) W36WI.

50

4 References

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”, Am. Iron and Steel Inst., Washington, D.C., AISI-S100.

Barth, K.E. et al (2005). “Evaluation of web compactness limits for singly and doubly symmetric steel I-girders”, Journal of Constructional Steel Research 61 2005 1411–1434.

Dinis, P.B., Camotim, D. (2006). “On the use of shell finite element analysis to assess the local buckling and post-buckling behavior of cold-formed steel thin-walled members”, III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5–8 June 2006.

Earls, C.J. (2001). “Constant moment behavior of high-performance steel I-shaped beams”, Journal of Const. Steel Research 57 (2001) 711–728.

Galambos, T.V., Ketter, R.L. (1959). “Columns under combined bending and thrust”, Journal of Engineering, Mechanics Division, ASCE 1959 85; 1–30.

Galambos, T.V. (1998). “Guide to Stability Design Criteria for Metal Structures”. 5th ed., Wiley, New York, NY, 815-822.

Jung, S., White, D.W. (2006). “Shear strength of horizontally curved steel I-girders—finite element analysis studies”, Journal of Constructional Steel Research 62, 2006: 329–342.

Kim, S., Lee, D. (2002). “Second-order distributed plasticity analysis of space steel frames”, Engineering Structures 24, 2002: 735–744.

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.

Schafer, B.W., Seif, M. (2008). “Comparison of Design Methods for Locally Slender Steel Columns” SSRC Annual Stability Conference, Nashville, TN, April 2008.

51

Seif, M., Schafer, B.W. (2007). “Cross-section Stability of Structural Steel.” American Institute of Steel Construction, Progress Report No. 1. AISC Faculty Fellowship, July 2007.

Seif, M., Schafer, B.W. (2008). “Cross-section Stability of Structural Steel.” American Institute of Steel Construction, Progress Report No. 2. AISC Faculty Fellowship, April 2008.

Seif, M., Schafer, B.W. (2009a). “Elastic Buckling Finite Strip Analysis of the AISC Sections Database and Proposed Local Plate Buckling Coefficients” Structures Congress, Austin, TX, April 2009.

Seif, M., Schafer, B.W.(2009b). “Finite element comparison of design methods for locally slender steel beams and columns” SSRC Annual Stability Conference, Phoenix, AZ, April 2009.

Seif, M., Schafer, B.W. (2009c). “Cross-section Stability of Structural Steel.” American Institute of Steel Construction, Progress Report No. 3. AISC Faculty Fellowship, April 2009.

Shifferaw, Y., and Schafer, B. W. (2008). "Inelastic bending capacity in cold-formed steel members." Report to American Iron and Steel Institute – Committee on Specifications, July 2008.

White, D.W. (2008). “Unified flexural resistance equations for stability design of steel I-section members: Overview.” ASCE, Journal of Structural Engineering, 134 (9) 1405-1424.

52

Appendix A : NRC Research Proposal

This Appendix shows a copy of a research proposal titled “Multi-scale

Structural Stability under Realistic Fire Loading”. The proposal was submitted to

the National Research Council (NRC), as part of a post-doctoral fellowship

application, and it aims to extend this research where the effect of realistic fire

loading scenarios on locally slender structural steel members will be studied.

Multi-scale Structural Stability under Realistic Fire Loading

Summary: Stability is paramount in the performance of steel structures under

fire. The work proposed here advances a multi-scale approach whereby heat transport and

stability are addressed at the cross-section level, and then coupled to member-level models

to enable predictions of complete building systems under fire. The work is significantly

aided by recent advances in the efficient prediction of locally unstable hot- rolled steel

cross-sections that formed the proposer’s Ph.D. work, and will be validated with facilities

uniquely available at NIST.

A.1 Problem:

The catastrophic collapse of the World Trade Center buildings (WTC

Towers 1and 2, and Building 7) brought the nation’s attention to the

vulnerability of our structures to extreme loading conditions, especially fire. Fire

53

was a significant factor in the collapse of the twin towers, which survived the

aircraft impacts, but could not withstand such severe fires. It is noted that fire

was the sole cause of the collapse of WTC 7, which was the first recorded

collapse of a structure of that magnitude entirely because of fire [Usmani et al.

