April 2009 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 3 ‐ AISC Faculty Fellowship Cross‐section Stability of Structural Steel
Tom, Please find enclosed the third 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 (see Section 3) focuses on web‐flange interaction, and comparisons of the AISC, AISI – Effective Width, and AISI – Direct Strength design methods for columns and beams with slender cross‐sections. The results indicate excessive conservatism in existing AISC approaches and point towards potential ways forward using alternative methods. Sincerely,
Mina Seif ([email protected]) Graduate Research Assistant
Ben Schafer ([email protected]) Associate Professor
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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
3
learning objectives, step‐by‐step instructions, and complementary homework problems for students.
Papers from this research: 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.
4
Papers from this research: 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 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.
Papers from this research: 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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Table of Contents
Summary of Progress .......................................................................................................2
1 Introduction ...............................................................................................................7
2 Finite Element Reliability Analysis of Hot-Rolled W-Section Steel Columns .........10
2.1 Introduction and Motivation.........................................................................10 2.2 Objective and Methodology ..........................................................................10
2.2.1 Variables and Statistical Parameters..........................................................12 2.2.1.1 Section’s Thickness .............................................................................. 13 2.2.1.2 Yield Strength ....................................................................................... 13 2.2.1.3 Modulus of Elasticity............................................................................ 15 2.2.1.4 Poisson’s Ratio...................................................................................... 15 2.2.1.5 Geometric Imperfections ...................................................................... 15 2.2.1.6 Residual Stresses................................................................................... 17
2.2.2 Finite Element Modeling ...........................................................................19 2.3 Results and Comments......................................................................................21
2.3.1 Taylor Series ..............................................................................................21 2.3.2 Mont Carlo Simulation ..............................................................................26
2.4 Main Conclusion................................................................................................34 2.5 Study Extension Suggestions.............................................................................35
3 Finite Element Comparison of Design Methods for Locally Slender Steel Beams and
Columns .............................................................................................................................37
3.1 Introduction and Motivation ..............................................................................37 3.2 Design Methods and Equations .........................................................................38 3.2.1 Column Design Equations .....................................................................39 3.2.2 Beam Design Equations .........................................................................40
3.3 Parameter Study and Modeling..........................................................................45 3.3.1 Approach..................................................................................................45 3.3.2 Geometric Variation...................................................................................45 3.3.3 Finite Element Modeling ...........................................................................48
3.4 Results................................................................................................................48
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3.4.1 Columns ...................................................................................................49 3.4.2 Beams ........................................................................................................54
3.5 Discussion..........................................................................................................58 3.5.1 Columns ...................................................................................................58 3.5.2 Beams ........................................................................................................59 3.5.3 Overall ......................................................................................................60
3.6 Long Members Parameter Study .......................................................................61 3.6.1 Introduction .............................................................................................61 3.6.2 Initial Approach ......................................................................................61
3.7 Summary and Conclusions ................................................................................63
4 References..................................................................................................................65
Appendix A : Additional Column Results .........................................................................68
Appendix B : Additional Beam Results.............................................................................72
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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
a 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
8
arranged into a common set of notation. This common notation highlights the
central role of cross‐section stability in predicting member strength.
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.
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The first part of this document, Progress Report #3, provides a finite
element reliability analysis study on hot rolled W‐sectioned structural steel
columns. The study aimed to assess the influence of the variation of 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 columns strength beyond what one might typically
expect.
The second part of this report presents and discusses a nonlinear finite
element analysis parameter study (performed in ABAQUS) on W‐section stub
columns and short beams. The study aims 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 concluding part of this report discusses the extension of the parameter
study to include longer columns and beams, thus including global buckling
modes. This will be a further step towards the completion of a database that will
allow us to utilize the elastic buckling information, for cross‐sections with large
variations in element slenderness, to provide suggestions and improvements for
the DSM applicability to structural steel.
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2 Finite Element Reliability Analysis of Hot-Rolled W-Section Steel Columns
2.1 Introduction and Motivation
Nonlinear finite element analysis is used as a tool in this research for
predicting the ultimate strength of structural steel sections. Such analyses are
sensitive to variations in their inputs, in much the same way real columns are
influenced by variations in modulus, yield strength, residual stresses etc. To
develop a fuller understanding of the potential variations a formal reliability
analysis of structural steel columns was initiated. This study provides necessary
knowledge of the input parameters for use in subsequent nonlinear analysis.
Further, the reliability analysis itself gives insight on the relative importance of
variations in the parameters, across the possible parameters, i.e., which is more
influential expected geometric imperfection magnitudes, or variations in the
yield stress?
2.2 Objective and Methodology
The main objective of this work is to study the influence of the variation of
design parameters on the ultimate strength of W‐section steel stub columns;
further understanding, highlighting, and quantifying the uncertainties of
parameters that lead to greater variation in column strength than what one might
typically expect from a deterministic design perspective.
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This study is focused on W-section steel stub (short) columns, avoiding
global (flexural) buckling modes. The length of the columns was determined
according to the stub column definitions of SSRC (i.e., Galambos 1998). The W14
sections are chosen for the study, as they represent “common” sections for
columns in high-rise buildings. In particular, the W14x233 section is chosen to
represent the W14 group, as the dimensions of this section represents the average
dimensions of the W14 group. A sketch showing the dimensions of the W14x233
section is given in figure 2.1.
16.0”
1.72”
1.07”
15.9”
16.0”
1.72”
1.07”
15.9”
Figure 2‐1 Sketch showing the simplified geometric dimensions of a W14X233 section.
A set of six random variable parameters were chosen for the purpose of
this study. These parameters are: cross-section flange and web thicknesses, yield
strength, modulus of elasticity, Poisson’s ratio, imperfection scale factor, and
maximum residual stress value. The variables and their statistical parameters are
12
described in detail in the next section, section 2.1.1, and summarized in table 2.1
at the end of that section.
As a first step, a Taylor series approximation was found to linearize the
strength function. Finite element analysis was used to find the column’s strength
where all the parameters were set to their mean values, then the variation of the
strength with each variable was found by varying that variable while fixing the
others at their mean values and analyzing for the strength. Results of the Taylor
series approximation are presented in section 3.1 of this report.
A Monte Carlo simulation was then used to generate the random variables,
which were fed into the nonlinear finite element models. Due to the long
computational time and the limited time available for the project, only thirty
simulations were done as an initiation of this nonlinear finite element reliability
analysis, further work is obviously needed to develop a complete picture. The
generated random variables and the finite element analysis results are presented
in section 2.3.2 of this report.
2.2.1 Variables and Statistical Parameters The most commonly used statistical summary of steel properties (t, E, Fy,
etc) are those initially presented by Galambos, T.V. and Ravindra, M.K. (1978) for
the initial development of Load and Resistance Factor Design (LRFD) for steel.
More recently Galambos, T.V. (2003) updated and provided a similar summary.
Bartlett, F.M., et al (2003) provide their own summary, literature review, and
13
additional data which they used to come up with new summary statistics. The
selected distributions (PDF) and moments (mean, variance, etc.) for each random
variable is discussed in the following sections.
2.2.1.1 Section’s Thickness
ASTM A6/A6M-04b (2003) does not provide specifications on the allowable
tolerances for W-sections flange (tf) and web (tw) thicknesses. To study the effect
of their variation on the columns strength they are considered as perfectly
correlated random variables and modeled as normal distributions. The statistical
parameters are chosen to match those commonly used for fabrication random
parameters. The thicknesses mean values are taken 1.0 tf and 1.0 tw for the flange
and web thicknesses, respectively, and a COV of 0.05 is used.
