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Department of Civil and Environmental Engineering
Stanford University
Report No. 173 March 2010
Design and Behavior of Steel Shear Plateswith Openings as Energy-Dissipating Fuses
By
X. Ma, E. Borchers, A. Pena, H. Krawinkler,
S. Billington and G.G. Deierlein
The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus.
Address:
The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020
(650) 723-4150 (650) 725-9755 (fax) [email protected] http://blume.stanford.edu
©2010 The John A. Blume Earthquake Engineering Center
3
ABSTRACT
Eleven steel shear plates with openings were tested to investigate their behavior under
cyclic loading. Two types of steel shear plates were studied, the slit fuse with narrow
slits that divide the plate into rectangular links, and the butterfly fuse with diamond-
shape openings that create butterfly-shape links in the plate. All tested fuses
demonstrated fairly ductile behavior with full load – deformation hysteretic response
at least up to 5% shear deformation, followed by pinched behavior, and eventually
low-cycle fatigue induced fracture at a shear deformation of more than 20%, or 30%
for the more ductile butterfly fuses. Fuses with partial or full buckling restraining
device were also tested, which showed delayed buckling and earlier fracture compared
to specimens without buckling restraining.
Design equations to determine the nominal strength and stiffness of the fuses are
derived and examined by test results. A number of dimensionless parameters
including the ratio of tangent stiffness to initial stiffness, residual bending strength to
nominal strength, deformation at peak strength and ultimate deformation at fracture
are established to develop the backbone curve for estimating fuse strength. Three-
dimensional numerical models of two butterfly fuse specimens were created and were
able to simulate buckling and load – deformation response under cyclic loading with
fairly good accuracy. More research topics are recommended for future studies.
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ACKNOWLEDGEMENTS
This report is based on research supported by the National Science Foundation under
Grant No. CMMI-0530756. In-kind funding and materials were provided by Gregory
P. Luth & Associates, Inc. of Santa Clara, CA. Additional financial support was
provided by the John A. Blume Research Fellowship and the Stanford Graduate
Fellowship. Financial support from these institutions is greatly appreciated.
The advice and guidance provided by Professor Jerome Hajjar and graduate student
Matthew Eatherton (the University of Illinois at Urbana-Champaign), Professor Sarah
Billington (Stanford University), Gregory Luth and John Oliva (Gregory P. Luth &
Associates, Inc. of Santa Clara, CA) are gratefully acknowledged. Also acknowledged
is graduate student Nathan Canney (Stanford University) for proof reading part of the
manuscript.
The information and conclusions presented in this report are those of the authors and
do not necessarily reflect the views of the National Science Foundation or other
contributors.
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TABLE OF CONTENTS
Abstract ........................................................................................................................... 3
Acknowledgements ........................................................................................................ 4
List of Tables .................................................................................................................. 8
List of Illustrations ......................................................................................................... 9
Nomenclature ............................................................................................................... 13
1 Introduction .......................................................................................................... 15
1.1 Application of Fuses in the Controlled Rocking System ............................... 15
1.1.1 Shear Deformation Demand ............................................................... 16
1.1.2 Tension Due to Kinematic Constraint ................................................ 17
1.2 Review of Selected Previous Research .......................................................... 18
1.3 Objectives and Organization of Report .......................................................... 19
2 Fuse Design Equations ......................................................................................... 27
2.1 Slit Fuse .......................................................................................................... 27
2.2 Butterfly Fuse ................................................................................................. 29
2.2.1 Determination of Fuse Link Shape ..................................................... 29
2.2.2 Strength and Stiffness Equations ........................................................ 31
2.3 Buckling-Restrained Fuses ............................................................................. 33
3 Quasi-Static Cyclic Test ....................................................................................... 37
3.1 Test Setup ....................................................................................................... 37
3.1.1 Test Matrix ......................................................................................... 37
3.1.2 Loading Setup ..................................................................................... 38
3.1.3 Instrumentation ................................................................................... 39
3.1.4 Loading History .................................................................................. 40
3.1.5 Material Testing .................................................................................. 41
3.2 Test Result of Slit Fuse S12-36 ...................................................................... 42
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3.2.1 Examination of the Test Setup ........................................................... 42
3.2.2 Conventions ........................................................................................ 43
3.2.3 Load – Deformation Relationship ...................................................... 44
3.2.4 Outer and Inner Links ......................................................................... 45
3.2.5 Bending and Tension Mode Behavior ................................................ 45
3.2.6 Strut Pin Fracture ................................................................................ 48
3.3 Test Result of the Rest of the Slit Fuse Specimens ........................................ 48
3.3.1 S10-40 ................................................................................................. 48
3.3.2 S10-56 ................................................................................................. 50
3.3.3 S10-36W ............................................................................................. 51
3.3.4 S10-56BR ........................................................................................... 52
3.4 Test Result of Butterfly Fuse B10-36 ............................................................. 53
3.4.1 Load-Deformation Relationship ......................................................... 53
3.4.2 Pinned Column Mechanism in Tension Mode ................................... 54
3.4.3 Comparison with S12-36 .................................................................... 55
3.5 Test Result of the Rest of Butterfly Fuse Specimens ..................................... 56
3.5.1 B09-56 ................................................................................................ 56
3.5.2 B06-37 ................................................................................................ 56
3.5.3 B02-14 ................................................................................................ 57
3.5.4 B10-36W ............................................................................................ 58
3.5.5 B07-18W ............................................................................................ 59
3.6 Summary of Test Result ................................................................................. 60
3.6.1 Strength and Stiffness Equations ........................................................ 60
3.6.2 Characterization of Fuse Behavior ..................................................... 61
3.7 Numerical Modeling ....................................................................................... 63
3.7.1 Description of the Model .................................................................... 63
3.7.2 Analytical Case Studies ...................................................................... 64
4 Conclusions ......................................................................................................... 103
4.1 Performance of Slit and Butterfly Fuses without Buckling Restraining ...... 103
7
4.2 Performance of Fuses with Partial or Full Buckling Restraining ................. 105
4.3 Design of Slit and Butterfly Fuses ................................................................ 105
4.4 Numerical Modeling ..................................................................................... 106
4.5 Future Research Needs ................................................................................. 107
References ................................................................................................................. 109
Appendix: Figures of ECC/SCHPFRCC Panel Tests ........................................... 111
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LIST OF TABLES
Table 3.1 Test Matrix .............................................................................................. 66
Table 3.2 List of Channels for Data Recording ...................................................... 66
Table 3.3 Loading Protocol ..................................................................................... 67
Table 3.4 Tension Coupon Cross-Section Geometry.............................................. 67
Table 3.5 Material Test Result (plates A ~ E) ........................................................ 68
Table 3.6 Material Test Result Provided by CEL Consulting (plates F and G) ..... 68
Table 3.7 Summary of Fuse Test Results ............................................................... 68
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LIST OF FIGURES
Figure 1.1 Shear plate fuses in the controlled rocking system: (a) single frame; (b)
dual-frame ............................................................................................... 22
Figure 1.2 Deformation demand in the fuses: (a) single frame; (b) dual-frame ...... 22
Figure 1.3 Tensile deformation and uplifting ........................................................... 23
Figure 1.4 A generic shear plate with slits ............................................................... 23
Figure 1.5 Test of steel slit dampers: (a) fractured specimen; (b) hysteresis response
(Chan and Albermani, 2008) ................................................................... 23
Figure 1.6 Test of steel shear wall with slits: (a) test specimen; (b) hysteresis
response (Hitaka and Matsui, 2003) ....................................................... 24
Figure 1.7 Test of honeycomb dampers: (a) test specimen; (b) hysteresis response
(Kobori et al., 1992) ................................................................................ 24
Figure 1.8 Test of ADAS elements: (a) specimen section perpendicular to loading
direction; (b) specimen section parallel to loading direction; (c)
hysteresis response (Aiken et al., 1993).................................................. 25
Figure 2.1 Fuse geometry: (a) slit fuse; (b) butterfly fuse; (c) slit fuse restrained from
buckling; (d) butterfly fuse with partial buckling restraining ................... 34
Figure 2.2 Decomposition of fuse link displacement ................................................. 34
Figure 2.3 Loading condition and internal force distribution of a butterfly fuse link 35
Figure 3.1 Fuse test setup: (a) schematic; (b) photo ................................................. 69
Figure 3.2 More test setup details: (a) out-of-plane bracing; (b) actuator and
reaction column ....................................................................................... 70
Figure 3.3 Instrumentation scheme .......................................................................... 70
Figure 3.4 Instrumentation photos: (a) linear variable displacement transducers
(LVDT’s); (b) temposonic transducers; (c) strain gauges; (d) strain gauge
connection box ........................................................................................ 71
Figure 3.5 Data acquisition and control arrangement .............................................. 72
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Figure 3.6 Steel tension coupons: (a) rectangular subsize (for plates A, B and C); (b)
rectangular sheet (for plate D); (c) round standard (for plate E) ............ 73
Figure 3.7 Steel tension coupon test setup ............................................................... 73
Figure 3.8 Stress-strain relationships of all tested coupons*: (a) plate A; (b) plate B;
(c) plate C; (d) plate D; (e) plate E ......................................................... 74
Figure 3.9 Comparison of target, actuator and fuse displacement: (a) 1 ~ 23 s; (b)
20 ~ 52 s; (c) elastic deformation in the test rig ..................................... 75
Figure 3.10 Load - deformation relationship for specimen S12-36 ........................... 75
Figure 3.11 Photos of specimen S12-36 ..................................................................... 76
Figure 3.12 Photos of specimen S12-36 details: (a) yielding around slit end; (b) side
view of buckling; (c) crack initiation ..................................................... 77
Figure 3.13 Free body diagram of loading beam and the fuse: (a) bending mode
(small deformation); (b) tension mode (large deformation) ................... 77
Figure 3.14 Measured strut axial forces ..................................................................... 78
Figure 3.15 Fractured pin left in the strut pin hole in test of S10-40 ......................... 78
Figure 3.16 Load - deformation relationship.............................................................. 78
Figure 3.17 Photos of specimen S10-40 ..................................................................... 79
Figure 3.18 Load - deformation relationship.............................................................. 80
Figure 3.19 Fractured link ends of specimen S10-56 ................................................. 80
Figure 3.20 Photos of specimen S10-56: (a) beginning of test; (b) yielding evident
(γ=2.1%); (c) buckling initiation (γ=4%); (d) buckling evident and crack
initiation (γ=6.1%); (e) crack evident (γ=14.3%); (f) fracture – end of
test (γ=16.1%) ......................................................................................... 81
Figure 3.21 Load-deformation relationship: (a) S10-36W; (b) comparison with S10-
40............................................................................................................. 82
Figure 3.22 Photos of specimen S10-36W details: (a) buckling initiation; (b)
buckling evident; (c) end of test.............................................................. 82
Figure 3.23 Photos of specimen S10-36W ................................................................. 83
Figure 3.24 Load-deformation relationship: (a) S10-56BR; (b) comparison with S10-
56............................................................................................................. 84
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Figure 3.25 Photos of specimen S10-36W: (a) beginning of test; (b) buckling and
crack initiation (γ=5.9%); (c) crack evident (γ=7.5%); (d) fracture – end
of test (γ=12.1%); (e) cracks at slit ends ................................................. 85
Figure 3.26 Load-deformation relationship................................................................ 86
Figure 3.27 Photos of specimen B10-36: (a) beginning of test; (b) yielding evident
(γ=1.5%); (c) buckling initiation (γ=2.4%); (d) buckling evident
(γ=10.3%); (e) buckling at zero displacement (γ= 0%); (f) fracture – end
of test (γ=16.3%) ..................................................................................... 87
Figure 3.28 Photos of specimen B10-36 details: (a) flaking around quarter-height; (b)
buckling – top view; (c) buckling – side view; (d) crack initiation; (e)
fracture; (f) fractured surface .................................................................. 88
Figure 3.29 Pinned column mechanism of a fuse link ............................................... 89
Figure 3.30 Load-deformation relationship................................................................ 89
Figure 3.31 Photos of specimen B09-56 details: (a) beginning of test; (b) yielding
evident (γ=2.7%); (c) buckling evident (γ=6.7%); (d) buckling at positive
peak (γ=24.7%); (e) buckling at zero displacement; (f) fracture – end of
test (γ=26.7%) ......................................................................................... 90
Figure 3.32 Gouging of specimen B09-56 ................................................................. 91
Figure 3.33 Load-deformation relationship................................................................ 91
Figure 3.34 Photos of specimen B06-37 details: (a) beginning of test; (b) buckling
initiation (γ=10.5%); (c) buckling evident (γ=14.4%); (d) crack initiation
(γ=18.5%); (e) buckling at zero displacement (γ=0%); (f) fracture – end
of test (γ=22.5%) ..................................................................................... 92
Figure 3.35 Load-deformation relationship of specimen B02-14 .............................. 93
Figure 3.36 Photos of specimen B02-14 - I: (a) beginning of test; (b) γ=1.1%; (c)
γ=2.6%; (d) γ=3.6% ................................................................................ 93
Figure 3.37 Photos of specimen B02-14 - II: (a) γ=8.4%; (b) γ=14.4%; (c) out-of-
plane deflection (γ=22.0%); (d) crack initiation (γ=24.0%); (e) crack
evident (γ=26.0%); (f) fracture- end of test (γ=26.0%) .......................... 94
12
Figure 3.38 Photos of specimen B02-14 details: (a) buckling; (b) crack at quart-
height; (c) fractured surface .................................................................... 95
Figure 3.39 Load-deformation relationship: (a) B10-36W; (b) comparison with B10-
36............................................................................................................. 95
Figure 3.40 Photos of specimen B10-36W: (a) beginning of test; (b) buckling
initiation (γ=2.4%); (c) buckling evident (γ=20.2%); (d) end of test – no
fracture (γ=20.2%); (e) side view of buckling; (f) gouging .................... 96
Figure 3.41 Load-deformation relationship of specimen B07-18W .......................... 97
Figure 3.42 Photos of specimen B10-36W: (a) beginning of test; (b) buckling
initiation (γ=6.0%); (c) buckling evident (γ=11.9%); (d) buckling at zero
displacement (γ=0%); (e) welding failure – end of test (γ=18.0%); (f)
crack initiation ........................................................................................ 98
Figure 3.43 Characterization of fuse behavior ........................................................... 99
Figure 3.44 Regression of fuse parameters: (a) tangent stiffness; (b) residual bending
strength; (c) shear deformation at peak strength; (d) ultimate shear
deformation at fracture ............................................................................ 99
Figure 3.45 Abaqus model of a butterfly fuse .......................................................... 100
Figure 3.46 Simulated buckling of butterfly fuse B06-37: (a) γ 0; (b) γ 23% ......... 100
Figure 3.47 Comparison of test and analysis load – deformation relationship ........ 101
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NOMENCLATURE
A rocking frame bay width based on column centerline
AF amplification factor
a width of fuse link middle section
B spacing between dual rocking frames based on column centerlines
Beff effective width of the fuse.
b width of fuse link end section
c distance between fuse end and edge of fixture
E Young’s modulus of steel
h distance between bolt lines
Ilink moment of inertia of one fuse link cross-section
k stiffness of the fuse
kB bending stiffness of fuse links
kR stiffness of band zone
L length of fuse links
Mc moment at the band segment
Mp plastic moment of the link cross-section
My yield moment of the link cross-section
n number of fuse links
Qp fuse nominal strength
Qu ultimate fuse strength at fracture
Qy fuse yield strength
t thickness of fuse plate
α fuse link end rotation angle
γ fuse shear deformation
γL fuse link length based shear deformation
Δ horizontal displacement of the loading beam
Δu fuse shear displacement at fracture
θ rocking frame uplift ratio
14
σy yield stress of steel
φ curvature of the band segment
15
CHAPTER 1
INTRODUCTION __________________________________________________________________________________
1.1 Application of Fuses in the Controlled Rocking System
The experimental and analytical study of steel energy dissipating fuses presented in
this report is part of a collaborative research project to develop the controlled rocking
system which is described in Eatherton (2010) and Ma (2010). Two alternative
configurations of the controlled rocking system are shown in Figure 1.1. In both
configurations, the system comprises steel braced frames that are allowed to uplift and
rock upon the column base, post-tensioning (PT) strands or rods that provide
overturning resistance and self-centering capability, and replaceable structural fuses to
dissipate energy. During the rocking, the frames and PT strands are designed to
remain elastic, while damage is concentrated in the replaceable fuses.
