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by
Sepehr Dara
2010
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The Thesis Committee for Sepehr Dara
Certifies that this is the approved version of the following thesis:
Guidelines for Preliminary Design of Beams in
Eccentrically Braced Frames
APPROVED BY
SUPERVISING COMMITTEE:
Supervisor:
Michael D. Engelhardt
Todd Helwig
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Guidelines for Preliminary Design of Beams in
Eccentrically Braced Frames
by
Sepehr Dara BS
Thesis
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science in Engineering
The University of Texas at Austin
May 2010
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Acknowledgements
I am grateful to Dr. Engelhardt for his support throughout my research. His vast
knowledge of the subject and his patience in guiding me truly helped me to overcome the
difficult parts of the project.
I also would like to thank Dr. Helwig for reading a draft of my thesis and
providing constructive feedback.
May 2010
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Abstract
Guidelines for Preliminary Design of Beams in
Eccentrically Braced Frames
Sepehr Dara, MSE
The University of Texas at Austin, 2010
Supervisor: Michael D. Engelhardt
Seismic-resistant steel eccentrically braced frames (EBFs) are designed so that
that yielding during earthquake loading is restricted primarily to the ductile links. To
achieve this behavior, all members other than the link are designed to be stronger than the
link, i.e. to develop the capacity of the link. However, satisfying these capacity design
requirements for the beam segment outside of the link can be difficult in the overall
design process of an EBF. In some cases, it may be necessary to make significant
changes to the configuration of the EBF in order to satisfy beam design requirements. If
this discovery is made late in the design process, such changes can be costly.
The overall goal of this research was to develop guidelines for preliminary design
of EBFs that will result in configurations where the beam is likely to satisfy capacity
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design requirements. Simplified approximate equations were developed to predict the
axial force and moment in the beam segment outside of the link when link ultimate
strength is developed. These equations, although approximate, provided significant
insight into variables that affect capacity design of the beam. These equations were then
used to conduct an extensive series of parametric studies on a wide variety of EBF
configurations. The results of these studies show that the most important variables
affecting beam design are 1) the nondimensional link length, 2) the ratio of web area to
total area for the wide flange section used for the beam and link, 3) the angle between the
brace and the beam, and 4) the flexural stiffness of the brace relative to the beam.Recommendations are provided for selection of values for these variables in preliminary
design.
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Table of Contents
LIST OF TABLES ......................................................................................................... xi
LIST OF FIGURES ...................................................................................................... xiii
CHAPTER 1 - INTRODUCTION AND BACKGROUND ........................................... 1
1.1. OVERVIEW ................................................................................................. 1
1.2. DEFINITION OF EBF .................................................................................... 1
1.3. I NELASTIC RESPONSE AND ENERGY DISSIPATION .......................................... 4
1.4. EBF BEHAVIOR AND DESIGN REQUIREMENTS ............................................... 5
1.4.1. Link plastic rotation angle .............................................................. 5
1.4.2. Forces in links and beams ............................................................... 7
1.4.3. Shear vs. flexural yielding links ...................................................... 9
1.4.4. Link nominal shear strength.......................................................... 10
1.4.5. Post-yielding behavior of links ..................................................... 11
1.5. DESIGN OF BEAM SEGMENT OUTSIDE OF THE LINK .................................... 14
1.6. R ESEARCH OBJECTIVES AND ORGANIZATION OF THESIS............................. 16
CHAPTER 2 - EBF FORCE ANALYSIS ................................................................... 18
2.1. ULTIMATE LINK SHEAR AND LINK END MOMENT ......................................... 18
2.2. APPROXIMATE ANALYSIS OF FORCES IN THE BEAM AND BRACE FOR EBF
WITH SHEAR LINKS .................................................................................. 20
2.3. APPROXIMATE A NALYSIS OF FORCES AND MOMENTS IN THE BEAM AND
BRACE FOR EBF WITH MOMENT LINKS ..................................................... 33
2.4. OBSERVATIONS AND REMARKS .................................................................. 35
CHAPTER 3 - YIELDING IN THE BEAM OUTSIDE OF THE LINK DUE TOCOMBINED AXIAL FORCE AND BENDING MOMENT .......................... 39
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3.1. AISC EQUATIONS FOR STRENGTH UNDER COMBINED FORCES ..................... 39
3.2. BEAM AXIAL FORCE AND MOMENT............................................................. 40
3.2.1 Shear links..................................................................................... 40
3.2.2 Moment links. ............................................................................... 41
3.3. YIELDING UNDER COMBINED AXIAL FORCE AND THE MOMENT .................... 42
3.3.1 Shear links..................................................................................... 43
3.3.2 Moment links ................................................................................ 44
3.4. PRELIMINARY EVALUATION OF VARIABLES AFFECTING BEAM YIELDING ...... 45
3.4.1. Variation of Aw/Ag ....................................................................... 45
3.4.2. Variation of Z/Ag .......................................................................... 46
3.4.3. Variation of α ............................................................................... 46
3.4.4. Variation of η ............................................................................... 46
3.4.5. Variation of β ............................................................................... 47
3.4.6. Preliminary investigation of an EBF with a shear link................... 48
3.4.7. Preliminary investigation of an EBF with a moment link .............. 54
3.4.8. Conclusions from the preliminary investigation ............................ 58
3.5. PARAMETRIC STUDY OF BEAM YIELDING .................................................... 59
3.5.1. Effect of η on EBFs with shear links ............................................. 61
3.5.2. Effect of η on EBFs with moment links ........................................ 71
3.5.3. Conclusions on the effect of η ...................................................... 82
3.5.4. Effect of Aw/Ag on EBFs with shear links ................................... 83
3.5.5. Effect of Aw/Ag on EBFs with moment links ................................ 86
3.5.6. Conclusions on the effect of Aw/Ag ............................................... 89
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3.5.7. Effect of Z/Ag on EBFs with shear links ....................................... 90
3.5.8. Effect of Z/Ag on EBFs with moment links ................................... 93
3.5.9. Conclusions on the effect of Z/Ag ................................................. 95
3.5.10. Evaluating the effect of β ............................................................ 95
3.5.11. Conclusions on the effect of β..................................................... 98
3.5.12. Effect of α for an EBF with shear links ....................................... 99
3.5.13. Effect of α on an EBF with moment links ................................. 101
3.5.14. Conclusions on the effect of α................................................... 103
CHAPTER 4 - STABILITY OF THE BEAM OUTSIDE THE LINK UNDERCOMBINED AXIAL FORCE AND BENDING ........................................... 104
4.1. AISC EQUATIONS FOR NOMINAL AXIAL AND FLEXURAL STRENGTH FOR
STABILITY ............................................................................................. 106
4.1.1. Nominal compressive strength .................................................... 106
4.1.2. Flexural strength ......................................................................... 106
4.2. BEAM STRENGTH BASED ON STABILITY .................................................... 107
4.2.1. Basis for Beam Stability Analysis ............................................... 108
4.2.2. Axial compressive strength of the beam based on stability .......... 110
4.2.3. Flexural strength of the beam based on stability .......................... 115
4.2.4. Combination of axial and flexural strength ................................. 120
4.3. CONCLUSIONS ........................................................................................ 129
CHAPTER 5 - SUMMARY, CONCLUSIONS AND DESIGNRECOMMENDATIONS .............................................................................. 131
5.1. SUMMARY .............................................................................................. 131
5.2. CONCLUSIONS ........................................................................................ 132
5.3. DESIGN RECOMMENDATIONS ................................................................... 134
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5.4. ADDITIONAL RESEARCH NEEDS................................................................ 137
REFERENCES...................................................................................................... 138
VITA ................................................................................................................... 139
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List of Tables
Table 3.1: Effect of varying η for EBF with beam section W10X12 ........................ 62
Table 3.2: Effect of varying η for EBF with beam section W10X112 ...................... 62
Table 3.3: Effect of varying η for EBF with beam section W12X14 ........................ 63
Table 3.4: Effect of varying η for EBF with beam section W12X152 ...................... 63
Table 3.5: Effect of varying η for EBF with beam section W14X22 ........................ 64
Table 3.6: Effect of varying η for EBF with beam section W14X145 ...................... 64
Table 3.7: Effect of varying η for EBF with beam section W16X26 ........................ 65
Table 3.8: Effect of varying η for EBF with beam section W16X100 ...................... 65
Table 3.9: Effect of varying η for EBF with beam section W18X35 ........................ 66
Table 3.10: Effect of varying η for EBF with beam section W18X143 ...................... 66
Table 3.11: Effect of varying η for EBF with beam section W21X44 ........................ 67
Table 3.12: Effect of varying η for EBF with beam section W21X147 ...................... 67
Table 3.13: Effect of varying η for EBF with beam section W24X55 ........................ 68
Table 3.14: Effect of varying η for EBF with beam section W24X146 ...................... 68
Table 3.15: Effect of varying η for EBF with beam section W10X12 ........................ 72
Table 3.16: Effect of varying η for EBF with beam section W10X112 ...................... 72
Table 3.17: Effect of varying η for EBF with beam section W12X14 ........................ 73
Table 3.18: Effect of varying η for EBF with beam section W12X152 ...................... 73
Table 3.19: Effect of varying η for EBF with beam section W14X22 ........................ 74
Table 3.20: Effect of varying η for EBF with beam section W14X145 ...................... 74
Table 3.21: Effect of varying η for EBF with beam section W16X26 ........................ 75
Table 3.22: Effect of varying η for EBF with beam section W16X100 ...................... 75
Table 3.23: Effect of varying η for EBF with beam section W18X35 ........................ 76
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Table 3.24: Effect of varying η for EBF with beam section W18X143 ...................... 76
Table 3.25: Effect of varying η for EBF with beam section W21X44 ........................ 77
Table 3.26: Effect of varying η for EBF with beam section W21X147 ...................... 77
Table 3.27: Effect of varying η for EBF with beam section W24X55 ........................ 78
Table 3.28: Effect of varying η for EBF with beam section W24X146 ...................... 78
Table 3.29: Effect of varying Aw/Ag for EBF with shear link ................................... 84
Table 3.30: Effect of varying Aw/Ag for EBF with moment link .............................. 87
Table 3.31: Effect of varying Z/Ag for EBF with shear link ...................................... 91
Table 3.32: Effect of varying Z/Ag for EBF with moment link ................................. 93Table 3.33: Effect of varying β ................................................................................. 96
Table 3.34: Effect of varying α for EBF with shear link ............................................ 99
Table 3.35: Effect of varying α for EBF with moment link ..................................... 101
Table 4.1: Properties of Sections for EBF Beam Stability Analysis ....................... 109
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List of Figures
Figure 1.1: Typical arrangements of EBFs ................................................................. 2
Figure 1.2: Inelastic action location in MRFs, EBFs and CBFs .................................. 4
Figure 1.3: Inelastic mechanism for EBF under lateral load ........................................ 4
Figure 1.4: Link plastic rotation of EBF with a link at the middle of the beam............ 6
Figure 1.5: Link plastic rotation of EBF with one link next to the column .................. 6
Figure 1.6: Link plastic rotation of EBF with two links next to the columns ............... 7
Figure 1.7: Distribution of forces in the link and the beam outside the link ................. 8
Figure 1.8: The links free body diagram ..................................................................... 9
Figure 1.9: Effect of link length on the inelastic behavior of the link ........................ 10
Figure 1.10: Variation of link nominal shear strength with link length........................ 11
Figure 1.11: Typical experimental response of a link subjected to cyclic shear ........... 12
Figure 2.1: Geometric properties of an EBF with the link at the middle of the
beam ...................................................................................................... 20
Figure 2.2: Free body diagram of brace connection panel ......................................... 21
Figure 2.3: Moment diagram of the beam and the brace ........................................... 22
Figure 2.4: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.15 ......................................................................................... 27
Figure 2.5: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.125 ....................................................................................... 28
Figure 2.6: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.10 ......................................................................................... 28
Figure 2.7: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.075 ....................................................................................... 29
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Figure 2.8: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.05 ......................................................................................... 29
Figure 2.9: Variation of Pbeam /Py for the EBF configuration with constant value
of e/ L = 0.025 ....................................................................................... 30
Figure 2.10: Variation of Pbeam /Py for the EBF with constant value of
Aw/Ag=0.39 (W16x50) ......................................................................... 30
Figure 2.11: Effect of the change in the link length on the axial load in the beam
outside the link ...................................................................................... .35
Figure 2.12: Effect of changing e and Ibeam/Ibrace on the moment in the beamoutside the link ....................................................................................... 36
Figure 2.13: Aw/Ag vs. Depth for the common AISC sections ................................... 38
Figure 2.14: Z/Ag vs. Depth for the common AISC sections ...................................... 38
Figure 3.1: Variation of Aw/Ag for all of AISC W-Shapes ....................................... 45
Figure 3.2: Variation of Aw/Ag for more common AISC W-Shapes (W8-W24) ...... 46
Figure 3.3: Variation of Z/Ag for all of AISC W-sections ........................................ 47
Figure 3.4: Variation of Z/Ag for more common AISC W-sections (W8-W24) ........ 48
Figure 3.5: EBF with shear link ................................................................................ 49
Figure 3.6: SAP2000 model for EBF with shear link ................................................ 51
Figure 3.7: Symmetric lateral loading on EBF with shear link .................................. 51
Figure 3.8: SAP2000 shear diagram for EBF with shear link .................................... 52
Figure 3.9: SAP2000 moment diagram for EBF with shear link ............................... 53
Figure 3.10: SAP2000 axial force diagram for EBF with shear link............................ 53
Figure 3.11: EBF with moment link ........................................................................... 54
Figure 3.12: SAP2000 model for EBF with moment link ........................................... 55
Figure 3.13: Symmetric lateral loading on EBF with moment link ........................... 56
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Figure 3.14: SAP2000 shear diagram for EBF with moment link ............................... 56
Figure 3.15: SAP2000 moment diagram for EBF with moment link ........................... 57
Figure 3.16: SAP2000 axial force diagram for EBF with moment link ....................... 57
Figure 3.17: Changing η and keeping other variables constant ................................... 61
Figure 3.18: Effect of changing η on the yield function for the beam in an EBF with
a shear link ............................................................................................. 69
Figure 3.19: Variation of axial yield function for an EBF with a shear link ................ 70
Figure 3.20: Variation of bending yield function for an EBF with a shear link ........... 71
Figure 3.21: Effect of changing η on the yield function for the beam in an EBF witha moment link ........................................................................................ 79
Figure 3.22: Variation of axial force yield function for an EBF with a moment link ... 80
Figure 3.23: Variation of bending yield function for an EBF with a moment link ....... 81
Figure 3.24: Changing Aw/Ag and keeping the other variables constant .................... 83
Figure 3.25: Effect of changing Aw/Ag on the yield function in the beam for an
EBF with a shear link ............................................................................. 85
Figure 3.26: Effect of changing Aw/Ag on the axial yield function in the beam for
an EBF with a shear link ........................................................................ 85
Figure 3.27: Effect of changing Aw/Ag on the bending yield function for an EBF
with a shear link ..................................................................................... 86
Figure 3.28: Effect of changing Aw/Ag on the yield function in the beam for an
EBF with a moment link ........................................................................ 88
Figure 3.29: Effect of changing Aw/Ag on the axial yield function for an EBF with
a moment link ........................................................................................ 88
Figure 3.30: Effect of changing Aw/Ag on the f bending function for an EBF with a
moment link ........................................................................................... 89
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Figure 3.31: Changing Z/Ag and keeping the other variables constant ........................ 90
Figure 3.32: Effect of changing Z/Ag on the yield function in the beam for an EBF
with a shear link ..................................................................................... 91
Figure 3.33: Effect of changing Z/Ag on the f axial yield function for an EBF with
a shear link ............................................................................................. 92
Figure 3.34: Effect of changing Z/Ag on the bending yield function for an EBF
with a shear link ..................................................................................... 92
Figure 3.35: Effect of changing Z/Ag on the yield function in the beam for an EBF
with a moment link ................................................................................ 94Figure 3.36: Effect of changing Z/Ag on the axial yield function for an EBF with a
moment link ........................................................................................... 94
Figure 3.37: Effect of changing Z/Ag on the bending yield function for an EBF
with a moment link ................................................................................ 95
Figure 3.38: Changing β and keeping the other variables constant .............................. 96
Figure 3.39: Effect of changing β on the yield in the beam ......................................... 97
Figure 3.40: Effect of changing β on the axial yield function in the beam ................... 97
Figure 3.41: Effect of changing β on the bending yield function in the beam .............. 98
Figure 3.42: Changing α and keeping the other variables constant .............................. 99
Figure 3.43: Effect of changing α on the yield function in the beam for an EBF with
a shear link ........................................................................................... 100
Figure 3.44: Effect of changing α on variation of the axial yield function for an
EBF with a shear link ........................................................................... 100
Figure 3.45: Effect of changing α on variation of the bending yield function for an
EBF with a shear link ........................................................................... 101
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Figure 3.46: Effect of changing α on the yield function in the beam for an EBF with
a moment link ...................................................................................... 102
Figure 3.47: Effect of changing α on the variation of the axial yield function for an
EBF with a moment link ...................................................................... 102
Figure 3.48: Effect of changing α on the variation of the bending yield function for
an EBF with a moment link .................................................................. 103
Figure 4.1: Axial compressive strength of W24 sections with Lx =200 inches and
variable Ly ........................................................................................... 111
Figure 4.2: Axial compressive strength W21 sections with Lx =200 inches andvariable Ly ........................................................................................... 111
Figure 4.3: Axial compressive strength of W18 sections with Lx =200 inches and
variable Ly ........................................................................................... 112
Figure 4.4: Axial compressive strength of W16 sections with Lx =200 inches and
variable Ly ........................................................................................... 112
Figure 4.5: Axial compressive strength of W14 sections with Lx =200 inches and
variable Ly ........................................................................................... 113
Figure 4.6: Axial compressive strength of W12 sections with Lx =200 inches and
variable Ly ........................................................................................... 113
Figure 4.7: Axial compressive strength of W10 sections with Lx =200 inches and
variable Ly ........................................................................................... 114
Figure 4.8: Normalized beam compressive strength ............................................... 116
Figure 4.9: Flexural strength of W24 sections as a function of unbraced length ...... 117
Figure 4.10: Flexural strength of W21 sections as a function of unbraced length ...... 117
Figure 4.11: Flexural strength of W18 sections as a function of unbraced length ...... 118
Figure 4.12: Flexural strength of W16 sections as a function of unbraced length ...... 118
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Figure 4.13: Flexural strength of W14 sections as a function of unbraced length ...... 119
Figure 4.14: Flexural strength of W12 sections as a function of unbraced length ...... 119
Figure 4.15: Flexural strength of W10 sections as a function of unbraced length ...... 120
Figure 4.16: Normalized beam flexural strength ....................................................... 121
Figure 4.17: Interaction diagrams for W24x146 ....................................................... 122
Figure 4.18: Interaction diagrams for W24x55 ......................................................... 122
Figure 4.19: Interaction diagrams for W21x147 ....................................................... 123
Figure 4.20: Interaction diagrams for W21x44 ......................................................... 123
Figure 4.21: Interaction diagrams for W18x143 ....................................................... 124Figure 4.22: Interaction diagrams for W18x35 ......................................................... 124
Figure 4.23: Interaction diagrams for W16x100 ....................................................... 125
Figure 4.24: Interaction diagrams for W16x26 ......................................................... 125
Figure 4.25: Interaction diagrams for W14x145 ....................................................... 126
Figure 4.26: Interaction diagrams for W14x22 ......................................................... 126
Figure 4.27: Interaction diagrams for W12x136 ....................................................... 127
Figure 4.28: Interaction diagrams for W12x14 ......................................................... 127
Figure 4.29: Interaction diagrams for W10x112 ....................................................... 128
Figure 4.30: Interaction diagrams for W10x12 ......................................................... 128
Figure 4.31: Relationship between Aw/Ag and bf /d for rolled W-shapes .................. 130
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1
CHAPTER 1
INTRODUCTION AND BACKGROUND
1.1. OVERVIEW
This thesis describes the results of research conducted on the behavior and design of the
beam segment outside of the link in seismic resistant steel eccentrically braced frames
(EBFs). EBFs are a lateral force resisting system that can be used to resist earthquake
loading in steel buildings. Design of the beam segment outside of the link often poses
significant difficulties in the overall design process for EBFs. The research conducted for
this thesis investigated design problems with the beam segment and evaluated potential
solutions.