2.2.1.2 Yield Strength
The yield strength is the variable that most affects the column response and
strength, as will be shown in the results section 2.3 of this report. Galambos, T.V.
and Ravindra, M.K. (1978) present different statistical parameters for flange and
web yield strengths, where the web yield strength has a 5% higher mean value
than that of the flange. However, the values they presented for the flange yield
strength statistical parameters are commonly used for the whole section. That
common practice was used, for example, by Buonopane, S.G. and Schafer, B.W.
(2006) who also pointed out that some more recent data report even slightly
lower values.
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For the purposes of this study, Galambos and Ravindra flange yield
strength statistical parameters are used for the whole section. A normal
distribution for the yield strength values, with a mean value of 1.05 fy and a COV
of 0.05 is used.
The material model used is similar to that of Barth, K.E. et al. (2005). Figure
2.2 shows the idealized stress-strain curve used for the analysis. The nominal
material model is shown in the figure. In the models, E and fy are treated as
random variables, but E’, Est, εst, and fu are treated as deterministic.
fu= 65
fy= 50
Engi
neer
ing S
tres
s (ks
i)
Engineering Strainyε stε
Slope, E =29000
Slope, Est=720Slope, Est=720
Slope, E’=145
=0.011
fu= 65
fy= 50
Engi
neer
ing S
tres
s (ks
i)
Engineering Strainyε stε
Slope, E =29000
Slope, Est=720Slope, Est=720
Slope, E’=145
=0.011
Figure 2‐2 Idealized engineering stress‐strain curve used for analysis.
15
2.2.1.3 Modulus of Elasticity
The modulus of elasticity, E, is modeled as a normal distribution with a
mean of 1.0E, and a COV of 0.06 as recommended by Galambos, T.V. and
Ravindra, M.K. (1978) and Galambos, T.V. (2003), with a nominal modulus value
of 29000 ksi.
2.2.1.4 Poisson’s Ratio
Poisson’s ratio does not have a significant effect on column strength as will
be shown in the results, but the effect of its variation is studied for the
completeness of this work. Again, the Poisson ratio, v, is modeled via a normal
distribution with a mean of 0.3, and a COV of 0.03 as recommended by
Galambos, T.V. and Ravindra, M.K. (1978) and Galambos, T.V. (2003).
2.2.1.5 Geometric Imperfections
Initial geometrical imperfections have a great effect on the nonlinearity of
column response and its resistance. The focus of this study is on stub columns,
therefore global out-of-plumbness and out-of-straightness imperfections are
ignored and only local imperfections are taken into consideration. Some guides,
e.g. ASTM A6/A6M-04b (2003) show limits for manufacture imperfections.
However, it is common in the technical literature, e.g. Kim and Lee (2002), to
introduce an initial web out of flatness of d/150 and an initial tilt in the
compression flanges of bf /150. Figure 2.3 shows a typical local buckling mode,
16
and the imperfections used for the analysis. More details on how these
imperfections were applied are found in section 2.2.2 of this report.
Since the commonly used imperfection magnitudes of bf /150 are usually
lower than the allowable variation specified by the ASTM A6/A6M-04b (2003),
for the purposes of this study, a uniform distribution was chosen for the initial
geometric imperfection values, randomly varying between zero and 2*bf/150,
with a mean value of the bf /150.
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 2‐3 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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2.2.1.6 Residual Stresses
The residual stresses in hot rolled steel are developed due to the impact of
temperature gradients during manufacturing, and observed values vary
significantly. Those locked-in stresses can have a significant effect on the
resistance of the column. A variety of residual stress distributions for hot-rolled
W-sections can be found in literature. Szalai, J. and Papp, F. (2005) studied and
compared the commonly used distributions: Young’s parabolic distribution, the
ECCS linear distribution, and Galambos and Ketter’s constant linear distribution.
They also proposed a new distribution. For the purposes of this work the classic
and commonly used distribution of Galambos, T.V. and Ketter, R.L. (1959), as
shown in Figure 2.4, is employed. Hall, D.H. (1981) pointed out that this constant
linear distribution better matches the available experimental data for residual
stresses measured on American W-sections.
As pointed out by Buonopane, S.G. (2008), data on statistics of residual
stresses are limited, and the common practice of using 30% of the section’s yield
strength, fy, is just a typical value and not a maximum one. Buonopane used
different uniform distributions with values randomly varying between zero and
a peak stress. For the purposes of this study, a similar uniform distribution was
chosen for the residual stress values, randomly varying between zero and 60% of
the section’s yield strength, with a mean value of 0.3 fy.
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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).
Table 2.1 summarizes all the random variables and their statistical
parameters and distributions.
Table 2‐1 The chosen random variables and their statistical parameters and distributions
Random Variable Distribution Mean COV Standard Deviation
Thickness Factor Normal 1.00 t 0.05 0.05 t
Yield Strength Normal 1.05 fy (=52.5 ksi) 0.10 5.25
Modulus of Elasticity Normal 1.00 E (=29000 ksi) 0.06 1740 ksi
Poissonʹs Ratio Normal 1.00 v (=0.3) 0.03 0.009
Imperfection Scale Factor Uniform 0.106 (=bf/150) 0.2737 0.029
Residual Stress Uniform 0.3 fy (=15 ksi) 0.5774 0.173 fy (=8.65 ksi.)
19
2.2.2 Finite Element Modeling The finite element analysis was conducted using ABAQUS. As mentioned
above, the analysis is conducted on stub (short) columns. All columns are
modeled with pin-pin boundary conditions, and loaded via incremental
displacement at the ends.
Based on previous work, Seif, M. and Schafer, B.W. (2009b) and as detailed
in section 3 of progress report #2, we selected the two dimensional S4 shell
element over the other available elements available in the ABAQUS element
library. The S4 elements have six degrees of freedom per node and adopt bilinear
interpolation for the displacement and rotation fields, incorporate finite
membrane strains, and their shear stiffness is yielded by “full” integration of the
element. Also, according to previous results, the mesh density was chosen to
have five elements on each unstiffened section member (flange) and ten elements
on each stiffened section member (web) and an aspect ration close to 1.0 in the
longitudinal direction.
As discussed in section 2.2.1.2, the material model used is similar to that of
Barth, K.E. et al. (2005). It 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
20
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 2.2 is converted to a true stress-strain curve for the
analysis.
The imperfections are defined for the finite element analysis by linearly
superposing a buckling eigen mode from a previous buckling analysis. An elastic
buckling analysis was performed on the W14x233 stub column section with its
variables at their mean values. The first buckling mode, which is a local mode
(shown in figure 2.3), is then introduced to the model as an initial geometric
imperfection. The buckling mode introduced is scaled by the variable
imperfection scaling factor.
The residual stresses defined in section 2.2.1.6 are applied for the finite
element analysis in terms of initial stress conditions along the columns
longitudinal direction, and given as the average value across the element at its
center, which is a common practice (e.g. Jung, S., White, D.W. (2006)).
The finite element modeling described was presented in more detail in
sections 4.2.3 through 4.2.6 of progress report #2, and will be used for the finite
element parametric study presented in section 3 of this report.
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2.3 Results and Comments
2.3.1 Taylor Series
The finite element analysis was used to find a Taylor series expansion
approximation for the column strength, R as follows:
Let the column strength, R, be a general nonlinear function of the random
variables Xi, i =1 to n. Mathematically,
)..,.........,( 21 nXXXfR = (2.1)
To calculate the mean and variance of R, we can use a Taylor series
expansion approximation of R to linearize the strength function as follows:
TOHXRXXXXXfR Xevaluated
n
ii
iin ..)(),.....,( *@1***
2*1 +
∂∂
−+= ∑ = (2.2)
Where the Xi* values are the “design point values”, which are the values
about which the R function is linearized.
