April 2009 AISC - Department of Civil Engineering Faculty Fellowship...April 2009 To: Tom ......

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

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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

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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.

9

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,

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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.

21

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

22

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)


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)


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)


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)


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

)


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

)


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

)


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)


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)


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 frequencyNormal PDF

4000 4200 4400 4600 4800 5000 52000

0.2

0.4

0.6

0.8

1

1.2

1.4


Com

ulat

ive

Freq

uenc

y or

Fx(s

treng

th)


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


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

AISC (2005). “Specification for Structural Steel Buildings”, American Institute of Steel Construction, Chicago, IL. ANSI/ASIC 360-05.

AISI (2007). “North American Specification for the Design of Cold-Formed Steel Structures”, Am. Iron and Steel Inst., Washington, D.C., AISI-S100.

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.

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

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)


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)


AISI (kf FSM)


Figure A. 5 Column study results for W36FR considering alternative design methods where fcrl (from


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)


AISI (kf FSM)


Figure A. 6 Column study results for W36WI considering alternative design methods where fcrl (from


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

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