This chapter presents an introduction and background information on seismic-resistant
steel EBFs. Previous research on the beam segment outside of the link is also
summarized. Finally, the objectives and scope of the research is described.
1.2. DEFINITION OF EBF
An eccentrically braced frame (EBF) is a type of steel framing system including beams,
columns and braces, where these members are arranged in a manner where at least one
end of each brace is connected to isolate a segment of the beam called a link. EBFs are
typically used as a lateral force resisting system for earthquake loading. The design intent
for a seismic resistant EBF is to provide high ductility under earthquake loading by
yielding of the link. An overview of EBF behavior and previous research is available in
Popov and Engelhardt (1988). Design requirements for seismic resistant EBFs in the US
are specified by the AISC Seismic Provisions for Structural Steel Buildings (AISC 2005).
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This document will be referred to herein as the AISC Seismic Provisions. The AISC
Seismic Provisions includes an extensive commentary that provides additional
information on behavior, design and past research on EBFs. The background information
provided on EBFs in the remainder of this chapter is taken from these two sources, with
additional references as noted.
When an EBF is subject to lateral load, the link transmits high shear, high bending
moment, and typically low levels of axial force. Consequently, links will normally
experience shear and/or flexural yielding during an earthquake. Other members of an
EBF, including the braces, the columns and the beams segments outside of the links are
intended to remain essentially elastic during an earthquake. Several possible
arrangements of EBFs are shown in Figure 1.1.
Fig. 1.1 – Typical arrangements of EBFs
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Fig. 1.1 – Continued
EBFs resist lateral load through a combination of frame action and truss action. They can
be viewed as a hybrid system between moment resisting frames (MRF) and
concentrically braced frames (CBF). EBFs provide high levels of ductility similar to
MRFs by concentrating inelastic action in the link, which can be designed and detailed
for highly ductile response. Locations where the inelastic action occurs in MRFs, EBFs
and CBFs are highlighted in Figure 1.2. At the same time, EBFs can provide high levels
of elastic stiffness, similar to that provided by CBFs, so the code drift requirements can
be met economically.
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Fig. 1.2 – Inelastic action location in MRFs, EBFs and CBFs
1.3. INELASTIC RESPONSE AND ENERGY DISSIPATION
As mentioned above inelastic action during an earthquake is intended to occur within the
link of an EBF. Figure 1.3 shows the inelastic mechanism for an EBF. The link can
experience very large inelastic rotations. As will be discussed later a well-designed and
detailed link should be able to sustain a cyclic inelastic rotation up to ±0.08 rad.
Fig. 1.3 – Inelastic mechanism for EBF under lateral load
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1.4. EBF BEHAVIOR AND DESIGN REQUIREMENTS
As described above, the intended behavior of an EBF subject to earthquake loading is
that yielding occurs within the ductile link while the other frame elements remain elastic.
To achieve this behavior, the links must be the weakest elements in the frame and the
braces, columns and the beam segment outside the links should therefore be necessarily
stronger than the links. It can be said that links are the fuse elements of an EBF (Popov
and Engelhardt 1988).
1.4.1. Link plastic rotation angle
The available ductility of a link is often described by its plastic rotation capacity. The
plastic rotation of a link can be denoted as γ p. A goal of EBF design is that the link plastic
rotation capacity exceeds the plastic rotation demand of an earthquake. In EBF design,
the link plastic rotation can be related to the plastic story drift angle, θ p, by the geometry
of a rigid plastic mechanism. Figures 1.4 through 1.6 show the rigid plastic mechanism
for three common EBF geometries with the total beam span denoted as L and the link
length denoted as e. Equation 1-1 presents the relationship between link plastic rotation
angle and the plastic story drift angle for mechanisms in Figures 1.4 and 1.5 and Equation
1-2 presents relationship between link plastic rotation angle and the plastic story drift
angle for the mechanism in Figure 1.6. Note that as the ratio of span length to link length
(L/e) increases, the link rotation angle also increases for a given plastic story drift angle.
Consequently, large values of L/e can result in excessive plastic rotation demands on the
link. The configuration shown in Figure 1.6, with two links in each level, places only
one-half the plastic rotation demand on the link as compared to the other configurations.
PP θe
L
γ = (Eq. 1-1)
PP θ2e
Lγ = (Eq. 1-2)
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Fig. 1.4 – Link plastic rotation of EBF with a link at the middle of the beam
Fig. 1.5 - Link plastic rotation of EBF with one link next to the column
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Fig. 1.6 - Link plastic rotation of EBF with two links next to the columns
1.4.2. Forces in links and beams
Figure 1.7 shows qualitatively the distribution of moment, shear and axial force in the
link and beam segments outside of the link in an EBF subjected to lateral load. Two
common EBF configurations are shown; one with the link at mid-span and the other with
the link connected to the column. The link is generally subject to high shear along its full
length, high end moments and low axial force. Yielding within the link can be shear
yielding, flexural yielding or a combination of shear and flexural yielding. Yielding of
links and the close relationship to link length will be discussed in greater detail below.
Of particular interest in Figure 1.7 are the forces in the beam segment outside of the link.
As shown in the figure, the beam segment has a high bending moment immediately
adjacent to the link. This is because the high moment at the end of the link must be
resisted primarily by the beam segment. The figure shows a drop in moment between the
end of the link and the adjoining beam segment. This drop in moment represents the
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portion of the link end moment transferred to the brace, assuming the connection between
the brace and the link can transfer moment.
In addition to high moment, the beam segment outside of the link is also typically
subjected to high axial force. The brace in an EBF also sees high axial force, and the
horizontal component of the brace axial force will generate high axial force in the beam
segment. Finally, as shown in Figure 1.7, the shear in the beam segment outside of the
link is generally small. Consequently, the force environment for the beam segment
outside of the link is dominated by high axial force and high moment. Since earthquake
loads are cyclic, the beam segment outside of the link experiences both axial tension and
axial compression. Designing the beam segment for these high moments and axial forces
can be difficult, which is described in greater detail later.
Fig. 1.7 – Distribution of forces in the link and the beam outside the link
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1.4.3. Shear vs. flexural yielding links
The length of the link is the key parameter that controls the inelastic behavior of the link.
Figure 1.8 shows a free body diagram of a link subjected to a constant shear and equal
and opposite end moments.
Fig. 1.8 – The links free body diagram
From static equilibrium, the link shear, link end moment and link length are related by
the following equation.
V
2Me = (Eq. 1-3)
Assuming no strain hardening in the link, no shear-flexure interaction and no influence of
the concrete deck on the top of the link, the ultimate value of shear and moment in the
link will be the respective plastic shear capacity (V p) and plastic moment capacity (M p) of
the link. These two values can be obtained from the following two equations.
wf ywy p )t2t(d 0.6FA0.6FV −==
(Eq. 1-4)
y p ZFM = (Eq. 1-5)
The influence of strain hardening on link behavior is discussed later.
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If the value of link length is equal to 2M p/V p, shear and flexural yielding will occur
simultaneously in the link. If the value of link length is less than 2M p/V p shear yielding
will occur in the link, and if the value of link length is greater than 2M p/V p flexural
yielding will occur in the link. These three cases are illustrated in Figure 1.9.
Fig. 1.9 – Effect of link length on the inelastic behavior of the link
1.4.4. Link nominal shear strength
According to the AISC Seismic Provisions, the nominal shear strength of the link is taken
using the following equation.
)e
2M,V(minV
p
pn = (Eq. 1-6)
The nominal shear strength refers to the shear that first causes significant yield of the
link. Figure 1.10 shows the variation of link nominal shear strength versus the link
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length. Link length is shown normalized with respect to ratio M p/V p of the link cross-
section. For a link length e less than2M p/V p, shear yielding occurs prior to flexural
yielding and the nominal shear strength is simply equal to the plastic shear capacity of the
link, V p. Once the link length exceeds2M p/V p, flexural yielding will occur at the link ends
prior to shear yielding. For this case, the nominal shear strength of the link is taken as the
shear that is in equilibrium with link end moments that are equal to M p. This results in a
nominal link shear strength that is equal to2M p/e.
Fig. 1.10 - Variation of link nominal shear strength with link length
1.4.5. Post-yielding behavior of links
Experiments have shown that links can exhibit a high degree of strain hardening. Figure
1.11 shows the results of a cyclic loading experiment on a shear yielding link. As shown
by these results, the ultimate shear strength of the link can be significantly higher than the
nominal shear strength due to strain hardening. Past researchers have suggested that the
ultimate shear strength of a link can be on the order of 1.4 to 1.5 times the nominal
strength. For design purposes, the AISC Seismic Provisions specify that the link ultimate
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5 6
V ( k i p s )
e / (Mp/Vp)
Vn = Vp
Vn = 2Mp/e
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strength be computed as 1.25 times its expected nominal shear strength. The expected
nominal shear strength is computed by taking the nominal strength (Eq. 1-6) and
multiplying by the factor R y. The R y factor, which is specified as 1.1 for ASTM A992
steel, accounts for the fact that the actual yield stress of the steel used for the link is likely
higher than the specified yield strength.
Fig. 1.11 - Typical experimental response of a link subjected to cyclic shear (Okazaki and
Engelhardt 2007)
Due to the significant strain hardening that occurs in links, combined shear and flexural
yielding will occur over a range of link length. Based on observations from experiments
and also based on the AISC Seismic Provisions, the inelastic response of the link will be
dominated by shear yielding if e ≤ 1.6M p/V p, inelastic response of the link will be
dominated by flexural yielding if e ≥ 2.6M p/V p, and finally, combined shear and flexural
yielding will dominate the inelastic response of the link if 1.6M p/V p ≤ e ≤ 2.6M p/V p.
-150
-100
-50
0
50
100
150
-0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10
V ( k i p s )
(rad)
Vn
Vult
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1.5. DESIGN OF BEAM SEGMENT OUTSIDE OF THE LINK
The basic design approach for seismic resistant EBFs is that the yielding under
earthquake loading should be restricted primarily to the links, as the links are the most
ductile elements of the frame. In design, this is achieved by designing the braces,
columns and the beam segments outside of the link for the maximum forces generated in
these members by the fully yielded and strain hardened links. That is, the braces, columns
and beams segments are designed to develop the fully strain hardened capacity of the
links (Engelhardt and Popov 1989).
The AISC Seismic Provisions follow this capacity design approach for the braces,
columns and beam segments. The AISC Seismic Provisions specify that the brace should
be designed for the forces corresponding to a link shear of equal to 1.25 R yVn. These
same provisions specify that the columns and beams segments outside of the links should
be designed for the forces corresponding to a link shear of 1.1 R yVn. As discussed earlier,
review of an extensive experimental database showed that ultimate link strength was, on
average, equal to about 1.4 times the measured yield strength. Consequently, the AISC
specified link strain hardening factors of 1.25 for brace design and 1.1 for beam and
column design appear quite low. Reasons for these low strain hardening factors, and the
use of different strain link strain hardening factors for different members are discussed in
the commentary to the AISC Seismic Provisions.
Regardless of the link strain hardening factor used for design, the basic approach involves
first estimating the link ultimate shear strength and link ultimate end moments. Analysis
is then conducted to determine the forces generated in the braces, columns and beam
segments by the link ultimate shear and end moment. The braces, columns and beam
segments are then designed for these forces.
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When implementing this capacity design approach, the brace and column sections can be
sized as needed to resist the capacity design forces generated by the link. The beam
segment outside of the link, however, poses a special difficulty. This is because the link
and the beam segment outside of the link are normally the same member. Consequently,
if the beam segment does not have adequate strength to resist the capacity design forces
generated by the link, increasing the size of the beam segment often will not be helpful.
This is because as the size of the beam segment is increased, the size of the link is also
increased, and the capacity design forces on the beam segment are consequently also
increased. Thus, if a larger section is chosen for the beam segment, the design forces on
the beam segment increase. In some cases, it may in fact be impossible to choose a
section for the beam segment that will satisfy capacity design requirements (Engelhardt et
al 1991).
As discussed earlier, the beam segment outside of the link must resist high cyclic axial
forces and high cyclic bending moments and therefore must be designed as a beam-
column. Limit states for the beam include yielding and instability. Due to the high axial
compression forces that can occur in the beam combined with bending moment, stability
limit states will generally govern the design of the beam.
Behavior of the beam segment outside of the link has been studied in experiments
(Engelhardt and Popov 1992). Experiments were conducted on EBF subassemblies in
which the beam segment outside of the link did not fully satisfy capacity design
requirements. These experiments showed that limited yielding of the beam segment
outside of the link, in the region of the brace connection, was not detrimental to the
strength or ductility of the EBF. However, instability in the beam segment outside of the
link resulted in a large loss of strength and ductility. Two forms of instability were
observed in these tests. One form was local buckling of the beam segment immediately
outside of the link. The other form was lateral torsional buckling of the beam segment
outside of the link. Either form of instability was detrimental to EBF performance.
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A brief analytical study of capacity design forces in the beam segment outside of the link
was conducted by Engelhardt et al (1991). This study examined the factors that affect the
magnitude of the axial force and bending moment developed in the beam when the link
has achieved its ultimate strength. This study concluded there are a number of EBF
configurations where it is impossible to satisfy capacity design requirements for the
beam. One the other hand, there were a number of EBF configurations where the beam
satisfied capacity design requirements.
1.6. RESEARCH OBJECTIVES AND ORGANIZATION OF THESIS
Satisfying capacity design requirements for the beam segment outside of the link can be
difficult in the overall design process of an EBF. As noted above, satisfying capacity
design requirements for the beam segment may be impossible for certain EBF
configurations. For these cases, it may be necessary to make significant changes to the
configuration of the EBF in order to satisfy beam design requirements. If this discovery is
made late in the design process, major changes to the EBF configuration can be costly. In
some cases, where the configuration of the EBF cannot be changes, it may be necessary
to strengthen the beam with cover plates or other costly measures.
The overall objective of the research described in this thesis is to identify EBF
configurations that will make satisfying capacity design requirements for the beam
segment difficult or impossible. The goal is to provide guidance to designers on the
factors that affect capacity design of the beam segment and how to best configure an EBF
at the preliminary design stage to minimize difficulties with the beam segment.
In Chapter 2 of this thesis, simplified approximate equations are developed to predict the
axial force and moment in the beam segment outside of the link when link ultimate
strength is developed. These equations, although approximate, are useful at the
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preliminary design stage to estimate capacity design forces for the beam. In Chapter 3,
the approximate beam force equations developed in Chapter 2 are combined with a beam
strength analysis for a limit state of a fully yielded cross-section under combined bending
and axial force. This analysis identifies EBF configurations where the beam segment will
yield prior to the development of the full capacity design forces. In Chapter 4, the
approximate beam force equations developed in Chapter 2 are combined with a beam
strength analysis based on a limit state of buckling under combined bending and axial
force. This analysis identifies EBF configurations where the beam segment will buckle
prior to the development of the full capacity design forces. Finally, Chapter 5 summarizes
results of this research and provides recommendations on EBF configurations that will
avoid difficulties in satisfying capacity design requirements for the beam segment outside
of the link.
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CHAPTER 2
EBF FORCE ANALYSIS
As described in Chapter 1, satisfying capacity design requirements in the beam outside
the link in eccentrically braced frames can be difficult for certain EBF configurations. In
this chapter, a simplified analysis is conducted to estimate the axial force and moment
that develops in the beam when the ultimate strength of the link is developed. The
purpose of this analysis is to gain insight into the key design variables that influence the
forces developed in the beam and identify potentially problematic EBF configurations. In
addition to computing forces in the beam, the axial force and moment developed in the
brace is also examined, since capacity design forces in the beam and brace are closely
related.
In the first part of this chapter, assumptions made on ultimate link strength are described.
This is followed by approximate analyses of capacity design forces in the beam and brace
when the link ultimate strength is controlled by shear. Finally, an approximate analysis is
conducted to estimate capacity design forces in the beam and brace when the linkultimate strength is controlled by flexure.
2.1. ULTIMATE LINK SHEAR AND LINK END MOMENT
As described in Chapter 1, the capacity design philosophy for EBFs requires that the
beam outside the link and the brace be designed to withstand the forces generated by the
fully yielded and strain hardened links. For the purposes of the analysis conducted in this
thesis, it is assumed that that ultimate shear strength of a link is 1.4 times the plastic shear
capacity V p. Similarly, it is also assumed that the ultimate link end moment is 1.4 times
the plastic moment capacity M p. This assumption is based on a review of experimental
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data on link strength by Okazaki and Engelhardt (2007). Note that the analysis presented
in this chapter can be easily modified for other assumed values of link ultimate strength,
if desired. It is further assumed that link end moments are equal when the link ultimate
strength is developed. This assumption is accurate when the link is in the middle of the
beam. It is also reasonable when the link is next to the column and the link length is
greater than about M p/V p (Popov and Engelhardt, 1988).
Based on the assumptions described above, the link ultimate shear and end moment are
computed as follows:
pultLink V1.4VV =≤ (Eq. 2-1)
pultLink M1.4MM =≤ (Eq. 2-2)
Where for a link constructed of a W-shape:
ZFM y p = (Eq. 2-3)
wf byWyP t)2t(d F0.6AF0.6V −== (Eq. 2-4)
Based on equilibrium, a link simultaneously develops a shear of 1.4V p and end moments
of 1.4M p for the following link length:
P
P
P
P
ult
ult
link
link
V
M2
V1.4
M1.42
V
M2
V
M2e ==≤= (Eq. 2-5)
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β is defined as the non-dimensional link length and is computed as follows:
)V
M(
eβ
p
p
= (Eq. 2-6)
Thus, for β ≤ 2, the link ultimate strength is controlled by shear. Links with β ≤ 2 are
referred to as shear links in this study. Similarly, for links with β ≥ 2, link ultimate
strength is controlled by flexure, and these are referred to as moment links in this study.
2.2. APPROXIMATE ANALYSIS OF FORCES IN THE BEAM AND BRACE FOR EBF WITH
SHEAR LINKS
Figure 2.1 shows the geometric properties of an EBF with the link in the middle of the
beam. In the analysis that follows, it is assumed that only lateral load acts on the EBF.
That is, it is assumed that gravity loads on the EBF are negligible.
Fig. 2.1 – Geometric properties of an EBF with the link at the middle of the beam
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Figure 2.2 shows a free body diagram of the brace connection panel. The brace
connection panel is the region where the brace attaches to the beam in Figure 2.1. The
axial force, moment, and shear in the section of the beam where the centerlines coincide
are also shown. The direction of axial force, shear and bending moment are chosen to be
consistent with the sign convention shown in the figure. The sign of the forces in the
equations that follow are based on these directions. However if the lateral load is in a
direction shown in the figure some of the forces will be in the opposite direction and this
can be realized from the sign next to each force. If the sign next to each force is positive,
the corresponding force is in the direction shown and if the sign is negative the force is in
opposite direction shown in figure.
Fig. 2.2 – Free body diagram of brace connection panel
The following equation can be derived from Figure 2.2 by summing moments about pointA (intersection of the beam and brace).
BraceBeamLink MMM += (Eq. 2-7)
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Analysis shows that moments at the far end of beam and brace are typically small, and
they will be assumed to be zero. Figure 2.3 shows the assumed moment diagram for the
beam and brace.
Fig. 2.3 – Moment diagram of the beam and the brace
The moment in the link is distributed to the beam and brace according to the relative
bending stiffness of these members. Assuming the beam and brace remain elastic and
second order geometric effects are not significant, the following two equations can be
used to estimate the moment in the beam and the brace based on the moment in the link.
In these equations α is the angle between the beam and the brace.