2004]. Fire is an extreme, low-probability high-consequence, structural loading

event, and it costs the U.S. economy about 270 billion dollars per year. It cost

about 317 billion dollars in 2006, which is near 3% of the U.S. gross domestic

product [Hall 2009]. Already, the federal government, as the largest single

owner of buildings in the U.S., requires all buildings greater than three stories to

be evaluated for the potential of progressive collapse. General Services

Administration (GSA), Department of Defense (DoD), and Department of State

(DoS) currently require such evaluation for their buildings design [see e.g. Senate

Report 107-57].

At the present time, there is a lack of understanding of the performance of

structures as complete systems comprised of components and connections under

extreme loading conditions such as realistic, uncontrolled fires. Current

specifications for the design of steel structures do not require structural

engineers to design for fire loading conditions. The practice for assessing the fire

resistance of a structure is based on the Standard Fire Tests [ASTM E119] which

has not changed much since introduced in 1917. This deficiency arises from the

lack of (i) science-based measurement tools for the evaluation of structural

systems’ performance under realistic fire loads, and (ii) validated data from full

54

scaled experiments under real fire exposures. It is postulated that a fuller

understanding of the problem will lead to the development and implementation

of standards and tools that explicitly consider realistic fire loading for both the

design of new buildings and assessment and retrofit of existing ones.

A.2 Background:

Steel sections become locally slender in the event of fire. The

performance of structural steel in fires is characterized by its thermal and

mechanical properties. Thermal properties are necessary to predict effects of

temperature rise in steel resulting from fire exposure and the resulting thermal

expansion. Prediction of mechanical behavior requires the stress-strain

relationship of steel at elevated temperatures and may be represented by such

parameters as elastic modulus, yield and ultimate strengths, and creep behavior.

The yield strength, fy, sustains its value up to temperatures of about 750 0F (399

0C) before degrading, while the elastic modulus, E, starts degrading dramatically

from temperatures as low as 200 0F (93 0C) [according to Appendix 4 of the

American Institute of Steel Construction (AISC) 2005 Manual of Steel

Construction]. During the initial stage of a fire, the elastic modulus will degrade

while the yield strength will remain fixed. The local stability slenderness limits of

structural steel sections depend on the ratio √( E/ fy) as defined in Table B4.1 of

AISC’s Manual of Steel Construction. Reducing E at a fixed fy has a similar effect

on the local stability of a section as increasing fy. Typically, locally slender cross-

55

sections are avoided in the design of hot-rolled steel structural elements, but in

the event of fire steel sections become locally slender. Recent work by Seif and

Schafer [2009a] addressed the issue of increasing the yield strength (or

equivalently degrading the elastic modulus in a fire case) where a series of

simple empirical equations were developed and used to construct a table which

is essentially a proposed alternative to AISC’s Table B4.1. Other work by Seif and

Schafer [2009b] investigated through a Finite Element Analysis (FEA) parametric

study how different steel design specifications handle sections with high local

slenderness (again, this is the case of sections under fire). Results showed that

AISC is overly conservative for most of the cases which could be uneconomical.

Such results show that there is much more to explore on the road to developing

design standards that could yield economical, yet safe structures that are to

perform under severe fire events and other hazards that could lead to

progressive collapse of the structure.

The problem is HARD to solve. Complex behavior of structural systems

performing at their ultimate limit state is not well understood and is usually not

addressed when designing structures. However, fire loading conditions are

considered extreme yet possible events that could lead to an unpredictable

response of the structure. Accordingly, analysis of such systems under fire

conditions requires detailed modeling for the accurate prediction of the

performance of the system as a whole at, or near, its ultimate limit state of

collapse. It is not easy to develop robust modeling techniques to accurately

56

compensate for the effects of thermal expansion in addition to the severe effects

of degraded material and mechanical properties at elevated temperatures. Thus,

validating such techniques against full scale tests of systems under real fire

exposure is necessary. Currently, some experimental data are available for

components subject to elevated temperatures. However, experimental data on

the performance of structures as complete systems, including connections and

members, subject to realistic building fire conditions and scenarios is almost non-

existent.