For the purposes of this study, the random variables, Xi, are the variables
defined in section 2.2.1: section thicknesses, tf and tw, yield strength, fy, modulus
of elasticity, E, Poisson’s ratio, v, Imperfection scale factor, Imp., and residual
stress, RS. The design point values are chosen to be the mean values of the
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random variables. The term: ),.....,( **2
*1 nXXXf is the finite element analysis result
for the column strength while using the random variables mean values, and
equals 4644.53 kips. The higher order terms, H.O.T., are neglected while using
the linearized version of R. Finally, the strength variations with the random
variables Xi are computed for each random variable by running a finite element
analysis twice; at Xi + σi and Xi ‐ σi of that variable while fixing the rest of the
variables at their mean values and computing the slope,iX
R∂∂ .
The column strengths resulting from the finite element analysis for the
cases described are given in table 2.2, and their load‐displacement responses are
shown in Figure 2.5. Those results yield the following Taylor series expansion
approximation for the strength, R:
R ≈ 4644.5 + 3119 (tf‐1.72) + 5013.7(tw‐1.07) – 2.1(fy‐52.5) + 0.0024(E‐29000) ‐
29.4(v‐0.3) ‐129.5(Imp‐0.106) – 2.21(RS‐8.65) (2.3)
For easier comparison, Equation 2.3 can be standardized by
definingi
iii
XXσ
μ−= for each random variable, and plugging it back into the
equation which becomes:
23
R ≈ 4644.5 + 268.23 ft + 268.23 wt – 11.03 yf + 4.176 E – 0.26 ν – 3.76 IMP
– 19.11 RS (2.4)
Since equation 2.3 is a linear equation of the random variables, the
variance of R could be found using:
∑ ==
n
i xiR ia
1222 σσ (2.5)
Where ai’s are the random variable coefficients in the equation. That yields
a standard deviation, Rσ , of 380 kips, and as mentioned above, the column
strength has a mean value , Rμ , of 4644.5 kips , and so a COV of about 0.08.
24
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Thickness
Original (mean values)Thicknesses + σt
Thicknesses - σt
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Yield stress
Original (mean values)fy + σfy
fy - σfy
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Young's modulus
Original (mean values)E + σE
E - σE
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Poisson's ratio
Original (mean values)ν + σνν - σν
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Imperfetion factors
Original (mean values)Imp. factor + σImp
Imp. factor - σImp
0 0.5 1 1.5 20
1000
2000
3000
4000
5000
Displacement, in
Load
, kip
s
Residual stresses
Original (mean values)Risidual stress + σRS
Risidual stress - σRS
Figure 2‐5 Load‐displacement relationships from the finite element analysis for varying each random variable while fixing the rest of the variables at their mean values.
25
Table 2.2 column strengths resulting from the finite element analysis for the cases described
Variable Strength (kips)
Difference from original (%)
Range for strength ± St. Dev.
Standard Deviation
Slope for Taylor series
Original (Means) 4644.53
+ St. Dev. 4908.23 5.68 3119.012* Thicknesses ‐ St. Dev. 4371.76 ‐5.87
536.47 0.05 t 5013.738**
+ St. Dev. 4609.94 ‐0.74 Yield strength
‐ St. Dev. 4632.26 ‐0.26 ‐22.32 5.25 ‐2.12571
+ St. Dev. 4648.41 0.08 Young modulus
‐ St. Dev. 4640.11 ‐0.10 8.3 1740 0.002385
+ St. Dev. 4644.26 ‐0.01 Poisson ratio
‐ St. Dev. 4644.79 0.01 ‐0.53 0.09 ‐2.94444
+ St. Dev. 4640.82 ‐0.08 Imperfection factor ‐ St. Dev. 4648.33 0.08
‐7.51 0.029 ‐129.483
+ St. Dev. 4624.55 ‐0.43 Residual stress
- St. Dev. 4662.81 0.39 ‐38.26 8.66 ‐2.20901
* slope for the flange thickness
** slope for the web thickness
Results show that the variations of the thicknesses, which directly vary the
cross‐section’s area, have a great direct effect on the column’s strength, where the
strength is almost directly proportional with the thicknesses variation. Figure 3.1
shows that varying the yield strength is the main parameter affecting the
column’s response and failure modes. It is noticed from the load‐displacement
relationships, that increasing and decreasing the yield strength both lead to
columns with lower (slightly) ultimate loads. However, one must be careful to
consider the difference between the squash load (Agfy) of the column and the
ultimate load (Agfu). In the models fy is varied, but fu is not – thus the load at yield
26
varies significantly in the figures but all models essentially converge to Agfu at
failure.
It is also noticed from the figure that varying the modulus of elasticity or
the imperfection scaling factor had very small effect on the column strength.
Also, varying the Poisson ratio had zero effect on the results. Extra finite element
analysis were performed at Poisson ratios of one tenth of the mean values, and
results still showed almost no difference in the column strength or response
(about 0.2%). Since elastic buckling has little role to play in these extremely
compact columns this observation is consistent with expectations.
2.3.2 Mont Carlo Simulation
Matlab was used to generate random variables for a Monte Carlo
simulation of the selected column. Thirty simulations were generated and
analyzed. (This essentially provides proof of concept only as more simulations
would be required for a fuller statistical treatment of the output). Relative
frequency histograms and PDF fits for the generated data are shown in Figure
2.6, while Figure 2.7 shows cumulative frequency histograms and CDF fits for
the generated data. The values of these random variables for each simulation are
given in table 2.3.
27
0.9 0.95 1 1.05 1.10
5
10
15
20
Thickness variation factor, (*tf and *tw )
Rel
ativ
e Fr
eque
ncy
or p
x(thic
knes
ses)
Relative Frequency Histogram and PDF Fit
40 45 50 55 60 650
0.05
0.1
0.15
0.2
Yield strength, ksi
Rel
ativ
e Fr
eque
ncy
or p
x(Fy)
Relative Frequency Histogram and PDF Fit
2.6 2.8 3 3.2 3.4
x 104
0
2
4
6
8x 10-4
Young modulus
Rel
ativ
e Fr
eque
ncy
or p
x(E)
Relative Frequency Histogram and PDF Fit
0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.320
20
40
60
80
100
120
140
Poisson ratio
Rel
ativ
e Fr
eque
ncy
or p
x( ν)
Relative Frequency Histogram and PDF Fit
0.06 0.08 0.1 0.12 0.140
10
20
30
40
50
Imperfection variation factor
Rel
ativ
e Fr
eque
ncy
or p
x(Imp)
Relative Frequency Histogram and PDF Fit
0 0.1 0.2 0.3 0.4 0.5 0.60
1
2
3
4
Residual stress factor, (fy)
Rel
ativ
e Fr
eque
ncy
or p
x(RS
)
Relative Frequency Histogram and PDF Fit
Figure 2‐6 Relative frequency histograms and PDF fits for the generated random variables.