In general, any form of structural element with adequate ductility and energy
dissipating capability can be employed as a fuse. Shear plate type fuses shown in
Figure 1.1 are adopted for this study for their compatibility with both the single and
dual frame configurations, and their simplicity in design and installation. In the dual-
frame configuration, fuses are installed between the two frames and mobilized by
relative movement between the two inner columns. This configuration has the
flexibility for the fuses to be distributed along the height of the building, which is
potentially more suitable for tall buildings requiring more fuses. The single frame
configuration has the fuse connected to the frame through a vertical transfer strut. As
the frame uplifts, the bottom of the fuses are pulled up, imposing shear deformation in
the fuses. The single frame has the advantage of simplicity and compactness (less
16
floor space), since fewer components are required. Locating the fuses in the first story
also makes them more accessible for installation and replacement. However, the
space constraint will limit the number and size of fuses. Therefore this configuration
is more suitable for low buildings where fewer fuses are needed.
Experimental testing in the development of the controlled rocking system consists of
quasi-static test of the fuse subassembly at Stanford University, quasi-static and hybrid
test of the dual and single frames at the University of Illinois at Urbana-Champaign
(UIUC), and a shaking table test of the single frame system at Hyogo Earthquake
Engineering Research Center (E-Defense) in Japan. The work presented in this report
is on the fuse testing at Stanford. Details about the overall project can be found in
Eatherton (2010) which focuses on the dual-frame configuration, and Ma (2010)
which has an emphasis on the single-frame configuration.
To provide fuses with sufficient ductility, it is necessary to first understand
deformation demand imposed on the fuses by the rocking frames as discussed in the
following two sections.
1.1.1 Shear Deformation Demand
Due to the smaller width of the fuse compared to width of the rocking frame, an uplift
angle θ of the rocking frame will lead to amplified shear deformation γ in the fuse as
shown in Figure 1.2. Assuming rigid body motions of the frames, the relationship
between the frame uplift ratio and the fuse shear drift can be expressed by an
amplification factor defined below
( )( )
+
=systemframesingle2systemframe-dual
eff
eff
BABBA
AF (1.1)
θγ ⋅= AF (1.2)
17
where AF = amplification factor; γ = fuse shear deformation; θ = rocking frame uplift
ratio; A = rocking frame bay width based on column centerline; B = spacing between
dual rocking frames based on column centerlines; Beff = effective width of the fuse.
The rocking systems examined in this project typically have an AF of 6 for the dual-
frame configuration (Eatherton, 2010), and 10 for the single frame (Ma, 2010), which
means shear deformation demand for the fuse is roughly 20% ~ 30% for an uplift ratio
of 3%. To ensure adequate ductility, the fuse ideally needs to have a deformation
capacity of more than 30% under cyclic loading.
1.1.2 Tension Due to Kinematic Constraint
Under large shear deformation, tension field action is expected to occur in the fuses
due to the effect of geometric constraint, as shown in Figure 1.2. For instance, in the
dual-frame configuration, kinematic analysis shows the relationship between tension
deformation in the fuse, εt, and frame uplift ratio, θ, can be approximated by the
following equation assuming rigid body rotation of the frame.
1)cos1)(1(21 −−++= ϑε
BA
BA
t
(1.3)
The equation shows tension deformation is larger when the distance between the two
frames is shorter (larger A/B ratio). In the test of the dual rocking frames at UIUC
(Eatherton, 2010), a ratio, A/B = 2.5, is adopted which results in the relationship
between fuse tension strain and rocking frame uplift angle shown in Figure 1.3. It can
be read from the plot that tension strain will exceed the yield limit when the uplift
ratio reaches 2%. In the single frame system, since the horizontal distance between
the moving vertical edges of the fuse is kept constant, the tension deformation is even
more profound. Consequently, the shear fuse is in fact subjected to significant tension
deformation in addition to the shear deformation. This loading condition needs to be
considered in the experimental design and analysis of fuse behavior.
18
1.2 Review of Selected Past Research
Extensive research has been conducted in the past on utilizing shear capacity of steel
plates to dissipate energy. Among the many design approaches to achieve high
ductility, one of the simple and cost-effective solutions is to cut openings or slits in a
steel plate, which divides the plate into a number of parallel links as shown in Figure
1.4. Compared to the original solid plate, the links are more flexible and primarily
exhibit flexural mode behavior, rather than the predominant shear mode behavior of
the original solid plate. Moreover, each individual link is more compact than the
original plate in terms of link width-to-thickness ratio, which improves buckling
resistance of the shear plate.
Numerous studies have been reported on designs of steel plates with slits or openings.
Nakamura et al. (1997) and Chan and Albermani (2008) investigated steel slit dampers
made of relatively thick plates, mostly with link width-to-thickness ratios between 1
and 2. In both studies, the dampers exhibited stable hysteretic behavior up to a peak
shear deformation between 10% and 20%. The dampers finally fractured due to low
cycle fatigue with an ultimate strength twice as much as the yield strength (Figure
1.5). Hitaka and Matsui (2003) studied relatively thin steel slit walls that are used as
shear walls in buildings (Figure 1.6). Since these walls are designed to occupy a full
story height, they are generally much larger and more slender compared to the slit
dampers. The walls also have less shear drift demand due to limit on story drift. It
was found that by limiting the link width-to-thickness ratio to 10, the steel shear walls
with slits can sustain roughly 2.5% shear drift without buckling.
Kobori et al. (1992) altered the shape of the links by cutting honeycomb-shaped
openings in steel plates, which left butterfly-shaped links (Figure 1.7). Such geometry
led to more distributed yielding along the length of the links because of the
resemblance of its geometry to the moment diagram of beams deforming in double
curvature. The damper links were made quite stocky to avoid buckling, with a width-
19
to-thickness ratio of 2.5 and length to thickness ratio of 6.7. Test result showed the
dampers sustained 30% shear deformation without degradation. Fatigue behavior of
the honeycomb dampers was extensively studied by Tagami et al. (1999) and
characterized using Manson-Coffin equations. Even better alignment between
bending capacity and moment distribution can be achieved by the ADAS device which
is composed of hourglass-shaped steel plates placed perpendicular to the loading
direction (Bergman and Goel, 1987). Aiken et al. (1993) conducted dynamic tests on
ADAS elements and showed that they can be safely designed for displacement ranges
up to 10 times the yield displacement (Figure 1.8).
In theory, among the reviewed designs, the ADAS configuration makes the best use of
material in terms of distributing yielding through flexural behavior of the links.
However, the slit and honeycomb dampers are easier to manufacture, and if well
designed, can possess equivalent energy absorption capacity. Besides, their planar
geometry may be more suitable in applications with space constraint. Although many
schemes of steel shear plates with openings have been studied as summarized in the
review, certain gaps still exist that may lead to new development of shear plate fuses.
For instance, the honeycomb damper study (Kobori et al., 1992) only presented one
shape design of the butterfly shaped links. Yet variations in length, width or spacing
of the links may be explored to provide more flexibility for design. Moreover, no
study was found on thinner butterfly links which may be adequate for application with
smaller ductility demand.
1.3 Objectives and Organization of Report
As stated earlier, fuses employed in the controlled rocking system need to be capable
of withstanding more than 30% shear deformation, under combined shear and tension
loading. Previous research has shown steel plates with slits or openings can
potentially provide a simple and cost-effective solution. Two types of fuse geometries
are chosen in this study to be further investigated, including plates with straight slits
20
referred to as the slit fuses, and plates with “honeycomb” or “diamond” shaped
openings referred to as the butterfly fuses which are named after the “butterfly”
shaped links. Slit and butterfly fuses studied in previous research belong to one of the
following two categories:
Thin slit fuses under relatively small shear deformation (up to 3%)
Thick slit and butterfly fuses under large shear deformation (10% ~ 30%)
where thin fuses are defined as having a link width-to-thickness ratio of 10 or larger,
and thick fuses as having a link width-to-thickness ratio of about 2. This study
attempts to fill some gaps on this subject by investigating fuses falling into the
following categories:
Thin slit fuses under large shear deformation
Butterfly fuses with various combinations of geometry and thickness
Fuses partially or fully restrained from buckling
The objective of this investigation is to (1) verify the performance of slit and butterfly
fuses through testing; (2) investigate the influence of several design variables
including width-to-thickness ratio, length-to-thickness ratio, and the effect of
restraining buckling; 3) validate numerical modeling techniques; (4) develop design
methods. This report is composed of five chapters which are organized as follows.
• Chapter 1 introduces the background and objectives of this study, and reviews
selected past studies relevant to the current work.
• Chapter 2 explains the design equations that govern the strength, stiffness and
geometry of the fuses.
• Chapter 3 presents the preparation, setup, and results of the cyclic quasi-static test
of the fuses, and numerical modeling of selected test specimens.
• Chapter 4 summarizes the major findings, conclusions and recommendations from
this study and suggests future research needs.
21
Five shear panels made of fiber cementitious composites were also tested in this
project to explore the potential application of high-performance cementitious materials
in the controlled rocking system. Tables and figures describing the test setup and
highlights of test results are included in Appendix A.
22
(a) (b)
Figure 1.1 Shear plate fuses in the controlled rocking system: (a) single frame; (b)
dual-frame
(a) (b)
Figure 1.2 Deformation demand in the fuses: (a) single frame; (b) dual-frame
A B A
γ
Beff
θθγ
A
Beff
23
Figure 1.3 Tensile deformation and uplifting
Figure 1.4 A generic shear plate with slits
(a) (b)
Figure 1.5 Test of steel slit dampers: (a) fractured specimen; (b) hysteresis response
(Chan and Albermani, 2008)
24
(a) (b)
Figure 1.6 Test of steel shear wall with slits: (a) test specimen; (b) hysteresis
response (Hitaka and Matsui, 2003)
(a) (b)
Figure 1.7 Test of honeycomb dampers: (a) test specimen; (b) hysteresis response
(Kobori et al., 1992)
25
(a) (b)
(c)
Figure 1.8 Test of ADAS elements: (a) specimen section perpendicular to loading
direction; (b) specimen section parallel to loading direction; (c)
hysteresis response (Aiken et al., 1993)
26
27
CHAPTER 2
FUSE DESIGN EQUATIONS __________________________________________________________________________________
2.1 Slit Fuse
A generic slit fuse is shown in Figure 2.1a. The fuse links can be viewed as a set of
beams in parallel with fixed ends. Strength and stiffness equations for the fuse are
derived assuming double curvature flexural behavior of these beams. The regions
outside the dash lines shown in Figure 2.1a are to be clamped by fixture devices such
as two angles sitting back to back. These regions are hence assumed to be part of the
rigid boundary and neglected when deriving the stiffness equations. The distance
between the end of the links and the edges of fixtures is denoted as c, and the solid
region with height c is referred to as the band zone. The band zones do not affect
strength of the fuse which is governed by flexural capacity of the links. However, they
do affect the stiffness of the fuse especially when c is not negligible relative to L.
Yield strength, Qy, corresponding to yielding of the exterior fiber at the link end
sections due to bending stress, and plastic strength, Qp, corresponding to plastic hinge
formation at the link end sections can be calculated by the following equations based
on the simple relationship between shear force and bending moment.
Ltnb
LM
nQ yyy 3
2 2 σ== (2.1)
Ltnb
LM
nQ ypp 2
2 2 σ== (2.2)
where My = yield moment of the cross-section (= b2tσy/6), Mp = plastic moment of the
cross-section (= b2tσy/4), σy = yield stress of steel, n = number of links, b = width of
28
fuse link end section, t = thickness of fuse plate, L = length of fuse links. Given the
large plastic deformation expected in a fuse, Qp is considered more representative of
fuse strength than the initial yield strength, Qy. Therefore fuse nominal strength refers
to the plastic fuse strength, Qp.
As shown in Figure 2.2, under shear load, displacement over the entire effective height
(L + 2c) of a fuse link is assumed to consist of two components, the bending
component, δB, and chord rotation component, δR. The bending component comes
from flexural deformation over the fuse link length, L, assuming rigid ends. The chord
rotation component is assumed to result from the rotation angle, α, at each end of the
link due to flexibility of the band zones. Although the band zones run continuously
above and below fuse links, it is assumed for stiffness calculation that each end of a
link is only restrained by a band segment with width, b, and this segment is separated
from the rest of the band zone. A unit shear load is applied and the bending and
rotation components of the displacement can be separately derived as follows.
3
3
3
33
1212
12 EtbL
EtbL
EIL
linkB ===δ (2.3)
LR αδ =
(2.4)
where E = Young’s modulus of steel, and Ilink = moment of inertia of one fuse link
cross-section (= tb3/12). The rotation angle, α, can be calculated by the following
equation.
33
62122
EtbcLc
EtbLc
EILc
EIMc
linklink
c =====ϕα
(2.5)
where φ = curvature of the band segment and Mc = moment at the band segment (= L/2
under a unit shear load). It then follows that,
3
26EtbcL
R =δ
(2.6)
29
Eqs. (2.3) and (2.6) can be rearranged to obtain the fuse stiffness term corresponding
to each displacement component assuming a total shear load equal to the number of
links, n, is applied to the fuse.