αCosII
IM
L
I
L
I
L
I
MM brace beam
beamLink
brace
brace
beam
beam
beam
beam
Link Beam+
=
+
= (Eq. 2-8)
brace beam
braceLink
brace
brace
beam
beam
brace
brace
Link Brace
IαCos
I
IM
L
I
L
I
L
I
MM+
=
+
= (Eq. 2-9)
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To estimate the axial force in the beam and the brace, it is assumed that the axial force in
the link is negligible.
0PLink ≈ (Eq. 2-10)
Equilibrium of forces in the horizontal and vertical direction for the free body diagram in
Figure 2.2 results in two following equations:
0αSinVPαCosP brace beam brace =++ (Eq. 2-11)
0VαCosVVαSinP beam bracelink brace =−−+ (Eq. 2-12)
And moment equilibrium of the link results in:
e
M2V link
link −= (Eq. 2-13)
As was noted before and shown in Figure 2.3, moment at the far end of the beam and
brace is assumed to be zero. Based on this assumption, Eq. 2-14 and Eq. 2-15 represent
the respective equilibrium relationship in the beam and the brace. In Eq. 2-14 L beam is the
length of the beam outside the link and in Eq. 2-15, L brace is the length of the brace.
beam
beam beam
L
MV = (Eq. 2-14)
brace
brace brace
L
MV = (Eq. 2-15)
Replacing the moment in the beam and the brace from Eq. 2-8 and Eq. 2-9 into Eq. 2-14
and Eq. 2-15 respectively, gives:
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αCosII
I
L
MV
brace beam
beam
beam
Link beam
+= (Eq. 2-16)
brace beam
brace
brace
Link brace
IαCos
I
I
L
MV
+
= (Eq. 2-17)
Rearrangement of Eq. 2-12 gives the following equation.
αSin
V
αtan
V
αSin
VP link brace beam
brace −+=
And by replacing the values of shear from Equations 2-13, 2-16 and 2-17 results in:
αSine
M2
IαCos
I
I
αtanL
M
αCosII
I
αSinL
MP link
brace
beam
brace
brace
link
brace beam
beam
beam
link brace +
+
++
=
This value for the axial load in the brace can be inserted in the Eq. 2-11. Also the value
for the shear in the brace given by Eq. 2-17 can also be inserted in Eq. 2-11, resulting in:
αSinVαCosPP brace brace beam −−=
]
IαCos
I
I
αSin
L
M[]
αtane
M2
IαCos
I
I
αCos
αtanL
M
αCosII
I
αtanL
M[
brace beam
brace
brace
link link
brace beam
brace
brace
link
brace beam
beam
beam
link
+
−+
+
++
−=
]
IαCos
I
I
αtanαSinL
1
e
2
IαCos
I
I
αCos1L
1
αCosII
I
L
1[
αtan
M
brace beam
brace
brace brace
beam
brace
brace brace beam
beam
beam
link
+
++
+
+
+
−=
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]e
2
CosαII
I
CosααSin
L
1
CosαII
I
αCos
L
1
CosαII
I
L
1[
tanα
M
brace beam
brace
2
beam brace beam
brace
3
beam brace beam
beam
beam
link +
++
++−=
+
This simplifies to the following equation:
)e
2
L
1(
αtan
MP
beam
link beam +−= (Eq. 2-18)
If the link is in the middle of the beam (configuration shown in Figure 1.5), then:
e)(Le
L
αtan
M2)
e
2
eL
2(
αtan
MP
2
eLL link link
beam beam−
−=+−
−=→−
= (Eq. 2-19)
The following two equations are alternative expressions for Eq. 2-19.
)tanα
h2L(
L
h2
M2P
tanα
h2Le link
beam
−
−=→−= (Eq. 2-19a)
Or
he
LMP
eL
2htanα link beam −=→
−= (Eq. 2-19b)
The value of axial force in the beam outside the link can be also presented in terms of
shear in the link by inserting the value of shear from Eq. 2-13 into Eq. 2-19:
2h
LV
eL
L
αtan
VP link
link beam =
−= (Eq. 2-20)
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If the link is a shear link, which means that e ≤ 2M p/V p or β ≤ 2 , then the shear in the
link reaches its ultimate value under the lateral load while the end moments in the link are
less than or equal to their ultimate value. Therefore by replacing the ultimate value of
shear from Eq. 2-1 into Eq. 2-20 it can be concluded that:
eL
L
αtan
V1.4P
p
beam−
−= (Eq. 2-21)
The value of V p from Eq. 2-4 can be inserted into Eq. 2-21, giving:
eL
L
αtan
AA
P0.84
eL
L
αtan
)AF(0.61.4P
W
g
y
Wy
beam−
−=−
−= (Eq. 2-22)
Finally, the axial force in the beam given by Eq. 2-22 can be normalized by the axial
yield strength of the beam, Py = AFy, resulting in:
p
p
g
W
y
beam
VM2e,
L
e1
1αtan
0.84AA
PP ≤
−
−= (Eq. 2-23)
Equation 2-23 shows that the axial force ratio P/Py in the beam outside the link is
dependent to the following variables:
- The ratio of the beam section web area to the beam section gross area: Aw/Ag
- The angle between the beam and the brace: α
- The ratio of the link length to the total span length: e/L
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Assuming these variables are changed independently of one another, Figures 2.4 to
Figure 2.9 show the effect of each variable on the axial force ratio P/P y in the beam
outside the link. For each figure the value of e/L is constant and decreases from each
figure to the next. In each case, the link length has been chosen to correspond to a shear
link, i.e. β ≤ 2. Each figure includes 3 diagrams with 3 different values of Aw/Ag. Note
that, in reality, the variables noted above are usually not independent of one another.
Consider, for example, the variables of α and e. When the total length of the beam and
height of the frame are fixed, changing the link length e will result in a change in α, the
angle between the beam and the brace. Despite these interdependencies, Figures 2.4 to
2.9 still provide important insights into the factors that affect the axial force ratio in the
beam. Figure 2.10 shows the effect of changing the link length for a specific wide flange
section. Discussion of the plots in Figures 2.4 to 2.10 is provided later in Section 2.4.
Fig. 2.4 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag = 0.53 (W21x44)
Aw/Ag = 0.42 (W16x36)
Aw/Ag = 0.28 (W12x45)
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Fig. 2.5 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.125
Fig. 2.6 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag=0.48 (W21x57)Aw/Ag=0.39 (W16x50)
Aw/Ag=0.28 (W12x45)
0
0.1
0.2
0.3
0.4
0.5
0.60.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag=0.48 (W21x57)
Aw/Ag=0.39 (W16x50)
Aw/Ag=0.28 (W12x45)
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Fig. 2.7 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.075
Fig. 2.8 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag=0.48 (W21x57)
Aw/Ag=0.39 (W16x50)
Aw/Ag=0.28 (W12x45)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag=0.48 (W21x57)
Aw/Ag=0.39 (W16x50)
Aw/Ag=0.28 (W12x45)
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Fig. 2.9 – Variation of Pbeam /P y for the EBF configuration with constant value of e/ L = 0.025
Fig. 2.10 – Variation of Pbeam /P y for the EBF with constant value of Aw/Ag=0.39 (W16x50)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
Aw/Ag=0.48 (W21x57)
Aw/Ag=0.39 (W16x50)
Aw/Ag=0.28 (W12x45)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 25 30 35 40 45 50 55 60 65 70
A b s ( P b e a m /
P y )
α (degree)
e/L = 50/400 = 0.125
e/L = 40/400 = 0.10
e/L = 30/400 = 0.075
e/L = 20/400 = 0.05
e/L = 10/400 = 0.025
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The same basic approach used above to estimate the axial force in the beam can be used
to estimate the axial force in the brace. Equation 2-11 can be rearranged as follows by
replacing the value of axial load in the beam from Equation 2-21:
αtanVαCos
PP brace
beam brace −−=
αtan
IαCos
I
I
L
M
eL
L
αSin
AF0.84
brace beam
brace
brace
Link Wy
+
−−
=
αCosαSinαCosII
I
L
M
eL
L
αSin
AF0.84
brace beam
brace
beam
Link Wy
+−−=
αCosαSinαCosII
I
2
eL2
.eV
eL
L
αSin
AF0.84
brace beam
brace
Link
Wy
+−+
−=
For a shear link, the ultimate value of shear in the link (1.4V p) can be substituted in the
last equation, resulting in the following expression:
αCosαSinαCosII
I
eL
e)AF(0.61.4
eL
L
αSin
AF0.84
brace beam
braceWyWy
+−−
−=
α)CosαSinαCosII
I
L
e
αSin
1(
L
e1
1AF0.84
brace beam
braceWy
+−
−
=
p
p
brace beam
brace
g
W
y
brace
V
M2e,α)CosαSin
αCosIII
Le
αSin1(
L
e1
0.84AA
PP ≤
+−
−
= (Eq. 2-24)
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If the shear in the brace is assumed to be negligible in Equation 2-11, then:
And therefore:
p
p
g
W
y
brace
V
M2e,
L
e1
1
αSin
0.84
A
A
P
P≤
−
= (Eq. 2-25)
Expressions can also be developed for moment in the beam and in the brace, relative to
the plastic moment capacity of the beam. From Equations 2-3 and 2-4 it can be concluded
that:
P
Link
P
Link
V
Vβ
2
1
M
M= (Eq. 2-26)
And the resulting expression for the shear links is as follows:
0.7βM
M
P
Link = (Eq. 2-27)
Inserting this expression of the moment in the link into Equation 2-8 and Equation 2-9
results in the two following equations:
eL
L
αSin
AF0.84
αCos
P
P
Wy beam
brace−=−=
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Cosα
I
I
I
I
β0.7αCosII
Iβ0.7
M
M
brace
beam
brace
beam
brace beam
beam
P
beam
+
=+
= (Eq. 2-28)
And
1Cosα
1
I
I
1β0.7
ICosα
1I
Iβ0.7
M
M
brace
beam brace beam
brace
P
brace
+
=
+
= (Eq. 2-29)
These equations show the values for the ratio of the moment in the beam and the brace to
the plastic moment capacity of the beam section.
2.3. APPROXIMATE ANALYSIS OF FORCES AND MOMENTS IN THE BEAM AND BRACE FOR
EBF WITH MOMENT LINKS
If the link is a moment link, which means that e > 2M p/V p or β > 2 then the moment in
the link reaches its ultimate value while the shear in the link will be less than or equal to
its ultimate value. Therefore in the equations resulting equilibrium at the brace–beam–
link intersection, instead of inserting the ultimate value for shear, the ultimate value for
the moment will be inserted.
The value of axial load in the beam can then be written as follows:
he
LZFy1.4
he
LM1.4
he
LMP Plink beam −=−=−=
And therefore the ratio of axial force in the beam to Py will be:
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p
p
gy
beam
V
M2e,
h
1
L
e
1.4
A
Z
P
P>−= (Eq. 2-30)
And the axial force in the brace, by neglecting the shear in the brace, can be written as:
α Cos
PP beam
brace −=
p
p
gy
brace
V
M2e,
e)(Le
L
αSin
2.8
A
Z
P
P>
−= (Eq. 2-31)
Also, expressions for the moment in the beam and the moment in the brace can be
developed in terms of the moment capacity of the beam and link. As noted earlier, for
moment links, the moment in the link reaches its ultimate value and therefore:
1.4M
M
P
Link = (Eq. 2-32)
Replacing this relationship into Equations 2-8 and 2-9 produces the following expression:
CosαI
I
I
I
1.4M
M
brace
beam
brace
beam
P
beam
+
= (Eq. 2-33)
And
1Cosα
1
I
I
11.4
M
M
brace
beamP
brace
+
= (Eq. 2-34)
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2.4. OBSERVATIONS AND REMARKS
If the values of story height, h, and bay width , L, are constant and only the link length is
changed, then based on Equations 2-19 and 2-20, it is possible to relate axial force ratio
in the beam, P/Py, to the non-dimensional link length, β. The result is shown in Figure
2.11. This figure shows that for shear links, since the ultimate value of the shear in the
link is constant, then the axial load in the beam outside the link is also constant. This is
because axial force in the beam is proportional to the shear in the link, as demonstrated
by Equation 2-20. When the link is a moment link, then the ultimate value of moment in
the link is a constant. For this case, the link shear will reduce as link length increases, and
the axial force in the beam will therefore also reduce as link length increases. This
relationship is apparent from Equation 2-19b.
Fig. 2.11 - Effect of the change in the link length on the axial load in the beam outside the link
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 0.5 1 1.5 2 2.5 3 3.5 4
A b s ( P b e a m
/ P y ) / ( L / h )
β=e/(Mp/Vp)
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Using Equations 2-28 and 2-33, Figure 2.12 shows the effect of changing the link length
and relative stiffness of the beam to the stiffness of the brace on the moment generated in
the beam. The dashed line in this graph represents the ratio of the moment in the link to
the moment capacity of the beam, which increases linearly by increasing the link length
for shear links and is constant for moment links. As can be seen from this plot, by
increasing the stiffness of the beam relative to the brace, the beam attracts more moment
from the end of the link. Also it can be concluded that whether the link is a shear or
moment link, the moment in the beam increases as the link length increases.
Fig. 2.12 - Effect of changing e and I beam /I brace on the moment in the beam outside the link
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500 4.000
A b s ( M b
e a m /
M P )
β=e/(Mp/Vp)
I beam / I brace = 1.82
I beam / I brace = 5.14
I beam / I brace = 7.07
I beam / I brace = 3.54
I beam / I brace = 0.51
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As Figures 2.4 through 2.9 show, the angle α between the beam and the brace has a large
effect on the axial force in the beam. A small angle between the brace and the beam
results in very large axial force in the beam. These figures also show that for shear links,
using sections with higher values of Aw/Ag for the beam results in higher axial load in the
beam. Note that sections with lower Aw/Ag ratios are sections commonly used as
columns. These sections offer an additional benefit of increased axial buckling capacity
in the beam. For wide flange sections listed in the AISC Manual, Figure 2.13 plots the
depth of each section against the value of Aw/Ag. Note that values of Aw/Ag range from
about 0.2 to 0.5 for most rolled wide flange shapes. For each depth, the sections with
higher values of Aw/Ag are generally the lighter sections.
For the moment links, instead of Aw/Ag of the beam, Z/Ag of the beam section affects the
axial load in the beam. By increasing this value, the axial load in the beam increases. As
Figure 2.14 shows, choosing sections with lower depth and for each specific depth
choosing a section with a lower weight will be beneficial for decreasing the axial force in
the beam outside the link, when moment links are used.
Within the assumptions made for this analysis, for both shear link and moment links, the
brace section properties does not have any effect on the axial load in the beam. This can
be useful for choosing the appropriate section for the brace because it only affects the
moment in the beam. As indicated by Figure 2.12, choosing a stiffer brace decreases the
moment demand on the beam outside the link.
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Fig. 2.13 - Aw/Ag vs. Depth for the common AISC sections
Fig. 2.14 - Z/Ag vs. Depth for the common AISC sections
0
0.1
0.2
0.3
0.4
0.5
0.6
5 10 15 20 25 30
A w / A g
Depth (in)
0
2
4
6
8
10
12
5 10 15 20 25 30
Z / A g ( i n )
Depth (in)
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CHAPTER 3
YIELDING IN THE BEAM OUTSIDE OF THE LINK DUE TO
COMBINED AXIAL FORCE AND BENDING MOMENT
In Chapter 2, approximate equations were derived for the axial force and bending
moment in the beam when link ultimate strength is developed. In this chapter, the
strength of the beam under the combined effect of axial force and bending moment is
considered. For this chapter, it is assumed that instability of the beam is prevented by
proper bracing and that local buckling does not control strength. As a result, the strengthof the beam will be controlled by yielding under combined axial force and bending
moment. In the next chapter, instability of the beam under combined axial force and
moment is considered. For either the case of yielding or instability, the strength of the
beam is computed using equations in the AISC Specification (AISC, 2005) for members
subjected to combined axial force and bending moment.
3.1. AISC EQUATIONS FOR STRENGTH UNDER COMBINED FORCES
In the 2005 AISC Specification (AISC, 2005), Equations H1-1a and H1-1b define the
nominal strength of a member under combined axial force and bending moment. These
equations are reproduced below.
For 0.2P
P
c
r ≥ : 1.0)M
M
M
M(
9
8
P
P
cy
ry
cx
rx
C
r ≤++ (Eq. 3-1) [Eq. H1-1a]
For 0.2P
P
c
r < : 1.0)M
M
M
M(
2P
P
cy
ry
cx
rx
C
r ≤++ (Eq. 3-2) [Eq. H1-1b]
In these equations:
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Pr is required axial compressive strength.
Pc is available axial compressive strength.
Mr is required flexural strength.
Mc is available flexural strength.
More detailed definitions of each term are available in the AISC Specification.
3.2. BEAM AXIAL FORCE AND MOMENT.
Equations 2-28 and 2-33 are the final equations to calculate the moment in the beam for
shear and moment, and are in terms of α and β. Since it is desired to evaluate the effect ofthese variables by developing equations representing the combination of axial force and
bending moment, Equations 2-23 and 2-30 which represent the axial force in the beam
are rearranged so they include these two variables. It should be noted that the
contributions of geometric configurations of an EBF, including the length of link e, the
span length L, the story height h and the angle between the beam and the brace α, are
related to each other. A change in one of these variables results in changes in others.
3.2.1. Shear links
From Chapter 2, expressions for the shear links were as follows:
L
e1
1
αtan
0.84
A
A
P
P
g
W
y
beam
−
−= (Eq. 2-23)
Considering the EBF configuration in Figure 1.5 and rearrangement of Equation 2-23
produces the following expression:
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)V
Mβ
h
1
tanα
20.42(
A
A)
h
e
tanα
20.42(
A
A
tanα
2h
etanα
2h
tanα
0.84
A
A
P
P
p
p
g
w
g
w
g
w
y
beam +−=+−=+
−=
)0.6A
Zβ
h
1
tanα
20.42(
A
A
wg
w +−=
And therefore:
βh
0.7
A
Z
tanα
0.84
A
A
P
P
gg
w
y
beam −−= (Eq. 3-3)
The following equation represents the moment in the beam
αCosI
I
I
I
β0.7αCosII
Iβ0.7
M
M
brace
beam
brace
beam
brace beam
beam
P
beam
+
=+
= (Eq. 2-28)
3.2.2. Moment links
For moment links, the axial force in the beam is:
h
1
L
e
1.4
A
Z
P
P
gy
beam−= (Eq. 2-30)
Rearranging this equation results in:
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)h
1
tanα
A0.6
Zβ
2(1.4
A
Z)
h
1
tanα
V
Mβ
2(1.4
A
Z
h
etanα
2h
e
1.4
A
Z
P
P
W
g
P
Pggy
beam +−=+−=+
−=
And therefore:
h
1.4
A
Z
tanαβ
1.68
A
A
P
P
gg
W
y
beam−−= (Eq. 3-4)
The moment in the beam was previously derived as:
αCosI
I
I
I
1.4M
M
brace
beam
brace
beam
P
beam
+
= (Eq. 2-33)
3.3. YIELDING UNDER COMBINED AXIAL FORCE AND THE MOMENT
As described earlier, in this chapter the strength of the beam is computed assuming
stability limit states do not control. Thus, the development of a fully yielded cross section
under combined axial force and bending moment will control beam strength. Therefore,
in Equations 3-1 and 3-2, the strength terms Pc and Mc can be taken as:
yc PP = (Eq. 3-5)
pc MM = (Eq. 3-6)
And also it is assumed that the beam is subjected to the moment only in the plane of the
frame and therefore in Equations 3-1 and 3-2 the value of the moment about the weak
axis is taken zero.
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0M ry = (Eq. 3-7)
Therefore Equations 3-1 and 3-2 simplify to the following equations:
ForP
0.2P
beam
y
≥ : 1.0M
M
9
8
P
P
p
beam
y
beam ≤+ (Eq. 3-8)
ForP
0.2P
beam
y
< : 1.0M
M
2P
P
p
beam
y
beam ≤+ (Eq. 3-9)
Since the axial force in the beam will normally be greater than 0.20 Py, Equation 3-8 can
be used below to evaluate beam strength.