28
0.9 0.95 1 1.05 1.10
0.2
0.4
0.6
0.8
1
Thickness variation factor, (*tf and *tw )
Com
ulat
ive
Freq
uenc
y or
Fx(th
ickn
esse
s) Comulative Frequency Histogram and CDF Fit
45 50 55 60 650
0.2
0.4
0.6
0.8
1
Yield strength, ksi
Com
ulat
ive
Freq
uenc
y or
Fx(F
y)
Comulative Frequency Histogram and CDF Fit
2.6 2.7 2.8 2.9 3 3.1 3.2 3.3
x 104
0
0.2
0.4
0.6
0.8
1
Young modulus, ksi
Com
ulat
ive
Freq
uenc
y or
Fx(E
)
Comulative Frequency Histogram and CDF Fit
0.285 0.29 0.295 0.3 0.305 0.31 0.315 0.320
0.2
0.4
0.6
0.8
1
Poisson ratio
Com
ulat
ive
Freq
uenc
y or
Fx( ν
)
Comulative Frequency Histogram and CDF Fit
0.06 0.08 0.1 0.12 0.140
0.2
0.4
0.6
0.8
1
Imperfection variation factor
Com
ulat
ive
Freq
uenc
y or
Fx(Im
p)
Comulative Frequency Histogram and CDF Fit
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
0.6
0.8
1
Residual stress factor, (*fy)
Com
ulat
ive
Freq
uenc
y or
Fx(R
S)
Comulative Frequency Histogram and CDF Fit
Figure 2‐7 Cumulative frequency histograms and CDF fits for the generated random variables.
29
The finite element reliability analysis results for all simulations are
presented in the form of load‐displacement relationships in Figure 3.8. The
column ultimate strengths from the load‐displacement curves are given in table
2.3, along with the values of the random variables used for each simulation.
0 0.5 1 1.5 2 2.50
1000
2000
3000
4000
5000
6000
Displacement, in.
Forc
e, k
ips
original (means)Simulation# 01Simulation# 02Simulation# 03Simulation# 04Simulation# 05Simulation# 06Simulation# 07Simulation# 08Simulation# 09Simulation# 10Simulation# 11Simulation# 12Simulation# 13Simulation# 14Simulation# 15Simulation# 16Simulation# 17Simulation# 18Simulation# 19Simulation# 20Simulation# 21Simulation# 22Simulation# 23Simulation# 24Simulation# 25Simulation# 26Simulation# 27Simulation# 28Simulation# 29Simulation# 30
Figure 2‐8 Load‐displacement relationships from the finite element analysis for all the simulations.
30
Table 2.3 Ultimate strengths and the values of the random variables used for each simulation
Simulation Strength Thickness variation factor
tf (in.)
tw (in.) fy (ksi) E (ksi) v
Imp. scaling factor
R.S. (*fy)
1 4379.91 0.967 1.664 1.035 44.04 29101.94 0.298 0.109 0.496
2 4431.11 0.971 1.670 1.039 51.07 27424.40 0.298 0.139 0.406
3 4647.05 1.014 1.744 1.085 48.32 30534.12 0.297 0.142 0.125
4 4069.20 0.903 1.553 0.966 54.12 29903.23 0.298 0.135 0.191
5 4347.73 0.933 1.606 0.999 51.68 33436.79 0.302 0.088 0.080
6 4824.31 1.038 1.786 1.111 54.72 30459.73 0.288 0.101 0.403
7 4319.97 0.945 1.626 1.012 56.53 29583.00 0.299 0.131 0.343
8 4663.41 0.989 1.702 1.059 52.89 28209.57 0.293 0.067 0.102
9 4660.45 0.986 1.697 1.056 48.36 28618.18 0.307 0.067 0.089
10 4748.04 1.026 1.764 1.097 58.58 29270.20 0.298 0.083 0.286
11 4238.57 0.948 1.630 1.014 43.74 27425.55 0.303 0.108 0.545
12 4766.89 1.035 1.779 1.107 54.74 27223.40 0.301 0.154 0.331
13 4357.35 0.944 1.623 1.010 54.84 28241.46 0.298 0.127 0.020
14 4344.14 0.931 1.601 0.996 54.59 31328.49 0.306 0.087 0.032
15 4636.88 0.996 1.713 1.065 52.52 30861.13 0.310 0.085 0.483
16 4429.30 0.973 1.674 1.041 50.66 30520.25 0.288 0.141 0.271
17 4285.78 0.949 1.632 1.016 52.24 29080.89 0.303 0.147 0.230
18 4700.69 1.013 1.742 1.083 48.25 30406.15 0.293 0.120 0.474
19 4553.91 0.973 1.674 1.041 52.90 31421.33 0.297 0.081 0.219
20 4897.85 1.051 1.807 1.124 56.90 29673.00 0.303 0.065 0.319
21 4524.67 0.988 1.700 1.058 48.57 29693.11 0.315 0.140 0.427
22 4694.09 1.030 1.771 1.102 58.71 32998.69 0.303 0.115 0.523
23 5115.67 1.093 1.880 1.170 50.64 30254.54 0.282 0.151 0.197
24 4087.93 0.904 1.555 0.967 43.33 27558.35 0.321 0.062 0.390
25 4281.59 0.939 1.615 1.005 51.18 28881.59 0.301 0.115 0.585
26 4963.76 1.065 1.831 1.139 60.23 29096.90 0.297 0.084 0.046
27 4251.78 0.941 1.619 1.007 64.25 29103.28 0.306 0.139 0.352
28 4685.40 0.997 1.714 1.066 51.14 29593.55 0.314 0.075 0.248
29 4686.66 1.012 1.740 1.083 58.76 26162.51 0.300 0.100 0.185
30 4901.29 1.042 1.792 1.115 47.89 29964.88 0.313 0.095 0.158
31
The Monte Carlo results are consistent with the Taylor Series in that
thickness is the variable most strongly correlated with the ultimate strength.
Again, ultimate strength in the models is largely converging to Agfu, and fu is
deterministic. Cursory examination of the load at first yield (Figure 2‐8) indicates
that fy in addition to t plays a stronger role at the lower load level.
The column strength using the specified mean values for the variables is
4644.53 kips. The mean value of the columns strength for the 30 simulations is
4549.8 kips, which is lower than the strength using the mean values, giving a bias
factor of about 0.98. The standard deviation from the simulation is 264.162 kips,
resulting in a a COV of 0.058 ‐ 70% the value of that calculated from the Taylor
series expansion. For this simple problem the first‐order Taylor Series
approximation is providing reasonable results.
Figure 2.9 shows the relative and cumulative histograms for the column
strengths for all the simulations. Figure 2.10 shows the column strengths results
along with the mean and standard deviations and the nominal strength using the
mean values. Finally, Figure 2.11 shows a normal distribution probability plot for
the results.
32
4000 4200 4400 4600 4800 5000 52000
0.5
1
1.5
2
2.5
3
3.5
4x 10-3
Column maximum strength
Rel
ativ
e Fr
eque
ncy
or p
x(stre
ngth
)
Relative Frequency Histogram and PDF Fit
Relative frequencyNormal PDF
4000 4200 4400 4600 4800 5000 52000
0.2
0.4
0.6
0.8
1
1.2
1.4
Column maximum strength
Com
ulat
ive
Freq
uenc
y or
Fx(s
treng
th)
Comulative Frequency Histogram and CDF Fit
Cumulative frequencyNormal CDF
Figure 2‐9 Relative and cumulative frequency histograms for the resulting column strengths.
33
4000 4200 4400 4600 4800 5000 52000
0.5
1
1.5
2
2.5
3
3.5
4x 10
-3
Column maximum strength
Rel
ativ
e Fr
eque
ncy
Relative Frequency Histogram
Stengths mean Nominal strength
0 5 10 15 20 25 300
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
Simulation number
Col
umn
max
imum
stre
ngth
0 5 10 15 20 25 303800
4000
4200
4400
4600
4800
5000
5200
Simulation number
Col
umn
max
imum
stre
ngth
Column strengthsStrengths meanStengths mean + σStengths mean - σNominal strength
Figure 2‐10 Column strengths results from simulations and the results statistical parameters.