3
3
LEtbnnk BB == δ (2.7)
2
3
6cLEtbnnk RR == δ
(2.8)
where kB = bending stiffness of fuse links, kR = stiffness of band zone. The total
stiffness of the fuse, k, can be obtained considering a fuse link and the band segments
are equivalent to two springs in series.
B
RBRB
RB kkkkk
kkk+
=+
=1
1
(2.9)
Substituting Eqs. (2.7) and (2.8) into the above equation leads to
3
611
611
+=
+=
LbnEt
Lck
Lck B
(2.10)
2.2 Butterfly Fuse
Derivation of equations for the butterfly fuse is more complicated because the cross-
sectional properties of a link are not constant, which necessitates integration over the
height of a link. Prior to establishing strength and stiffness equations, criteria should
be developed to determine the shape of the butterfly links.
2.2.1 Determination of Fuse Link Shape
Figure 2.3 illustrates a generic butterfly link and its internal force diagrams under a
shear load, Q. The logic for a butterfly-shaped link is to better align bending capacity
with the shape of the moment diagram, which results in larger section at the ends
where moment is the maximum and smaller section in the middle where moment is
zero. Such alignment leads to more distributed yielding, as opposed to the
30
concentrated yielding at the ends of rectangular links in a slit fuse. Ideally, the links
should be proportioned such that all the points along the edges of the link yield
simultaneously, which is expressed by the following equation:
yxI
xwxM σ=)(
2/)()( (2.11)
where M(x), w(x), I(x) = bending moment, width and moment of inertia of a section
that is at distance x from the horizontal centerline of a link (Figure 2.3). Satisfaction
of such relationship results in an hourglass-shaped link with curved edges, which is
rather complicated for design and fabrication. A butterfly shape with straight edges
may provide a satisfactory compromise between perfect simultaneous yielding and
simplicity.
The shape of a butterfly link is determined by the ratio of width of the middle section
to width of the end sections, a/b, as shown in Figure 2.1b. The proposed approach to
determine the a/b ratio is based on considerations for fracture resistance. Two types of
locations on a link are considered vulnerable to fatigue and the ensuing fracture: a
plastic hinge where deformation concentrates, and a joint of two edges with different
curvatures which is prone to stress concentration. Both situations tend to result in
strain localization and ultimately fracture. For slit fuses with rectangular links, it is
reported that fractures occur at link ends (Hitaka and Matsui, 2003, Chan and
Albermani, 2008), which is understandable because the end of a rectangular link has
both abrupt change of curvature and plastic hinge formation, making it the most
vulnerable cross-section. To decouple the compounding factors, location of plastic
hinges should be away from the joints with curvature change. A butterfly link has
curvature change at the end sections, as well as at the waist. Plastic hinges therefore
should be away from both the end and middle sections, leaving the quarter-height
point between the end and middle sections a choice for the location of a plastic hinge.
The following derivation finds the a/b ratio that leads to first plastic hinge formation at
quarter-height points of the links. It starts by locating the point with maximum
bending stress.
31
The bending moment, width and moment of inertia of a cross section at height x are
calculated as follows:
x
LMxM 02)( = (2.12)
ax
Labxw +
−=
)(2)(
(2.13)
tax
LabtxwxI
33 )(2
121)(
121)(
+
−==
(2.14)
where all variables are defined in Figures 2.1b and 2.3. Stress at the exterior fiber of a
cross section is found as
[ ]2
0
)(2
1122
)()()(
xaLxabLtMxw
xIxM
+−==σ (2.15)
Having dσ/dx = 0 leads to the location and magnitude of the maximum bending stress.
2
,)(2
3 0max
Lab
axwhenatab
Mm −=
−=σ (2.16)
Let xm = L/4, and it is found
31
=ba (2.17)
2.2.2 Strength and Stiffness Equations
Yield strength can be found by equating the maximum stress to yield stress.
yatab
M σσ =−
=)(2
3 0max (2.18)
It follows that
3)(2
0yatab
Mσ−
= (2.19)
Latabn
LMnQ y
y 3)(42 0 σ−
== (2.20)
Substituting a/b = 1/3 into the above equation leads to
32
Ltnb
Q yy
σ2
278
= (2.21)
As stated earlier, nominal strength is defined as the load that causes the first plastic
hinge to form at the same cross-section where yielding first occurs.
Latabn
QQ yyp
σ)(223 −
== (2.22)
When a/b = 1/3, the nominal strength becomes
Ltnb
Q yp
σ2
94
= (2.23)
Unit load method is used to derive the equation for bending stiffness, kB, of the
butterfly fuse. A unit shear load is applied at the top of a butterfly link. Internal work
is then integrated over the height of the link to find displacement due to bending, δB.
[ ]∫∫ +−==
−
2/
0 3
22/
2/
2
/)(2122
)()( LL
LB dxaLxab
xEt
dxxEI
xMδ (2.24)
The integration function with symbolic expressions in the programming software
Matlab (Matlab, 2010) is used to find the following integrated form.
tEbba
Lba
ba
ba
B33
3
)1(2
)3)(1(ln23
−
−−+
=δ (2.25)
Stiffness of butterfly fuses can then be derived the same way as that of the slit fuses.
3
3
33
47.0)3)(1(ln23
)1(21
≈
−−+
−==
LbnEt
LtnEb
ba
ba
ba
ba
nkB
B δ (2.26)
2
3
61
cLEtbnnk
RR ==
δ
(2.27)
3
47.08.211
11
+=
+=
LbnEt
Lck
kkk B
RB (2.28)
33
2.3 Buckling-Restrained Fuses
Alternative details are also studied to understand the influence of restraining buckling
on fuse behavior. Two pairs of back-to-back channels are installed covering the upper
and lower portion of the slit fuse links to restrain their out-of-plane deformation as
shown in Figure 2.1c. The welded-end connection shown in Figure 2.1d is designed
by research collaborators from Greg P. Luth and Associates, Inc. This scheme is
motivated by the potential application of fuses in coupled concrete walls, where heavy
steel plates with bolt holes may be precast into reinforced concrete walls and fuses can
be later bolted to these plates. This detail also provides partial bucking-restraining
effect since the top and bottom portions of the fuse links are braced on one side by the
rigid end plates.
34
(a) (b)
(c) (d)
Figure 2.1 Fuse geometry: (a) slit fuse; (b) butterfly fuse; (c) slit fuse restrained
from buckling; (d) butterfly fuse with partial buckling restraining
Figure 2.2 Decomposition of fuse link displacement
35
Figure 2.3 Loading condition and internal force distribution of a butterfly fuse link
36
37
CHAPTER 3
QUASI-STATIC CYCLIC TEST __________________________________________________________________________________
3.1 Test Setup
3.1.1 Test Matrix
Parameters of the test specimens are summarized in Table 3.1 and definition of the
variables can be found in Chapter 2. The S series contains five slit fuse specimens,
while the B series includes six butterfly fuse specimens. In addition to fuse type, the
key parameters investigated here are b/t and L/t ratios which indicate relative thickness
and slenderness of the links respectively. These two ratios are important factors
related to failure mechanism of the fuse. Hitaka and Matsui (2003) reported that for
slit fuse, stable hysteretic behavior can be sustained up to 3% shear drift if the b/t ratio
is limited to 20. Since this study is aimed at 20% drift, the b/t ratio is set to be below
or slightly above 10 to improve the ductility. Motivation for parameter selection of
each specimen is explained below.
Specimen S12-36 and S10-40 have approximately the same parameters except that
S10-40 has more links (n = 13) and therefore is wider overall. The purpose of these
two specimens is to study the influence of overall aspect ratio of the fuse plate on
global shear buckling. S10-56 can be compared to S10-40 to illustrate the influence of
the L/t ratio. The last two S series specimens are tested to investigate alternative
connection details and the effect of restraining buckling.
38
The B series starts with a control specimen B10-36 that can be compared to B09-56
and B06-37 to illustrate the influence of the L/t and b/t ratios, respectively. B02-14 is
a much thicker fuse for which low cycle fatigue is expected to be the dominant failure
mechanism. The remaining two specimens investigate the influence of welded end
details and partial buckling restraining.
Sequence number in Table 3.1 indicates the actual testing sequence of the specimens.
The eleven fuses were cut from a total of 6 plates, denoted as plates A ~ F. Laser
cutting was used in most of the cases primarily for economic consideration. Water
cutting was only used when the thickness of the plate exceeded the capacity of the
laser cutting provider. Dimension ws is the diameter of rounded slit ends, or the ends
of openings in the case of the butterfly fuse. After fabrication of the first three
specimens, ws was increased to make it easier for laser cutting, as well as improve the
smoothness of the slit ends. Specimens 4 ~ 7 were designed by Gregory P. Luth &
Associates, Inc. (GPLA). They used a different criterion to decide the shape of
butterfly links and therefore used an a/b ratio not equal to 0.33.
3.1.2 Loading Setup
All fuses were tested in a steel frame with a horizontal actuator as shown in Figure
3.1. At each side of the fuse, pin-ended struts connect the loading beam to the
foundation beam. In addition to ensuring that the loading beam stays horizontal
during testing, the struts also balance vertical tension force developed in the fuse at
large drift level. Without the struts, tension force will be smaller because the loading
beam can move down without resistance, which reduces vertical stretch in the fuse.
The top loading beam was braced out of plane by a frame of vertical and diagonal
angles as shown in Figure 3.2a. Connection of the actuator is shown in Figure 3.2b.
39
3.1.3 Instrumentation
A schematic diagram of the instrumentation layout is shown in Figure 3.3. In addition
to the actuator’s built-in displacement transducer and load cell, two temposonic
transducers (T1 and T2) and 5 linear variable displacement transducers (L1 ~ L5) were
used to measure deformation of the fuses and detect any anomalous movement in the
test setup. For clarity, these transducers are shown to be in front of the fuse in Figure
3.3. In the actual test, they were attached to the back side of the fuse. Strain gauges
(SG1 ~ SG8) were attached to the vertical struts to measure axial forces as well as out-
of-plane bending deformation. Most instruments were attached to a separate base
column that was ballasted to the floor. T1 was fixed to the top fuse connection angle,
and its readings very closely matched those of the actuator. T2 was attached to the
fuse just outside of the connection angle to capture any slip between the angle and the
specimen. A similar instrument pair, L3 and L4 were used at the bottom of the fuse.
In early tests, L1 and L2 were placed at the ends of the slits to determine if shear in the
continuous band zones above and below the slits contributed to the overall deflection.
These instruments were anchored to the connection angles, and their absolute
displacement could be determined by adding the readings from T1 and L4,
respectively. After the first three tests, it was realized having the bands outside of the
slits is not an efficient way to utilize the material. Thus most of the remaining
specimens were designed to have links almost as long as the distance between the top
and bottom angles, or the distance between top and bottom welding lines in the case of
fuses with welded ends. L1 and L2 were hence eliminated in those tests. Another
instrument L5 was used in early tests to capture vertical deformation. It was later
found that very little information could be derived from this measurement, and L5 was
eliminated too. Figure 3.4 shows photos of the actual instruments. Table 3.2
summarizes all the data channels.
Figure 3.5 shows the control system which comprised an MTS controller, a control
computer, and a data acquisition computer with the LabView data acquisition software
40
installed. The MTS controller received commands from the control computer, and
sent signals to the actuator. It also received back displacement and force readings
from the actuator. Since the controller can only accommodate six input channels and
does not have strain gauge channels, the LabView system was necessary to
synchronize and record all data readings including actuator readings sent from the
MTS controller.
3.1.4 Loading History
The loading protocol for the steel fuse test shown in Table 3.3 was created based on
the AISC code loading sequence for link-to-column connections in eccentrically
braced frames (AISC, 2005). This loading protocol was adopted because of the
similarity in the function of links in eccentrically braced frames and fuses in the dual
rocking frames. Both types of elements are the primary source of hysteretic energy
dissipation in their corresponding structural systems.
The AISC protocol was modified by adding three extra sets of six low-amplitude
cycles preceding the prescribed loading procedure. The intent was to capture the
behavior of the fuses in the elastic range, since most of the fuses were predicted to
yield before a shear drift ratio of 0.00375 radians. Shear deformation is defined as
h∆
=γ (3.1)
where Δ = horizontal displacement of loading beam; h = distance between bolt lines (=
508 mm (20 in), Figure 2.1a). This definition for shear deformation ensures the same
displacement history for all specimens since h is a constant. However, it needs to be
noticed that actual shear deformation occurs primarily within the height of the links.
Link length based shear deformation is thus defined as
LL∆
=γ (3.2)
where L = length of links. For specimens with different link length, L, the loading
history based on γ results in different γL. In such cases, γL can be used to compare fuse
41
deformation capacities. Conversion between the two measures can be done by the
following equation:
γγLh
L = (3.3)
The shape of the displacement history is defined as a sinusoidal curve, with lower
loading rates occurring at the points of peak displacements. For the majority of the
loading cycles, the loading rate was set such that one complete cycle took 50 seconds.
Beyond the 0.003 radian displacement level, the loading rate was set at an average of
1.3 mm/s to aid in observation of the test and to ensure that the test remained quasi-
static.
The first fuse test used higher loading rates that resulted in some inaccuracies in the
test procedure. After this test, the actuator was tuned, and the loading rates were
adjusted to the levels shown in Table 3.3.
3.1.5 Material Testing
A total of seven steel plates were used to manufacture the fuse specimens. To
determine basic material properties, uniaxial tensile tests were performed. Plates A~E
were tested at Stanford University, while plates F and G were tested by CEL
Consulting who was commissioned by GPLA.
The tension test at Stanford followed ASTM Standard E8 (ASTM, 2003). Tension
coupon types were determined according to the thickness of each plate (Figure 3.6).
The MTS 89-kN (20-kip) servo-hydraulic load frame was used to conduct the test with
displacement control. Figure 3.7 shows the setup of a typical test. An extensometer
with 25 mm (1 inch) gauge length was attached to the coupon to measure
displacement. Data channels included actuator displacement and force, and
extensometer displacement. Cross sections of the coupons were measured before and
after the test to determine reduction in area. For cylinder coupons, diameters in two
42
perpendicular directions were measured to find the average diameter. Table 3.4
summaries the cross-sectional geometry of each coupon.
Engineering stress-strain relationships for all the coupons are plotted in Figure 3.8.
Yield stress was obtained following the 0.2% offset method (ASTM, 2003). During
the test, the extensometer was removed from the coupon once necking started in order
to protect the equipment. Two coupons were tested for each plate and the results are
summarized in Table 3.5. Results provided by CEL Consulting are given in Table 3.6.