3.3.1. Shear links
Replacing the expression for the axial load from Equation 3-3 and bending moment from
Equation 2-28 into Equation 3-8 results in the following equation:
1)
αCosI
I
I
I
β0.79
8(β)
h
0.7
A
Z
αtan
0.84
A
A(
brace
beam
brace
beam
gg
W ≤
+
++
The ratio of the beam stiffness to the brace stiffness is defined using the variable η, as
follows:
brace
beam
I
Iη = (Eq. 3-10)
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Therefore
1)Cosαη
η
β(0.62β)h
0.7
A
Z
αtan
0.84
A
A
( gg
W≤
+++
(Eq. 3-11)
3.3.2. Moment links
Replacing the expression for the axial load from Equation 3-4 and bending moment from
Equation 2-33 into Equation 3-8 results in the following equation:
1)
αCosI
II
I
1.498()
h1.4
AZ
tanαβ1.68
AA(
brace
beam
brace
beam
gg
W≤
+++
Substituting η results in:
1)αCosη
η(1.24)
h
1.4
A
Z
tanαβ
1.68
A
A(
gg
W≤
+++ (Eq. 3-12)
As Equations 3-11 and 3-12 show, the following variables affect yielding of the beam
under combined axial force and bending moment:
h,η,α,β,A
Z,
A
A
gg
W
To reduce the number of variables in the evaluation of beam strength below, it will be
assumed that the story height, h, is equal to 130 inches.
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3.4. PRELIMINARY EVALUATION OF VARIABLES AFFECTING BEAM YIELDING
In this section, typical ranges of values for the variables Aw/Ag, Z/Ag, β, α and η are
discussed. This is followed by analysis of a simple EBF using the structural analysis
computer program SAP2000.
3.4.1. Variation of Aw /Ag
Figures 3.1 and 3.2 plot the values of Aw/Ag for rolled W-Shapes in the AISC Manual.
Figure 3.1 covers all W-Shapes whereas Figure 3.2 covers only W8 through W24
sections, as these are most commonly used in EBFs. As these two diagrams show, the
range of Aw/Ag is limited to 0.2 to 0.5, with average of around 0.35 for all the sections
and around 0.30 for the more common sections.
Fig. 3.1 – Variation of Aw/Ag for all of AISC W-Shapes
0
0.1
0.2
0.3
0.4
0.5
0.6
0 200 400 600 800 1000
A w
/ A g
Nominal Wt (lb/ft)
Average (Aw/Ag) = 0.34
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Fig. 3.2 – Variation of Aw/Ag for more common AISC W-Shapes (W8-W24)
3.4.2. Variation of Z/Ag
Figure 3.3 and Figure 3.4 plot the values of Z/Ag
for all W-Shapes (Fig. 3.3) and the most
common W-Shapes (Fig. 3.4). The range of variation is between 2 and 16 for all of theW-Shapes with the average of around 8.5. The range of Z/Ag values is between 3 and 10
for the more common sections with the average of around 6.
3.4.3. Variation of α
Usually the angle between the beam and the brace in EBFs varies between 30 degrees to
60 degrees. As is discussed later, in this range it is more desirable to choose higher
values.
3.4.4. Variation of η
In EBF beam design, it can be advantageous to provide a stiff brace, i.e., to use a small
value of η. A stiffer brace will reduce the moment on the beam by attracting a greater
0
0.1
0.2
0.3
0.4
0.5
0.6
0 100 200 300 400 500 600 700 800 900
A w / A g
Nominal Wt (lb/ft)
Average (Aw/Ag) = 0.31
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share of the link end moment to the brace. However, the requirement that the brace and
beam centerlines intersect at the end of the link will often require that the depth of the
brace be chosen less than the depth of the beam to accommodate the brace-beam
connection. As a result, the value of η will typically be greater than 1. The range of
variation for η in later analysis will be taken as 1 to 2.
3.4.5. Variation of β
Providing short shear links, i.e. low values of β, is generally preferred in EBF designing.
However, if links are too short, plastic rotation demands on the link can exceed the link
plastic rotation capacity. Therefore, in the analysis that follows, the value of β for EBFs
with shear links are taken in the range of 1 to 2. For the EBFs with moment links, thevalue of β is greater than 2.
Fig. 3.3 – Variation of Z/Ag for all of AISC W-sections
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500 600 700 800 900
Z / A g
Nominal Wt (lb/ft)
Average (Z/Ag) = 8.58
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Fig. 3.4 – Variation of Z/Ag for more common AISC W-sections (W8-W24)
3.4.6. Preliminary investigation of an EBF with a shear link
In this section, analysis of beam yielding is conducted for a simple single story EBF. The
frame geometry and members are chosen within the typical range of values discussed
above. The potential for beam yielding is assessed using Equation 3-11, which is based
on a number of approximation and simplifications. The frame is then analyzed using the
structural analysis computer program SAP2000 (Computers and Structures, 2010). Forces
predicted in the beam from the SAP2000 analysis are then used to check for yielding in
the beam. The purpose of the SAP2000 analysis is to evaluate the accuracy of Equation
3-11 for predicting beam yielding.
The EBF with a shear link that is used for this analysis is shown in Figure 3.5. The beam
and link is a W16x57 section, the columns are W16x77 sections and the braces are
W10x88 sections. The steel for all members is ASTM A992 with a specified yield stress
of 50 ksi.
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700 800 900
Z / A g
Nominal Wt (lb/ft)
Average (Z/Ag) = 6.33
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Fig. 3.5 – EBF with shear link
For this frame, the values of the key variables appearing in Equation 3-11 are as follows:
1.2927.2
35β,1.42
534
758η,45α,5.79
A
Z,0.38
A
A
gg
W=======
These values are within the typical ranges for these variables discussed above.
Substituting these values in Equation 3-11 results in:
10.89[0.523]0.04][0.319
)]Cos451.42
1.4229)([(0.62)(1.)(1.29)]
130
0.7(5.79)()
tan45
0.84[(0.38)(
1)Cosαη
ηβ(0.62β)
h
0.7
A
Z
αtan
0.84
A
A(
gg
W
<=++=
+++
≤+
++
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This result predicts that no yielding will occur in the beam under the combined axial
force and moment developed in this member when the link reaches its ultimate shear
strength of 1.4V p.
In the next step, the frame shown in Figure 3.5 is analyzed using SAP2000. A linear
elastic analysis is conducted. Although the post-elastic behavior of the frame is of interest
in this analysis, a linear elastic analysis can still be used to predict the forces in the beam
segments outside of the link, the braces and the columns. This is because the frame is
designed to restrict yielding to the link, whereas the beams, braces and columns are
designed to remain elastic. The approach used is to amplify the external lateral load on
the frame until the shear force in the link is equal to its ultimate value of 1.4V p. The link
end moments corresponding to this ultimate shear is determined solely by equilibrium for
the case where the link is at mid-span and the end moments are consequently equal in
magnitude. Since the relationship between link shear and link end moment is determined
from equilibrium, the correct link end moments can be predicted by SAP2000. Finally,
since the members outside of the link remain elastic, the SAP2000 elastic analysis can
correctly predict the forces generated in the beams, braces and columns by the ultimate
link shear and end moment. Note that while an elastic analysis can be used in this manner
to correctly compute forces in the members outside of the link, the analysis will not
correctly predict frame displacements, since the inelastic deformation of the link is not
modeled. Figure 3.6 shows the SAP2000 frame model and member sizes.
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Fig. 3.6 – SAP2000 model for EBF with shear link
Next, the frame is subjected to a symmetric lateral loading as shown in Figure 3.7. The
value of the lateral loads has been chosen so that the shear in the link reaches its ultimate
value of 1.4Vp.
Fig. 3.7 – Symmetric lateral loading on EBF with shear link
Based on the beam section properties:
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k.in525050105ZFM
kips190506.340.6F0.6AV
kips8405016.8FAP
ksi50F
y p
yw p
ygy
y
=×==
=××==
=×==
=
The ultimate shear and ultimate moment value in the link is given as follows:
k.in26552
26635
2
e.VM
266k 1.4VV
link link
plink
=×
==
==
By subjecting the beam to the loading shown in Figure 3.7, the shear diagram shown in
Figure 3.8 is computed by SAP2000:
Fig. 3.8 – SAP2000 shear diagram for EBF with shear link
The shear in the link is 266 kips, which is equal to its ultimate value of 1.4V p. Figure 3.9
shows the moment diagram for the frame from the SAP2000 analysis.
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Fig. 3.9 – SAP2000 moment diagram for EBF with shear link
The end moment in the link computed by the SAP2000 analysis, is 4655k.in, which
agrees with the previous equilibrium calculation. The maximum moment in the beam
outside the link from the SAP2000 analysis is k.in2979M beam = .
Figure 3.10 shows the axial loads in the frame members from the SAP2000 analysis.
Fig. 3.10 – SAP2000 axial force diagram for EBF with shear link
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The maximum axial force in the beam outside the link is k 302P beam = . By substituting the
beam axial force and moment from the SAP2000 analysis and the cross-section strength
values into Equation 3-8 results in:
186.050.036.05250
2979
9
8
840
302≤=+=+
3.4.7. Preliminary investigation of an EBF with a moment link
To evaluate beam yielding in an EBF with a moment link, the frame shown in Figure
3.11 will be analyzed. As with the previous frame, the beam and link is a W16x57, thecolumns are W16x77 sections and the braces are W10x88 sections.
Fig. 3.11 – EBF with moment link
For this frame, the values of the key variables appearing in Equation 3.12 are as follows:
2.572.27
70β,1.42η,45α,5.79
A
Z,0.38
A
A
gg
W ======
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These values are within the typical ranges for these variables discussed earlier.
Substituting these values in Equation 3-12 results in:
W
g g
A 1.68 Z 1.4η( ) (1.24 ) 1
Aβ tanα A h η Cos α
1.68 1.4 1.42[(0.38)( ) (5.79)( )] [(1.24)( )]
2.57tan45 130 1.42 Cos45
[0.25 0.06] [0.83] 1.14 1
+ + ≤+
+ ++
= + + = >
This result predicts that yielding will occur in the beam under the combined axial force
and moment developed in this member when the link reaches its ultimate flexural
strength of 1.4M p.
As before, this same frame is analyzed using SAP2000 to predict the axial force and
bending moment in the beam when the link ultimate strength is achieved. The following
figure shows the members used in the SAP2000 model.
Fig. 3.12 – SAP2000 model for EBF with moment link
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The frame is then subjected to a symmetric lateral loading as shown in Figure 3.13. The
value of the loads have been chosen in a way that the end moment in the link reaches its
ultimate value of 1.4M p, which is equal to Mlink = 1.4 × 5250 in-k = 7350 in.k. Based on
equilibrium of the link, the corresponding link shear is 2Mlink /e = 2 × 7350/70 = 210 kips.
Fig. 3.13 – Symmetric lateral loading on EBF with moment link
By subjecting the beam to the loading shown in Figure 3.13, the SAP2000 analysis shows
the shear diagram for the frame in Figure 3.14.
Fig 3.14 – SAP2000 shear diagram for EBF with moment link
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The shear in the link is 210 kips, which agrees with the equilibrium calculation above.
Figure 3.15 shows the moment diagram in the frame members from the SAP2000
analysis.
Fig 3.15 – SAP2000 moment diagram for EBF with moment link
The moment in the link is 7341 k.in which is very close to its ultimate value. The
maximum moment in the beam outside the link computed in the SAP2000 analysis is
k.in7654M beam = .
Fig 3.16 – SAP2000 axial force diagram for EBF with moment link
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Figure 3.10 shows the axial forces in the members from the SAP2000 analysis. The
maximum axial force in the beam outside the link is k 266P beam = . By substituting the
beam axial force and moment from the SAP2000 analysis and the cross-section strength
values into Equation 3-8 results in:
113.181.032.05250
4765
9
8
840
266>=+=+
3.4.8. Conclusions from the preliminary investigation
The analysis conducted in this section showed very good agreement for the forces
computed in the beam from the SAP2000 analysis and forces computed using the
approximate equations derived in Chapter 2. Note that the SAP model was developed to
match the general assumptions made in Chapter 2 that the moment at the brace to column
connection and beam-to-column connection is zero. However, within these assumptions,
the two analyses agreed very well. This provides some confidence that the approximate
analysis in Chapter 2, although based on a number of simplifications, provides reasonable
results.
In an actual EBF design, an engineer would need to compute the capacity design forces in
the beam for the specific conditions of that frame. The approach used here with SAP2000
can be used for such an analysis. Thus, the SAP model could reflect conditions different
than though assumed here. For example, rigid connections might be used at the brace-to-
column connection and at the beam-to-column connection, lateral loads might not be
applied symmetrically to the frame, gravity loads on the EBF may be significant, etc. For
such cases, the capacity design forces will differ somewhat from those computed using
the approximate equations in Chapter 2. Nonetheless, the capacity design forces
computed using the approximate equations derived in Chapter 2 cover very common EBF
design conditions, and therefore provide a useful basis for conducting parametric studies
to guide preliminary design.
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3.5. PARAMETRIC STUDY OF BEAM YIELDING
In this section, a parametric study is conducted to evaluate the effect of each geometric
variable in Equations 3-11 and 3-12 on the development of yielding in the beam. The
values of these variables are changed within the ranges discussed earlier in this chapter.
While the value for one variable is changed, the other variables are kept constant, to the
extent possible. However, since some of the variables are interrelated, it will not always
be possible to change only one variable at a time.
To evaluate the strength of the beam outside the link compared to its yield strength under
combined axial force and moment, yield functions are defined by a simple modification
to Equations 3-11 and 3-12. The yield functions are defined by taking the portion of the
equation to the left of the inequality sign and subtracting 1. The resulting yield functions,
represented by the symbol Φ, is given by Equations 3-13 and 3-14.
If the yield function is equal to zero, then the terms in the interaction equation add to
unity. That is, a value of the yield function equal to zero indicates the beam has just
achieved its full strength based on yielding. A negative value of the yield function
indicates that the forces in the beam are below a level that causes full cross-section
yielding, i.e. that the beam is essentially elastic. A value for the yield function greaterthan zero indicates that the yield strength of the beam has been exceeded.
For shear links:
1)Cosαη
ηβ(0.62β)
h
0.7
A
Z
αtan
0.84
A
A(
gg
WShort −
+++=Φ (Eq. 3-13)
This equation is basically a rearrangement of Equation 3-11.
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And for moment links:
1)αCosη
η(1.24)h1.4
AZ
tanαβ1.68
AA(
gg
WLong −
+++=Φ (Eq. 3-14)
This equation is a rearrangement of Equation 3-12.
Note that the subscript “short” is used for shear links, and the subscript “long” is used for
moment links. These terms are used rather than “shear” and “moment” to avoid confusion
with the forces related yield functions described below.
Two additional yield functions have been defined to compare the relative contributions of
axial force and bending moment towards yielding of the beam.
For shear links:
βh
0.7
A
Z
αtan
0.84
A
A)(
gg
WShortaxial +=Φ (Eq. 3-15)
Cosαη
ηβ0.62)( Short bending
+=Φ (Eq. 3-16)
And for moment links:
h
1.4
A
Z
tanαβ
1.68
A
A)(
gg
WLongaxial +=Φ (Eq. 3-17)
αCosηη1.24)( Long bending
+=Φ (Eq. 3-18)
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For both shear and moment links, Φaxial + Φ bending -1 = Φ. Note also that the axial yield
function is also equal to P/Py in the beam and the bending yield function is equal to
(8/9)×(M/M p) in the beam.
The effect of each of the following variables are evaluated to determine the effect on the
yield function: Aw/Ag, Z/Ag, β, α and η. Note that it is not possible to independently
vary Aw/Ag and Z/Ag, since these variables are related. However, by being selective in
choosing the beam sections, it is possible to keep one of these almost constant while the
other is changed.
3.5.1 Effect of η on EBFs with shear links
To evaluate the effect of η, for specific beam section, the brace section is changed to give
different values of η. Figure 3.17 shows qualitatively how the value of η so is changed
while other variables remain constant.
Fig. 3.17 – Changing η and keeping other variables constant
Tables 3.1 to 3.14 show the values for the yield function for EBFs with shear links as η is
varied.