34
4000 4200 4400 4600 4800 5000 5200
0.01
0.05
0.1
0.25
0.5
0.75
0.9
0.95
0.99
Data
Pro
babi
lity
Probability plot for Normal distribution
Figure 2‐11 Normal distribution probability plot for the simulated column strengths.
2.4 Main Conclusion
This work indicates the ability to perform reliability analysis on structural
steel shapes using nonlinear finite element analysis as the predictive engine.
Further, the results indicate that even simple first order Taylor Series expansions
may provide useful information about the statistical variation in the strength and
the influence of the input variables. In the analyses it is observed that variation in
the yield stress had the greatest impact on the shape of the load‐deformation
response, while variation in the thickness was by far the most influential variable
on strength. Though thickness is shown here to be a strong indicator of ultimate
strength in these models, two mitigating factors should be considered: one the
35
models assumed the flange and web are positively correlated (both reducing the
same) in reality the total area is approximately fixed and they are actually
negative correlated, second the ultimate strength considered here was Agfu in
many cases and fu was deterministic, thus leaving only variations in Ag to
influence the strength.
2.5 Study Extension Suggestions
• Extend the work initiated herein on the W14 sections to include a wider
range of W‐sections and other section types. Non‐typical sections with a
wide range of slenderness can be also studied.
• Expand this finite element Reliability study, to include studying:
‐ The effect of different column lengths.
‐ The effect of different material models; elastic regions, strength, and
strain hardenings.
‐ The effect of different types, shapes, and scales of initial local and
global geometric imperfections.
‐ The effect of using different residual stress distributions.
• Further study the effect of the variation of the web and flange thicknesses
on the column strength, using different approaches such as: fixing the
36
section area, negatively correlating the thicknesses (increasing one will
decrease the other).
• Study the effect of correlating the randomness of residual stresses to the
randomness of the sections yield strength.
37
3 Finite Element Comparison of Design Methods for Locally Slender Steel Beams and Columns
3.1 Introduction and Motivation
Typically, locally slender (noncompact or slender) cross‐sections are
avoided in the design of hot‐rolled steel structural members. However, this
strategy becomes inefficient with the advent of high and ultra‐high yield strength
steels, as the increased yield stress drives even standard shapes from locally
compact to locally slender. The effect of increasing the yield strength on local
buckling is a topic that has seen some study in recent years (see e.g., Earls 1999).
Initial analysis of the AISC provisions (Schafer and Seif 2008) indicates geometric
regions where the existing AISC design approach may be excessively
conservative, and other regions where it may be moderately unconservative.
Efficient and reliable strength predictions are needed for hot‐rolled steel cross‐
sections to take advantage of more locally slender sections. Since the design of
locally slender cross‐sections is common practice in cold‐formed steel, the two
design methods in current use for cold‐formed steel are compared with the AISC
approach herein.
The objective of this study is to describe and analyze a series of nonlinear
finite element analyses used to compare different design methods for locally
slender, braced, steel beams and columns.
38
The nonlinear finite element analysis parameter study, using ABAQUS, is
initiated for the purpose of understanding and highlighting the parameters that
lead to the divergence between the capacity predictions of the different design
methods. Particular attention is placed on understanding the regimes where the
AISC methods give divergent results, from either the other Specifications, or the
nonlinear finite element analysis results.
3.2 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.
To aid the comparison of the available methods the design strength
formulas, for locally slender W‐section beams and columns, are provided in a
common notation. The resulting design expressions highlight the prominent role
of elastic cross‐section stability as the key parameter for strength prediction.
39
3.2.1 Column Design Equations Previous work reported in Progress Report #1 provided and examined the
design expressions for unbraced columns with locally slender cross-sections
(Schafer and Seif 2008). Design expressions for stub (braced) columns were
provided in section 4.1 of Progress Report #2, and summarized here for
completeness. The expressions are shown in Table 3.1 in a manner that provides
a focus on the local buckling strength predictions alone, and may be readily
compared to the subsequently conducted nonlinear finite element analysis.
The column design equations of Table 3-1 have been previously detailed
(Schafer and Seif 2008), summary observations were: (a) local plate buckling (fcrh,
fcrb, fcrl) and yield stress (fy) are the key parameters for determining the strength,
regardless of the method; (b) AISC and AISI-Effective Width use local plate
buckling solutions for the isolated flange (fcrh) and web (fcrb), while DSM uses the
local bucking solution for the cross-section as a whole (fcrl); (c) AISC ignores post-
buckling capacity in slender unstiffened elements (see the Qs expression).
40
Table 3-1 Braced column strength for locally slender I-shaped section
AISC
ys
crhAht
ff
ff
crh
a
ycrbff
ycrbyff
ycrb
s
ygasn
fQf
ff..
ff.Q
ff.
fff..
ff.
Q
fAQQP
g
w
y
crh
y
crh
y
crb
y
crb
=
⎪⎩
⎪⎨⎧
≤⎟⎠
⎞⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ −−−
>=
⎪⎪
⎩
⎪⎪
⎨
⎧
≤
<<−
≥
=
=
2 if 16019011
2 if 01
if 11
2 if 5904151
2 if 01
53
53
AISI (Effective Width Method)
⎪⎩
⎪⎨⎧
<⎟⎠⎞
⎜⎝⎛ −
≥=ρρ=
⎪⎩
⎪⎨⎧
<⎟⎠⎞
⎜⎝⎛ −
≥=ρρ=
+=
=
ycrhff
ff
ycrh
hhe
ycrbff
ff
ycrb
bbe
wefeeff
yeffn
f.f.
f.fhh
f.f.
f.fbb
thtbA
fAP
y
crh
y
crh
y
crb
y
crb
22 if2201
22 if1 where
22 if2201
22 if1 where
4
DSM (Direct Strength Method, AISI Appendix 1)
ll
l
l
ll
crgcr
ygy
ycry
.
PP
.
PP
ycry
n
fAP
fAP
P.PP.
P.PPP
y
cr
y
cr
=
=
⎪⎩
⎪⎨
⎧
<⎟⎠⎞⎜
⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞⎜
⎝⎛−
≥=
661 if1501
661 if4040
3.2.2 Beam Design Equations The beam design equations are shown in
Table . They provide the strength prediction for laterally braced I-shaped
cross-sections which may be locally slender.
bebe
bebe
½he
½he
bebe
bebe
½he
½he
41
Table 3-2 Braced beam strength for locally slender I-section
AISC
( )nfnwn M,MM min=
WLB ycrh M.M 32≥ pnw MM =
ycrhy MMM. >>32
crhy MM ≥ ypgnw MRM =
( ) 11751 3001200 ≤−−= + crhyFE
aa
pg M/M.Ryw
w
10≤== flangewebffww A/Atb/hta
FLB ycrb M.M 96≥ pnf MM =
ycrby MMM. >>96
y*y MM = if ycrh MM ≥ else nw*y MM =
crby MM ≥ crbxcrbnf fSMM ==
AISI (Effective Width Method)
yeffn fSM =
effeffeff y/IS = , ∫= effeff dAyI 2 , ∫∫= effeffeff dA/ydAy
⎪⎩
⎪⎨⎧
<⎟⎠⎞
⎜⎝⎛ −
≥=ρρ=
ycrbff
ff
ycrb
bbe f.f.
f.fbb
y
crb
y
crb 22 if2201
22 if1 where
⎪⎩
⎪⎨⎧
<⎟⎠⎞
⎜⎝⎛ −
≥=ρρ=
ycrhff
ff
ycrh
hhe f.f.
f.fhh
y
crh
y
crh 22 if2201
22 if1 where
41 /hh e= , 42 /hh e= , for 4≥b/h
DSM (Direct Strength Method – AISI Appendix 1)
ycr M.M 661>l ( ) ⎟⎠⎞
⎜⎝⎛ −−−=
y
crMM
yppn .MMMM l6611
ycr M.M 661≤l y
.