3.2 Test Result of Slit Fuse S12-36
3.2.1 Examination of the Test Setup
Before reviewing fuse behavior, instrument readings are analyzed to examine the test
setup. Redundant displacement readings at multiple locations are first compared. The
difference between the actuator displacement reading and the measured displacement
at the top angle (actuator displacement - T1) varied between 1 ~ 3 mm and was
proportional to the actuator force. This might have come from closing of gaps in the
actuator-loading beam connection, as well as slip between loading beam and top
angles. The fact that it is proportional to actuator force indicates there might be elastic
deformation in the test rig. Potential sources include elastic deformation of the
reaction column and the loading beam. Difference was also observed between top
angles and the fuse (T1 - T2) which was very small at first and then increased to about
2 mm at larger displacement levels. Difference between the fuse and the bottom
angles (L3 – L4) was similar. Such differences caused discrepancy between actuator
displacement and the actual shear drift that was experienced by the fuse. Actual drift
of the fuse is calculated as the relative displacement between fuse top edge and bottom
edge, or Δ = T2 – L3. In the cases when L3 reading is erroneous, which happened in
tests No. 1, 2, 9 and 11 (S12-36, S10-40, B09-56, B02-14), L4 is used instead to
calculate fuse drift.
43
Another source of inaccuracy comes from the actuator response. Figures 3.9a and b
compares target displacement, actuator displacement and fuse drift in two loading
segments. It is evident the actuator constantly missed target by 1~2 mm when
displacement was less than 20 mm, which may be attributed to the lack of sensitivity
of the actuator at small displacements. The discrepancy between actuator
displacement and fuse drift is also notable. This is due to the inaccuracies across
multiple connection interfaces explained in the previous paragraph. Total difference
between prescribed loading history and fuse drift was about 5 mm during most part of
this test. Figure 3.9c shows the load-displacement relationship with different
displacement measures at the beginning of the test. Slope of the load-actuator
displacement curve is smaller as a result of the imperfection in the test rig. The curve
does not have flat portions that are typical when there are slips, which means slip is
not a significant source of discrepancy between the actuator displacement and fuse
drift. In subsequent tests, the discrepancy was observed to be between 2 to 4 mm
varying from test to test. Such difference is insignificant considering the large
displacement inputs which usually went above 70 mm,
3.2.2 Conventions
In the discussion of test results of this specimen and all other specimens, load-
deformation relationship and photos are used to illustrate observations from each test.
Load is normalized by nominal strength Qp (Eq. 2.2), and net displacement Δ
experienced by the fuse is divided by fuse bolt-to-bolt height, h = 508 mm (Figure
2.1), to calculate shear deformation, γ (Eq. 3.1). The calculation of Δ has been
explained in Section 3.2.1. When comparing the behavior of two different fuses, the
fuse link length-based deformation measure, γL = Δ/L, is used instead of γ, which
provides a consistent basis for comparing actual drift angle in the fuse links regardless
of the generally different link length. Load and displacement towards the left in
Figure 3.1 is assumed as positive. Photos are selected to show the key deformation
44
stages experienced by a fuse: yielding, buckling, crack initiation and fracture.
Displacement written on the labels in the photos refers to target values according to
the loading history, whereas drift shown in the caption of each figure are based on
actual measured values. When referring to a specific loading cycle, for instance 58P,
the number indicates the cycle number (first column in Table 3.3) and “P” or ”N”
means the positive or negative peak. Conventions described here also apply in the
discussion of subsequent tests.
3.2.3 Load-Deformation Relationship
Load-deformation relationship obtained from the test is shown in Figure 3.10, while
photos showing the progression of fuse deformation are shown in Figures 3.11 and
3.12. The fuse demonstrated stable hysteretic behavior at relatively small drift level,
and remained elastic in the first 36 loading cycles. Nonlinear behavior was observed
at about 0.27% drift, after which the hysteretic loops began expanding as the fuse
deformation increased and load exceeded fuse nominal strength. Quantitative
comparison between predicted and observed stiffness and strength is presented in
section 3.6.1. During this stage, yielding pattern can be observed by looking at the
lines radiating from the ends of the slits due to flaking of the mill scale (Figures 3.11b
and 3.12a). These lines indicate high stress level and concentrated yielding in these
regions where plastic hinges are expected to form.
No notable change in behavior was observed between repeated cycles at the same drift
level. The only exception occurred during the loading at 1.78% shear deformation
whereby a small drop in strength was observed in one of the repeating loading loops.
This was due to an accidental stop of the test when the actuator hit a mistakenly set
interlock of 12.5 mm. The test was restarted from the beginning of cycle No. 51. The
links began to buckle out of plane at 1.78% drift when 6 mm out-of-plane deflection
was measured at the lower portion of the first link from left (Figure 3.11c). After
buckling, fuse strength continued increasing and peaked at 2.41% drift when buckling
45
became evident and caused the strength to drop (Figures 3.11d and 3.12b). Fracture
had initiated at some of the slit ends by the time fuse reached 5.71% drift (Figures
3.11e and 3.12c). From there, cracks propagated horizontally with each cycle, and
significant pinching was observed from the hysteretic loops. By 9.75% deformation,
some cracks were 25 mm long, and the test was finally stopped at 12.35% drift when
three links completely fractured at their top ends (Figure 3.11f).
3.2.4 Outer and Inner Links
It is interesting to note the two links at the left and right edges had the most notable
buckling, yet the smallest cracks (Figure 3.11f). The more severe buckling
deformation can be explained by the change of boundary conditions at the edges. The
inner links are restrained by the continuous band zones above and below the links.
For the edge links the restraining effect is weaker because the bands do not extend
outside the links. As a result, buckling of the edge links extended into the band zones,
which reduced concentration of deformation at the ends of slits and alleviated the
propagation of cracking. Such evidence also hints that restraining buckling might
aggravate low cycle fatigue, which will be discussed in more details in test S10-56BR.
3.2.5 Bending and Tension Mode Behavior
Analysis of mechanics reveals that the fuse behavior can be characterized by two
distinct modes: bending mode and tension mode. The fuse starts by exhibiting
bending mode behavior. As shear deformation increases and buckling occurs, it
transitions to tension mode behavior. Accompanying this transition is the change of
force distribution in the pinned struts. When deformation is relatively small, the fuse
links are like parallel beams with fixed ends deforming in double curvature. Figure
3.13a shows a free body diagram including the loading beam and the fuse cut at its
middle height. Since middle height point is the inflection point of the link, moment is
zero at this section. Shear force Qb resulting from bending capacity is equal to load Q
46
applied by the actuator, and the two together causes the overturning moment which is
balanced by axial forces in the struts, N1 and N2. From equilibrium of moment, it
follows that
Q
dQeNN 43.021 ≈== (3.4)
Force pattern in the two struts during the bending mode behavior is that they are equal
in magnitude and opposite in sign, with one pushing the loading beam up and the other
one holding it down. After fuse buckling, flexural strength of the links is reduced and
tension capacity becomes the primary source of lateral resistance and consequently
tension mode becomes dominant. As shown in Figure 3.13b, shear force Qb is smaller
since bending capacity is reduced by buckling. Meanwhile, tension force T in the fuse
grows large as a result of the vertically stretched links at large shear deformation. Tx,
the horizontal component of the tension force, T, thus becomes another source of shear
force to resist external load Q.
xb TQQ += (3.5)
As shear deformation increases and buckling becomes more severe, bending capacity
keeps decreasing while tension force increases and gradually replaces the bending
contribution to become the dominant source of shear force. Behavior of the struts is
also significantly affected by this change. The increase of Ty, the vertical component
of T, causes both of the struts to be in compression. Relationship between forces in
the struts, fuse and the external load can be derived from the following equilibrium.
=−×
=+
QedTdN
TNN
y
y
21
21 (3.6)
which leads to
−≈−=
+≈+=
QTd
QeTN
QTd
QeTN
yy
yy
43.05.02
43.05.02
2
1 (3.7)
47
Comparing Eqs. (3.4) and (3.7), it is evident that after the system switches to tension
mode, N1 becomes larger in compression and N2 turns from tension into a smaller
compressive force if external load Q is not too large (0.43Q < 0.5Ty). Eq. (3.7) is
generally applicable for both behavior modes and can be used to explain the change in
strut force pattern. When drift is small, Ty ≈ T ≈ 0, and N1 and N2 are equal in
magnitude and opposite in sign, which agrees with Eq. (3.4). Note the equations do
not look the same because in deriving Eq. (3.4), a tensile N2 is considered positive,
whereas in Eq. (3.7) the opposite is assumed. As drift becomes larger, Ty increases.
As a result, compressive force N1 becomes larger and tensile force N2 becomes smaller
and smaller and eventually reaches zero when 0.5Ty = 0.43Q. When the fuse
transitions to tension mode as Ty gets even larger, N1 keeps increasing and N2 turns
into a compression force.
Force pattern change in the struts can serve as evidence to the mode transition of the
fuse. Figure 3.14 shows forces in the struts calculated from strain gauge reading. The
selected force histories correspond to the transition from bending to tension mode.
Positive force in struts refers to tension, and positive actuator force is towards the left.
Left and right struts correspond to N1 and N2, respectively. Prior to the 35th second, N1
and N2 moved in opposite direction and their magnitude was proportional to actuator
force. The ratio of strut force to actuator force also roughly agreed with the prediction
by Eq. (3.4). Although not shown in the plot, it is found at lower drift the agreement
was even better. Such observation exemplifies the force pattern in bending mode.
After the 35th second, the tension mode patter began to dominate. In the first upward
half-cycle of actuator force, left strut force, N1, remained negative and kept increasing
in absolute value. Right strut force, N2, as predicted, was first in tension, but quickly
started dropping to zero and turned into compression as the actuator force reached its
peak. This patter persisted from then on.
Mode transition also explains characteristics of the load-deformation curve shown in
Figure 3.10. Until 3.86% drift cycle, the fuse was in bending mode, and stable
48
hysteretic behavior was observed. Since the 5.71% loop, it gradually transitioned to
tension mode. During a loading cycle, strength remained low and flat through most of
the loop and only picked up at large drifts. In the flat portion, it was the remaining
bending capacity that constituted shear force. At peak drifts, contribution from
tension grew due to the combination of larger tension force and more inclination of the
links.
3.2.6 Strut Pin Fracture
When the test was paused at the positive peak displacement of the 56th cycle, a very
loud “bang” was heard. This “bang” caused abrupt changes in strain gauge readings,
especially at the lower portion of the left strut. It was later found in the second test
that both of the pins in the left strut were fractured. It is suspected that the lower left
pin fractured in this test when the “bang” was heard. Evidence to this is that peak
strength is relatively low at large drifts compared to a similar specimen S10-40. If the
pin had fractured, the struts could take less compression force and therefore tension in
the fuse would be smaller.
Displacement sensor L3 and strain gauge 7 were not functioning during the test.
Strain gauge 4 had a huge spike when the loud “bang” occurred at positive peak of
56th cycle. After that its reading seemed erroneous.
3.3 Test Result of the Rest of Slit Fuse Specimens
3.3.1 S10-40
This fuse has similar geometric parameters compared with S12-36. The links are 25
mm longer and the fuse plate is wider. One of the objectives of this test is to study the
influence of fuse aspect ratio on global shear buckling of the plate.
49
During this test, two pins connecting the left strut to the loading frame were found
fractured. The lower left one is believed to have fractured in the test of specimen S12-
36, and the upper left one fractured in this test when a loud “bang” sound was heard.
The fractured pin in the pin hole is shown in Figure 3.15. The test was stopped and all
the four 25 mm diameter pins were replaced by 38 mm ones. The struts were also
modified to accommodate this change. The rest of the test was resumed and finished
on a different day.
Figure 3.16a shows the load-deformation relationship, and Figure 3.17 shows
deformation of the fuse. Behavior similar to that of specimen S12-36 was observed.
The left edge link first started deflecting out-of-plane at 0.77% drift. The measured
out-of-plane deflection was less than 2 mm. Buckling became visible from the next
drift level 1.1% (Figure 3.17c). The two edge links had the most notable buckling
compared to inner links because of the change in boundary conditions. Despite
apparent buckling, strength kept increasing until 2.8% drift (53P, Figure 3.17d).
Crack initiated when the fuse was at the negative peak of the same cycle (Figure
3.17e). One cycle later (54N), the upper left pin in the left strut fractured. Actuator
was brought to zero-load and the test was stopped, which is why the load-deformation
curve ended when it crossed zero force returning from -4.1% drift. When the test was
resumed after replacing the fractured pins, the fuse apparently switched to tension
mode. The curve started pinching. The fuse had less and less strength when it crossed
the zero displacement point, which indicates decreasing contribution to shear force
from bending capacity. On the other hand, contribution from the horizontal
component of tension force was increasing and over-compensated the loss of bending
contribution. The net effect was increased strength at peak drifts. As cracks grew
longer, the reduction of cross-sectional area began to affect tension force and the total
shear strength started decreasing again at 13% drift (58P). Test was stopped at 15%
drift when 10 out of the 13 links totally fractured at the ends (Figure 3.17f).
50
Load – deformation relationships of this specimen and specimen S12-36 are compared
in Figure 3.16b. Note in this figure, the fuse link length based deformation measure,
γL = Δ/L, is used instead of γ. Overall, the two fuses experienced the same stages
before the fatigue induced failure. Peak strengths during tension mode behavior was
larger in specimen S10-40, which is why it is suspected one pin already fractured in
test S12-36. It is again observed in this test that the two edge links had the smallest
cracks, while their buckling extended into the band zones above and below the links.
This reconfirmed the finding from test S12-36 that less restraining on buckling
alleviates low cycle fatigue. Global shear buckling of the entire plate was not
observed in either of the two tests. The fuse still behaved as a group of individual
links, and the change of overall aspect ratio did not show any apparent effect on the
behavior.
3.3.2 S10-56
This specimen examines the influence of L/t ratio by comparison with specimen S10-
40. Load – deformation relationships of the two specimens are shown in Figure 3.18,
while Figures 3.19 and 3.20 show photos of specimen S10-56 during the test. Similar
behavior to previous tests was observed despite the longer links. Stable hysteretic
behavior was sustained for many cycles after plastic hinge formation. Buckling did
not appear until 4.0% drift (54P) when links on the left side started deflecting out-of-
plane. The first measured Out-of-plane deflection was 3 mm. It is so small that
buckling can hardly be seen in the photo (Figure 3.20c). In the same cycle at the
negative peak drift (54N), all links began to buckle. Next cycle saw crack initiation at
the ends of some slits and evident buckling with drift reaching 6.1% (Figure 3.20d).
Out-of-plane deflection increased to 10~15 mm, and cracks were about 1~2 mm long.