NIST is the place to solve. The best place to address and conduct the

proposed research is the Building and Fire Research Laboratory (BFRL) at the

National Institute of Standards and Technology (NIST) for many reasons.

The “Fire Resistance Design and Rehabilitation of Structures” project is part

of the “Structural Performance under Multi-Hazards” program within BFRL’s

national strategic priority: “Measurement Science for Disaster Resilient

Structures and Communities”. BFRL is uniquely qualified to supervise this

project because of its long history of investigations in the field of structural

failures. Over the years, they have used state of the art computational tools to

analyze the failures of complex multi-story structures. Additionally, their

analytical and experimental work on structural response to fire pushed the state-

of-the-art through their comprehensive investigation of the fire-induced collapse

of the World Trade Center; WTC 1, WTC 2, and WTC 7 [see e.g. NIST NCSTAR

57

1A 2008]. Furthermore, BFRL houses the Fire Research Division that has a

considerable expertise in the characterization and simulation of a building fire

environment, and with strong collaborations with the Structures Group at BFRL.

In 2009, based on the national demand to address the effects of fire on

structures after the 9/11 tragedy, NIST was granted 22 million dollars to

construct a National Structural Fire Resistance Laboratory (NSFRL). The NSFRL

aims to be a unique testing laboratory that is capable of testing full scale

structural systems under realistic fire and loading conditions. It is postulated that

such testing will enhance understanding of structural system performance under

fire conditions, and will provide data for computer models to enable the

development of performance- based design methods. Such research could save

thousands of lives and billions of dollars in property damage.

Tools for success are now available. For many reasons, now is the perfect

time to conduct this research; (i) NIST’s recent investigation reports on the WTC

(Towers 1 and 2, and Building 7) have focused the nation’s attention on the

importance of understanding and predicting the complex behavior of structures

under realistic fire scenarios, the attention and demand that lead to funding and

establishing the NSFRL, (ii) the anticipated completion of the NSFRL by early

2012, making now the optimum timing to conduct research that will lead to a

comprehensive testing program that will fully utilize the NSFRL, and (iii) the

work of this project demands detailed and advanced structural modeling to

58

capture the nonlinear behavior of the structure as a system at, or beyond, its

ultimate capacity limit state under severe fire loading scenarios. Today’s high

performance computation tools and parallel processing capabilities have

advanced the analytical capabilities of Computational Fluid Dynamics and Finite

Element Analysis to a stage that is capable of analyzing such complex structures

under extreme conditions.

A.3 Methodology and approach:

In order to accomplish the main objective of designing structural systems

that can perform safely and adequately under realistic fires and other hazards,

the proposed work is planned around the following tasks: (i) develop improved

design provisions for steel sections which due to elevated temperatures lose their

cross-section stability, and (ii) develop detailed and simplified validated three-

dimensional models for steel-framed structural system behavior up to collapse

under fire conditions. These tasks will provide the measurement science to

enable the development of performance-based design procedures for structures

under fire conditions.

Multi-scale approach. On the local element scale, it is planned to continue

efforts on researching cross-sectional stability of locally slender structural steel

elements by including fire loading. The beneficial post-buckling reserve strength

that exists in local buckling modes will be studied. The effect of various non-

linear material models will be considered for degrading mechanical properties at

59

elevated temperatures. ABAQUS and ANSYS software packages will be used to

conduct the non-linear Finite Element Analysis (FEA) of the work. The modeling

effort will include hot-rolled steel sections typically used for building beams and

columns. Different modeling techniques will be investigated using both two-

dimensional shell elements and three-dimensional solid elements at different

mesh densities to reach the optimal cost-efficient realistic methods. Loading will

include fire scenarios such as those recently identified by the International Code

Council (ICC) and the National Fire Protection Association (NFPA).

