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 (mina.seif@jhu.edu) Graduate Research Assistant Ben Schafer (schafer@jhu.edu) Associate Professor 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 2
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. 3
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. Progress Report #3 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.
4
Progress Report #4 April 2009 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.
5
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
4
Introduction........................................................................................................45
References..................................................................................................................51
6
Appendix A : NRC Research Proposal..............................................................................53
7
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. 8
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 9
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 10
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. 11
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 12
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‐
13
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. Progress Report #3
Local modes
1.2
Progress Report #4
Local-Global modes interaction
1
0.8
Pn / Py
stub
0.6
λc
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
λc
Figure 2-1 Effect of global column slenderness on predicted strength, and regions covered in the
parametric studies of progress reports #3 and #4
14
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.,
15
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,
16
•
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
W14x233
W14FI
W14FR
W36x330
W36FR
W36WI
bf/2tf
h/tw
h/bf
tf/tw
4.62
varied
varied
4.54
varied
Fixed
13.35
fixed
varied
35.15
varied
varied
0.90
fixed
fixed
2.13
fixed
fixed
1.61
varied
fixed
1.81
fixed
varied
17
35
f y = 50 ksi (345 MPa)
30
k =1.2
f
k =0.5
25
bf /2tf
λrf
k =0.6
f
k =6.0
w
w
20
k =0.5
f
k =29
W14FI
w
W14FR
15
k =0.5
f
k =29
w
k =0.9
f
k =2.1
10
w
λpf
k =0.6
f
k =5.0
5
0
w
0
k =0.6
f
k =5.6
w
k =0.5
f
k =27
W36FR
w
k =0.1
f
k =36
W36WI
w
λpw 100
50
k =0.05
f
k =36
w
λrw
150
h/tw
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.
18
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:
λc =
fy
f creuler
=
fy
KL
r π 2E
(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 backcalculated to maintain the specified λc values.
19
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:
λe =
My
(2-2)
M cre
Fixing λe, the critical buckling moment, Mcre, is calculated for each section.
Mcre is also defined as follows:
M cre = Cb
π 2 EI y GJ
L2
+
π 2 EI yπ 2Cw
L4
(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.
20
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.
21
Engineering Stress (ksi)
fu= 65
fy= 50
Slope,
Est=720
Slope,
E’=145
Slope,
E =29000
εy
ε st =0.011
Engineering Strain
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).
22
-
-
-
σ c = 0 .3 f y
⎛
⎞
bf t f
⎟
+ σ t = σ c ⎜⎜
⎟
+
(
−
2
)
b
t
t
d
t
f
f
w
f
⎝
⎠
-
-
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
23
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.
1000
900
800
700
Load
600
500
400
300
x
Local
200
Global
100
0
0
10
x
1
2
10
3
10
Half wave length
10
L
Figure 2-5 Typical CUFSM curve where local and global buckling modes are determined.
24
d /150
(b)
bf /150
(a)
bf
bf /150
d
d /150
(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. 25
L /1000
(b)
(a) (a)
L /1000
(c)
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
Pn/Py
W14FI
W14FR
1
1
0.5
0.5
0
0
0.5
1
0
1.5
0
0.5
Pn/Py
W36FI
1
0.5
0.5
0
0.5
1
Lambdac
1.5
W36WI
1
0
1
0
1.5
0
0.5
1
Lambdac
1.5
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 ,
A
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.