At this moment strength dropped for the first time. In the subsequent cycles, tension
mode behavior became apparent and there was moderate strength increase between
8% and 10% shear deformation. During these cycles cracks appeared in most of the
links and some cracks grew to 5~6 mm long. Cracks grew much longer (25 mm) in
51
the 14% cycle (Figure 3.20e) and a clear reduction of strength can be seen. Finally at
the negative peak of the 16% cycle (60N), two links completely fractured and the test
was ended (Figures 3.19 and 3.20f).
Comparing load-deformation relationship of specimens S10-56 and S10-40 (Figure
3.18b, the two exhibited very similar response prior to reaching the first peak strength
point at about 5.7% drift ratio. Difference in L/t ratio did not have an apparent effect.
After the peak, both switched to tension mode. S10-40 continued significant increase
in peak strength, whereas S10-56 did not have much increase. Such difference means
there was a larger tension contribution in the case of S10-40. Two factors affected the
contribution of tension force to shear strength, the amount of tensile force and the
inclination angle of the links, both of which are related to length of the links. Given a
certain drift, tensile deformation in a shorter link is larger, and so is its inclination
angle. Therefore S10-40 was more affected by tension force. This comparison shows
that for fuses with the same b/t ratio, L/t ratio has little impact on their behavior when
shear drift is small and bending mode is dominant. However, the one with smaller L/t
ratio can reach higher strength once the fuse is in tension mode.
3.3.3 S10-36W
This specimen has similar geometric parameters as specimen S10-40. The key
difference is it is welded to two heavy backing plates that are bolted to the test rig
(Figure 3.23a). The backing plates can be considered rigid relative to the fuse. In
addition to connecting fuse to the test rig, they also provide partial out-of-plane
bracing. Another difference in fuse geometry is that the slits are 13 mm wide, as
opposed to 3 mm wide for previous specimens. This change was made to have a
bigger rounded end so that stress concentration may be reduced at slit ends. Besides,
it also makes it easier for laser cutting to pass the slit ends.
52
Load – deformation relationships are shown in Figure 3.21, and photos shown in
Figures 3.22 and 3.23. Buckling started at 2.4% drift (51P, Figures 3.22a, 3.23c).
Measured out-of-plane deflection was about 5 mm at the lower left side. The
hysteresis curve was still stable until the next amplitude (53P), after which the loading
loop began to shrink upon reloading. Buckling became notable at this moment
(Figures 3.22b, 3.23d). Meanwhile crack initiation was observed at link ends. At
negative peak of the same loading loop, out-of-plane deflection grew to 15 mm. In the
next two cycles, cracks did not grow much. Links clearly were gouging against the
backing plates which kept them from twisting towards the back side of the plates. In
the 56th cycle, cracks started growing rapidly and soon became 10 mm long (Figure
3.23e). Marks from gouging were visible on the backing plates. Significant cracking
caused strength to drop in the subsequent cycles. After two cycles, the far left and
right links fractured through and the test was ended (Figures 3.22c and 3.23f).
Compared to specimen S10-40 (Figure 2.21b), specimen S10-36W sustained about the
same amount of deformation before the fracture failure. However, S10-36W
dissipated more energy since it had less strength degradation. Buckling occurred in
specimen S10-36W at 2.4% deformation, slightly later than the unbraced S12-36
which buckled at 1.8% shear deformation. However, crack initiation started earlier (at
3.4% drift for S10-36W v.s. 5.7% drift for S12-36). After buckling, the braced
specimen S10-36W had more increase in peak strength, which may be due to the
friction force between the links and the backing plates. Overall, the backing-plate
connection is a viable design alternative which is especially convenient for connecting
fuses to concrete walls via the end plates. Its buckling restraining effect can also delay
strength degradation and achieve more energy dissipation.
3.3.4 S10-56BR
This specimen is a replicate of S10-56, except that channels were installed to restrain
buckling of the fuse links (Figure 3.25a). Load-deformation relationships are shown
53
in Figure 3.24 and photos shown in Figure 3.25. Behavior of this specimen was
essentially the same as the unrestrained S10-56 up to 4.0% deformation. At this point
buckling was observed in S10-56 and strength dropped in the next cycle, whereas in
S10-56BR, buckling was restrained and strength kept increasing. At the next peak,
crack initiation was observed in both cases at slit ends (Figure 3.25b). Strength of the
restrained specimen then plummeted as cracks rapidly grew. Three cycles later five of
the thirteen links totally fractured at 12.1% drift (Figure 3.25d). On the other hand,
the crack propagation was much slower in the unrestrained specimen and more than
80% of its peak strength was sustained for 4 more cycles after crack initiation.
The contrast displays two effects of restraining buckling. First of all it can delay
strength degradation, although this effect does not have a large impact for a fuse with
20% expected deformation. More importantly, restraining buckling affects low-cycle
fatigue. Since links cannot buckle, strain concentration is more severe at link ends.
Consequently crack propagation is accelerated and the fuse experiences a more brittle
failure once cracks appear. From the point of view of improving ductility, restraining
buckling of the fuse is not desirable.
3.4 Test Result of Butterfly Fuse B10-36
This is the first butterfly fuse tested. Its geometric parameters are close to S12-36 and
therefore can demonstrate the difference between slit and butterfly fuses. The
specimen was designed by GPLA who followed different principles to determine the
geometry of fuse links. The a/b ratio for the links is 0.4 which is close to the 0.33
value recommended in this study.
3.4.1 Load-Deformation Relationship
Load-deformation relationships are shown in Figure 3.26, and photos of the fuse
shown in Figures 3.27 and 3.28. As the deformation increased, flaking of mill scale
54
was observed. Different from slit fuses which had flaking around slit ends, this
specimen had flaking concentrated around quarter height level of the links (Figure
3.28a). This agrees with the prediction of yielding and formation of plastic hinges at
quarter-height sections. More flaking also started appearing along the height of the
links, which indicates distributed yielding. At a drift of 2.4% (P51), links started
buckling. Bottom part of the links deflected out-of-plane for about 2 mm (Figure
3.27c). In the second cycle at the same drift level, strength began to deteriorate.
Following buckling began the transition from bending to tension mode. The fuse
started deriving lateral strength from the horizontal component of tension force in the
links. As links had more tensile deformation, and became more inclined, tension
contribution became larger and eventually offset the loss of bending capacity due to
buckling. Peak strength exceeded the pre-buckling maximum strength at 6.5% drift
(P55). Similar to the slit fuses, significant pinching accompanied the tension mode
behavior. Buckling became so severe that the links turned 90 degrees and became
perpendicular to the loading direction (Figure 3.27d). Cracks initiated at link middle
sections at a drift of 14.3% (Figure 3.28d). One cycle later at 16.3% drift, almost all
links fractured at middle sections and the test was ended (Figures 3.27f, 3.28e and
3.28f).
3.4.2 Pinned Column Mechanism in Tension Mode
Figure 3.29 conceptually illustrates the behavior of a fuse link throughout a loading
cycle. The link starts as a beam deforming in double curvature as shown in part a.
When drift gets larger, lateral-torsional buckling first occurs and the link transforms
into three joined pieces as shown in part b. The quarter height sections become hinges
with limited bending resistance, and the central portion behaves as a column with
pinned ends. As the loading moves more to the left, the central column elongates to
accommodate most of the drift, while the two end portions remain vertical (Figure
3.27 d). When the loading direction is reversed after reaching a peak, the central
column is in compression and provides resistance to the external load as shown in part
55
c. As the shear load grows, the central column experiences flexural buckling as shown
in part d, and consequently, strength of the fuse drops abruptly. This is seen in the
load - deformation relationship in Figure 3.26a, where after unloaded to zero force, the
strength first increases and then drops suddenly due to the flexural buckling of the
links. After the flexural buckling, the link becomes a mechanism and the only lateral
strength comes from the limited bending capacity of the two hinges shown in part e.
This stage corresponds to the long flat portion of a loading cycle. As top end of the
link moves more to the right, the central portion is again in tension and fuse strength
starts picking up again (part f). After reaching the peak and loading is reversed,
central column buckles in compression and the cycle is repeated. It is not surprising
cracks form at the middle section, because it undergoes repeated tension and large
amount of rotation due to flexural buckling.
3.4.3 Comparison with S12-36
Figure 3.26b compares load - deformation relationships of the butterfly fuse and slit
fuse S10-40. Evidently the butterfly fuse sustained more deformation and was able to
withstand higher load at large drifts. Two factors contributed to such differences.
First of all, location of hinges affected the amount of tension contribution. Forming
hinges at quarter height sections of the butterfly links made the tension portion of a
link shorter and more inclined than that in a slit fuse, which led to more tension
contribution to shear strength. Secondly, distributed yielding reduced strain
concentration at a particular point of the link, and delayed crack initiation (14.3% drift
for B10-36 vs. 5.7% drift for S10-40). Butterfly fuses therefore can be expected to be
more ductile than slit fuses.
56
3.5 Test Result of the Rest of Butterfly Fuse Specimens
3.5.1 B09-56
Specimen B09-56 has a b/t ratio almost identical to that of B10-36, and an L/t ratio
roughly 50% larger. Comparison between these two will illustrate the effect of L/t
ratio on fuse performance. Figure 3.30 shows the load-deformation relationships, and
Figure 3.31 illustrates the progression of deformation. Buckling started at a drift of
4.8%, and became evident at 6.7% drift (Figure 3.31c). The pinned-column
mechanism is again observed in Figures 3.31d and 3.31e. At zero displacement, some
of the links buckled in opposite direction and gouged against each other, which left
clear marks on the surface (Figures 3.31e and 3.32). Cracks appeared at middle height
sections of all links at 24.7% drift and one link fractured in the next cycle at 26.7%
drift (Figure 3.31f).
The overall progress of behavior was similar to specimen B10-36. Comparison of
load-deformation relationship shows that the two specimens sustained about an equal
amount of deformation, although the total number of loading cycles experienced was
slightly different. The longer B09-56 had 65 cycles while B10-36 had 60. Buckling
appeared at approximately the same deformation. The increase in peak strength is
more profound in specimen B10-36, which may be attributed to the smaller link length
of specimen B10-36.
3.5.2 B06-37
This specimen differs from B10-36 in that its b/t ratio is half as big, which can be used
to illustrate the influence of b/t ratio. Load-deformation relationship and fuse
deformation is shown in Figures 3.33 and 3.34, respectively. Stable hysteretic
behavior was sustained into much larger deformation compared to fuse B10-36.
Buckling did not appear until 10.5% drift. Such improvement in stability can be
57
attributed to the smaller b/t ratio which improves buckling resistance. After buckling,
the behavior transitioned to tension mode that was again characterized by the same
pinned-column mechanism as shown in Figures 3.34d and 3.34e. Cracks appeared in
the 18.5% loading cycle at a small notch in the second link from the left which was an
initial defect of the plate. Two cycles later, two links fractured at the middle sections
at 22.5% drift.
Figure 3.33b demonstrates that the thicker specimen B06-37 has a larger range of
stable behavior due to the postponement of buckling. Ultimate deformation of B06-37
is slightly smaller. Overall, using smaller b/t ratio improved stability and energy
dissipation capability of the fuse.
3.5.3 B02-14
Design of this specimen was inspired by the work of Kobori et al. (1992) who tested
relatively thick dampers with similar geometry as butterfly fuses. Their design used a
b/t ratio of 2.5, and an L/t ratio of 6.7. Specimen B02-14 adopted a similar b/t ratio,
but relaxed the L/t ratio assuming the b/t ratio governs the thick fuse behavior. Figure
3.35 shows the load-deformation relationship which illustrates stable hysteretic
behavior throughout the test. No notable buckling was observed and bending mode
behavior dominated the whole test. Figures 3.36 and 3.37 show the progression of
fuse deformation. The specimen was white washed before the test. Flaking of white
wash started at quarter height level on the links as predicted by the design principles
explained in Chapter 2 (Figure 3.36b). Larger flaking then appeared at quarter height
level and gradually spread to the full height of the links (Figure 3.36 and 3.37). These
evidences again pointed to the advantage of distributed yielding in the butterfly fuse.
Slight out-of-plane deflection was observed at 22% drift (Figures 3.37c and 3.38a).
However, it did not grow into severe buckling as observed in thinner fuses. At 24.0%
drift, crack initiated on the left link at a point close to the quarter-height section where
plastic hinge was predicted to form (Figures 3.37d and 3.38b, N64). It started opening
58
up rapidly in the next cycle and fractured the link at 26.0% drift (Figures 3.37e, 3.37f
and 3.38c). Due to strain hardening, ultimate strength exceeded twice the design
strength.
The four butterfly fuse test showed that smaller b/t ratio is effective in delaying
buckling and can notably affect fuse behavior. When the fuse is thick enough,
buckling is practically prevented and bending behavior is the only dominant mode
until the ultimate failure due to low-cycle fatigue. The hysteresis loops therefore keep
expanding and more energy can be dissipated. The quarter height sections become
most vulnerable to fatigue because it is where plastic hinges form and hence are
subjected to the largest rotation angle. Thick butterfly fuses are more desirable for the
purpose of energy dissipation. However, the significant overstrength may pose
challenges for capacity-designed members that are connected to the fuse.
3.5.4 B10-36W
This specimen is identical to B10-36 except that it has the welded backing plate
connection (Figure 3.40a). Figure 3.39 shows the load-deformation relationship, and
Figure 3.40 illustrates the progression of deformation. The buckling restraining effect
was not as profound compared to the difference observed between the slit fuse pair
S10-36W and S10-40 (Figure 3.21). Buckling still caused the links to twist for more
than 90 degrees (Figures 3.40c ~ 3.40f). Links gouged against backing plates as they
were twisting (Figure 3.40f), resulting in frictional force that contributed to total shear
resistance. Yielding spread into the strips above and below the links, where “arcs” of
flaking mill scale can be seen (Figures 3.40c and 3.40d).
The load-deformation relationship of B10-36W and B10-36 (Figure 3.39b) shows that
the two specimens had almost the same response up to 2.4% drift when the fuses
started buckling. After buckling, the behavior was still similar. The strength of B10-
36W was 5% ~ 10% higher than that of B10-36, which may be attributed to the
59
friction force between links and the backing plate in B10-36W. Upon reloading,
strength of B10-36W gradually leveled off. It can be seen from Figure 3.40 that the
pinned-column mechanism was not so evident due to the interference of the backing
plates. Therefore, the central part of links did not experience a flexural buckling in
compression upon reloading, which explains the gradual leveling off of the load. The
most notable difference between the two specimens was that no cracks appeared in
B10-36W by the end of the test, which may also be attributed to the absence of
pinned-column mechanism due to the backing plates. Without the hinges, strain
localization at the middle sections was alleviated, which potentially prevented crack
initiation. Overall, the backing-plate connection was beneficial in delaying cracks,
although its influence on reducing strength degradation was not as notable.