Knowledge gained on the local cross-section level will then be coupled to

member-level models (columns, beams, and connections) to enable predictions of

complete building system performance subject to fire scenarios. Tools will be

developed to couple the NIST’s Fire Dynamics Simulator (FDS) with suitable

structural analysis codes through thermal analysis of steel sections with

fireproofing. The probabilistic distributions of all the variables will be integrated

into the models to set the structural reliability framework.

Detailed three dimensional modeling of a few typical steel-framed

buildings will be studied and validated through parametric studies for the

purpose of developing simplified models that can be used by designers for

evaluating structure performance up to collapse. The 3-D models will include

building framing (beams, girders, columns, and their connections). Also

included will be the composite floor slab. The models for steel sections will

60

include the stability considerations at elevated temperatures as outlined above.

A significant portion of this effort will be in developing an array of simplified

models of typical steel connections that are able to capture the predominant

connection behavior and failure modes under fire loads, including thermal

expansion effects and diminished mechanical properties with elevated

temperatures. The analyses will provide an insight into the system behavior of

the structure and identify possible failure mechanisms of the structure under fire

loading conditions.

Knowledge gained from computational analysis will be used to guide

developing an experimental plan for testing full scale structural systems under

three dimensional loads with real fires at the National Structural Fire Resistance

Laboratory (NSFRL). The experimental data will be used to validate the

computational models. The results of this work will contribute significantly to

the NIST research that aims to develop performance based design approaches

that consider fire loading as a basic design requirement.

Figure (1) shows a summary of the proposed multi-scale approach.

61

Figure A-1 Summary of the proposed multi-scale approach.

Element Level(flange, web, flange-web interaction…)

Local cross-sectional stability

Member Level (column, beam, detailed connections,….)

Development of simplified beam elements

Structural System Level(whole structure)

3D modeling & FE parameter study

- Propose experimental program for NSFRL- Contribute to the development of performance based design approach

MULTI-SCALE APPROACHFor structural stability under realistic fire loading

Preliminary findings:

In the event of fire, steel sections become locally slender, as mentioned above.

The design of locally slender steel cross-sections may be completed by a variety

of methods, three of which are examined here: (i) The AISC method, as

embodied in the 2005 AISC Specification, (ii) The AISI Effective Width Method

from the main body of the 2007 AISI Specification for cold-formed steel, and, (iii)

The Direct Strength Method as given in Appendix 1 of the 2007 AISI

Specification. To examine the impact of elevated temperatures and slenderness

on the local buckling mode, and the impact of web-flange interaction in W-

sections on the column’s capacity, the axial strength capacity of a W14x233

section was determined according to the procedures of the three design methods

at temperatures ranging from room temperature to 2000 ºF. The material

62

parameters at elevated temperatures provided in Appendix 4 of the AISC’s 2005

Manual of Steel Construction were used. Also, the effect of further increasing the

elements’ slenderness was examined through reducing their thickness. Figure (2)

shows the normalized column strength as predicted by the different steel design

specifications versus temperature for a W14x233: (a) original section, (b) section

with reduced flange thickness, (c) section with reduced web thickness, and (d)

section with reduced flange and web thicknesses at a fixed flange to web

thickness ratio.

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Nor

mal

ized

Stre

ngth

, Pn/P

y

AISCAISIDSM

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

AISCAISIDSM

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Temprature, oF

Nor

mal

ized

Stre

ngth

, Pn/P

y

AISCAISIDSM

0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.2

0.4

0.6

0.8

1

Temprature, oF

AISCAISIDSM

(a) (b)

(c) (d)

Figure A-2 Normalized column strength as predicted by the different design methods versus temperature for a W14x233: (a) original section, (b) section with reduced flange thickness, (c) section with reduced web thickness, and (d) section with reduced flange and web thicknesses at a fixed flange to web thickness ratio.

63

Preliminary results show divergence in the capacity predictions for the

three steel design specifications. These results point out the need to perform

nonlinear FEA parameter studies for the purpose of understanding and

highlighting the parameters that lead to the divergence between the capacity

predictions of the different design methods at elevated temperatures. Such

knowledge gained on the local cross-section level could then be coupled to

member-level models to enable predictions of complete building systems

performances subject to the fire scenarios.