MM
.
MM
n M.My
cr
y
cr4040
1501 ⎟⎠⎞⎜
⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞⎜
⎝⎛−= ll
( )6601
66070
.
.M/MM.MMM crhy
yppnw −
−−−=
( )38001
38070
..
.M/MM.MMM
crby*ynwnwnf −
−−−=
bebe
bb
h1
h2
42
Even a cursory examination of
Table reveals that AISC, AISI, and DSM use radically different methods to
provide the strength prediction of locally slender beams. The AISC method is
compiled from sections F2-F5 of the Specification (also see White 2008). Similar to
Schafer and Seif (2008) for columns, the λ limits (h/tw, bf/2tf) have been replaced
by fcrh and fcrb limits (actually Mcr limits where Mcr is just Sxfcr) as (a) this is more
general, (b) this allows for a comparison between different design methods in a
common notation, and (c) the centrality of elastic local buckling is made clear by
this change.
Compilation of the AISC Specification from sections F2-F5 is non-trivial. However, it is possible to divorce the web local buckling limit state and flange local bucking limit state from the individual F2-F5 sections. For the convenience of users of the AISC Specification, these expressions are provided in
notation similar to AISC in the following (also this makes clear the λ to Mcr conversions used and provided in
Table ). Mn is the minimum of Mnlw and Mnlf.
AISC Web local buckling
C: pww λ≤λ then pwn MM =l
NC: rwwpw λ<λ<λ then ( )pwrw
pwwyppwn MMMM
λ−λ
λ−λ−−=l
S: wrw λ≤λ then ypgwn MRM =l
( ) 11 3001200 ≤λ−λ−= + rwwaa
pg w
wR
10≤== flangewebffww A/Atb/hta
AISC Flange local buckling
43
C: pff λ≤λ then pfn MM =l
NC: rffpf λ<λ<λ then ( )pfrf
pff*ywnwnfn M.MMM
λ−λ
λ−λ−−= 70lll
y*y MM = if web NC or C rww λ<λ
wn*y MM l= if web S wrw λ≤λ
S: frf λ≤λ then fcrfn MM ll =
where λ’s are defined in Table B4.1 of AISC, and Mcrlf is the local buckling of
the flange, i.e., Sxfcr, and the k for fcr is defined in terms of h/tw.
General Expression Notes
Expressions have been limited to centerline models of an I-shaped doubly
symmetric section. In practice, AISC and AISI use slightly different k values to
determine fcrb and fcrh, in the parametric studies herein they are set equal. For
columns kf=0.7 and kw=5.0, for beams kf employs the AISC expression:
( )2
2
2
112 ⎟⎟⎠
⎞⎜⎜⎝
⎛
ν−
π==
btEkSfSM f
fxcrbxcrb , 7604350 .t/h
k.w
f ≤=≤
while kw=36 (refer to Seif and Schafer 2009a for a more detailed discussion
on the local plate buckling coefficients values underlining the AISC
specification). For DSM fcrl or Mcrl are determined from an elastic bucking finite
strip analysis (Schafer and Ádány 2006). For AISI (Effective Width) the neutral
axis shift has been ignored, for the studied sections the error is small. The DSM
expression for Mcrl>1.66My is based on recent work (Shifferaw and Schafer 2008)
and is under ballot at AISI at the time of this writing.
Definition of Key Variables
44
Ag = gross cross-sectional area
b = half of the flange width (bf=2b)
E = modulus of elasticity
fcrb = flange local buckling stress = kf[π2E/(12(1-ν2))](tf/b)2
fcrh = web local buckling stress = kw[π2E/(12(1-ν2))](tw/h)2
fcrl = cross-section local buckling stress, e.g. by finite strip analysis
fy = yield stress
h = centerline web height
kf = flange plate buckling coefficient
Mcrb = web local buckling moment = Sxfcrb
Mcrh = flange local buckling moment = Sxfcrh
Mcrl = cross-section local buckling moment = Sx fcrl, e.g. by finite strip
Mn = nominal flexural capacity
Mp = plastic moment
My = moment at first yield
Pn = nominal compressive capacity
Sx = gross section modulus about the x-axis
tf = flange thickness
tw = web thicknes
λf = b/tf also see AISC Table B4.1 for λfr, λfp
λw = h/tw also see AISC Table B4.1 for λwr, λwp
ν = Poisson’s ratio
45
3.3 Parameter Study and Modeling
3.3.1 Approach The purpose of the nonlinear finite element (FE) analysis parameter study
initiated 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.
The FE analysis is 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). 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.
3.3.2 Geometric Variation To examine the impact of slenderness in the local buckling mode, and the
impact of web-flange interaction in I-sections, four series of parametric studies
are performed under axial and bending loading:
46
• 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,
• 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 and Figure 3-1. Figure 3-1 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
47
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 3-1 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.
48
3.3.3 Finite Element Modeling The finite element modeling for this study is similar to that described in
section 2.2.2 of this report. Analysis was conducted using ABAQUS on the stub
columns and short beams with pin-pin boundary conditions, and loaded via
applying an incremental displacements and rotations till failure.
Again, the S4 elements were chosen according to results from previous
work shown by Seif, M. and Schafer, B.W. (2009b) and detailed in section 3 of
progress report #2. The mesh density was chosen to have five elements on each
unstiffened section member (flange) and ten elements on each stiffened section
member (web).
The Barth, K.E. et al. (2005) material model shown in Figure 2.2 and
described in section 2.2.2 is used. The initial geometric imperfections from elastic
buckling analysis, presented in that same section and shown in Figure 2.3, are
adopted. The Galambos and Ketter (1959) residual stresses distribution shown in
Figure 2.4 is implanted in the models in terms of initial stress conditions along
the members’ longitudinal direction.
3.4 Results
As discussed previously (see
Table 2.3 and Error! Reference source not found.) the parametric study is
broken into 4 groups: W14FI, W14FR, W36FR, and W36WI. Here the results of
49
the parametric study are presented for each group, including comparisons to the
AISC, AISI, and DSM design methods as summarized in Table 3-1 and
Table .
3.4.1 Columns Typical load displacement results for the nonlinear FE collapse analysis of
the stub column specimens are shown for the W36WI group in Error! Reference
source not found.. The thicker (recall tw is varied here) specimens develop
significant strain hardening (in compression!) while the thinner specimens
exhibit less displacement before collapse. The peak loads recorded, denoted with
a “*” in Error! Reference source not found., are in excess of the squash load
(Py=Agfy) due to the strain hardening. For comparison of design methods, this
strength was deemed unrealistic. As an alternative the peak load that was
recorded at a displacement equal to 3 times the gross yield displacement (Δ=3Δy ,
Δy=Py/E) denoted as a “o” in Error! Reference source not found. was utilized as
the predicted strength (Pn). This same procedure was employed for all the
parametric studies. Similar load displacement curves for the rest of the study
groups are presented in Appendix A.1.