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Table 3.1 – Effect of varying η for EBF with beam section W10X12
Beam W10X12
Ix-x Zx-x Z / Ag Aw Mp / Vp Aw/Ag
53.8 12.6 3.559 1.79 11.695 0.507
L 275.20Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.005 0.451 0.007 -0.542
W27x194 0.006 0.451 0.008 -0.541
e 15.20 W24x162 0.010 0.451 0.012 -0.537
W21x83 0.029 0.451 0.032 -0.517
α 45.00 W18x76 0.040 0.451 0.044 -0.505
W16x50 0.081 0.451 0.083 -0.466
L/h 2.12 W14x48 0.111 0.451 0.109 -0.440W12x40 0.175 0.451 0.160 -0.389
e/L 0.06 W10x39 0.257 0.451 0.215 -0.334
W8x31 0.489 0.451 0.329 -0.220
β 1.30 W6x16 1.676 0.451 0.567 0.018
Table 3.2 – Effect of varying η for EBF with beam section W10X112
Beam W10X112
Ix-x Zx-x Z / Ag Aw Mp / Vp Aw/Ag
716 147 4.468 6.719 36.461 0.204
L 307.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.077 0.203 0.080 -0.717
W27x194 0.091 0.203 0.092 -0.705
e 47.50 W24x162 0.138 0.203 0.132 -0.665
W21x83 0.391 0.203 0.288 -0.509
α 45.00 W18x76 0.538 0.203 0.349 -0.448
W16x50 1.086 0.203 0.489 -0.308
L/h 2.37 W14x48 1.479 0.203 0.546 -0.251
W12x40 2.332 0.203 0.620 -0.177
e/L 0.15 W10x39 3.425 0.203 0.670 -0.128W8x31 6.509 0.203 0.729 -0.069
β 1.30 W6x16 22.305 0.203 0.783 -0.014
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Table 3.3 - Effect of varying η for EBF with beam section W12X14
Beam W12X14
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
88.6 17.4 4.182 2.29 12.663 0.550
L 276.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.009 0.492 0.011 -0.497
W27x194 0.011 0.492 0.013 -0.496
e 16.50 W24x162 0.017 0.492 0.019 -0.489
W21x83 0.048 0.492 0.052 -0.456
α 45.00 W18x76 0.066 0.492 0.070 -0.439
W16x50 0.134 0.492 0.129 -0.379L/h 2.13 W14x48 0.183 0.492 0.166 -0.342
W12x40 0.288 0.492 0.234 -0.274
e/L 0.06 W10x39 0.423 0.492 0.303 -0.205
W8x31 0.805 0.492 0.430 -0.078
β 1.30 W6x16 2.760 0.492 0.643 0.135
Table 3.4 - Effect of varying η for EBF with beam section W12X152
Beam W12X152
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag1430 243 5.436 9.483 42.708 0.212
L 315.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.155 0.216 0.145 -0.639
W27x194 0.181 0.216 0.165 -0.619
e 55.50 W24x162 0.276 0.216 0.227 -0.557
W21x83 0.781 0.216 0.423 -0.361
α 45.00 W18x76 1.075 0.216 0.486 -0.298
W16x50 2.169 0.216 0.608 -0.176
L/h 2.43 W14x48 2.954 0.216 0.650 -0.134
W12x40 4.65 0.216 0.700 -0.084
e/L 0.18 W10x39 6.842 0.216 0.730 -0.054
W8x31 13 0.216 0.764 -0.020
β 1.30 W6x16 44.548 0.216 0.793 0.009
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Table 3.5 - Effect of varying η for EBF with beam section W14X22
Beam W14X22
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
199 33.2 5.115 2.9969 18.46352 0.461
L 284.00Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.021 0.424 0.024 -0.552
W27x194 0.025 0.424 0.028 -0.548
e 24.00 W24x162 0.038 0.424 0.042 -0.535
W21x83 0.108 0.424 0.107 -0.469
α 45.00 W18x76 0.149 0.424 0.141 -0.436
W16x50 0.301 0.424 0.241 -0.335
L/h 2.18 W14x48 0.411 0.424 0.296 -0.280
W12x40 0.648 0.424 0.385 -0.191
e/L 0.08 W10x39 0.952 0.424 0.462 -0.114
W8x31 1.809 0.424 0.579 0.003
β 1.30 W6x16 6.199 0.424 0.723 0.147
Table 3.6 - Effect of varying η for EBF with beam section W14X145
Beam W14x145
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1710 260 6.088 8.5816 50.49563 0.200
L 325.60Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.185 0.211 0.168 -0.621
W27x194 0.217 0.211 0.190 -0.599
e 65.60 W24x162 0.330 0.211 0.257 -0.532
W21x83 0.934 0.211 0.458 -0.330
α 45.00 W18x76 1.285 0.211 0.520 -0.269
W16x50 2.594 0.211 0.633 -0.156
L/h 2.50 W14x48 3.533 0.211 0.671 -0.117W12x40 5.570 0.211 0.715 -0.074
e/L 0.20 W10x39 8.181 0.211 0.741 -0.047
W8x31 15.54 0.211 0.770 -0.018
β 1.30 W6x16 53.271 0.211 0.795 0.006
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Table 3.7 - Effect of varying η for EBF with beam section W16X26
Beam W16X26
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
301 44.2 5.755 3.7525 19.63136 0.488
L 285.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.033 0.451 0.036 -0.514
W27x194 0.038 0.451 0.041 -0.508
e 25.50 W24x162 0.058 0.451 0.061 -0.488
W21x83 0.164 0.451 0.152 -0.397
α 45.00 W18x76 0.226 0.451 0.195 -0.354
W16x50 0.457 0.451 0.316 -0.233
L/h 2.20 W14x48 0.622 0.451 0.377 -0.172
W12x40 0.980 0.451 0.468 -0.081
e/L 0.09 W10x39 1.440 0.451 0.540 -0.009
W8x31 2.736 0.451 0.640 0.091
β 1.30 W6x16 9.377 0.451 0.749 0.200
Table 3.8 - Effect of varying η for EBF with beam section W16X100
Beam W16X100
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1500 200 6.734 8.792 37.91088 0.296
L 309.30Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.163 0.296 0.151 -0.553
W27x194 0.191 0.296 0.171 -0.533
e 49.30 W24x162 0.290 0.296 0.235 -0.470
W21x83 0.820 0.296 0.433 -0.271
α 45.00 W18x76 1.128 0.296 0.496 -0.209
W16x502.276 0.296 0.615 -0.089L/h 2.38 W14x48 3.099 0.296 0.656 -0.048
W12x40 4.886 0.296 0.704 0.000
e/L 0.16 W10x39 7.177 0.296 0.734 0.030
W8x31 13.636 0.296 0.767 0.062
β 1.30 W6x16 46.729 0.296 0.794 0.090
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Table 3.9 - Effect of varying η for EBF with beam section W18X35
Beam W18X35
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
510 66.5 6.456 5.055 21.92549 0.490
L 288.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.055 0.457 0.059 -0.484
W27x194 0.065 0.457 0.068 -0.475
e 28.50 W24x162 0.099 0.457 0.099 -0.444
W21x83 0.279 0.457 0.228 -0.315
α 45.00 W18x76 0.383 0.457 0.283 -0.259
W16x50 0.774 0.457 0.421 -0.121
L/h 2.22 W14x48 1.054 0.457 0.482 -0.060
W12x40 1.661 0.457 0.565 0.023
e/L 0.10 W10x39 2.440 0.457 0.625 0.082
W8x31 4.636 0.457 0.699 0.157
β 1.30 W6x16 15.888 0.457 0.772 0.229
Table 3.10 - Effect of varying η for EBF with beam section W18X143
Beam W18x143
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
2750 322 7.648 12.3078 43.60379 0.292
L 316.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.298 0.299 0.239 -0.462
W27x194 0.349 0.299 0.266 -0.435
e 56.50 W24x162 0.531 0.299 0.345 -0.356
W21x83 1.502 0.299 0.546 -0.155
α 45.00 W18x76 2.067 0.299 0.599 -0.102
W16x50 4.172 0.299 0.687 -0.014
L/h 2.43 W14x48 5.681 0.299 0.714 0.013
W12x40 8.957 0.299 0.745 0.044e/L 0.18 W10x39 13.157 0.299 0.762 0.061
W8x31 25 0.299 0.781 0.080
β 1.30 W6x16 85.669 0.299 0.797 0.096
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Table 3.11 - Effect of varying η for EBF with beam section W21X44
Beam W21X44
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
843 95.4 7.338 6.93 22.94372 0.533
L 289.80Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.092 0.499 0.092 -0.409
W27x194 0.107 0.499 0.106 -0.395
e 29.80 W24x162 0.163 0.499 0.151 -0.350
W21x83 0.461 0.499 0.318 -0.183
α 45.00 W18x76 0.634 0.499 0.381 -0.120
W16x50 1.279 0.499 0.519 0.018
L/h 2.23 W14x48 1.742 0.499 0.573 0.072
W12x40 2.746 0.499 0.640 0.139
e/L 0.10 W10x39 4.033 0.499 0.685 0.184
W8x31 7.664 0.499 0.737 0.236
β 1.30 W6x16 26.262 0.499 0.784 0.283
Table 3.12 - Effect of varying η for EBF with beam section W21X147
Beam W21x147
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
3630 373 8.634 14.256 43.60737 0.33
L 316.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.394 0.337 0.288 -0.375
W27x194 0.461 0.337 0.317 -0.345
e 56.50 W24x162 0.702 0.337 0.400 -0.262
W21x83 1.983 0.337 0.592 -0.070
α 45.00 W18x76 2.729 0.337 0.638 -0.025
W16x50 5.508 0.337 0.712 0.049
L/h 2.43 W14x48 7.5 0.337 0.734 0.072
W12x40 11.824 0.337 0.758 0.095
e/L 0.18 W10x39 17.368 0.337 0.772 0.109
W8x31 33 0.337 0.786 0.124
β 1.30 W6x16 113.084 0.337 0.798 0.136
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Table 3.13 - Effect of varying η for EBF with beam section W24X55
Beam W24X55
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1360 135 8.282 8.92305 25.21559 0.547
L 292.80Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.148 0.518 0.139 -0.343
W27x194 0.173 0.518 0.159 -0.324
e 32.80 W24x162 0.263 0.518 0.219 -0.263
W21x83 0.743 0.518 0.413 -0.069
α 45.00 W18x76 1.023 0.518 0.477 -0.005
W16x50 2.064 0.518 0.601 0.119
L/h 2.25 W14x48 2.810 0.518 0.644 0.162
W12x40 4.430 0.518 0.695 0.213
e/L 0.11 W10x39 6.507 0.518 0.727 0.245
W8x31 12.364 0.518 0.763 0.281
β 1.30 W6x16 42.368 0.518 0.793 0.311
Table 3.14 - Effect of varying η for EBF with beam section W24X146
Beam W24X146
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
4580 418 9.720 14.638 47.59302 0.340
L 321.90Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.497 0.354 0.333 -0.313
W27x194 0.582 0.354 0.364 -0.282
e 61.90 W24x162 0.885 0.354 0.448 -0.198
W21x83 2.502 0.354 0.629 -0.017
α 45.00 W18x76 3.443 0.354 0.669 0.023
W16x50 6.949 0.354 0.732 0.086
L/h 2.48 W14x48 9.462 0.354 0.750 0.104
W12x40 14.918 0.354 0.770 0.124
e/L 0.19 W10x39 21.913 0.354 0.781 0.135
W8x31 41.636 0.354 0.793 0.147
β 1.30 W6x16 142.679 0.354 0.802 0.156
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For each one of the above tables a curve is presented in Figure 3.18. Section properties of
the beams are also shown in the figure. The angle between the beam and the brace is 45
degrees and the value of β is 1.3 for all the cases.
Fig. 3.18 – Effect of changing η on the yield function for the beam in an EBF with a shear link
-0.800
-0.600
-0.400
-0.200
0.000
0.200
0.400
0.600
0 1 2 3 4 5 6
y i e l d f u n c t i o n ( Ф )
η
W10X12 (Aw/Ag=0.51 , Z/Ag=3.60)
W14X22 (Aw/Ag=0.46 , Z/Ag=5.12) W18X35 (Aw/Ag=0.49 , Z/Ag=6.46)
W16X22 (Aw/Ag=0.49 , Z/Ag=5.76)
W12X14 (Aw/Ag=0.55 , Z/Ag=4.18)
W21X44 (Aw/Ag=0.53 , Z/Ag=7.34)
W24X55 (Aw/Ag=0.55 , Z/Ag=8.28)
W21X147 (Aw/Ag=0.33 , Z/Ag=8.63)
W18X143(Aw/Ag=0.29 , Z/Ag=7.65)
W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W24X146 (Aw/Ag=0.34 , Z/Ag=9.72)
W14X145 (Aw/Ag=0.20 , Z/Ag=6.09)
W12X152 (Aw/Ag=0.21 , Z/Ag=5.44)
W10X112 (Aw/Ag=0.20 , Z/Ag=4.47)
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Figures 3.19 and 3.20 show the corresponding values of the axial yield function and the
bending yield function.
Fig. 3.19 – Variation of axial yield function for an EBF with a shear link
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0 1 2 3 4 5 6
Ф
a x i a l
η
W10X12 (Aw/Ag=0.51 , Z/Ag=3.60)
W12X14 (Aw/Ag=0.55 , Z/Ag=4.18)
W14X22 (Aw/Ag=0.46 , Z/Ag=5.12)
W16X22 (Aw/Ag=0.49 , Z/Ag=5.76)
W18X35 (Aw/Ag=0.49 , Z/Ag=6.46)
W21X44 (Aw/Ag=0.53 , Z/Ag=7.34)
W24X55 (Aw/Ag=0.55 , Z/Ag=8.28)
W24X146 (Aw/Ag=0.34 , Z/Ag=9.72)W21X147 (Aw/Ag=0.33 , Z/Ag=8.63)
W18X143(Aw/Ag=0.29 , Z/Ag=7.65) W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W14X145 (Aw/Ag=0.20 , Z/Ag=6.09)
W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W10X112 (Aw/Ag=0.20 , Z/Ag=4.47)
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Fig. 3.20 - Variation of bending yield function for an EBF with a shear link
As Figure 3.19 shows, the axial force ratio P/Py in the beams considered in this analysis
never goes below 0.2, which confirms the assumption made in using Equation 3.8.
Observations f rom this analysis are made after the next section related to the effect of η
on EBFs with moment links.
3.5.2. Effect of η on EBFs with moment links
Tables 3.15 to 3.28 show the values of the yield function as the value of η is varied for
each specific beam section using different types of bracing. The link length is chosen so
the value of β is greater than 2.
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
0 1 2 3 4 5 6
Ф
b e n d i n g
η
W10x12
W10x112
W12x14
W12x152
W14x22
W14x145
W16x26
W16x100
W18x35
W18x143
W21x44
W21x147
W24x55
W24x146
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Table 3.15 - Effect of varying η for EBF with beam section W10X12
Beam W10X12
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
53.8 12.6 3.559 1.795 11.695 0.507
L 290.40Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.005 0.366 0.010 -0.624
W27x194 0.006 0.366 0.012 -0.622
e 30.40 W24x162 0.010 0.366 0.018 -0.616
W21x83 0.029 0.366 0.049 -0.584
α 45.00 W18x76 0.040 0.366 0.067 -0.567
W16x50 0.081 0.366 0.128 -0.505
L/h 2.23 W14x48 0.111 0.366 0.168 -0.465
W12x40 0.175 0.366 0.246 -0.388
e/L 0.10 W10x39 0.257 0.366 0.331 -0.303
W8x31 0.489 0.366 0.507 -0.127
β 2.60 W6x16 1.676 0.366 0.872 0.238
Table 3.16 - Effect of varying η for EBF with beam section W10X112
Beam W10X112
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
716 147 4.468 6.719 36.461 0.204
L 354.90Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.077 0.180 0.123 -0.697
W27x194 0.091 0.180 0.142 -0.679
e 94.90 W24x162 0.138 0.180 0.203 -0.617
W21x83 0.391 0.180 0.442 -0.378
α 45.00 W18x76 0.538 0.180 0.536 -0.284
W16x50 1.086 0.180 0.751 -0.069
L/h 2.73 W14x48 1.479 0.180 0.839 0.019W12x40 2.332 0.180 0.952 0.131
e/L 0.27 W10x39 3.425 0.180 1.028 0.208
W8x31 6.509 0.180 1.118 0.298
β 2.60 W6x16 22.305 0.180 1.202 0.382
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Table 3.17 - Effect of varying η for EBF with beam section W12X14
Beam W12X14
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
88.6 17.4 4.182 2.29 12.66376 0.550
L 292.90Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.009 0.401 0.017 -0.582
W27x194 0.011 0.401 0.019 -0.580
e 32.90 W24x162 0.017 0.401 0.029 -0.570
W21x83 0.048 0.401 0.079 -0.520
α 45.00 W18x76 0.066 0.401 0.107 -0.492
W16x50 0.134 0.401 0.198 -0.401
L/h 2.25 W14x48 0.183 0.401 0.255 -0.344
W12x40 0.288 0.401 0.359 -0.240
e/L 0.11 W10x39 0.423 0.401 0.465 -0.134
W8x31 0.805 0.401 0.660 0.061
β 2.60 W6x16 2.760 0.401 0.987 0.388
Table 3.18 - Effect of varying η for EBF with beam section W12X152
Beam W12X152
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1430 243 5.436 9.483 42.708 0.212
L 371.00Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.155 0.196 0.223 -0.581
W27x194 0.181 0.196 0.254 -0.551
e 111.00 W24x162 0.276 0.196 0.349 -0.456
W21x83 0.781 0.196 0.651 -0.153
α 45.00 W18x76 1.075 0.196 0.748 -0.056
W16x50 2.169 0.196 0.935 0.131
L/h 2.85 W14x48 2.954 0.196 1.001 0.196
W12x40 4.657 0.196 1.077 0.272
e/L 0.30 W10x39 6.842 0.196 1.124 0.320
W8x31 13 0.196 1.176 0.372
β 2.60 W6x16 44.548 0.196 1.221 0.416
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Table 3.19 - Effect of varying η for EBF with beam section W14X22
Beam W14X22
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
199 33.2 5.115 2.996 18.463 0.461
L 308.00Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.021 0.353 0.037 -0.610
W27x194 0.025 0.353 0.043 -0.604
e 48.00 W24x162 0.038 0.353 0.064 -0.582
W21x83 0.108 0.353 0.165 -0.481
α 45.00 W18x76 0.149 0.353 0.217 -0.430
W16x50 0.301 0.353 0.371 -0.275
L/h 2.37 W14x48 0.411 0.353 0.456 -0.191
W12x40 0.648 0.353 0.593 -0.053
e/L 0.16 W10x39 0.952 0.353 0.712 0.065
W8x31 1.809 0.353 0.892 0.245
β 2.60 W6x16 6.199 0.353 1.113 0.467
Table 3.20 - Effect of varying η for EBF with beam section W14X145
Beam W14x145
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1710 260 6.088 8.581 50.495 0.200
L 391.20Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.185 0.196 0.258 -0.546
W27x194 0.217 0.196 0.292 -0.513
e 131.20 W24x162 0.330 0.196 0.395 -0.409
W21x83 0.934 0.196 0.706 -0.099
α 45.00 W18x76 1.285 0.196 0.800 -0.004
W16x50 2.594 0.196 0.974 0.170
L/h 3.01 W14x48 3.533 0.196 1.033 0.229
W12x40 5.570 0.196 1.100 0.296
e/L 0.34 W10x39 8.181 0.196 1.141 0.337
W8x31 15.545 0.196 1.186 0.382
β 2.60 W6x16 53.271 0.196 1.224 0.419
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Table 3.21 - Effect of varying η for EBF with beam section W16X26
Beam W16X26
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
301 44.2 5.755 3.752 19.631 0.488
L 311.00Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.033 0.378 0.055 -0.567
W27x194 0.038 0.378 0.064 -0.558
e 51.00 W24x162 0.058 0.378 0.094 -0.528
W21x83 0.164 0.378 0.234 -0.388
α 45.00 W18x76 0.226 0.378 0.301 -0.321
W16x50 0.457 0.378 0.487 -0.135
L/h 2.39 W14x48 0.622 0.378 0.580 -0.042
W12x40 0.980 0.378 0.720 0.098
e/L 0.16 W10x39 1.440 0.378 0.832 0.210
W8x31 2.736 0.378 0.985 0.363
β 2.60 W6x16 9.377 0.378 1.153 0.531
Table 3.22 - Effect of varying η for EBF with beam section W16X100
Beam W16X100
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1500 200 6.734 8.792 37.910 0.296
L 358.60Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.163 0.264 0.232 -0.504
W27x194 0.191 0.264 0.264 -0.473
e 98.60 W24x162 0.290 0.264 0.361 -0.375
W21x83 0.820 0.264 0.666 -0.071
α 45.00 W18x76 1.128 0.264 0.762 0.026
W16x50 2.276 0.264 0.946 0.210
L/h 2.76 W14x48 3.099 0.264 1.010 0.273
W12x40 4.886 0.264 1.083 0.347
e/L 0.27 W10x39 7.177 0.264 1.129 0.393
W8x31 13.636 0.264 1.179 0.443
β 2.60 W6x16 46.729 0.264 1.222 0.485
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Table 3.23 - Effect of varying η for EBF with beam section W18X35
Beam W18X35
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
510 66.5 6.456 5.055 21.925 0.490
L 317.00Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.055 0.387 0.090 -0.523
W27x194 0.065 0.387 0.104 -0.509
e 57.00 W24x162 0.099 0.387 0.152 -0.462
W21x83 0.279 0.387 0.351 -0.263
α 45.00 W18x76 0.383 0.387 0.436 -0.177
W16x50 0.774 0.387 0.648 0.035
L/h 2.44 W14x48 1.054 0.387 0.742 0.129
W12x40 1.661 0.387 0.870 0.256
e/L 0.18 W10x39 2.440 0.387 0.961 0.348
W8x31 4.636 0.387 1.076 0.463
β 2.60 W6x16 15.888 0.387 1.187 0.574
Table 3.24 - Effect of varying η for EBF with beam section W18X143
Beam W18x143
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
2750 322 7.648 12.307 43.603 0.292
L 373.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.298 0.271 0.368 -0.361
W27x194 0.349 0.271 0.410 -0.318
e 113.50 W24x162 0.531 0.271 0.532 -0.197
W21x83 1.502 0.271 0.843 0.114
α 45.00 W18x76 2.067 0.271 0.924 0.195
W16x50 4.172 0.271 1.060 0.331
L/h 2.87W14x48
5.681 0.271 1.103 0.374W12x40 8.957 0.271 1.149 0.420
e/L 0.30 W10x39 13.157 0.271 1.177 0.448
W8x31 25 0.271 1.206 0.477
β 2.60 W6x16 85.669 0.271 1.230 0.501
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Table 3.25 - Effect of varying η for EBF with beam section W21X144
Beam W21X44
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
843 95.4 7.338 6.93 22.943 0.533
L 319.60Brace η Фaxial Фbending
Ф=Фa+Фb-
1
h 130.00 W30x191 0.092 0.424 0.142 -0.434
W27x194 0.107 0.424 0.163 -0.413
e 59.60 W24x162 0.163 0.424 0.232 -0.344
W21x83 0.461 0.424 0.489 -0.087
α 45.00 W18x76 0.634 0.424 0.586 0.010
W16x50 1.279 0.424 0.799 0.222
L/h 2.46 W14x48 1.742 0.424 0.882 0.306
W12x40 2.746 0.424 0.986 0.410
e/L 0.19 W10x39 4.033 0.424 1.055 0.479
W8x31 7.664 0.424 1.135 0.559
β 2.60 W6x16 26.262 0.424 1.207 0.631
Table 3.26 - Effect of varying η for EBF with beam section W21X147
Beam W21x147
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
3630 373 8.634 14.256 43.607 0.33
L 373.50Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.394 0.306 0.444 -0.250
W27x194 0.461 0.306 0.490 -0.204
e 113.50 W24x162 0.702 0.306 0.618 -0.076
W21x83 1.983 0.306 0.914 0.220
α 45.00 W18x76 2.729 0.306 0.985 0.291
W16x50 5.508 0.306 1.099 0.405
L/h 2.87 W14x48 7.5 0.306 1.133 0.439W12x40 11.824 0.306 1.170 0.476
e/L 0.30 W10x39 17.368 0.306 1.191 0.497
W8x31 33 0.306 1.214 0.520
β 2.60 W6x16 113.084 0.306 1.232 0.538
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Table 3.27 - Effect of varying η for EBF with beam section W24X55
Beam W24X55
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
1360 135 8.282 8.923 25.215 0.547
L 325.60Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.148 0.443 0.214 -0.343
W27x194 0.173 0.443 0.244 -0.314
e 65.60 W24x162 0.263 0.443 0.336 -0.221
W21x83 0.743 0.443 0.635 0.078
α 45.00 W18x76 1.023 0.443 0.733 0.176
W16x50 2.064 0.443 0.924 0.366
L/h 2.50 W14x48 2.810 0.443 0.991 0.433
W12x40 4.430 0.443 1.069 0.512
e/L 0.20 W10x39 6.507 0.443 1.118 0.561
W8x31 12.364 0.443 1.173 0.616
β 2.60 W6x16 42.368 0.443 1.220 0.662
Table 3.28 - Effect of varying η for EBF with beam section W24X146
Beam W24X146
IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag
4580 418 9.720 14.638 47.593 0.340
L 383.80Brace η Фaxial Фbending Ф=Фa+Фb-1
h 130.00 W30x191 0.497 0.325 0.512 -0.163
W27x194 0.582 0.325 0.560 -0.115
e 123.80 W24x162 0.885 0.325 0.690 0.014
W21x83 2.502 0.325 0.967 0.291
α 45.00 W18x76 3.443 0.325 1.029 0.353
W16x50 6.949 0.325 1.125 0.450
L/h 2.95 W14x48 9.462 0.325 1.154 0.478W12x40 14.918 0.325 1.184 0.508
e/L 0.32 W10x39 21.913 0.325 1.201 0.526
W8x31 41.636 0.325 1.219 0.544
β 2.60 W6x16 142.679 0.325 1.234 0.558
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For each one of the above tables a diagram is presented in Figure 3.21. Note that β is 2.6
for all the cases.