Pn/Py
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Pn/Py
W36FR
0
AISC
1
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
(fy /fcrl)0.5
1
2
3
W36WI
2
3
(fy /fcrl)0.5
Figure 2-9 Results of column parametric study for 4 study groups (stub)
29
Pn/Py
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Pn/Py
1
AISC
1 AISI
0.5
0
DSM
123
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
Figure 2-10 Results of column parametric study for 3 design methods (stub)
Pn/Py
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Pn/Py
W36FR
0
AISC
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
0.5
1
2
3
W36WI
1
2
3
0.5
(fy /fcrl)
(fy /fcrl)
Figure 2-11 Results of column parametric study for 4 study groups (λc=0.9)
30
Pn/Py
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Pn/Py
1
AISC
1
0.5
AISI
0
123
DSM
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
Figure 2-12 Results of column parametric study for 3 design methods (λc=0.9)
Pn/Py
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Pn/Py
W36FR
AISC0
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
0.5
1
2
3
W36WI
1
2
3
0.5
(fy /fcrl)
(fy /fcrl)
Figure 2-13 Results of column parametric study for 4 study groups (λc=1.5)
31
Pn/Py
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Pn/Py
1
1
0.5 AISC
0 AISI
123
DSM
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
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
A
A
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
Mn/Mp
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Mn/Mp
W36FR
0
AISC
1
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
1
(fy /fcrl)0.5
2
3
W36WI
2
3
(fy /fcrl)0.5
Mn/Mp
Figure 2-16 Results of beam parametric study for 4 study groups (short)
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Mn/Mp
1
AISC
1
AISI
0.5
0
DSM
123
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
Figure 2-17 Results of beam parametric study for 3 design methods (short)
34
Mn/Mp
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Mn/Mp
W36FR
0
AISC
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
1
2
3
W36WI
1
(fy /fcrl)0.5
2
3
(fy /fcrl)0.5
Mn/Mp
Figure 2-18 Results of beam parametric study for 4 study groups (λe=0.6)
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Mn/Mp
1
AISC
1
0.5
0AISI
123
DSM
0.5
ABAQUS
0
1
2
3
0.5
(fy /fcrl)
Figure 2-19 Results of beam parametric study for 3 design methods (λe=0.6)
35
Mn/Mp
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Mn/Mp
W36FR
0
AISC
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
1
2
3
W36WI
1
(fy /fcrl)0.5
2
3
(fy /fcrl)0.5
Mn/Mp
Figure 2-20 Results of beam parametric study for 4 study groups (λe=1.34)
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Mn/Mp
1
AISC
1
0.5 AISI
0
123
DSM
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
Figure 2-21 Results of beam parametric study for 3 design methods (λe=1.34)
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
Fcr =
Cbπ 2 E
⎛ Lb ⎞
⎜ ⎟
⎝ rts ⎠
2
⎛ Jc ⎞ ⎛ Lb ⎞
1 + 0.078 ⎜
⎟⎜ ⎟
⎝ S x ho ⎠ ⎝ rts ⎠
2
(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 lateraltorsional buckling stress (Eq. F2-4) is utilized – it is clear the use of this
approximation must be done with care.
AISC
AISI
DSM
ABAQUS
(a)
1
0.8
0.8
0.6
0.6
Mn/Mp
Mn/Mp
1
0.4
0.4
0.2
0.2
0
0.5
1
1.5
2
2.5
0
3
0.5
(b)
0.5
1
1.5
2
2.5
3
(fy/fcrl)0.5
(fy/fcrl)
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
Mn/Mp
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Mn/Mp
W36FR
0
AISC
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
1
2
3
W36WI
1
(fy /fcrl)0.5
2
3
(fy /fcrl)0.5
Mn/Mp
Figure 2-24 Results of beam parametric study for 4 study groups (λe=0.6)
1
1
0.5
0.5
0
1
2
0
3
1
2
3
0.5
(fy /fcrl)
Mn/Mp
1
AISC
AISI
1
0.5
0
DSM
123
ABAQUS
0.5
0
1
2
3
0.5
(fy /fcrl)
Figure 2-25 Results of beam parametric study for 3 design methods (λe=0.6)
40
Mn/Mp
W14FI
W14FR
1
1
0.5
0.5
0
1
2
3
Mn/Mp
W36FR
0
AISC
AISI
DSM
ABAQUS
1
1
0.5
0.5
0
1
2
0
3
1
2
3
W36WI
1
(fy /fcrl)0.5
2
3
(fy /fcrl)0.5
Mn/Mp
Figure 2-26 Results of beam parametric study for 4 study groups (λe=1.34)
1
1
0.5
0.5
0
1
2
0
3
1
2
3
(fy /fcrl)0.5
Mn/Mp
1
AISC
1 AISI
0.5
0
123
DSM
ABAQUS
0.5
0
1
2
3
(fy /fcrl)0.5
Figure 2-27 Results of beam parametric study for 3 design methods (λe=1.34)
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 localglobal interaction is underway.