The cross-section fire degradation and local stability models will feed a

custom frame element which can then be used for whole building modeling. It is

noted that a fully coupled (thermal and mechanical) shell element model of a

building is not computationally efficient, thus a multiple scale approach will be

considered. It is planned to develop multi-scale tools that can be evaluated and

validated in a one-of-a-kind way with the testing capabilities that will be

available at NIST.

A.4 New techniques:

Presently, consideration of fire as a loading condition is not required in

design practice in the USA. Current practice for assessing the fire resistance of a

structure is prescriptive and does not take into consideration realistic fire

scenarios or structural system performance. This proposed work aims to provide

the data necessary to develop improved design provisions for steel sections,

64

which due to elevated temperatures may lose their cross-sectional stability, as

well as to standardize performance-based design methods that evaluate

structural system performance under realistic fires. For the first time, fire loading

conditions will be considered as a standard design condition. Also, for the first

time, full scale structures loaded in three dimensions will be experimentally

tested and analyzed under realistic fires up to collapse. Large-deformation

analysis approaches will be used for structural systems under fire, static, and

dynamic loads up to collapse. This will help identify fire-induced system failure

mechanisms, which are a result of many local failure mechanisms.

A.5 Expected results and significance:

The main objective of this proposed work is to develop an improved

understanding of the performance of structural systems subject to extreme,

realistic, and uncontrolled fires for the purpose of increasing the safety of both

existing and future structures. The most significant expected results are

summarized as follows:

• On the local element level: Accurate and robust modeling techniques for

cross‐sectional stability of locally slender structural steel elements

including fire loading and the beneficial post‐buckling reserve strength

that exists in local buckling modes will be studied.

65

• On the member level: Knowledge gained on the local cross‐section level will

then be coupled to member‐level models to enable predictions of complete

building system performance subject to fire scenarios.

• On the structure level: Detailed three dimensional modeling of typical steel‐

framed buildings will be studied and validated through parametric

studies for the purpose of developing simplified models that can be used

by designers for evaluating structure performance up to collapse.

• Knowledge gained from computational analysis will be used to guide

developing an experimental plan for testing full scale structural systems

under real fires at the NSFRL. The results will contribute to the NIST

research that aims to develop performance‐based design approaches that

consider fire loading as a basic design requirement.

A.6 References:

ASTM Standard E119, (2009), “Fire Standards and Flammability Standards”, ASTM International, West Conshohocken, PA, 2009, DOI: 10.1520/E0119-09C. Hall J. R. (2009), “Total Cost of Fire in the United States”, National Fire Protection Association, NFPA. NIST (2008), “Final Report on the Collapse of World Trade Center Building 7”, Federal Building and Fire Safety Investigation of the World Trade Center Disaster, NIST NCSTAR 1A, Gaithersburg, MD: National Institute of Standards and Technology.

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NIST (2009), “Best Practice Guidelines for Structural Fire Resistance Design of Concrete and Steel Buildings”, NISTIR 7563, Draft for public comments, Gaithersburg, MD: National Institute of Standards and Technology. Seif, M., Schafer, B.W. (2009a), “Elastic buckling finite strip analysis of the AISC sections database and proposed local plate buckling coefficients”, ASCE’s Structures Congress Proceedings, May 2009. Seif, M., Schafer, B.W. (2009b), “Finite element comparison of design methods for locally slender steel beams and columns”, SSRC Annual Stability Conference Proceedings, April 2009, p. 69-90. U.S. Senate Congress Committee an Appropriations (2002), “Treasury and General Government Appropriation Bill”, 107th Congress, 1st session, Senate Report 107-57. Usmani, A.S., Chung, Y.C., Torero, J.L. (2003), “How Did the WTC Towers Collapse: A New Theory”, Fire Safety Journal, v. 38, issue 6, p. 501-533.

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Documents

April 2010 AISC - Department of Civil Engineering Faculty Fellowship Progress... · April 2010 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 4 ‐ AISC