For the W36WI group at a displacement slightly past the chosen peak loads,
Figure 3.3 provides the axial deformed shape for the original W36x330 section, as
well as those of a thinner section and a thicker section (recall tw is varied here). It
is clear that the thinner section experiences significant web local buckling. For
50
the same sections, Figure 3.4 shows the membrane longitudinal stress. The figure
shows that for the thinner section, all the load is carried through the flanges. For
the original and thicker sections, although fully compact according to the AISC
definition, the stress is not uniformly carried in the webs – suggesting
noncompact behavior and a stress distribution more consistent with typically
assumed effective width distributions. Finally, for the same sections, Figure 3.5
shows the membrane plastic strain. The thinner section only experiences plastic
strain in the flanges, while the original section experiences a fair amount in the
web, indicating its ability to dissipate greater energy, and helps to indicate why
the falling branch of the load-displacement curve does not start until the section
undergoes significant deformation for that section.
51
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2000
4000
6000
8000
10000
12000
Figure 3-2 Load-displacement for W36WI column study
Figure 3-3 Axial deformed shape for three samples from the W36WI study group
52
Figure 3-4 Membrane Longitudinal stresses for three samples from the W36WI study group
Figure 3-5 Membrane plastic strain for three samples from the W36WI study group
The ABAQUS predicted capacities are compared to the AISC, AISI, and
DSM methods in Error! Reference source not found. and Error! Reference
source not found.. Error! Reference source not found. presents the results for
53
each of the 4 parameter studies. Error! Reference source not found. presents all 4
studies against each of the design methods. The horizontal axis in all figures is
the elastic local slenderness of the cross-section: √fy/fcrl. Where fcrl is determined
for each cross-section from a finite strip analysis. Note, for an elastic buckling
analysis fcrb=fcrh= fcrl; however for AISC and AISI the strength prediction uses
assumed k’s to determine fcrb and fcrh thus fcrb≠fcrh≠ fcrl for AISC and AISI. If in
AISC and AISI fcrb and fcrh are forced to the fcrl value, the strength predictions are
changed – an example comparison for this case is provided in Error! Reference
source not found. for the W14FI study. Similar plots for the rest of the study
groups are presented in Appendix A.2.
1 2 30
0.5
1
W14FI
1 2 30
0.5
1
W14FR
1 2 30
0.5
1
W36FR
1 2 30
0.5
1
W36WI
AISCAISIDSMABAQUS
Figure 3-6 Results of column parametric study for 4 study groups
54
1 2 30
0.2
0.4
0.6
0.8
1
1 2 30
0.2
0.4
0.6
0.8
1
1 2 30
0.2
0.4
0.6
0.8
1
AISCAISIDSMABAQUS
AISC AISI
DSM
Figure 3-7 Results of column parametric study for 3 design methods
0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
ABAQUS
Figure 3-8 Column study results for W14FI considering alternative design methods where fcrl (from
FSM) replaces fcrb and fcrh in AISC and AISI
55
3.4.2 Beams The moment-rotation results for the W14FI parametric study group of
braced (short) beams is provided in Error! Reference source not found.. Similar
to the columns, the rotation at 3θy (where θy=MyL/2EI) was taken as a maximum
allowable rotation, and the Mn at this value used as the predicted strength of the
beams when comparing with design methods. The load displacement curves for
the rest of the study groups are presented in Appendix B.1.
For the W14FI group at a rotation immediately past the chosen peak
moments, Figure 3.10 provides the axial deformed shape for the original
W14x233 section, as well as that of a thinner section. The thinner section
experiences flange local buckling, while the original section undergoes pure
flexure almost exclusively. For the same sections, Figure 3.11 shows the
membrane longitudinal stress.
56
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
4
Rotation, Rad.
Mn,
kip
s.in
Figure 3-9 Moment-rotation for W14FI beam study
Figure 3-10 Axial deformed shape for two samples from the W14FI study group
57
Figure 3-11 Membrane longitudinal stresses for two samples from the W14FI study group
The predicted capacities from the nonlinear collapse analysis in ABAQUS
are shown for each of the 4 parameter groups in Figure 3-12 and compared
against the design methods directly in Figure 3-13. The local slenderness √fy/fcrl
(or equivalently √My/Mcrl) is used as the horizontal axis, while the vertical axis is
the capacity normalized to the plastic moment, Mp.
58
1 2 30
0.5
1
W14FI
1 2 30
0.5
1
W14FR
1 2 30
0.5
1
W36FR
1 2 30
0.5
1
W36WI
AISCAISIDSMABAQUS
Figure 3-12 Results of beam parametric study for 4 study groups
1 2 30
0.2
0.4
0.6
0.8
1
1 2 30
0.2
0.4
0.6
0.8
1
1 2 30
0.2
0.4
0.6
0.8
1
2.82.9
AISCAISIDSMABAQUS
AISC
DSM
AISI
Figure 3-13 Results of beam parametric study for 3 design methods
59
3.5 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.
3.5.1 Columns AISI’s implementation of the Effective Width Method provides, by far, the
best prediction of the column capacity. Only in the W14FR study does any
method (DSM in this case) slightly outperform AISI.
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
slender flanges completely uneconomical.
DSM provides a reasonable approximation only when both the web and
flange slenderness are changed – when one element is much more slender than
another (for instance the flange in the W14FI study or the web in the W36WI
study) the DSM approach is overly conservative. For example, in the W36WI
group, the presumption that the local buckling of the web initiates a similar local
buckling in the flange does not occur. Examination of the deformed shape at
collapse shows that the deformation is primarily one of web local buckling with
little flange deformation. DSM’s assumption, driven from elastic stability
60
analysis, that member local buckling and member strength are one in the same is
not observed in this section.
It is worth noting that this phenomenon (where DSM provides overly
conservative predictions) exists in cold-formed steel members (which are of
constant thickness) when one element is significantly wider than another. In
these cases it has been found that although DSM is conservative, such sections
also have significant serviceability problems. Such sections with highly varying
element slenderness typically benefit from the inclusion of a longitudinal
stiffener which provides a significant boost to the elastic buckling of the slender
element and brings DSM predictions back in-line with observed capacities.
Nonetheless, this phenomenon needs further study before DSM can be fully
realized in structural steel.
3.5.2 Beams AISC predictions are excellent when the section is compact. For
noncompact flanges, AISC may be modestly unconservative (see W36FR, W14FR
results), while for slender flanges, as in columns, the AISC predictions are
excessively conservative. The web expressions appear to provide adequate
strength predictions for all ranges (see W36WI results), again as long as the
flange is compact. Transitions in the AISC expressions are not necessarily smooth
(this is noted in the AISC commentary) and large disincentives, inconsistent with
observed performance, are put on slender flanges.
61
AISI’s Effective Width Method is for beams, like columns, the overall best
performer in comparison with the finite element analysis results. However,
unlike in columns when all cases agreed well, the W14FI study shows the
effective width of the flange to be too conservative (presumably because
beneficial stabilization by the compact web is not accounted for). The W36
parameter study results are predicted quite well. Although AISI provides
expressions for inelastic reserve (i.e., when Mn>My), the expressions involved
and the solution are not in closed-form; therefore they have been ignored in
Table and the presented figures, and the AISI capacity is limited to My as
shown.
DSM’s accuracy for the beams meets or exceeds AISI and AISC except for
the W36WI case, where the method is progressively more conservative as the
web slenderness increases (see the discussion of DSM for columns above). Note,
multiple curves are presented for DSM in Figure 3-13 because of the
normalization to Mp (as opposed to My).
3.5.3 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
62
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
consistently conservative, and conceptually simple prediction method that is
worthy of further study.