Fig. 3.21 – Effect of changing η on the yield function for the beam in an EBF with a moment link
-0.800
-0.600
-0.400
-0.200
0.000
0.200
0.400
0.600
0.800
0 1 2 3 4 5 6
y i e l d f u n c t i o n ( Ф )
η
W10X12 (Aw/Ag=0.51 , Z/Ag=3.60)
W14X22 (Aw/Ag=0.46 , Z/Ag=5.12)
W18X35 (Aw/Ag=0.49 , Z/Ag=6.46)
W16X22 (Aw/Ag=0.49 , Z/Ag=5.76)
W12X14 (Aw/Ag=0.55 , Z/Ag=4.18)
W21X44 (Aw/Ag=0.53 , Z/Ag=7.34)
W24X55 (Aw/Ag=0.55 , Z/Ag=8.28)
W21X147 (Aw/Ag=0.33 , Z/Ag=8.63)
W18X143(Aw/Ag=0.29 , Z/Ag=7.65)
W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W24X146 (Aw/Ag=0.34 , Z/Ag=9.72)
W14X145 (Aw/Ag=0.20 , Z/Ag=6.09)
W12X152 (Aw/Ag=0.21 , Z/Ag=5.44)
W10X112 (Aw/Ag=0.20 , Z/Ag=4.47)
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Fig. 3.22 – Variation of axial force yield function for an EBF with a moment link
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0 1 2 3 4 5 6
Ф
a x i a l
η
W10X12 (Aw/Ag=0.51 , Z/Ag=3.60)
W12X14 (Aw/Ag=0.55 , Z/Ag=4.18)
W14X22 (Aw/Ag=0.46 , Z/Ag=5.12)
W16X22 (Aw/Ag=0.49 , Z/Ag=5.76)
W18X35 (Aw/Ag=0.49 , Z/Ag=6.46)
W21X44 (Aw/Ag=0.53 , Z/Ag=7.34)
W24X55 (Aw/Ag=0.55 , Z/Ag=8.28)
W24X146 (Aw/Ag=0.34 , Z/Ag=9.72)W21X147 (Aw/Ag=0.33 , Z/Ag=8.63)
W18X143(Aw/Ag=0.29 , Z/Ag=7.65) W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W14X145 (Aw/Ag=0.20 , Z/Ag=6.09)
W16X100 (Aw/Ag=0.30 , Z/Ag=6.73)
W10X112 (Aw/Ag=0.20 , Z/Ag=4.47)
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It should be noticed that some of the beams have the axial yield function less than 0.2,
which violates the initial assumption in using Equation 3.8. However, the value for the
yield function is still close to 0.2 for these sections.
Fig. 3.23 - Variation of bending yield function for an EBF with a moment link
0.000
0.200
0.400
0.600
0.800
1.000
1.200
0 1 2 3 4 5 6
Ф
b e n d i n g
η
W10x12
W10x112
W12x14
W12x152
W14x22
W14x145
W16x26
W16x100
W18x35
W18x143
W21x44
W21x147
W24x55
W24x146
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3.5.3. Conclusions on the effect of η:
Based on Figures 3.18 for shear links and Figure 3.21 for moment links, as the value of η
is reduced, the value of the yield function is reduced. For each specific beam, using a
stiffer brace results in a lower value for η. Specifically since the value of η is usually
between 1 and 2 and the portion of the diagrams in Figures 3.18 and 3.21 related to these
values is relatively steep, the effect of changing η can be significant. Changing η from 1
to 2 can affect the beam yield function around 0.1 for EBFs with shear links and around
0.2 for EBFs with moment links. As is discussed in next section, for a constant value of
η, beams with a higher value of Aw/Ag will have a higher yield function. This can also be
seen in Figures 3.18 and 3.21.
Figures 3.19 and 3.22 show that no matter what section is used for the brace, the value of
axial force in the beam does not change. Figures 3.20 and 3.23 show that η has a large
influence on the bending yield function. This is expected, since changing the brace
stiffness relative to the beam has a significant effect on the fraction of the link end
moment transferred to the beam and the brace. These figures also show that changing the
beam section, while keeping η constant, has almost no effect on the bending yield
function.
The analysis above also shows that EBFs with moment links, compared to EBFs with
shear links, have higher bending yield functions but similar axial yield functions.
Consequently, EBFs with moment links generally have higher total beam yield functions.
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3.5.4. Effect of Aw/Ag on EBFs with shear links
In this section, the effect of changing Aw/Ag on the beam yield functions for EBFs with
shear links are considered. As mentioned before, it is impossible to change Aw/Ag and
keep Z/Ag constant. However, if the depth of the section remains constant while the
weight of the section is changing, the value of Z/Ag will only change slightly while the
value of Aw/Ag changes significantly.
The other variables can be kept constant, thereby
allowing an evaluation of the effect of Aw/Ag on the beam yield function. Figure 3.24
qualitatively shows the changes made to the EBF configuration to change the value of
Aw/Ag.
Fig. 3.24 – Changing Aw /Ag and keeping the other variables constant
Table 3.29 shows the values of variables defining the configuration of the EBF. As can
be seen from the table, all of the sections chosen for the beams have approximately the
same depth so that the value of Z/Ag remains relatively constant.
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Table 3.29 – Effect of varying Aw /Ag for EBF with shear link
H
Beam &Brace Aw/Ag Z / Ag (Z / Ag)avg Mp / Vp130.00
α W12x87 0.218 5.156
4.83
39.263
45.00 W12x72 0.223 5.118 38.193
W12x58 0.231 5.082 36.630
β W12x50 0.276 4.924 29.658
1.30 W12x40 0.274 4.871 29.625
W12x26 0.343 4.862 23.563
η W12x19 0.485 4.434 15.232
1 W12x14 0.550 4.182 12.663
Beam &
Bracee L e/L Фaxial Фbending Ф=Фa+Фb-1
W12x87 51.04 311.05 0.164 0.218 0.472 -0.310
W12x72 49.65 309.66 0.160 0.221 0.472 -0.306
W12x58 47.62 307.63 0.154 0.228 0.472 -0.300
W12x50 38.56 298.57 0.129 0.266 0.472 -0.262
W12x40 38.51 298.53 0.129 0.264 0.472 -0.264
W12x26 30.63 290.64 0.105 0.323 0.472 -0.205
W12x19 19.80 279.81 0.070 0.441 0.472 -0.086
W12x14 16.46 276.47 0.059 0.496 0.472 -0.032
Figures 3.25, 3.26 and 3.27 plot the value of the yield function, the axial yield function
and the bending yield function for different values of Aw/Ag based the results in Table
3.29.
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Fig. 3.25 – Effect of changing Aw /Ag on the yield function in the beam for an EBF with a shear
link
Fig. 3.26 – Effect of changing Aw /Ag on the axial yield function in the beam for an EBF with a
shear link
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
-0.050
0.000
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
y i e l d f u n c t i o n ( Ф
)
Aw / Ag
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Ф
a x i a l
Aw / Ag
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Fig. 3.27 – Effect of changing Aw /Ag on the bending yield function for an EBF with a shear link
Some observations are made on effect of Aw/Ag on the yield functions in EBFs with shear
links after the next section related to the EBFs with moment links.
3.5.5. Effect of Aw /Ag on EBFs with moment links
Table 3.30 shows the values of the variables used for analysis of an EBF with a moment
link. Also listed are the yield functions computed for various values of Aw/Ag.
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Ф
b e n d i n g
Aw / Ag
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Table 3.30 - Effect of varying Aw /Ag for EBF with moment link
hBeam &
BraceAw/Ag Z / Ag (Z / Ag)avg Mp / Vp130.00
α W12x87 0.218 5.156
4.83
39.263
45.00 W12x72 0.223 5.118 38.193
W12x58 0.231 5.082 36.630
β W12x50 0.27 4.924 29.658
2.60 W12x40 0.274 4.871 29.625
W12x26 0.343 4.862 23.563
η W12x19 0.485 4.434 15.232
1 W12x14 0.550 4.182 12.663
Beam &
Bracee L e/L Фaxial Фbending Ф=Фa+Фb-1
W12x87 102.084 362.096 0.281 0.193 0.726 -0.080
W12x72 99.304 359.316 0.276 0.144 0.726 -0.129
W12x58 95.238 355.250 0.268 0.149 0.726 -0.124
W12x50 77.112 337.124 0.228 0.178 0.726 -0.094
W12x40 77.027 337.039 0.228 0.177 0.726 -0.096
W12x26 61.264 321.276 0.190 0.222 0.726 -0.051
W12x19 39.605 299.617 0.132 0.313 0.726 0.039
W12x14 32.925 292.937 0.112 0.355 0.726 0.082
Figures 3.28, 3.29 and 3.30 plot the value of yield function, the axial yield function and
the bending yield function for different values of Aw/Ag based the result in Table 3.30.
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Fig. 3.28 – Effect of changing Aw /Ag on the yield function in the beam for an EBF with a moment
link
Fig. 3.29 – Effect of changing Aw /Ag on the axial yield function for an EBF with a moment link
-0.100
-0.050
0.000
0.050
0.100
0.150
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
y i e l d f u n c t i o n ( Ф )
Aw / Ag
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Ф
a x i a l
Aw / Ag
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Fig. 3.30 – Effect of changing Aw /Ag on the f bending function for an EBF with a moment link
3.5.6. Conclusions on the effect of Aw /Ag
For EBFs with either shear links or moment links, increasing the value of A w/Ag results
in a higher value of the yield function. As noted earlier, for W-Shapes in the AISC
Manual, the value of Aw/Ag is within the range of 0.2 to 0.5. As Figures 3.25 and 3.28
show, changing Aw/Ag within this range can change the value of yield function around
0.3 for EBFs with shear links and around 0.15 for EBFs with moment links. It is also
clear from this analysis that Aw/Ag only affects the axial yield function and has no effect
on the bending yield function. Overall, however, it is clear from this analysis that it is
advantageous to use beam and link sections with low values of Aw/Ag.
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
Ф
b e n d i n g
Aw / Ag
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3.5.7. Effect of Z/Ag on EBFs with shear links
In this section, the effect of varying Z/Ag of the beam section on the beam yield function
is examined. As described earlier, since the value of Aw/Ag is directly related to the beam
section properties, changing the value of Z/Ag
also results in changing Aw/Ag. However,
the purpose of this analysis is to evaluate the effect of Z/Ag independently from the other
variables. The value of Aw/Ag is related with the weight of the section and for AISC W-
Shapes when the weight of the section increases the value of Aw/Ag for the section also
increases. Light sections for a given depth have Aw/Ag near the high value of 0.5 and
heavy sections have an Aw/Ag around the low value of 0.2. On the other hand, the value
of Z/Ag is more dependent on depth of the section and therefore sections with different
depths and about the same weight would have almost the same value of Aw/Ag and
different value of Z/Ag.
Figure 3.31 show qualitatively the manner in which the EBF configuration is changed
that all the variables are kept constant and only Z/Ag changes. Table 3.31 lists the values
of the variables used for this analysis and the resulting values of the yield functions.
Figures 3.25, 3.26 and 3.27 plot the values of the yield function, the axial yield function
and the bending yield function for different values of Z/Ag based on results in Table 3.31.
Fig. 3.31 – Changing Z/Ag and keeping the other variables constant
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Table 3.31 - Effect of varying Z/Ag for EBF with shear link
hBeam &
BraceAw/Ag (Aw/Ag)avg Z / Ag Mp / Vp130.00
α 45.00 W24x103 0.409
0.42
9.240 37.643
β W21x73 0.417 8 31.949
1.30 W18x71 0.401 7.019 29.122
η W16x36 0.418 6.037 24.041
1 W14x26 0.433 5.227 20.118
W12x22 0.459 4.521 16.403
W10x19 0.418 3.843 15.302
Beam &
Bracee L e/L Фaxial Фbending Ф=Фa+Фb-1
W24x103 48.94 308.95 0.158 0.420 0.472 -0.108
W21x73 41.53 301.55 0.137 0.411 0.472 -0.117
W18x71 37.86 297.87 0.127 0.404 0.472 -0.124
W16x36 31.25 291.27 0.107 0.397 0.472 -0.131
W14x26 26.15 286.17 0.091 0.392 0.472 -0.136
W12x22 21.32 281.34 0.075 0.387 0.472 -0.141
W10x19 19.89 279.91 0.071 0.382 0.472 -0.146
Fig. 3.32 – Effect of changing Z/Ag on the yield function in the beam for an EBF with a shear link
-0.160
-0.140
-0.120
-0.100
-0.080
-0.060
-0.040
2 3 4 5 6 7 8 9 10
y i e l d f u n c t i o n ( Ф )
Z / Ag
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Fig. 3.33 – Effect of changing Z/Ag on the f axial yield function for an EBF with a shear link
Fig. 3.34 – Effect of changing Z/Ag on the bending yield function for an EBF with a shear link
0.375
0.380
0.385
0.390
0.395
0.400
0.405
0.410
0.415
0.420
0.425
2 3 4 5 6 7 8 9 10
Ф
a x i a l
Z / Ag
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
2 3 4 5 6 7 8 9 10
Ф
b e n d i n g
Z/ Ag
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Observations regarding the effects of changing Z/Ag on the yield functions for the beam
in EBFs with shear links are made after the next section related to the EBFs with
moment link.
3.5.8. Effect of Z/Ag on EBFs with moment links
Table 3.32 lists the values of the variables used for analysis of EBFs with moment links.
Figures 3.35, 3.36 and 3.37 plot the values of the yield function, the axial yield function
and the bending yield function for different values of Z/Ag based the results in Table
3.32.
Table 3.32 - Effect of varying Z/Ag for EBF with moment link
hBeam &
BraceAw/Ag (Aw/Ag)avg Z / Ag Mp / Vp130.00
α
45.00 W24x103 0.409
0.42
9.240 37.64352
β W21x73 0.417 8 31.94912
2.60 W18x71 0.401 7.019 29.12219
η W16x36 0.418 6.037 24.04135
1 W14x26 0.433 5.227 20.11831
W12x22 0.459 4.521 16.40354
W10x19 0.418 3.843 15.30287
Beam &Brace
e L e/L Фaxial Фbending Ф=Фa+Фb-1
W24x103 97.873 357.885 0.273 0.372 0.726 0.098
W21x73 83.067 343.079 0.242 0.359 0.726 0.085
W18x71 75.717 335.729 0.225 0.348 0.726 0.075
W16x36 62.507 322.519 0.193 0.338 0.726 0.064W14x26 52.307 312.319 0.167 0.329 0.726 0.055
W12x22 42.649 302.661 0.140 0.321 0.726 0.048
W10x19 39.787 299.799 0.132 0.314 0.726 0.040
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Fig. 3.35 – Effect of changing Z/Ag on the yield function in the beam for an EBF with a moment
link
Fig. 3.36 – Effect of changing Z/Ag on the axial yield function for an EBF with a moment link
0.000
0.020
0.040
0.060
0.080
0.100
0.120
2 3 4 5 6 7 8 9 10
y i e l d f u n c t i o n ( Ф )
Z / Ag
0.310
0.320
0.330
0.340
0.350
0.360
0.370
0.380
2 3 4 5 6 7 8 9 10
Ф
a x i a l
Z / Ag
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Fig. 3.37 – Effect of changing Z/Ag on the bending yield function for an EBF with a moment link
3.5.9. Conclusions on the effect of Z/Ag
For EBFs with both shear links and moment links, increasing the value of Z/Ag results in
a higher value of the yield function. For more common W-Shapes in the AISC Manual,
the value of Z/Ag is within the range of 3 to 10. Changing Z/Ag in this range will change
the yield function around 0.05 in EBFs with shear links and around 0.07 in EBFs with
moment links. As with Aw/Ag, varying the value of Z/Ag only changes the axial yield
function and has no effect on the bending yield function.
3.5.10. Evaluating the effect of β
Links in EBFs with a value of β less than 2 are classified as shear links in this thesis.
Similarly, links with a value of β greater than 2 are classified as moment links. Figure
3.38 qualitatively illustrates the manner in which the EBF configuration is changed so
that only the value of β changes and the other variables remain constant. Table 3.32
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
2 3 4 5 6 7 8 9 10
Ф
b e n d i n g
Z/ Ag
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shows the value of the variables used for analysis of the effect of changing β and the
values of the resulting yield functions.
Fig. 3.38 – Changing β and keeping the other variables constant
The yielding functions of the beam for an EBF with a shear link were computed using
Equations 3.15 and 3.16. Similarly, the yield functions of the beam for an EBF with a
moment link have been computed using Equations 3.17 and 3.18.
Table 3.33 – Effect of varying β
Beam & Brace W14x82
β e L e/L Фaxial Фbending Ф=Фa+Фb-1Z / Ag 5.7917
Aw/Ag 0.2675
S h e a r L i nk
0.0 0.00 260.01 0 0.225 0.000 -0.775
0.4 14.43 274.44 0.053 0.237 0.145 -0.618
Mp / Vp 36.08 0.8 28.86 288.88 0.1 0.250 0.291 -0.460
1.2 43.30 303.31 0.143 0.262 0.436 -0.302
α 1.6 57.73 317.74 0.182 0.275 0.581 -0.144
45.00 2 72.16 332.17 0.217 0.287 0.726 0.013
M o
m e n t L i nk
2 72.16 332.17 0.217 0.287 0.726 0.013
η 2.4 86.59 346.60 0.25 0.250 0.726 -0.024
1 2.8 101.02 361.04 0.28 0.223 0.726 -0.051
3.2 115.46 375.47 0.307 0.203 0.726 -0.071
h 3.6 129.89 389.90 0.333 0.187 0.726 -0.086
130.00 4 144.32 404.33 0.357 0.175 0.726 -0.099
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Figures 3.39, 3.40 and 3.41 plot the variation of the yield function, the axial yield
function and the bending yield function based on the results in Table 3.33.
