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Engineering Structures 32 (2010) 2258–2267
Contents lists available at ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Shape optimization of metallic yielding devices for passive mitigation of
seismic energy
Kazem Ghabraie a, Ricky Chan b,∗, Xiaodong Huang b, Yi Min Xie b
aFaculty of Engineering and Surveying, University of Southern Queensland, Toowoomba Qld 4350, Australia
bSchool of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne VIC 3001, Australia
article info
Article history:
Received 24 December 2009
Received in revised form
31 March 2010
Accepted 31 March 2010
Available online 29 April 2010
Keywords:
Shape optimization
Energy dissipation
Bi-directional Evolutionary Structural
Optimization
Metallic damper
Cyclic tests
Earthquake resistant structure
abstract
Bi-directional Evolutionary Structural Optimization (BESO) is a well-established topology optimization
technique. This method is used in this paper to optimize the shape of a passive energy dissipater designed
for earthquake risk mitigation. A previously proposed shape design of a steel slit damper (SSD) device
is taken as the initial design and its shape is optimized using a slightly modified BESO algorithm. Some
restrictions are imposed to maintain simplicity and to reduce fabrication cost. The optimized shape shows
increased energy dissipation capacity and even stress distribution. Experimental verification has been
carried out and proved that the optimized shape is more resistant to low-cycle fatigue.
©2010 Elsevier Ltd. All rights reserved.
1. Introduction
In the last two decades development of energy dissipation
devices for mitigation of wind and earthquake has flourished.
Various types of passive, semi-active and active devices have been
proposed, tested and implemented [1]. With this technology, a
large portion of input energy from wind or earthquake excitations
is dissipated by designated devices. As a result, structural
responses are suppressed, and major structural elements can be
protected from damage. Particularly in earthquake applications,
metallic devices which utilize yield deformation of metals remain
among the most popular types selected by engineers. They are
reliable, inexpensive to fabricate, easy to install and maintain.
Metallic devices can be classified into flexural types, such as
hourglass shape ADAS [2], triangular shape TADAS [3]; shear types
such as YSPD [4] and axial types, such as the Buckling Retrained
Brace [5]. Devices are mainly designed to be incorporated into
lateral-load-resisting system in structural frames, but some are
developed to be installed between beam and columns [6].
∗Corresponding author. Tel.: +61 4 2120 1134; fax: +61 3 9639 0138.
E-mail addresses: kazem.ghabraie@usq.edu.au (K. Ghabraie),
ricky.chan@rmit.edu.au (R. Chan), xiaodong.huang@rmit.edu.au (X. Huang),
mike.xie@rmit.edu.au (Y.M. Xie).
Design of metallic devices requires several desirable engineer-
ing characteristics:
1. possessing sufficient elastic strength and stiffness such that
device is not excited to inelastic region under service loads;
2. having stable and large energy dissipative capability; and
3. having reasonable resistance against low-cycle fatigue.
With respect to low-cycle fatigue, current design standard in
the United States requires devices to undergo five fully reversed
cycles at maximum earthquake device displacement [7]. Generally,
in order to increase the resistance to low-cycle fatigue, stress
concentration has to be avoided.
Along with the revolutionary improvement of digital comput-
ers in recent decades, computational methods and numerical tech-
niques have established their place as invaluable engineering tools.
Among these, numerical optimization methods have attracted a
great number of researchers and have been improved a lot. Partic-
ularly, the state-of-the-art shape and topology optimization tech-
niques have been applied to a range of physical problems and
have been proved to yield much better results than experimen-
tal designs [8,9]. The Evolutionary Structural Optimization (ESO)
method, introduced by Xie and Steven [10] is a simple and effec-
tive topology optimization technique which can tackle shape op-
timization problems as well. This method iteratively improves the
design domain by removing its inefficient parts. A Bi-directional
0141-0296/$ – see front matter ©2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2010.03.028
K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267 2259
Fig. 1. The SSD device design proposed by Chan and Albermani [18].
version of the ESO method, called BESO, has been later proposed by
Querin et al. [11,12] and Yang et al. [13]. In BESO, besides removal
of inefficient parts, the efficient parts of the design domain will be
improved by adding more material next to them. Since its introduc-
tion, the BESO algorithm has been improved significantly [14]. The
improved BESO algorithm has been successfully applied to non-
linear problems [15,16]. This method is also capable of optimizing
both shape and topology of the designs [17].
In this paper, the BESO algorithm is modified to optimize the
shape of an existing steel slit damper device design (SSD). The
proposed algorithm applies some shape restrictions to the design
to make the final shape easily manufacturable. An efficient device
design should possess a high energy dissipation capability per
unit volume. To gain this, the proposed algorithm maximizes the
total plastic dissipation. It is also demonstrated that the optimum
design resulted from the proposed optimization algorithm, show
less stress concentration than the initial design.
In order to verify the numerical results and to address the
shortcomings of the numerical models, physical experiments
are carried out. It is demonstrated that experimental outcomes
support the numerical results.
2. Optimization
Chan and Albermani [18] have proposed a class of simple
designs for SSD devices supported by a series of experimental test
results. Fig. 1 shows the typical shape of the device. The size of the
slits (w) can be controlled by varying land b. In this paper, a new
class of design is proposed by optimizing the shape of the slits in
Fig. 1. To achieve this, a shape optimization algorithm based on the
BESO technique is proposed and utilized here. Some restrictions
are imposed to maintain the simplicity of the shape and hence
reduce its fabrication costs. These restrictions are discussed in
detail in Section 2.4.