44
3 Strain Distribution in Locally
Slender Structural Steel CrossSections
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 Qfactor 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 stressstrain 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.
Avg str at different sections for W14arstress34
Avg str at different sections for W14afstress34
0
0
-50
-50
(a)
(b)
thin
original
thick
thin
original
thick
-40 -20
0
0
-40 -20
0
-50
0
-50
Avg str at different sections for W36arstress34
Avg str at different sections for W36awstress34
0
0
-50
-50
(c)
0
-50
0
thin
original
thick
(d)
thin
original
thick
50
0
-50
-50
0
50
-50
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.
-3
0
x 10
-3
Avg str at different sections for W14arstrain34
0
-5
x 10
Avg str at different sections for W14afstrain34
-5
(a)
(b)
thin
original
thick
thin
original
thick
-3
0
x 10
-3
-0.01
0
0.01
0
x 10
-5
0
5
-3
x 10
-5
-5
-3
0
x 10
-3
Avg str at different sections for W36arstrain34
0
-5
x 10
Avg str at different sections for W36awstrain34
-5
(c)
(d)
thin
original
thick
thin
original
thick
-3
0
x 10
-3
-5
0
5
0
-3
x 10
-5
0
5
-3
x 10
x 10
-5
-5
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.
Avg str at different sections for W14arstress34
Avg str at different sections for W14afstress34
0
0
-50
-50
(a)
(b)
thin
original
thick
0
-50
0
thin
original
thick
50
-50
0
-50
0
50
-50
Avg str at different sections for W36arstress34
Avg str at different sections for W36awstress34
0
0
-50
-50
(c)
(d)
thin
original
thick
0
-50
0
thin
original
thick
50
0
-50
-50
0
50
-50
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.
-3
Avg str at different sections for W14arstrain34
0
-0.005
-0.01
0
-5
x 10
(a)
Avg str at different sections for W14afstrain34
(b)
thin
original
thick
thin
original
thick
-3
-0.02
0
-0.005
-0.01
-3
0
x 10
0
0.02
0
-5
x 10
-0.01
-3
Avg str at different sections for W36arstrain34
0
-5
-5
x 10
(c)
0
0.01
Avg str at different sections for W36awstrain34
(d)
thin
original
thick
thin
original
thick
-3
0
x 10
-3
-0.01 0 0.01
0
-5
-5
x 10
-5
0
5
-3
x 10
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
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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”,
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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 Isection members: Overview.” ASCE, Journal of Structural Engineering, 134 (9) 14051424.
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 nonexistent.
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 stateof-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 threedimensional 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 nonlinear 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 twodimensional 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
MULTI-SCALE APPROACH
For structural stability under realistic fire loading
Element Level
Member Level
(flange, web, flange-web interaction…)
(column, beam, detailed connections,….)
Local cross-sectional stability
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
Figure A-1 Summary of the proposed multi-scale approach.
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 Wsections 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.
(a)
(b)
AISC
AISI
DSM
Normalized Strength, P n/Py
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
200
400
600
800
1000
1200
1400
1600
1800
0
2000
(c)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
(d)
AISC
AISI
DSM
1
Normalized Strength, P n/Py
AISC
AISI
DSM
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
200
400
600
800
1000 1200
Temprature, oF
1400
1600
1800
AISC
AISI
DSM
1
0
2000
0
200
400
600
800
1000 1200
Temprature, oF
1400
1600
1800
2000
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.
66
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.
67