3.6 Long Members Parameter Study
3.6.1 Introduction The next step in this research is the extension of the parametric study to
long columns and beams where the locally slender cross-sections may interact
with global (flexural, lateral-torsional, etc.) buckling modes. Of particular interest
is the interaction of sections such as in the W36WI study where the sections are
predicted to have quite low capacities by DSM and show the potential for much
stronger interaction with global modes due to the abruptness of the moment-
rotation response. The available design methods treat these local-global
interactions with quite different approaches, and thus further study is needed in
this regime. In addition, studying the sensitivity to additional input parameters:
imperfections, residual stresses, and material stress-strain curve is needed. A
primary goal of this research remains to provide recommendations for the
extension of DSM to structural steel.
3.6.2 Initial Approach As a first step towards the completeness of this parameter study data base,
the same design groups described in section 3.3.2 are used. The members’
63
lengths are then determined, to achieve certain preset slenderness parameter
values. The slenderness parameter, λc, is defined in terms of the member’s length
and cross-section dimensions as follows:
( )( ) yygy
crec fr/KL
EfAKL/EI
PP
2
222 π=
π==λ (e.g., for flexural buckling)
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 length, L, is then back calculated to maintain a certain λc.
Modeling the members for the ABAQUS analysis is done in a similar
approach to that used for the shorter members and described in section 3.3.3. The
main difference is in the inclusion of initial geometric imperfections, where
global buckling modes are included at this time. Initial geometric imperfections
are added through linearly superposing both a scaled local and a scaled global
buckling eigen mode from a finite strip analysis using CUFSM (Schafer, B.W.,
Ádány, S. (2006)) performed on each section.
Figure 3.14 shows a sample of these buckling modes determined from the
CUFSM’s load versus half-wave-length curves. The pure local buckling mode is
that occurring at the inflection point and the global buckling mode is that
occurring at a half wave length equal to the member’s length. The local buckling
mode is scaled so that the maximum nodal displacement is equal to the larger of
bf /150 or d/150 as shown in Figure 2.3, while the global buckling mode is scaled
64
so that the maximum nodal displacement is equal to L/1000. Analysis and
interpretation of the results is currently underway.
100 101 102 1030
100
200
300
400
500
600
700
800
900
1000
Half wave length
Load
x
x
L
Local
Global
Figure 3-14 Sample of local and global buckling modes determined from CUFSM.
3.7 Summary and Conclusions
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
65
(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
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.
66
4 References
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AISI (2007). “North American Specification for the Design of Cold-Formed Steel Structures”, Am. Iron and Steel Inst., Washington, D.C., AISI-S100.
ASTM (2003). “Standard Specification for General Requirements for Rolled Structural Steel Bars, Plates, Shapes, and Sheet Piling”, ASTM A6/A6M-04b, American Society for Testing and Materials, West Conshohocken, PA.
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.
Bartlett, F.M., et al (2003). “Updating standard shape material properties database for design and reliability”, Engineering Journal / American Institute of Steel Construction, First Quarter, 2003; 2-14.
Buonopane, S.G. (2008). “Strength and reliability of steel frames with random properties”, Journal of Structural Engineering, ASCE, February 2008, 337-344.
Buonopane, S.G., Schafer, B.W. (2006). “Reliability of steel frames designed with advanced analysis”, Journal of Structural Engineering, ASCE, February 2006, 267-276.
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. (1999). “On the inelastic failure of high strength steel I-shaped beams”, Journal of Constructional Steel Research 49 (1999) 1–24.
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., Ravindra, M.K. (1978). “Properties of steel for use in LRFD”, Journal of the Structural Division, ASCE, September 1978, ST9; 1459-1468.
67
Galambos, T.V. (1998). “Guide to Stability Design Criteria for Metal Structures”. 5th ed., Wiley, New York, NY, 815-822.
Galambos, T.V. (2003). “Load and resistance factor design”, Engineering Journal / American Institute of Steel Construction, Third Quarter, 1981; 74-82.
Hall, D.H. (1981). “Proposed steel column strength criteria”, Journal of the Structural Division, ASCE, April 1981, ST4; 649-670.
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.
Nowak, A.S., Collins, K.R. (2000). “Reliability of Structures”, McGraw Hill, 2000, ISBN: 0-07-048163-6.
Salmon, C.G., Johnson, J.S. (1996). “Steel structures: design and behavior: emphasizing load and resistance factor design” HarperCollins.
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.
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.
68
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.
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.
Szalai, J., Papp, F. (2005). “A new residual stress distribution for hot-rolled I-shaped sections”, Journal of Constructional Steel Research 61, 2005: 845–861.
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69
Appendix A : Additional Column Results
A.1 Load‐Displacement Curves
Load-displacement results for the nonlinear FE collapse analysis of the stub
column specimens are shown in Figures A.1 through A.3 for the W14FI, W14FR,
and W36WR groups respectively. Similar to Figure 3.2, the peak loads recorded,
are denoted with a “*” and the alternative peak loads that were recorded at a
displacement equal to 3 times the gross yield displacement (Δ=3Δy , Δy=Py/E) are
denoted as a “o” in the figures.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1000
2000
3000
4000
5000
6000
7000
8000
Displacement, in.
Pn,
kip
s
Figure A. 1 Load-displacement for W14FI column study
70
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2000
4000
6000
8000
10000
12000
14000
Displacement, in.
Pn,
kip
s
Figure A. 2 Load-displacement for W14FR column study
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Displacement, in.
Pn,
kip
s
Figure A. 3 Load-displacement for W36FR column study
71
A.2 Load Displacement Curves
Figures A.4, A.5 and A.6 show results similar to those presented in Figure
3.8 for, the W14FR, W36FR, and the W36WI groups respectively. The figures
present the ABAQUS predicted capacities as well as those predicted by the
different design methods versus the elastic local slenderness of the cross-section:
√fy/fcrl, in addition to the AISC and the AISI predictions when the fcrb and fcrh are
forced to the fcrl.
0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
(fy/fcrl)0.5
Pn/
Py
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
ABAQUSABAQUS Limited
Figure A. 4 Column study results for W14FR considering alternative design methods where fcrl (from FSM) replaces fcrb and fcrh in AISC and AISI
72
0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
(fy/fcrl)0.5
Pn/
Py
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
ABAQUSABAQUS Limited
Figure A. 5 Column study results for W36FR considering alternative design methods where fcrl (from
FSM) replaces fcrb and fcrh in AISC and AISI
0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
(fy/fcrl)0.5
Pn/
Py
AISC (kf=0.7)
AISI (kf=0.7)
DSM (FSM)AISC (kf FSM)
AISI (kf FSM)
ABAQUSABAQUS Limited
Figure A. 6 Column study results for W36WI considering alternative design methods where fcrl (from
FSM) replaces fcrb and fcrh in AISC and AISI
73
Appendix B : Additional Beam Results
Moment-rotation results for the nonlinear FE collapse analysis of the short
beam specimens are shown in Figures B.1 through B.3 for the W14FR, W36FR,
and W36WI groups respectively. Similar to Figure 3.9, the peak moments
recorded, are denoted with a “*” and the alternative peak moments that were
recorded at a rotation equal to 3θy (where θy=MyL/2EI) are denoted as a “o” in
the figures.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
1
2
3
4
5
6
7x 10
4
Rotation, Rad.
Mn,
kip
s.in
Figure B. 1 Moment-rotation for W14FR beam study
74
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
0.5
1
1.5
2
2.5x 10
5
Rotation, Rad.
Mn,
kip
s.in
Figure B. 2 Moment-rotation for W36FR beam study
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40
2
4
6
8
10
12
14x 10
4
Rotation, Rad.
Mn,
kip
s.in
Figure B. 3 Moment-rotation for W36WI beam study