3.5.5 B07-18W
This specimen attempts to examine the combination of backing plates and a
moderately thick butterfly fuse. The load - deformation relationship is shown in
Figure 3.41, and progression of deformation shown in Figure 3.42. Buckling appeared
later at 6% drift (55P, Figure 3.42b) compared to thinner fuses, which is more likely
due to the smaller b/t ratio, rather than the presence of the backing plates. Stable
behavior was sustained until buckling became more severe at 8% drift (56P), after
which pinching started. At a drift of 17.1% and a load 1.5 times the nominal strength,
welding at the bottom of the fuse failed and caused the strength to immediately drop
by about 25%. The test was ended at the negative peak of the same cycle (60N, Figure
3.42e). It was then noticed cracks had initiated at quarter height surface of the links
(Figure 3.42f). Cracks did not have a chance to propagate before the welding failure.
At the negative peak of 57th cycle, it was noticed the actuator missed displacement
target by 5 mm. For the following 3 cycles, actuator still missed targets and it seemed
the actuator force was capped by 290 kN in the negative direction (pulling direction
for actuator). In the last cycle, since the fuse had deteriorated, the actuator was able to
60
reach the displacement target without exceeding the 290 kN cap. After the test it was
found a pressure limit set on the hydraulic pump in the pulling direction was the cause
of the load cap.
This test demonstrated the importance of accounting for overstrength when designing
connection details. This is especially important for slit and butterfly fuses because at
large drift, they typically can develop strength 1.5~2 times as much as the nominal
strength due to the combined effect of strain hardening and tension mode behavior.
The latter effect also subjects the connections to double shear in both horizontal and
vertical directions.
3.6 Summary of Test Results
3.6.1 Strength and Stiffness Equations
Key response parameters obtained from the test are summarized in Table 3.7. This
section focuses on the first six columns in the table to examine the strength and
stiffness design equations developed in Chapter 2. The subscript “t” indicates test
result. Yield strength from the test, Qyt, is determined by visually identifying the start
point of nonlinear behavior from the load – deformation curve of each specimen, and
plastic strength, Qpt, is simply assumed to be 1.5Qyt based on the relationship, Qp =
1.5Qy, which can be observed from Eqs. (2.1) and (2.2). Stiffness from the test, kt, is
calculated by finding the slope of the initial linear portion of a load – net displacement
curve. Predictions of nominal strength and stiffness based on design equations are
calculated using the nominal geometry (Table 3.1), mean values of yield stresses
obtained from material test (Tables 3.5 and 3.6), and an assumed Young’s modulus, E
= 200 kN/mm2 (29,000 ksi). Comparison between the predictions and test results are
shown in the columns under Qp/Qpt and k/kt in Table 3.7. The nominal strength
prediction has an error less than or close to 10% except for two specimens B09-56 and
B06-37. Error in the stiffness prediction is less than 15% except for three cases, S10-
61
56, S10-56BR, and B07-18W. Potential sources of error may include inaccuracies in
using the nominal geometry and Young’s modulus in the design equations. Besides,
the assumptions made in the derivation of stiffness equations to account for the
influence of band zones may also be further investigated to improve the accuracy.
3.6.2 Characterization of Fuse Behavior
The load – deformation curve of the butterfly fuse B06-37 is shown in Figure 3.43 to
represent the typical fuse hysteretic behavior which can be approximately enveloped
by a bilinear backbone curve. The tangent stiffness, k’, is defined by the following
equation.
yu
pu QQk
∆−∆
−=
5.1' (3.8)
where Qu = ultimate fuse strength at fracture, Δu = fuse shear displacement
corresponding to Qu, Δy = fuse shear displacement at fuse yielding. The nominal
strength Qp (=1.5Qy) is assumed to be reached at 1.5Δy. Note in Figure 3.43 the
horizontal axis is represented by fuse link length based shear deformation, γL = Δ/L,
while displacement should be used when calculating the stiffness measures, k and k’.
The shear deformation, γL,peak, corresponds to the peak strength of a fuse before its
hysteretic loops starts pinching due to buckling. The shear deformation, γL,u,
corresponds to the ultimate fuse strength at fuse fracture. Residual bending strength,
Qr, is the force on the last loading loop at zero deformation, which indicates the
strength from remaining bending capacity of fuse links. Residual bending strength can
be used to determine the rate of bending strength decrease between deformations γL,peak
and γL,u. With these parameters, the peak strength of a fuse at a certain deformation
can be determined, which is sufficient to characterize the behavior of a thick fuse such
as B02-14 that does not notably buckle before its facture failure, or a thin fuse prior to
its pinching. Parameters γL,peak and Qr provide important information about the
pinching behavior. However, more assumptions still need to be made in order to fully
characterize pinching of a degrading fuse, which warrants more investigation in future
studies.
62
For design and modeling purpose, it is desirable to be able to predict the characteristic
parameters. The parameters obtained from test results are summarized in Table 3.7.
Prediction of Qp and k is already discussed in the previous section. The remaining
four parameters are plotted against the fuse thickness measure, b/t, as shown in Figure
3.44. To simplify the situation, the four specimens with partial or full buckling
restraining are not included here. Moreover, specimen S10-40 is not represented in
Figures 3.44a that investigates the tangent stiffness ratio, kt’/kt, because the test was
conducted with a fractured pin (explained in Section 3.2.6) and cannot represent the
true kt’/kt under sound loading conditions. Specimen B02-14 is excluded from Figure
3.44b that investigates the residual bending strength ratio, Qrt/Qpt, because the
definition of residual bending strength does not apply to this relatively thick butterfly
fuse that did not buckle in the test. Figures 3.44c and 3.44d contain all seven
specimens that are not buckling restrained. (Figure 3.44c seems to show only six data
points because the points corresponding to S10-40 and S10-56 almost completely
overlap with each other.)
The tangent stiffness ratio and residual bending strength ratio show a linear correlation
with the b/t ratio. The linear equations based on regression are shown in the figures.
Specimen S10-56 in Figure 3.44a is excluded in the regression because it lies far away
from other data points. It can be told from Figure 3.18 that the hysteretic loops of
specimen S10-56 is quite different from the typical fuse behavior shown in Figure
3.43. The ultimate fuse strength did not exceed the peak strength prior to buckling,
which is the reason its tangent stiffness is exceptionally low compared to other
specimens. The relationship between shear deformation at peak strength and b/t ratio
can be approximated by an exponential curve determined from regression as shown in
Figure 3.44c. The ultimate shear deformation, however, does not show a strong
correlation with b/t ratio (Figure 3.44d). The fracture deformation is roughly above
30% for the butterfly fuses, and between 15% and 20% for the slit fuses. More
63
investigation on fatigue behavior of the fuses is necessary to understand the influence
factors on ultimate shear deformation.
3.7 Numerical Modeling
3.7.1 Description of the Model
Numerical modeling was conducted for two butterfly fuses B09-56 and B06-37 to
verify the modeling techniques which potentially can be employed in future studies to
analyze fuses with other untested geometries. The commercial software package
Abaqus (Abaqus, 2008) is used for the numerical modeling.
The entire test assembly is modeled to represent the actual kinematic conditions as
shown in Figure 3.45. Three-dimensional four-node shell elements are used to model
the fuse plate, which is capable of simulating out-of-plane deflection and buckling of
the fuse links. The two vertical struts are modeled by rigid truss elements whose top
ends are constrained to the “L” shaped rigid beam elements that represent the loading
beam. The top node of the L beam, referred to as the loading node, is vertically
aligned with the position of the actuator. Horizontal displacement history is applied at
the loading node to impose cyclic loading. The bottom edge of the fuse is fixed, while
the top edge is slaved to the loading node. The circles indicate pin connections.
Each fuse link is modeled by six and forty-six elements along the short and long edges
respectively. Six elements along the short edge are considered sufficient to fully
simulate the complicated twisting deformation of fuse links due to buckling. The
large number of elements along the long edge is to keep the shape of each element
approximately a square. The mesh is automated by the Abaqus software with seeds of
nodes planted along the edges in advance. The band zones above and below the links
are also modeled to account for their influence on stiffness.
64
A sine wave shaped initial out-of-plane deflection is assigned to each link to initiate
buckling. The coordinate in the out-of-plane direction of each node, z0, is defined by
the following equation.
=
LyLz πsin
1000 (3.9)
where y = vertical distance between a fuse link node and the bottom edge of the link.
An elastic-perfectly-plastic relationship is assumed as the material model for the shell
elements. The nominal Young’s modulus, E = 200 kN/mm2 (29,000 ksi), and yield
stress obtained from material test are used to define the material relationship.
3.7.2 Analytical Case Studies
Cyclic analysis of fuses B09-56 and B06-37 was conducted. Since the two fuses have
the same geometry, only thickness of the shell elements was modified to run analysis
of the second fuse. The simulated buckling of B06-37 is shown in Figure 3.46. The
buckled links look quite similar to the actual specimen at corresponding deformations
shown in Figures 3.34e and 3.34f, except that fracture is not considered by analysis.
Simulated buckling of fuse B09-56 also looks similar to the test specimen, which is
not repeatedly shown here. Figure 3.47 compares the load – deformation relationships
obtained from the test and the analysis. Note that the full displacement history was
not implemented in analysis. Overall, the salient characteristics of the hysteretic loops
are well captured by analysis result. A number of differences are noted when details
are examined. First of all, the peak strengths for B06-37 are smaller in the analysis
result, and the hardening portion of each loading loop is not as steep compared to test
result. These differences are understandable considering no material hardening is
included in the numerical model. Curiously, the opposite trend is observed for
specimen B09-56, which warrants analysis of more specimens to establish the general
trend. Secondly, the analysis loops are not as rounded compared to the test because of
the assumed bilinear relationship that does not include the Bauschinger effect. Thus
65
further improvement can be expected if a more sophisticated material model is
employed.
66
Table 3.1 Test Matrix
ID notation: First letter, “S”/“B”: Slit fuse/Butterfly fuse; two numbers: b/t and L/t ratios; ending
letters, “W”/“BR”: Welded end connection/Buckling-Restrained.
Table 3.2 List of Channels for Data Recording
ID L b t a c n b/t L/t a/b w s Plate Cutting Sequence(mm) (mm) (mm) (mm) (mm) (mm) Method No.
S12-36 229 73 6 N/A 76 7 11.5 36.0 N/A 3 A Laser 1S10-40 254 60 6 N/A 64 13 9.5 40.0 N/A 3 A Laser 2S10-56 356 60 6 N/A 13 13 9.5 56.0 N/A 3 B Laser 3S10-36W 229 64 6 N/A 76 6 10.0 36.0 N/A 13 F Laser 6S10-56BR 356 60 6 N/A 13 13 9.5 56.0 N/A 3 B Laser 8
B10-36 229 64 6 25 76 6 10.0 36.0 0.40 13 F Laser 4B09-56 356 57 6 19 13 7 9.0 56.0 0.33 13 C Laser 9B06-37 356 57 10 19 13 7 6.0 37.3 0.33 13 D Water 10B02-14 356 57 25 19 13 3 2.3 14.0 0.33 13 E Water 11B10-36W 229 64 6 25 76 6 10.0 36.0 0.40 13 F Laser 5B07-18W 229 89 13 30 76 3 7.0 18.0 0.34 13 G Water 7
Ch. No. Label Location Measurement Instrument Type Notes1 SG1 Left strut front up Strain Strain gauge2 SG2 Left strut front down Strain Strain gauge3 SG3 Left strut back up Strain Strain gauge4 SG4 Left strut back down Strain Strain gauge5 SG5 Right strut front up Strain Strain gauge6 SG6 Right strut front down Strain Strain gauge7 SG7 Right strut back up Strain Strain gauge8 SG8 Right strut back down Strain Strain gauge9 T1 Top angle Displacement Temposonic transducer10 T2 Fuse top edge Displacement Temposonic transducer11 L1 Fuse link top end Displacement LVDT Not used in test No. 3, 5~1112 L2 Fuse link bottom end Displacement LVDT Not used in test No. 3, 5~1113 L3 Fuse bottom edge Displacement LVDT14 L4 Bottom angle Displacement LVDT15 Force Actuator Force MTS controller output16 Disp Actuator Displacement MTS controller output17 L5 Fuse link vertical Displacement LVDT Not used in test No. 4~11
67
Table 3.3 Loading Protocol
Table 3.4 Tension Coupon Cross-Section Geometry
Note: t = thickness, d = diameter, w = width
Number of γ=∆/h Actuator
Displacement ΔLoad Rate
rCycle Time
tcycle
Cummulative Time
Cycles (%) (mm) (mm/s) (s) (min)1 - 6 6 0.0625 0.3 0.03 50 2.57 - 12 6 0.125 0.6 0.05 50 5.0
13 - 18 6 0.250 1.3 0.10 50 7.519 - 24 6 0.375 1.9 0.15 50 10.025 - 30 6 0.50 2.5 0.20 50 11.731 - 36 6 0.75 3.8 0.30 50 13.337 - 42 6 1.0 5.1 0.41 50 15.043 - 46 4 1.5 7.6 0.61 50 16.747 - 50 4 2.0 10.2 0.81 50 18.351 - 52 2 3 15.2 1.22 50 20.053 1 4 20.3 1.63 50 21.754 1 5 25.4 2.03 50 23.355 1 7 35.6 2.54 56 25.256 1 9 45.7 2.54 72 27.657 1 11 55.9 2.54 88 30.558 1 13 66.0 0.51 520 39.259 1 15 76.2 0.51 600 49.260 1 17 86.4 0.51 680 60.561 1 19 96.5 0.51 760 73.262 1 21 106.7 0.51 840 87.263 1 23 116.8 0.51 920 102.564 1 25 127.0 0.51 1000 119.265 1 27 137.2 0.51 1080 137.266 1 29 147.3 0.51 1160 156.5
Cycle #
Plate t or d w t or d w t or d1 w or d2 t or d1 w or d2
A 6.2 6.3 6.2 6.3 3.4 3.6 3.2 3.7B 6.0 6.4 6.1 6.4 3.3 3.7 3.2 3.6C 6.4 6.4 6.4 6.4 2.9 3.2 2.9 3.2D 9.5 13.0 9.5 13.0 6.2 8.9 5.2 7.4E 12.8 N/A 12.8 N/A 7.9 8.1 7.6 8.1
BeforeTest (mm) After Test (mm)Test 1 Test 2 Test 1 Test 2
68
Table 3.5 Material Test Result (plates A ~ E)
Table 3.6 Material Test Result Provided by CEL Consulting (plates F and G)
Table 3.7 Summary of Fuse Test Results
ID Qyt Qpt=1.5Qyt Qp/Qpt kt k/ kt kt' kt'/kt γLt,peak γLt,u Qrt/Qpt
kN kN kN/mm kN/mm
S12-36 103.6 155.4 1.13 86.1 1.13 0.95 1.1% 5.4% 5.4% 23.6%
S10-40 119.3 179.0 1.12 92.5 0.96 1.92 2.1% 5.7% 22.0% 26.9%
S10-56 100.8 151.2 1.01 51.0 1.31 0.32 0.6% 5.7% 14.6% 20.8%
S10-36W 59.5 89.3 1.03 60.3 0.90 1.33 2.2% 7.6% 16.6% 27.0%
S10-56BR 102.7 154.1 0.99 55.9 1.19 1.72 3.1% 8.5% 8.5% 47.5%
B10-36 59.0 88.5 0.99 46.8 0.95 0.95 2.0% 5.4% 36.2% 25.6%
B09-56 27.4 41.1 1.53 13.7 1.15 0.29 2.1% 9.6% 38.1% 35.0%
B06-37 47.3 71.0 1.25 21.2 1.12 0.73 3.4% 17.7% 29.1% 49.9%
B02-14 61.0 91.5 0.91 25.8 1.05 0.91 3.5% 34.3% 37.1% 162.0%
B10-36W 62.6 93.9 0.94 49.2 0.90 0.92 1.9% 5.4% 44.9% 22.5%
B07-18W 133.5 200.3 1.07 86.9 1.27 1.50 1.7% 17.8% 38.0% 30.7%
ThicknessPlate (mm) Test 1 Test 2 Mean Test 1 Test 2 Mean Test 1 Test 2 Mean
A 6 346 331 339 441 446 443 69% 70% 70%B 6 361 358 359 490 489 489 69% 71% 70%C 6 343 350 347 435 436 435 77% 78% 77%D 10 271 381 326 422 526 473 55% 69% 62%E 25 253 284 268 441 464 453 61% 88% 74%
Reduction of Area (%)σ y (MPa) σ u (MPa)
Thickness σy (MPa) σu (MPa)Plate (mm) Test 1 Test 2 Mean Test 1 Test 2 Mean Test 1 Test 2 Mean
F 6 277 268 273 387 374 381 32% 39% 36%G 13 355 370 362 520 529 524 26% 28% 27%
Elongation in 51 mm (%)
69
(a)
(b)
Figure 3.1 Fuse test setup: (a) schematic; (b) photo
Fuse specimen
Loading Beam
Fudation Beam
70
(a) (b)
Figure 3.2 More test setup details: (a) out-of-plane bracing; (b) actuator and
reaction column
Figure 3.3 Instrumentation scheme
Front: SG1, 2Back: SG3, 4
Front: SG5, 6Back: SG7, 8
71
(a) (b)
(c) (d)
Figure 3.4 Instrumentation photos: (a) linear variable displacement transducers
(LVDT’s); (b) temposonic transducers; (c) strain gauges; (d) strain gauge
connection box
72
Figure 3.5 Data acquisition and control arrangement
Data acquisition computer
MTS controller
Control computer
73
(a)
(b)
(c)
Figure 3.6 Steel tension coupons: (a) rectangular subsize (for plates A, B and C);
(b) rectangular sheet (for plate D); (c) round standard (for plate E)
Figure 3.7 Steel tension coupon test setup
Bottom fixed
Tension
Tension coupon
74
(a) (b)
(c) (d)
(e)
Figure 3.8 Stress-strain relationships of all tested coupons*: (a) plate A; (b) plate B;
(c) plate C; (d) plate D; (e) plate E *Test of plates F and G was conducted by CEL Consulting. No graph was provided in CEL’s report.
