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Copyright

by

Sepehr Dara

2010



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





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



iv

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



v

Abstract



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



vi

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.



vii

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



viii

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



ix

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



x

5.4. ADDITIONAL RESEARCH NEEDS................................................................ 137

REFERENCES...................................................................................................... 138

VITA ................................................................................................................... 139



xi

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
























xii






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



xiii

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


of e/ L = 0.125 ....................................................................................... 28


of e/ L = 0.10 ......................................................................................... 28


of e/ L = 0.075 ....................................................................................... 29



xiv


of e/ L = 0.05 ......................................................................................... 29


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



xv

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



xvi

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



xvii

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


variable Ly ........................................................................................... 112


variable Ly ........................................................................................... 112


variable Ly ........................................................................................... 113


variable Ly ........................................................................................... 113


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






xviii




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 4.21: Interaction diagrams for W18x143 ....................................................... 124Figure 4.22: Interaction diagrams for W18x35 ......................................................... 124









Figure 4.31: Relationship between Aw/Ag and bf /d for rolled W-shapes .................. 130



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).



2

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



3

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.



4

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



5

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)



6

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



7

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



8

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



9

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.



10

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



11

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



12

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



14

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.



15

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.



16

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



17

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.



18

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



19

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)



20

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



21

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)



22

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)



23

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:



24

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

+

++

+

+

+

−=



25

]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)



26

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



27

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)



28



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)



29



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)



30


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



31

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)



32

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−=−=



33

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:



34

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)



35

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)



36

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







37

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.



38

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)



39

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:



40

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:



41

)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:



42

)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.



43

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)



44

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.



45

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



46

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



47

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



48

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



49

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

<=++=

+++

≤+

++



50

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.



51

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:



52

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.



53

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



54

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 ======



55

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



56

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



57

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



58

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.



59

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.



60

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)



61

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.



62

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


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



63

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


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


Beam W12X152

IX-X ZX-X Z / Ag Aw Mp / Vp Aw/Ag1430 243 5.436 9.483 42.708 0.212


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



64


Beam W14X22


199 33.2 5.115 2.9969 18.46352 0.461


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


Beam W14x145


1710 260 6.088 8.5816 50.49563 0.200


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



65


Beam W16X26


301 44.2 5.755 3.7525 19.63136 0.488


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


Beam W16X100


1500 200 6.734 8.792 37.91088 0.296


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



66


Beam W18X35


510 66.5 6.456 5.055 21.92549 0.490


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


Beam W18x143


2750 322 7.648 12.3078 43.60379 0.292


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



67


Beam W21X44


843 95.4 7.338 6.93 22.94372 0.533


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


Beam W21x147


3630 373 8.634 14.256 43.60737 0.33


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



68


Beam W24X55


1360 135 8.282 8.92305 25.21559 0.547


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


Beam W24X146


4580 418 9.720 14.638 47.59302 0.340


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



69

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)



70

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)



71

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



72


Beam W10X12


53.8 12.6 3.559 1.795 11.695 0.507


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


Beam W10X112


716 147 4.468 6.719 36.461 0.204


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



73


Beam W12X14


88.6 17.4 4.182 2.29 12.66376 0.550


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


Beam W12X152


1430 243 5.436 9.483 42.708 0.212


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



74


Beam W14X22


199 33.2 5.115 2.996 18.463 0.461


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


Beam W14x145


1710 260 6.088 8.581 50.495 0.200


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



75


Beam W16X26


301 44.2 5.755 3.752 19.631 0.488


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


Beam W16X100


1500 200 6.734 8.792 37.910 0.296


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



76


Beam W18X35


510 66.5 6.456 5.055 21.925 0.490


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


Beam W18x143


2750 322 7.648 12.307 43.603 0.292


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



77


Beam W21X44


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


Beam W21x147


3630 373 8.634 14.256 43.607 0.33


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



78


Beam W24X55


1360 135 8.282 8.923 25.215 0.547


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


Beam W24X146


4580 418 9.720 14.638 47.593 0.340


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



79

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


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)



80

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)



81

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



82

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.



83

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.



84

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.



85

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



86

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



87

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 &


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.



88

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


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



89

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



90

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



91

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 &


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


Z / Ag



92

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



93

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



94

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


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



95

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



96

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



97

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


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



98

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



99

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



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


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



101

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



102

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


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



103

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



104

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



105

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



106

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]



107

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



108

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.



109

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



110

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.



111

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



112



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



113



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



114


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



115

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.



116

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



117

Fig. 4.9 –Flexural strength of W24 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



118



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



119



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



120


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



121

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



122

Fig. 4.17 – Interaction diagrams for W24x146


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



123



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


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



124



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=75in


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=125in

Lb=Ly=100in


Lb=Ly=25in & 0in

Yielding strength



125



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


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



126



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=25in & 0in

Lb=Ly=175in

Lb=Ly=100in

Yielding strength



127



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



128



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=50in

Lb=Ly=25in & 0in

Lb=Ly=200in

Yielding strength



129

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.



130

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



133

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.



134

• 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



135

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



136

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.



138

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



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.

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