Fig. 3.39 – Effect of changing β on the yield in the beam
Fig. 3.40 – Effect of changing β on the axial yield function in the beam
-0.900
-0.800
-0.700
-0.600
-0.500
-0.400
-0.300
-0.200
-0.100
0.000
0.100
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
y i e l d f u n c t i o n ( Ф )
β
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Ф
a x i a l
β
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Fig. 3.41 – Effect of changing β on the bending yield function in the beam
3.5.11. Conclusions on the effect of β
As discussed in Chapter 1, shear links provide the best overall stiffness, strength and
ductility for an EBF. Therefore, the use of EBFs with non-dimensional links lengths of β
less than 2, in which the link behavior is controlled by shear yield is advantageous. For
the EBFs with shear y links, decreasing the link length, which results in a lower value of
β, decreases the value of yielding function linearly. However, due to link rotation limits,
values of β less than 1 are generally not possible. As Figure 3.39 shows, for the range of
β within 1 to 2 the value of the yield function changes by about 0.4. This significant
variation is more due to the change in bending yield function than the axial yield
function.
For the EBFs with moment links, which have β greater than 2, increasing the link length
decreases the axial yield function. Since the moment yield function does not change with
β, the total yield function decreases. As Figure 3.39 shows for EBFs with moment links,
changing β does not have a large effect on the yield function.
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Ф
b e n d i n g
β
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3.5.12. Effect of α for an EBF with shear links
In this section, the effect of changing the angle between the beam and the brace, α, on the
beam yield function is considered. Figure 3.42 qualitatively shows the manner in which
the EBF configuration is changed so that α is varied but other variables do not change.
Fig. 3.42 – Changing α and keeping the other variables constant
Table 3.34 shows the value of variables used for the analysis of the effects of varying α.
As the value of β in this table shows, all these configurations for this analysis are for
EBFs with shear links. Figures 3.43, 3.44 and 3.45 plot the variation of the yield function,
the axial yield function and the bending yield function based on the results in Table 3.34.
Table 3.34 – Effect of varying α for EBF with shear link
S h or t L i nk
Beam &
Brace α L e/L Фaxial Фbending Ф=Фa+Фb-1h
W14x82 130.00
25.00 604.49 0.078 0.522 0.423 -0.055
Aw/Ag e 30.00 497.25 0.094 0.430 0.432 -0.138
0.268 46.90 35.00 418.24 0.112 0.362 0.443 -0.195
40.00 356.77 0.131 0.308 0.456 -0.235
Z / Ag β 45.00 306.92 0.153 0.265 0.472 -0.263
5.792 1.30 50.00 265.08 0.177 0.229 0.491 -0.280
55.00 228.97 0.205 0.198 0.512 -0.290
Mp / Vp η 60.00 197.03 0.238 0.170 0.537 -0.292
36.080 1 65.00 168.15 0.279 0.145 0.567 -0.288
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100
Fig. 3.43 – Effect of changing α on the yield function in the beam for an EBF with a shear link
Fig. 3.44 – Effect of changing α on variation of the axial yield function for an EBF with a shear
link
-0.350
-0.300
-0.250
-0.200
-0.150
-0.100
-0.050
0.000
10.00 20.00 30.00 40.00 50.00 60.00 70.00
y i e l d f u n c t i o n ( Ф )
α
0.000
0.100
0.200
0.300
0.400
0.500
0.600
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Ф
a x i a l
α
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Fig. 3.45 – Effect of changing α on variation of the bending yield function for an EBF with a
shear link
3.5.13. Effect of α on an EBF with moment links
Table 3.35 shows the value of variables used to evaluate variations in α on the beam yield
functions for EBFs with moment links. The value of β is greater than 2 and therefore all
these configurations are EBFs with moment links.
Table 3.35 – Effect of varying α for EBF with moment link
L on gL i nk
Beam &
Brace α L e/L Фaxial Фbending Ф=Фa+Фb-1h
W14x82 130.00
25.00 651.40 0.144 0.433 0.650 0.084
Aw/Ag e 30.00 544.16 0.086 0.362 0.665 0.026
0.268 93.81 35.00 465.14 0.101 0.309 0.682 -0.009
40.00 403.68 0.116 0.268 0.702 -0.029
Z / Ag β 45.00 353.82 0.133 0.235 0.726 -0.038
5.792 2.60 50.00 311.99 0.15 0.207 0.755 -0.03855.00 275.87 0.17 0.183 0.788 -0.029
Mp / Vp η 60.00 243.93 0.192 0.162 0.827 -0.011
36.080 1 65.00 215.06 0.218 0.143 0.872 0.015
0.300
0.350
0.400
0.450
0.500
0.550
0.600
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Ф
b e n d i n g
α
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Figures 3.46, 3.47 and 3.48 plot the variation of the yield function, the axial yield
function and the bending yield function for an EBF with a moment link based on the
result if Table 3.35.
Fig. 3.46 – Effect of changing α on the yield function in the beam for an EBF with a moment link
Fig. 3.47 – Effect of changing α on the variation of the axial yield function for an EBF with a
moment link
-0.060
-0.040
-0.020
0.000
0.020
0.040
0.060
0.080
0.100
10.00 20.00 30.00 40.00 50.00 60.00 70.00
y i e l d f u n c t i o n ( Ф )
α
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Ф
a x i a l
α
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Fig. 3.48 – Effect of changing α on the variation of the bending yield function for an EBF with a
moment link
3.5.14. Conclusions on the effect of α
The typical range for the angle between the beam and the brace in an EBF is 30 to 60
degrees. Within this range for the EBFs with shear links, increasing α results in a lower
yield function. Although the bending yield function increases with increasing α, the
larger rate of decrease in the axial yield function dominates, and so the overall yield
function decreases with increasing values of α. As Figure 3.43, shows changing α within
its typical range can change the yielding function around 0.15. For the EBFs with
moment links there is an optimum value at about 45 degree as Figure 3.46 shows.
Changing α would changes the yield function around 0.2 in this case.
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
10.00 20.00 30.00 40.00 50.00 60.00 70.00
Ф
b e n d i n g
α
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CHAPTER 4
STABILITY OF THE BEAM OUTSIDE THE LINK UNDER
COMBINED AXIAL FORCE AND BENDING
In Chapter 3, EBF configurations that can lead to yielding of the beam outside of the link
were investigated. The strength of the beam under combined axial force and moment was
defined by the development of a fully yielded cross-section. The development of a fully
yielded cross-section was assessed using the P-M interaction equations in Chapter H of
the 2005 AISC Specification. Accordingly, in these equations, the nominal axial strength
of the beam was taken as Py and the nominal flexural strength was taken as M p. No
resistance factors were included in the analysis.
In an actual EBF, the strength of the beam under combined axial force and moment will
often be less than that corresponding to a fully yielded cross-section due to instability in
the beam. That is, the strength of the beam will often be controlled by stability limit states
rather than the yield limit state. Very high levels of cyclic axial force, both tension and
compression are developed in the beam of an EBF under earthquake loading. At the same
time, the beam is subjected to very high cyclic bending moments. The combination of
large axial force and moment can cause instability of the beam. This can include local
buckling of the flange and web, and overall buckling between points of lateral support for
the beam. The critical case will normally occur when the beam axial force is
compressive. When bending moment is added for this case, one flange will be
particularly critical, where the compression due to bending adds to the compression due
to axial force. For typical EBF configurations where the brace is attached to the bottom
flange of the beam, compression due to bending and compression due to axial force are
additive in the top flange. Consequently, the top flange will normally be most critical for
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stability. However, depending on the relative magnitudes of the axial force and bending
moment, the bottom flange may also have high levels of compression stress.
In this chapter, a limited study is conducted to evaluate issues related to capacity design
of the beam outside of the link, as controlled by stability limit states. The 2005 AISC
Seismic Provisions requires that either a compact or seismically compact section be used
for the link and beam of an EBF. As a result, it is assumed that local buckling of the
flange or web will not control the strength of the beam. Consequently, the controlling
stability limit state will be overall member buckling between points of lateral support.
Since the beam resists both axial force and bending moment, the instability will normally
be in the form of lateral torsional buckling.
The strength of the beam based on member buckling is assessed using the P-M
interaction equations in Chapter H of the 2005 AISC Specification. The following section
reviews the calculation of the nominal axial strength and nominal flexural strength based
on stability limit states in the AISC Specification. As was the case in Chapter 3, no
resistance factors is included in the analysis. The required axial and flexural strength of
the beam is computed using the approximate equations developed in Chapter 2. Note that
in applying the P-M interaction equation, it is assumed that 2nd
order effects have a
negligible effect on the moment and axial force in the beam.
As with the analysis presented in Chapters 2 and 3, the objective of the analysis in this
chapter is to identify general trends that affect capacity design of beams in EBFs, to help
guide preliminary design. As such, a number of simplifications are used in the analysis.
4.1. AISC EQUATIONS FOR NOMINAL AXIAL AND FLEXURAL STRENGTH FOR STABILITY
The P-M interaction equation in Chapter H of the AISC Specification requires calculation
of the nominal axial compressive strength, Pn, and the nominal flexural strength, Mn, of a
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member. The nominal axial compressive strength of a member is defined in Section E3
and the nominal flexural strength is defined in Section F2 of the AISC Specification. Key
equations are summarized in the following sections.
4.1.1. Nominal compressive strength
The nominal compressive strength (Pn) of members without slender elements can be
calculated by the following equation:
gcr n AFP = (Eq. 4-1) [Eq. E3-1]
In this equation, the flexural buckling stress is calculated by the following equations.
WhenyF
E4.71
r
KL≤ : y
F
F
cr F0.658F e
y
= (Eq. 4-2) [Eq. E3-2]
WhenyF
E4.71
r
KL≥ : ecr
F877.0F = (Eq. 4-3) [Eq. E3-3]
Detailed definitions of all terms are provided in the AISC Specification.
4.1.2. Flexural strength
The nominal flexural strength ( M n) of doubly symmetric compact I-shape members about
their major axis can be calculated by taking the smallest value from the Equations 4-4, 4-
5 and 4-6.
4.1.2.1. Limit state of yielding
xy pn ZFMM == (Eq. 4-4) [Eq. F2-1]
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4.1.2.2. Limit state of lateral-torsional buckling
When p b LL ≤ : the limit state of lateral-torsional buckling does not apply
When r b p LLL ≤≤ :
−−−−= )
LLLL)(S0.7F(MMCM
pr
p bxy p p bn
(Eq. 4-5) [Eq. F2-2]
When r b LL > : xcr n SFM = (Eq. 4-6) [Eq. F2-3]
In these equations Lb is length between the points of lateral bracing, which are defined as
points braced against lateral displacement of the compression flange or points braced
against twist of the cross section. The value of L p and Lr can be calculated by Equations
F2-5 and F2-6 in AISC Specification. Also F cr can be calculated by Equation F2-4 in thespecification.
4.2. BEAM STRENGTH BASED ON SATBILITY
A critical design parameter controlling the buckling strength of a member subjected to
axial force and bending is the distance between lateral braces. In the following sections,
the effect of unbraced length on the axial compressive strength and on the flexural
strength of the beam outside of the link is evaluated independently. Afterward, for
selected specific beam sections and for different lateral bracing conditions, the value of
compressive strength and flexural strength of the beam are combined using the AISC P-
M interaction equation.
4.2.1. Basis for Beam Stability Analysis
In a typical EBF, the length of the beam from the end of the link to the column will
normally be in the range of about 12 to 15 feet (144 to 180 inches). For the purposes of
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this analysis, the length of the beam is conservatively taken as 200 inches. The AISC
Seismic Provisions require top and bottom flange lateral bracing at the end of the link.
Thus, it is assumed that, as a minimum, top and bottom flange lateral bracing is provided
at the beam ends, i.e. at the link and at the column. Consequently, the largest unbraced
length possible for the beam in the analysis below is 200 inches.
Between the ends of the beam, additional bracing is normally provided at the top flange
through connection to the floor slab. Thus, the top flange is normally continuously
braced. Additional bracing for the bottom flange may be provided at discrete locations,
normally through connection to floor beams or other bracing members. In some cases, no
additional bottom flange bracing is provided between the ends of the beam.
Assessing the unbraced length of the beam for the purposes of calculating P n and Mn can
sometimes be difficult due to the combination of continuous top flange bracing and
discrete bottom flange bracing. Further, under cyclic loading, it is possible to develop
compression in both top and bottom flanges. For the purposes of this analysis, several
simplifying assumptions will be made with regard to unbraced length. When computing
Pn for strong axis buckling, the effective unbraced length will be taken equal to the full
length of the beam, i.e. 200 inches. Consequently, it is assumed that the floor slab
provides no restraint to strong axis buckling. When computing Pn for weak axis buckling,
the unbraced length will be taken as the distance between bottom flange braces. Thus, the
beneficial effect of the top flange continuous bracing provided by the floor slab for
restraint of weak axis buckling is neglected. Finally, for lateral torsional buckling, the
unbraced length is also taken as the distance between bottom flange braces, again
neglecting the beneficial effect of the continuous top flange bracing. These assumptions
on unbraced length will generally be conservative and are also consistent with typical
design practice for EBFs.
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For the analysis of beam stability below, the W-Shape sections listed in Table 4.1 are
considered. All sections are assumed to be of A992 steel with a specified yield stress of
50 ksi. As shown in this table, sections ranging from W10 to W24 are included in the
analysis. For each depth, two sections are analyzed. One is a relatively heavy shape with
a low value of Aw/Ag. The other is a relatively light shape with a high value of Aw/Ag. As
discussed in Chapter 3, the Aw/Ag ratio of the beam and link has a large influence on the
capacity design strength demands on the beam. Recall that sections with lower values of
Aw/Ag were highly advantageous for reducing beam strength demands.
Table 4.1 – Properties of Sections for EBF Beam Stability Analysis
Section
W24 x 146 0.34 0.52 15.5
W 24 x 55 0.55 0.30 28.5
W 21 x 147 0.33 0.57 16.0
W 21 x 44 0.50 0.31 30.8
W18 x 143 0.29 0.57 17.9
W18 x 35 0.49 0.34 33.3
W16 x 100 0.30 0.61 19.2
W16 x 26 0.49 0.35 36.4
W14 x 145 0.20 1.05 12.9
W14 x 22 0.46 0.36 40
W12 x 136 0.21 0.93 16.1
W12 x 14 0.55 0.33 50.4
W10 x 112 0.20 0.91 19.2
W10 x 12 0.51 0.40 50.5
Also listed in Table 4.1 are two additional parameters that are often used to assess
resistance to out-of-plane buckling. The first is the ratio of flange width to depth of the
section, bf /d. In general, the higher the bf /d ratio, the greater is the resistance to lateral
torsional buckling and out-of-plane flexural buckling. The second parameter is the ratio
of unbraced length to flange width, L/bf . In general, lower values for this ratio are an
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indication of greater buckling resistance. For purposes of comparing the various sections
in Table 4.1, the L/bf ratio has been computed using an unbraced length of 200 inches.
4.2.2. Axial compressive strength of the beam based on stability
The axial compressive strength of the beam must be evaluated for flexural buckling about
both the weak axis and strong axis of the member. As described above, the unbraced
length of the beam for strong axis buckling is taken as 200 inches. For weak axis
buckling, the unbraced length is considered to vary from zero to 200 inches. The effective
unbraced length for weak axis buckling is denoted by the symbol Ly. By using Equation
4-1, the value of axial compressive strength was calculated for both the weak axis and
strong axis. In both cases, the effective length factor, K, was taken as 1.0. The smaller ofthe strong and weak axis strength values was then taken as the axial compressive
strength.
The axial compressive strength of the W-Shapes listed in Table 4.1 is plotted in Figures
4.1 to 4.7. For each section, the axial compressive strength is plotted against the weak
axis unbraced length Ly. In these plots, strong axis buckling controls in the regions of the
plot where the axial compressive strength remains constant with respect to Ly. Once the
strength begins to decrease with Ly, weak axis buckling controls.
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Fig. 4.1 – Axial compressive strength of W24 sections with L x =200 inches and variable L y
Fig. 4.2 – Axial compressive strength W21 sections with L x =200 inches and variable L y
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly (in)
W24x146
W24x55
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W21x147
W21x44
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Fig. 4.3 – Axial compressive strength of W18 sections with L x =200 inches and variable L y
Fig. 4.4 – Axial compressive strength of W16 sections with L x =200 inches and variable L y
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W18x143
W18x35
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W16x100
W16x26
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Fig. 4.5 – Axial compressive strength of W14 sections with L x =200 inches and variable L y
Fig. 4.6 – Axial compressive strength of W12 sections with L x =200 inches and variable L y
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W14x145
W14x22
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W12x136
W12x14
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Fig. 4.7 – Axial compressive strength of W10 sections with L x =200 inches and variable L y
For each beam section shown in Figures 4.1 to 4.7, the values of the axial compressive
strength can be normalized to the compressive yielding strength the beam, Py. This
permits an assessment of the effect of instability on the strength of the member. These
results are shown in Figure 4.8.
In Figure 4.8, the beam sections are qualitatively divided into two groups. One group is
the lighter beams of each depth. These beams lose their compressive capacity rapidly as
the weak axis unbraced length increases. As this figure shows, if weak axis lateral braces
are provided only at the ends of the beam, the weak axis unbraced length Ly is 200
inches, and the strength of the beam is less than 20-percent of P y. For these sections, if
weak axis lateral bracing is provided at fairly close intervals, say 40 to 60 inches, then anaxial compressive strength on the order of 70 to 80-percent of Py can be achieved. Based
on Table 4.1, these lighter beam sections generally have higher values for Aw/Ag, ranging
from about 0.35 to 0.45. They also have lower values of bf /d, ranging from about 0.3 to
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100 120 140 160 180 200
P n ( k i p s )
Ly(in)
W10x112
W10x12
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0.4. For an unbraced length of 200 inches, these sections also have relatively high values
ranging from about 30 to 50.
The other group of the beams in Figure 4.8 is the heavier sections for each depth. These
beams, even without any weak axis lateral bracing between their two ends, can sustain
60% to 80% of their yield capacity. If weak axis lateral bracing is provided, for example,
at say 8-ft intervals, these beams provide more than 90% of their yield capacity. For these
sections, the ratio of Aw/Ag ranges from about 0.2 to 0.35, the ratio bf /d ranges from
about 0.5 to 1, and the ratio of L/bf for an unbraced length of 200 inches ranges from 15
to 20.
4.2.3. Flexural strength of the beam based on stability
For each specific W-Shape considered in the previous section, flexural strength was
computed as a function of the unbraced length L b for lateral torsional buckling. The
flexural strength was taken as the minimum value from Equations 4-4, 4-5 and 4-6. The
expression to calculate the moment gradient factor C b is given by Equation F1-1 in the
2005 AISC specification. This factor is calculated for the unbraced lengths of the member
under consideration. For the beam in an EBF, the moment at the end of the beam attached
ot the column is normally close to zero. For this case, if the beam outside the link is not
laterally supported between its two ends then the value of C b will be equal to 1.67. As the
number of lateral supports gets larger between the two ends of the beam the value of C b
for the span closest to the link gets closer to 1. Here, conservatively the value of C b is
considered to be equal to 1 for all the cases considered.
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Fig. 4.8 – Normalized beam compressive strength
Figures 4.9 to 4.16 show the results of these calculations. As before, each figure shows
the flexural strength of a relatively light and relatively heavy section of the same nominaldepth. Figure 4.17 shows the beam flexural strength normalized by the plastic moment
capacity.