75
(a) (b)
(c)
Figure 3.9 Comparison of target, actuator and fuse displacement: (a) 1 ~ 23 s; (b)
20 ~ 52 s; (c) elastic deformation in the test rig
Figure 3.10 Load - deformation relationship for specimen S12-36
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γ = Δ / h (%)
Q /
Qp
S12-36Qp = 176 kN
76
(a) (b)
(c) (d)
(e) (f)
Figure 3.11 Photos of specimen S12-36: (a) beginning of test; (b) yielding evident
(γ=0.91%); (c) buckling initiation (γ=1.78%); (d) buckling evident
(γ=3.86%); (e) crack initiation (γ=5.71%); (f) fracture – end of test
(γ=12.35%)
77
(a) (b)
(c)
Figure 3.12 Photos of specimen S12-36 details: (a) yielding around slit end; (b) side
view of buckling; (c) crack initiation
(a) (b)
Figure 3.13 Free body diagram of loading beam and the fuse: (a) bending mode
(small deformation); (b) tension mode (large deformation)
78
Figure 3.14 Measured strut axial forces* *SG2 reading became erroneous after pin fracture, and SG7 did not work. Therefore SG1 and SG3 are
used to calculate force in left strut, and SG6 and SG8 are used for right strut.
Figure 3.15 Fractured pin left in the strut pin hole in test of S10-40
(a) (b)
Figure 3.16 Load - deformation relationship: (a) S10-40; (b) comparison with S12-36
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γ = Δ / h (%)
Q /
Qp
S10-40Qp = 201 kN
-30 -20 -10 0 10 20 30-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γL = Δ / L (%)
Q /
Qp
S10-40S12-36
79
(a) (b)
(c) (d)
(e) (f)
Figure 3.17 Photos of specimen S10-40: (a) beginning of test; (b) yielding evident
(γ=0.77% 1st cycle); (c) visible buckling (γ=1.1%); (d) buckling evident
(γ=2.8%); (e) crack initiation (γ=2.8%); (f) fracture – end of test
(γ=15%)
80
(a) (b)
Figure 3.18 Load - deformation relationship: (a) S10-56; (b) comparison with S10-40
Figure 3.19 Fractured link ends of specimen S10-56
-20 -10 0 10 20-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γ = Δ / h (%)
Q /
Qp
S10-56Qp = 152 kN
-30 -20 -10 0 10 20 30-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γL = Δ / L (%)
Q /
Qp
S10-56S10-40
81
(a) (b)
(c) (d)
(e) (f)
Figure 3.20 Photos of specimen S10-56: (a) beginning of test; (b) yielding evident
(γ=2.1%); (c) buckling initiation (γ=4%); (d) buckling evident and crack
initiation (γ=6.1%); (e) crack evident (γ=14.3%); (f) fracture – end of
test (γ=16.1%)
82
(a) (b)
Figure 3.21 Load-deformation relationship: (a) S10-36W; (b) comparison with S10-
40
(a) (b) (c)
Figure 3.22 Photos of specimen S10-36W details: (a) buckling initiation; (b)
buckling evident; (c) end of test
-15 -10 -5 0 5 10 15-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
S10-36WQp = 92 kN
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γL = Δ / L (%)
Q /
Qp
S10-36WS10-40
83
(a) (b)
(c) (d)
(e) (f)
Figure 3.23 Photos of specimen S10-36W: (a) beginning of test; (b) yielding evident
(γ=1.5%); (c) buckling initiation (γ=2.4%); (d) buckling evident and
crack initiation (γ=3.4%); (e) crack evident (γ=8.4%); (f) fracture – end
of test (γ=12.6%)
84
(a) (b)
Figure 3.24 Load-deformation relationship: (a) S10-56BR; (b) comparison with S10-
56
-15 -10 -5 0 5 10 15-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γ = Δ / h (%)
Q /
Qp
S10-56BRQp = 152 kN
-30 -20 -10 0 10 20 30-1.5
-1
-0.5
0
0.5
1
1.5
Shear deformation γL = Δ / L (%)
Q /
Qp
S10-56BRS10-56
85
(a) (b)
(c) (d)
(e)
Figure 3.25 Photos of specimen S10-36W: (a) beginning of test; (b) buckling and
crack initiation (γ=5.9%); (c) crack evident (γ=7.5%); (d) fracture – end
of test (γ=12.1%); (e) cracks at slit ends
86
(a) (b)
Figure 3.26 Load-deformation relationship: (a) B10-36; (b) comparison with S10-40
-20 -10 0 10 20-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B10-36Qp = 88 kN
-40 -20 0 20 40-2
-1
0
1
2
Shear deformation γL = Δ / L (%)
Q /
Qp
B10-36S10-40
87
(a) (b)
(c) (d)
(e) (f)
Figure 3.27 Photos of specimen B10-36: (a) beginning of test; (b) yielding evident
(γ=1.5%); (c) buckling initiation (γ=2.4%); (d) buckling evident
(γ=10.3%); (e) buckling at zero displacement (γ= 0%); (f) fracture – end
of test (γ=16.3%)
88
(a) (b)
(c) (d)
(e) (f)
Figure 3.28 Photos of specimen B10-36 details: (a) flaking around quarter-height; (b)
buckling – top view; (c) buckling – side view; (d) crack initiation; (e)
fracture; (f) fractured surface
89
(a) (b) (c) (d) (e) (f)
Figure 3.29 Pinned column mechanism of a fuse link
(a) (b)
Figure 3.30 Load-deformation relationship: (a) B09-56; (b) comparison with B10-36
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B09-56Qp = 63 kN
-40 -20 0 20 40-2
-1
0
1
2
Shear deformation γL = Δ / L (%)
Q /
Qp
B09-56B10-36
90
(a) (b)
(c) (d)
(e) (f)
Figure 3.31 Photos of specimen B09-56 details: (a) beginning of test; (b) yielding
evident (γ=2.7%); (c) buckling evident (γ=6.7%); (d) buckling at positive
peak (γ=24.7%); (e) buckling at zero displacement; (f) fracture – end of
test (γ=26.7%)
91
Figure 3.32 Gouging of specimen B09-56
(a) (b)
Figure 3.33 Load-deformation relationship: (a) B06-37; (b) comparison with B10-36
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B06-37Qp = 89 kN
-40 -20 0 20 40-2
-1
0
1
2
Shear deformation γL = Δ / L (%)
Q /
Qp
B06-37B10-36
92
(a) (b)
(c) (d)
(e) (f)
Figure 3.34 Photos of specimen B06-37 details: (a) beginning of test; (b) buckling
initiation (γ=10.5%); (c) buckling evident (γ=14.4%); (d) crack initiation
(γ=18.5%); (e) buckling at zero displacement (γ=0%); (f) fracture – end
of test (γ=22.5%)
93
Figure 3.35 Load-deformation relationship of specimen B02-14
(a) (b)
(c) (d)
Figure 3.36 Photos of specimen B02-14 - I: (a) beginning of test; (b) γ=1.1%; (c)
γ=2.6%; (d) γ=3.6%
-30 -20 -10 0 10 20 30-3
-2
-1
0
1
2
3
Shear deformation γ = Δ / h (%)
Q /
Qp
B02-14 Qp = 83 kN
94
(a) (b)
(c) (d)
(e) (f)
Figure 3.37 Photos of specimen B02-14 - II: (a) γ=8.4%; (b) γ=14.4%; (c) out-of-
plane deflection (γ=22.0%); (d) crack initiation (γ=24.0%); (e) crack
evident (γ=26.0%); (f) fracture- end of test (γ=26.0%)
95
(a) (b) (c)
Figure 3.38 Photos of specimen B02-14 details: (a) buckling; (b) crack at quart-
height; (c) fractured surface
(a) (b)
Figure 3.39 Load-deformation relationship: (a) B10-36W; (b) comparison with B10-
36
-20 -10 0 10 20
-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B10-36WQp = 88 kN
-50 0 50
-2
-1
0
1
2
Shear deformation γL = Δ / L (%)
Q /
Qp
B10-36WB10-36
96
(a) (b)
(c) (d)
(e) (f)
Figure 3.40 Photos of specimen B10-36W: (a) beginning of test; (b) buckling
initiation (γ=2.4%); (c) buckling evident (γ=20.2%); (d) end of test – no
fracture (γ=20.2%); (e) side view of buckling; (f) gouging
97
Figure 3.41 Load-deformation relationship of specimen B07-18W
-20 -10 0 10 20-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B07-18WQp = 214 kN
98
(a) (b)
(c) (d)
(e) (f)
Figure 3.42 Photos of specimen B10-36W: (a) beginning of test; (b) buckling
initiation (γ=6.0%); (c) buckling evident (γ=11.9%); (d) buckling at zero
displacement (γ=0%); (e) welding failure – end of test (γ=18.0%); (f)
crack initiation
99
Figure 3.43 Characterization of fuse behavior
(a) (b)
(c) (d)
Figure 3.44 Regression of fuse parameters: (a) tangent stiffness; (b) residual bending
strength; (c) shear deformation at peak strength; (d) ultimate shear
deformation at fracture
-40 -20 0 20 40-150
-100
-50
0
50
100
150
Shear deformation γL = Δ / L (%)
Q (k
N)
B06-37Qp = 89 kN
Qr
Qp
γL,peak γL,u
k’
k
y = -0.0022x + 0.0426
0%
1%
2%
3%
4%
0 2 4 6 8 10 12Tang
ent s
tiffn
ess
ratio
kt' /
kt
Ratio b/t
S10-56
y = -0.0528x + 0.7922
0%
10%
20%
30%
40%
50%
60%
4 5 6 7 8 9 10 11 12Res
idua
l ben
ding
stre
ngth
Q
rt / Q
pt
Ratio b/t
y = 0.5939e-0.226x
0%
10%
20%
30%
40%
0 2 4 6 8 10 12She
ar d
efor
mat
ion
at p
eak
stre
ngth
γL,
peak
Ratio b/t
0%
10%
20%
30%
40%
50%
0 2 4 6 8 10 12Ulti
mat
e S
hear
def
orm
atio
n γ L
,u
Ratio b/t
Butterfly fuseSlit fuse
Figure 3
Figure 3
3.45 Abaqu
(
3.46 Simul
us model of
(a)
lated bucklin
10
a butterfly f
ng of butterf
00
fuse
fly fuse B06-
(b)
-37: (a) γ = 0
0; (b) γ = 23%
%
101
(a) (b)
(c) (d)
Figure 3.47 Comparison of test and analysis load – deformation relationship: (a)
B09-56 test; (b) B09-56 analysis; (c) B06-37 test; (d) B06-37 analysis
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B09-56Qp = 63 kN
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B09-56, Abaqus analysis
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B06-37Qp = 89 kN
-30 -20 -10 0 10 20 30-2
-1
0
1
2
Shear deformation γ = Δ / h (%)
Q /
Qp
B06-37, Abaqus analysis
102
103
CHAPTER 4
CONCLUSIONS __________________________________________________________________________________
A total of eleven steel shear plates with openings were tested to investigate their
behavior as energy-dissipating fuses. Five fuse specimens are steel plates with slits
that divide the plate into rectangular links, and six specimens have diamond-shaped
openings that create butterfly-shaped links in the plate (Figure 2.1). Design equations
and a simple back-bone curve are proposed for fuse design and characterization of
fuse behavior. The commercial software Abaqus is used to model two butterfly fuse
specimens to verify numerical modeling techniques. Major findings and
recommendations from this study are presented as follows. When describing fuse
shear deformation, the link length based measure, γL = Δ/L, is used in this chapter.
4.1 Performance of Slit and Butterfly Fuses without Buckling Restraining
All five tested slit fuses are designed with a b/t ratio of about 10. The slit fuses
without partial or full buckling restraining sustained ideal hysteretic behavior up to at
least 5% shear deformation, after which the hysteretic loops are severely pinched as a
result of lateral-torsional buckling of fuse links. Prior to buckling, the fuse response
can be characterized by double-curvature bending of the fuse links. After the
buckling, the response is dominated by tension field action due to stretch of the links.
During the tension mode behavior, fuse shear strength remains low for the most part of
a loading loop, yet can reach a peak strength sometimes even higher than the
maximum strength experienced before buckling (e.g., Figure 3.16a). Ultimately, the
fuse loses a significant portion of its strength due to fracture at the ends of one or more
links resulting from low-cycle fatigue. The tested slit fuses fractured at a shear
104
deformation beyond 20%. The maximum strength experienced before fracture failure
is roughly between 1.3Qp and 1.6Qp.
The tested butterfly fuses have more variations in their geometry, with the b/t ratio
ranging between 2 and 10, and L/t ratio ranging between 14 and 56. Overall, the b/t
ratio is found to have the largest influence on fuse behavior. Relatively thin butterfly
fuses, defined as having a b/t ratio equal to about 10, experience a similar transition
from bending to tension mode behavior as the thin slit fuses, although the butterfly
fuses tend to have larger deformation capacity and reach higher peak strength during
the tension mode behavior. For the two thin butterfly fuses tested, full hysteretic loops
were sustained up to 5% and 10% shear deformation, respectively. And the ultimate
fracture occurred at more than 35% shear deformation. The maximum strength occurs
always during tension mode behavior, and reaches nearly 2Qp before the fracture. The
thick butterfly fuse, defined as having a b/t ratio equal to about 2, maintained bending
mode behavior until a fracture at 37% shear deformation, and reached a maximum
strength of 2.3Qp.
Overall, both the slit and butterfly fuses demonstrated good energy dissipation
capability and fracture resistance. For conventional applications with low to moderate
shear deformation demand, such as a shear wall whose expected shear deformation is
usually less than 3%, both types of fuse can deliver good hysteretic performance. For
application in the controlled rocking system as explained in Chapter 1, pinching is
expected with thin slit or butterfly fuses at large deformation, while uncompromised
performance can still be achieved with a thick butterfly fuse. Generally, the butterfly
fuse has more resistance to buckling and fracturing compared to the slit fuse, while the
latter is slightly easier to detail. The current cutting technology can sufficiently handle
the manufacturing of either type.
105
4.2 Performance of Fuses with Partial or Full Buckling Restraining
Four fuses were tested with partial or full buckling restraining. For the slit fuses,
partial buckling restraining delayed bucking by about 2% shear deformation compared
to its counterpart specimen without buckling restraining. And full buckling restraining
completely prevented buckling. However, these benefits came at the cost of incurring
fracture at a shear deformation about 5% smaller. For the butterfly fuses, restraining
buckling had less notable influence on the fuse behavior, probably due to the reduced
contact area between fuse links and the backing plates since the butterfly links neck
towards its center. For both types, fuse strength with buckling restraining is 5% ~
10% higher due to the friction force between fuse links and the backing plates.
Overall, the benefits of restraining buckling are limited, especially considering the
more complicated detailing requirements. However, the partial buckling restraining
detail with the welded backing plates may be convenient for connection with concrete
walls, whereby the backing plates can be precast in the walls.
4.3 Design of Slit and Butterfly Fuses
Design equations to determine the nominal strength and stiffness of the fuses are
proposed (Eqs. (2.2) and (2.10) for the slit fuse, and Eqs. (2.23) and (2.28) for the
butterfly fuse). Comparison with test results shows that the design equations generally
predicted test result with an error less than 15% for most of the specimens. To
characterize hysteretic behavior of the fuse under large deformation, a number of
parameters including the ratio of tangent stiffness to initial stiffness, k’/k, residual
bending strength ratio, Qr/Qp, deformation limit for pinching behavior, γL,peak, and
deformation at fracture, γL,u, are employed and their correlation with the b/t ratio is
demonstrated in Figure 3.44. These parameters can be used to estimate deformation
capacity of the fuse, as well as peak strength at a given deformation demand. To fully
characterize the pinching fuse behavior, more investigation is still needed to establish
the deterioration parameters.
106
The design of a fuse may start with a given geometry limit on the width and height of
the fuse, together with ductility and strength demand, and sometimes even a stiffness
demand. The ductility demand is typically given as a displacement which should be
interpreted in the context of the geometry limit. By altering the length of fuse links,
the deformation demand in terms of shear deformation γL can be adjusted. If the
demand of γL is moderate, for instance below 5%, both the thin slit and butterfly fuses
with the b/t ratio at about 10 can be adequate options. With larger demand of γL, both
types of fuses can still be used if pinching can be tolerated, although the butterfly fuse
may be preferred because of its higher resistance to buckling and fracturing. If
pinching behavior is undesirable, the thick butterfly fuse with the b/t ratio
approximately below 2.5 and L/t ratio below 14 should be adopted. These
recommendations are based on observations made about the tested fuse specimens. A
parametric study on the various design parameters may be warranted to establish more
comprehensive guidelines on the selection of parameters.
With a chosen fuse type, length of links, and b/t ratio, the thickness of the fuse plate
can be altered to achieve the desirable number of links and stiffness. The tangent
stiffness can be used to estimate fuse strength at the largest expected shear
deformation. The strength is important for checking the capacity designed members
that connect with the fuse.
4.4 Numerical Modeling
A three-dimensional model with the fuse represented by shell elements is found to be
able to simulate the buckling behavior and load – deformation response with fairly
good accuracy (Figures 3.46 and 3.47). Further improvement of the model can be
made by adopting more sophisticated material model that can approximate the
Bauschinger effect and include strain hardening. The limitation of the model is its
complicated mesh that requires quite an amount of manual work, even with the aid of
107
some automation functions provided in the Abaqus software. Besides, it takes several
hours to run a simulation with the test loading protocol on a computer with standard
capacity. A simple fuse model that employs beam and truss elements along with
rotational springs to model the bending and tension behavior of fuse links was created
in the open-sourced software OpenSees (Mazzoni, et al., 2009). Sufficiently accurate
results can be achieved with this model with significant reduction in computational
cost. More details about the OpenSees model can be found in Ma (2010).
4.5 Future Research Needs
A number of observations made about the test result and numerical analysis result
have been suggested for further investigations in the discussions in Chapter 3. In
addition, the following areas may also be the topics for future research in order to
more thoroughly understand the fuse behavior and develop design guidelines.
Quantified correlation between geometry limits and performance. Specimens with
certain combinations of b/t ratio and L/t ratio were studied in the current research.
More design cases with more varying combinations of b/t and L/t ratios need to be
analyzed in order to fully quantify the correlation between these geometric parameters
and fuse performance, mainly the occurrence of buckling and low-cycle fatigue.
Prediction of low-cycle fatigue failure by numerical tools. Although numerical
models are rarely used to directly simulate the physical fatigue and fracture
phenomena of steel structural members, criteria for predicting low-cycle fatigue
employing numerical analysis have been proposed, e.g., Kanvinde and Deierlein
(2008). These criteria may be used to numerically analyze the fuse fatigue behavior
and develop design guidelines.
Characterization of pinching behavior. The pinched thin fuses may find applications
in situations where a certain amount of strength degradation may be beneficial to
108
protect other connected members, or preserve self-centering capability of the structural
system (Ma, 2010). It is desirable to be able to characterize the pinching and
incorporate it in numerical models. More studies are needed to first establish a set of
parameters that characterizes the pinching behavior, and then identify the correlation
between these parameters and fuse design parameters.
109
REFERENCES
Abaqus (2008). Abaqus analysis user's manual, version 6.8.
Aiken, I., Nims, D., Whittaker, A., and Kelly, J. (1993). “Testing of passive energy dissipation systems.” Earthquake Spectra, Vol. 9, No. 3, Earthquake Engineering Research Institute, California.
AISC. (2005). “Seismic Provisions for Structural Steel Buildings”, American Institute of Steel Construction, Chicago, p. 81.
ASTM Standard E8. (2003) “Standard test methods for tension testing of metallic materials.” American Society for Testing and Materials, pp. 62-79.
Bergman, D. M., and Goel, S. C. (1987). “Evaluation of cyclic testing of steel plate devices for added damping and stiffness.” Report No. UMCE87-10, the University of Michigan.
Chan, R. W. K., and Albermani, F. (2008). “Experimental study of steel slit damper for passive energy dissipation.” Engineering Structures, 30 (2008), pp. 1058-1066.
Eatherton, M. (2010). “Large-scale cyclic and hybrid simulation testing and development of a controlled-rocking steel building system with replaceable fuses.” Ph.D. Dissertation, University of Illinois at Urbana-Champaign.
Hajjar, J., Eatherton, M., Gregory, D., Ma, X., Peña, A., Krawinkler, H., and Billington, S. (2008). “Controlled rocking of steel frames as a sustainable new technology for seismic resistance in buildings.” International Association of Bridge & Structural Engineering Symposium.
Hitaka, T., and Matsui, C. (2003). “Experimental study on steel shear wall with slits.” J. Struct. Eng., ASCE, Vol. 129, No.5, pp. 586-595.
Kanvinde, A. M., Deierlein, G. G. (2008). “Validation of cyclic void growth model for fracture initiation in blunt notch and dogbone steel specimens.” J. Struct. Eng., ASCE, Vol. 134, No. 9.
Kobori, T., Miura, Y., Fukusawa, E., Yamada, T., Arita, T., and Takenake, Y. (1992). “Development and application of hysteresis steel dampers.” Proc. 11th World Conference on Earthquake Engineering, pp. 2341-2346.
Ma, X. (2010). “Seismic design and behavior of self-centering braced frame with controlled rocking and energy-dissipating fuses.” Ph.D. Dissertation, Stanford University.
110
Matlab. (2010). MATLAB, version R2010a, the MathWorks, Inc.
Mazzoni, S., McKenna, F., Scott, M. H., and Fenves, G. L. (2009). Open system for earthquake engineering simulation user command-language manual, OpenSees version 2.0, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA.
Nakamura. H., Ono, Y., Sugiyama, S., Wada, A., Baba, M., and Miyabara, T. (1997). “Passive control system using steel slit damper, part 1: test of steel slit damper”, Summaries of Technical Papers of Annual Meeting AIJ, Kanto, Japan, pp. 843-844 (in Japanese).
Richards, R. W., and Uang, C. –M. (2006). “Testing protocol for short links in eccentrically braced frames.” J. Struct. Eng., ASCE, Vol. 132, No. 8, pp. 1183-1191.
Tagami, J., et al. (1999). “Low cycle fatigue test of steel plate dampers with honeycomb shaped openings.” (part1-3), Summaries of Technical Papers of Annual Meeting AIJ, Chugoku, Japan, pp. 791-796 (in Japanese).
111
APPENDIX: Figures of ECC/SCHPFRCC Panel Tests
Figure A.1 Test setup
Table A.1 Test Matrix
ID Width Height Thickness Grouted* Date Cast Date Tested
mm mm mm
ECC**-full 610 381 95 Yes 2007/03/19 2007/06/01
ECC-half 610 381 76 No 2007/02/27 2007/03/21
ECC-quarter 610 381 95 Yes 2007/03/08 2007/05/24
SCHPFRCC***-full 610 381 95 Yes 2007/04/06 2007/05/28
SCHPFRCC-half 610 381 76 Yes 2007/02/22 2007/04/21
*Grouted: Grout filled in the gap between the panel and loading and foundation beams to prevent
rocking of the panel
**ECC: Engineered cementitious composites
***SCHPFRCC: Self-consolidating high performance fiber reinforced cenmentitious composites
hbolt
Shear force, Q
Displacement, Δ
Shear deformationγ = Δ/hbolt
Panelspecimen
112
(a)
(b)
Figure A.2 Drawing of specimens: (a) full panel; (b) half panel
609.
6 m
m (2
4 in
)
635.0 mm (25 in)
63.5 mm (2.5 in)
38.1
mm
(1.5
in)
φ22.
2 m
m (7
/8 in
) hol
es
@ 7
6.2
mm
(3 in
) o.c
.
#5 steel rebar used, area = 200 mm2 (0.31 in2)
609.
6 m
m (2
4 in
)
635.0 mm (25 in)
63.5 mm (2.5 in)
38.1
mm
(1.5
in)
φ22.
2 m
m (7
/8 in
) hol
es
@ 7
6.2
mm
(3 in
) o.c
.
#5 steel rebar used, area = 200 mm2 (0.31 in2)
113
Figure A.3 Drawing of quarter panel specimen
609.
6 m
m (2
4 in
)
635.0 mm (25 in)
63.5 mm (2.5 in)
38.1
mm
(1.5
in)
φ22.
2 m
m (7
/8 in
) hol
es
@ 7
6.2
mm
(3 in
) o.c
.
#5 steel rebar used, area = 200 mm2 (0.31 in2)
114
Figure A.4 Load – deformation relationship of ECC – full (1 kip = 4.45 kN)
(a) (b)
(c) (d)
Figure A.5 Photos of ECC – full: (a) γ = 0; (b) γ = -3%; (c) γ = -5%; (d) γ = -9%
115
Figure A.6 Load – deformation relationship of ECC – half (1 kip = 4.45 kN)
(a) (b)
(c) (d)
Figure A.7 Photos of ECC – half: (a) γ = 0; (b) γ = 1.5%; (c) γ = 11%; (d) γ = -11%
116
Figure A.8 Load – deformation relationship of ECC – quarter (1 kip = 4.45 kN)
(a) (b)
(c) (d)
Figure A.9 Photos of ECC – quarter: (a) γ = 0; (b) γ = -3%; (c) γ = -4%; (d) γ = -9%
117
Figure A.10 Load – deformation relationship of SCHPFRCC – full (1 kip = 4.45 kN)
(a) (b)
(c) (d)
Figure A.11 Photos of SCHPFRCC – full: (a) γ = 0; (b) γ = -3%; (c) γ = -5%; (d) γ = -
9%
118
Figure A.12 Load – deformation relationship of SCHPFRCC – half (1 kip = 4.45 kN)
(a) (b)
(c) (d)
Figure A.13 Photos of SCHPFRCC – half: (a) γ = 0; (b) γ = 1.5%; (c) γ = 4%; (d) γ =
-9%