0
10
20
30
40
50
60
70
80
90
100
110
0 20 40 60 80 100 120 140 160 180 200
P n
/ P y ( % )
Ly (in)
W12x14
W10x12
W14x22
W16x26
W18x35
W21x44
W24x55
W16x100
W18x183
W10x112
W12x136
W14x145
W21x147
W24x146
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Fig. 4.9 –Flexural strength of W24 sections as a function of unbraced length
Fig. 4.10 –Flexural strength of W21 sections as a function of unbraced length
0
5000
10000
15000
20000
25000
30000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W24x146
W24x55
0
5000
10000
15000
20000
25000
30000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W21x147
W21x44
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Fig. 4.11 –Flexural strength of W18 sections as a function of unbraced length
Fig. 4.12 –Flexural strength of W16 sections as a function of unbraced length
0
5000
10000
15000
20000
25000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W18x143
W18x35
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W16x100
W16x26
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Fig. 4.13 –Flexural strength of W14 sections as a function of unbraced length
Fig. 4.14 –Flexural strength of W12 sections as a function of unbraced length
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W14x145
W14x22
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20 40 60 80 100 120 140 160 180 200
M n (
k . i n )
Lb (in)
W12x136
W12x14
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Fig. 4.15 –Flexural strength of W10 sections as a function of unbraced length
As before, the beam sections plotted in Figure 4.16 can be divided into two groups: the
lighter sections (with high Aw/Ag, low bf /d and high L/bf ) and the heavier sections (with
low Aw/Ag, high bf /d and low L/bf ). As with axial compressive capacity, the lighter
sections lose flexural strength rapidly as the unbraced length is increased. The heavier
sections, on the other hand, maintain flexural strength levels close to M p even for large
unbraced lengths.
4.2.4. Combination of axial and flexural strength
For each of the W-Shapes considered above, their strength under combined axial
compression and bending was computed using the AISC interaction equation (reproduced
as Equations 3-1 and 3-2 in this thesis). The results are plotted in the form of interaction
diagrams in Figures 4.17 to 4.30. As before, the unbraced length for strong axis flexural
buckling was taken as 200 inches. For each section, a family of interaction curves are
plotted for various values of unbraced length for weak axis flexural buckling, Ly, and
0
2000
4000
6000
8000
10000
12000
0 20 40 60 80 100 120 140 160 180 200
M n ( k . i n )
Lb (in)
W10x112
W10x12
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various unbraced lengths for lateral torsional buckling, L b. For each interaction diagram,
it was assumed that Ly is equal to L b. For each section, the dashed interaction diagram
corresponds to the yield limit state, for which Pn is equal to Py and Mn is equal to M p.
Fig. 4.16 – Normalized beam flexural strength
0
10
20
30
40
50
60
70
80
90
100
110
120
0 20 40 60 80 100 120 140 160 180 200
M n / M p ( % )
Lb (in)
W12x14
W10x12
W14x22
W16x26
W18x35
W21x44
W24x55
W18x143
W21x147
W24x146
W16x100
W14x145
W12x136
W10x112
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Fig. 4.17 – Interaction diagrams for W24x146
Fig. 4.18 – Interaction diagrams for W24x55
0
500
1000
1500
2000
2500
0 5000 10000 15000 20000 25000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in & 25in & 0in
Yield strength
0
100
200
300
400
500
600
700
800
900
0 1000 2000 3000 4000 5000 6000 7000 8000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in
Yielding strength
Lb=Ly=25in & 0in
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Fig. 4.19 – Interaction diagrams for W21x147
Fig. 4.20 – Interaction diagrams for W21x44
0
500
1000
1500
2000
2500
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100inLb=Ly=75in
Lb=Ly=50in & 25in & 0in
Yield strength
0
100
200
300
400
500
600
700
0 1000 2000 3000 4000 5000 6000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in
Lb=Ly=25in & 0in
Yielding strength
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Fig. 4.21 – Interaction diagrams for W18x143
Fig. 4.22 – Interaction diagrams for W18x35
0
500
1000
1500
2000
2500
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125inLb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in & 25in & 0in
Yield strength
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000 3500
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175inLb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75inLb=Ly=50in
Lb=Ly=25in & 0in
Yielding strength
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Fig. 4.23 – Interaction diagrams for W16x100
Fig. 4.24 – Interaction diagrams for W16x26
0
200
400
600
800
1000
1200
1400
1600
0 2000 4000 6000 8000 10000 12000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in & 25in & 0in
Yield strength
0
50
100
150
200
250
300
350
400
450
0 500 1000 1500 2000 2500
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in
Lb=Ly=25in & 0in
Yielding strength
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Fig. 4.25 – Interaction diagrams for W14x145
Fig. 4.26 – Interaction diagrams for W14x22
0
500
1000
1500
2000
2500
0 2000 4000 6000 8000 10000 12000 14000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in & 100in & 75in & 50in & 25in & 0in
Yield strength
0
50
100
150
200
250
300
350
0 200 400 600 800 1000 1200 1400 1600 1800
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=75inLb=Ly=50in
Lb=Ly=25in & 0in
Lb=Ly=175in
Lb=Ly=100in
Yielding strength
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Fig. 4.27 – Interaction diagrams for W12x136
Fig. 4.28 – Interaction diagrams for W12x14
0
500
1000
1500
2000
2500
0 2000 4000 6000 8000 10000 12000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in & 75in & 50in & 25in & 0in
Yield strength
0
50
100
150
200
250
0 100 200 300 400 500 600 700 800 900 1000
P n ( k i p s )
Mn (k.in)
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in
Lb=Ly=75in
Lb=Ly=50in
Lb=Ly=25in & 0in
Lb=Ly=200in
Yielding strength
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Fig. 4.29 – Interaction diagrams for W10x112
Fig. 4.30 – Interaction diagrams for W10x12
0
200
400
600
800
1000
1200
1400
1600
1800
0 1000 2000 3000 4000 5000 6000 7000 8000
P n ( k i p s )
Mn (k.in)
Lb=Ly=200in
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100in & 75in & 50in & 25in & 0in
Yield strength
0
20
40
60
80
100
120
140
160
180
200
0 100 200 300 400 500 600 700
P n ( k i p s )
Mn (k.in)
Lb=Ly=175in
Lb=Ly=150in
Lb=Ly=125in
Lb=Ly=100inLb=Ly=75in
Lb=Ly=50in
Lb=Ly=25in & 0in
Lb=Ly=200in
Yielding strength
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4.3. CONCLUSIONS
To satisfy capacity design requirements, the beam outside of the link must be capable of
sustaining high levels of combined axial force and bending moment. As shown in
Chapter 3, there are many configurations of EBFs that will generate axial force and
bending moment in the beam that exceed the strength of the beam based on a fully
yielded cross-section. Consequently, if stability considerations significantly reduce the
strength of the beam below its yield strength, satisfying beam capacity design
requirements will become increasingly problematic for many EBF configurations.
The analysis conducted in this chapter, although based on a number of simplifying
assumptions, has demonstrated that for typical beam lengths in EBFs, instability will
significantly reduce the strength of the beam for sections with relatively high values of
Aw/Ag, on the order of 0.45 to 0.55. As suggested by the sections listed in Table 4.1, and
based on the analysis of typical rolled shapes in Chapter 3, sections with high values of
Aw/Ag tend to be the lighter section for each depth category. Initial sizing of a link section
for shear will often lead a designer to these lighter shapes. However, since the beam
section is the same as the link section, the use of these lighter shapes will make satisfying
capacity design requirements for the beam very difficult, if not impossible.
On the other hand, the heavier sections, characterized by low values of Aw/Ag, can
develop strength levels close to their full yield strength. While these sections may be
considered inefficient for the link, their use will greatly facilitate satisfying capacity
design requirements for the beam.
The analysis shown in this chapter suggests that the ratios bf /d and L/bf are both good
indicators of the degree to which stability may control the strength of the beam. Sections
with higher values of bf /d, typically greater than 0.5 were capable of developing a greater
fraction of the full yield strength for the unbraced spans typical for beams in EBFs.
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Similarly, the ratio of L/bf was also a good indicator potential for a significant loss of the
full yield strength due to instability.
Figure 4.31 is a plot of bf /d versus Aw/Ag for all of the rolled W-Shapes listed in the
AISC Manual. From this plot, it is clear that there is a strong correlation between these
two parameters. That is, sections with low values of Aw/Ag also have relatively high
values of bf /d. Consequently, it appears that the ratio Aw/Ag provides a good indication of
the degree to which instability may control the strength of the beam outside of the link.
Sections that exhibit high buckling strength tend to have relatively low value of Aw/Ag.
As demonstrated in Chapter 3, sections with low values of Aw/Ag are more likely to
satisfy capacity design requirements when beam strength is controlled by yielding.
Consequently, combining the results of Chapters 3 and 4, it is clear that the use of
sections with low values of Aw/Ag for the beam and link is advantageous in avoiding
problems in satisfying capacity design requirements for the beam in EBFs.
Fig. 4.31 – Relationship between Aw /Ag and b f /d for rolled W-shapes
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2
A w
/ A g
bf /d
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CHAPTER 5
SUMMARY, CONCLUSIONS AND DESIGN RECOMMENDATIONS
5.1. SUMMARY
This thesis has described the results of research on the design of the beam outside of the
link in seismic-resistant steel eccentrically braced frames. EBFs are designed so that
yielding during earthquake loading is restricted primarily to the ductile links. To achieve
this behavior, all members other than the link are designed to be stronger than the link,
i.e. to develop the capacity of the link. These members are therefore designed for theforces generated by the fully yielded and strain hardened links.
Satisfying capacity design requirements for the beam segment outside of the link can be
difficult in the overall design process of an EBF, because the link and the beam are
typically the same wide flange member. Consequently, if the beam segment does not
have adequate strength to resist the capacity design forces generated by the link,
increasing the size of the beam segment may not be helpful. This is because as the size of
the beam is increased, the size of the link is also increased, and the capacity design forces
on the beam are consequently also increased. Thus, if a larger section is chosen for the
beam, the design forces on the beam increase. In some cases, it may in fact be impossible
to choose a section for the beam segment that will satisfy capacity design requirements.
If this discovery is made late in the design process, costly changes to the EBF
configuration may be necessary. In some cases, where the configuration of the EBF
cannot be changed, it may be necessary to strengthen the beam with cover plates or take
other costly measures.
The overall goal of this research was to develop guidelines for preliminary design of
EBFs that will result in configurations where the beam is likely to satisfy capacity design
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requirements. More specifically, the objectives were to identify the key design variables
that affect capacity design of the beam, to identify which variables have the largest effect
on capacity design of the beam, and to suggest values for these variables for use in
preliminary design.
A brief introduction and background information on EBFs was provided in Chapter 1. In
Chapter 2, simplified approximate equations were developed to predict the axial force
and moment in the beam segment outside of the link when link ultimate strength is
developed. These equations, although approximate, provide significant insight into
variables that affect the levels of axial force and moment in the beam. In Chapter 3, the
approximate beam force equations developed in Chapter 2 were combined with a beam
strength analysis for a limit state of a fully yielded cross-section under combined bending
and axial force. A series of parametric studies were then conducted to indentify EBF
configurations where the beam segment will yield prior to the development of the full
capacity design forces. In Chapter 4, the approximate beam force equations developed in
Chapter 2 were combined with a beam strength analysis based on a limit state of buckling
under combined bending and axial force. This analysis identified factors that affect the
ability of the beam to resist capacity design forces without buckling.
5.2. CONCLUSIONS
The results of this study show that the following variables have an impact on capacity
design of the beam.
• The non-dimensional link length, β, where β = e/(M p/V p).
Smaller values of β tend to reduce strength demands on the beam. This is because
link end moments decrease with β, and therefore the moment in the beam
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decreases with β. Consequently, the use of short shear yielding links is preferred.
• The ratio of web area to total area, Aw/Ag of the wide flange section used for the
beam and link.
The value of Aw/Ag for all rolled wide flange shapes ranges from about 0.2 to to
0.55. Smaller values of Aw/Ag tend to reduce strength demands on the beam.
Sections with smaller values of Aw/Ag also are generally capable of sustaining
high combined axial compression and bending without buckling.
• The angle between the brace and the beam, α.
Larger values of α tend to reduce strength demands on the beam. This is because
the horizontal component of the brace axial force, which is proportional to link
shear, is generally transferred directly to the beam. Consequently, as α increases,
beam axial force decreases.
• The ratio of the beam to brace moment of inertia, η = I beam/I brace.
Smaller values of η tend to reduce strength demands on the beam. As η decreases,
the flexural stiffness of the brace increases relative to the flexural stiffness of the
beam. This, in turn, results in a greater fraction of the link end moment transferred
to the brace and away from the beam. That is, smaller values of η will reduce the
bending moment in the beam. Note that this conclusion presumes that the brace is
connected to the beam using a fully-restrained moment resisting connection,
which is common practice in EBF design. If the brace is connected to the beam
using a nominally pinned connection, the value of β is infinity and the beam will
then need to resist the entire link end moment. This, in turn, will increase thedifficulty of satisfying capacity design requirements for the beam.
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• The ratio of plastic section modulus to total area, Z/Ag of the wide flange section
used for the beam and link.
The value of Z/Ag for all rolled wide flange shapes ranges from about 3 to to 10.
Smaller values of Z/Ag tend to reduce strength demands on the beam.
The parametric studies showed that the relative importance of these key design variables
on capacity design of the beam were in the order listed above. That is, the
nondimensional link length β had the largest impact on beam design, the ratio of web
area to total area, Aw/Ag, was the second most important variable, and so on. The ratio of
plastic section modulus to total area, Z/Ag, had the smallest effect on capacity design of
the beam.
5.3. DESIGN RECOMMENDATIONS
This section provides some general recommendations for preliminary design of EBFs that
are intended to preclude problems in satisfying beam capacity design requirements. These
recommendations are based on the analyses conducted for this research combined with
judgment on other factors that may affect EBF design. Recommendations are providedfor preliminary selection of the design variables discussed above. However,
recommendations are not provided for Z/Ag, as this variable had relatively small impact
on beam capacity design.
1. Nondimensional link length, β.
The nondimensional link length β, where β = e/(M p/V p) has a significant impact on
capacity design of the beam. As indicated by the analysis, the smaller the value of β,
the smaller the strength demands on the beam. Whereas small values of β are
beneficial for beam design, small values of β may be detrimental in other aspects of
the EBF design. For example, smaller values of β generally correspond to smaller
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values of actual link length, e. Link rotation demands increase as the ratio of link
length to span length, e/L, decreases. Consequently, very small values of β may cause
difficulties in satisfying link rotation limits. In addition, as noted in the commentary
of the AISC Seismic Provisions, links with small values of β, say less than about 1.0,
have shown very high degrees of overstrength in experiments. Higher overstrength
will increase capacity design forces on the beam. Based on these competing design
requirements, it is recommended that values of β in the range of about 1.1 to 1.3 be
chosen for preliminary design. That is, it is recommended that link length e be chosen
in the range of about 1.1 M p/V p to 1.3 M p/V p.
2. Ratio of web area to total area, Aw/Ag of the wide flange section used for the beam
and link.
This section property has a large influence on capacity design of the beam. Wide
flange sections with smaller values of Aw/Ag reduce strength demands on the beam
and also provide greater resistance to buckling. Additionally, sections with low values
of Aw/Ag tend to have high values of M p/V p. High values of M p/V p are beneficial
because this allows the use of higher values of absolute link length, e, for small
values of nondimensional link length, β. This, in turn, results in higher values of e/L,which reduces link rotation demands. Consequently, using sections with small values
of Aw/Ag offers numerous benefits. Based on this research, it is recommended that
wide flange sections with Aw/Ag less than about 0.3 be chosen for preliminary design
of the beam and link. Note that sections satisfying this recommendations tend to be
the sections commonly used for column applications, which are the W8s, W10s,
W12s, and W14s.
3. The angle between the brace and the beam, α.
Configuring an EBF with small values for the angle α between the beam and brace
can lead to large strength demands on the beam. Consequently, larger values of α are
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preferred to avoid problems with capacity design of the beam. However, large values
of α will typically lead to a significant reduction in the lateral stiffness of an EBF,
thereby increasing the difficulty in satisfying code specified drift limitations and in
satisfying link rotation limits. Based on these competing design requirements, it is
recommended that values of the brace-beam angle α be chosen in the range of about
40 to 50 degrees for preliminary design.
4. The ratio of the beam to brace moment of inertia, η = I beam/I brace.
Providing a brace that has a relatively high flexural stiffness is advantageous for the
beam, as the brace attracts a larger fraction of the link end moment and thereby
reduces the moment that must be resisted by the beam. Consequently, lower values of
η are advantageous for capacity design of the beam. However, there are generally
practical limitations that preclude the use of very small values of η. This is because as
the brace moment of inertia increases, the depth of the brace also generally increases.
As brace depth increases, it becomes increasing difficult to satisfy the EBF design
requirement that the brace and beam centerlines intersect at the end of the link or
inside of the link. Based on these competing design requirements, it is recommended
that values of η in the range of about 1 to 1.5 be chosen for preliminary design. As
noted above, this recommendation presumes that a fully-restrained moment resisting
connection be provided between the brace and the beam.
The recommendations provided above are intended only for preliminary design. In the
final design, a detailed analysis of the beam for capacity design forces is still required.
However, using the recommendations provided above for preliminary design should help
mitigate problems with satisfying beam capacity design requirements in the detailed
design phase for an EBF
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5.4. ADDITIONAL RESEARCH NEEDS
The research conducted herein, as well as normal design practice treat the beam outside
of the link as a bare steel member. However, a composite concrete floor slab is normally
present. Consequently, the beam outside of the link is, in reality, a composite member.
Composite action is likely to provide a significant increase in both the axial and flexural
strength of the member. Thus, taking advantage of composite action will likely be highly
beneficial for satisfying capacity design requirements for the beam. However, at present,
there is little information available on methods to design the beam outside of the link as a
composite member. Research is needed on the behavior of composite beams under large
cyclic axial force and bending to provide such design guidance. This includes research on
the effective width of the slab, including shear lag effects in the slab near the link end of
the beam and near the column end of the beam; requirements for shear connectors,
including the effects of cyclic loading on shear connector strength; and bracing
requirements for the beam bottom flange.
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REFERENCES
AISC (2005). “Seismic Provisions for Structural Steel Buildings.” Standard ANSI/AISC341-05, American Institute of Steel Construction, Inc., Chicago, IL.
Engelhardt, M.D. and Popov, E.P. (1989). "On Design of Eccentrically Braced Frames,"
Earthquake Spectra, Volume 5, No. 3, pp. 495-511.
Engelhardt, M.D., Tsai, K.C., and Popov, E.P. (1991). "Stability of Beams in
Eccentrically Braced Frames," Proceedings: US-Japan Seminar on Cyclic Buckling of
Steel Structures and Structural Elements Under Dynamic Loading Conditions, Osaka,
Japan.
Popov, E.P. and Engelhardt, M.D. (1988). "Seismic Eccentrically Braced Frames,"
Journal of Constructional Steel Research, Volume 10, pp. 321-354.
Okazaki, T., and Engelhardt, M.D. (2007). "Cyclic loading behavior of EBF links
constructed of ASTM A992 steel," Journal of Constructional Steel Research Volume 63,
pp. 751–765
Computers and Structures. (2010). SAP2000. Retrieved April 1st, 2010, from
http://www.csiberkeley.com/ product_SAP.html
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VITA
Sepehr Dara was born in Tehran, Iran on July 21, 1986, the son of Farrokh Dara
and Behdokht Eskandari. He received his high school diploma in 2004 from Allameh
Helli high school in Tehran, Iran. After high school, he attended the University of Tehran
in 2004. He received his Bachelor of Science in Engineering degree from the University
of Tehran in 2008. In August 2008, he entered graduate school at the University of Texas
at Austin.
Permanent address: 415 W 39th St. Apt314
Austin, TX, 78751
This Thesis was typed by the author.