2.1. Numerical modeling
For numerical modeling, the flanges are considered solid and a
plane stress rectangular mesh is used to model the web. A uniform
web thickness of t=8 mm is considered overall the design except
for the elements on the far left and right sides of the domain. These
elements which are in the vicinity of the flanges are modeled using
thicker elements to simulate fillets (Fig. 2).
For the sake of fabrication, the holes are prevented from being
too wide by setting the two strips of 15 mm width on the left
and right sides as non-designable elements. Fig. 2 illustrates the
designable and non-designable domains.
The left side of the model is fixed and a uniform vertical
displacement is applied to the right side of the model. The loading
cycle consists of three steps: an upward displacement of 10 mm,
Fig. 2. The designable and non-designable domains.
followed by a downward displacement of 20 mm, and finally an
upward displacement of 10 mm up to the original location (Fig. 2).
In this manner the elastic strain energy will be zero after a full
cycle and the total strain energy would be equal to the total plastic
dissipation.
2.2. Problem statement
To optimize the shape of the SSD, the total plastic energy
dissipation is considered as the objective function which is to
be maximized. In order to prevent the optimization algorithm
from catching the extreme full or empty domain designs, it is
necessary to include an additional constraint to restrict the amount
of usable material [19]. Here we use a volume constraint which
forces the algorithm to use a certain amount of material in the
design domain. Alternatively, one can impose a restriction on the
maximum force instead of using a volume constraint [15]. The
optimization problem can be expressed as
max
x1,x2,...,xN
EP
subject to V=¯
V
and shape restrictions
(1)
2260 K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267
where EPis the total plastic dissipation; Vand ¯
Vare the actual and
target volumes; xi-s are the design variables and Nis the number of
elements in the design domain. Shape restrictions are fully covered
in Section 2.4. In the BESO algorithm, design variables are binary
values with x=1 indicating the presence of material in the i-th
element and x=0 representing a void in the location of the i-th
element.
2.3. Sensitivity analysis
To evaluate the effect of adding or removing an element during
the optimization process, one needs to perform a sensitivity
analysis.
Because the loading sequence consists of a full cycle, the total
plastic dissipation will be equal to the total strain energy. Hence,
for this case, we can write
EP=ET=If·du,(2)
where ETis the total strain energy; and fand uare nodal force and
displacement vectors respectively. Using the trapezoidal method
for numerical integration, this definition can be rewritten as
EP=lim
n→∞ "1
2
n
X
i=1
(uT
i−uT
i−1)(fi+fi−1)#.(3)
The shape sensitivities of the nonlinear systems have been
calculated for different types of problems by Huang and Xie [16]
Buhl et al. [20], Jung and Gea [21]. Here we briefly describe the
sensitivity analysis of the problem (1) based on [16].
To solve the nonlinear equilibrium system, an iterative
procedure is commonly used to eliminate the residual force. The
residual force vector, r, is defined as the difference between the
external and internal force vectors. The equilibrium can thus be
expressed as
r=f−ˆ
f=0.(4)
The internal force vector ˆ
fis defined as
ˆ
f=
N
X
e=1ZVe
CT
eBTσdV=
N
X
e=1
CT
epe(5)
with Cedenoting the matrix that transforms the local nodal values
of the e-th element to global nodal values; Bbeing the matrix that
transforms a change in displacement into a change in strain; and σ
representing the local element stress vector.
In order to calculate the sensitivities of the objective function,
EP, with respect to a design variable, x, we rewrite the (3) by adding
an adjoint vector λmultiplied by a zero function
EP=lim
n→∞ "1
2
n
X
i=1
(uT
i−uT
i−1)(fi+fi−1)−λT
i(ri+ri−1)#.(6)
Now, differentiating (6) with respect to x, one can obtain
∂Ep
∂x=lim
n→∞"1
2
n
X
i=1
(uT
i−uT
i−1)∂fi
∂x+∂fi−1
∂x
+1
2
n
X
i=1 ∂uT
i
∂x−∂uT
i−1
∂x!(fi+fi−1)
−λT
i∂ri
∂x+∂ri−1
∂x#.(7)
The system of concern is subject to a gradual change in
displacement at certain nodes. At those degrees of freedom where
the displacement is explicitly defined, ∂uj
∂x=0. Everywhere else,
fj=0. The second term in the above equation, thus, cancels out.
Further, by considering (4), the above equation can be simplified
to
∂Ep
∂x=lim
n→∞"1
2
n
X
i=1
(uT
i−uT
i−1)∂fi
∂x+∂fi−1
∂x
−λT
i∂fi
∂x+∂fi−1
∂x+λT
i ∂ˆ
fi
∂x+∂ˆ
fi−1
∂x!#.(8)
To eliminate the unknown terms in (8), the adjoint equation is
defined as
λi=ui−ui−1(9)
from which the adjoint vector is readily calculable. Now, using (9)
in (8), one can get
∂Ep
∂x=lim
n→∞ "1
2
n
X
i=1
(uT
i−uT
i−1) ∂ˆ
fi
∂x+∂ˆ
fi−1
∂x!#.(10)
The validity of this sensitivity analysis is demonstrated through a
simple analytical problem by Huang and Xie [16].
Using a linear approximation, one can write
∂EP
∂x≈1EP
1x(11)
and
∂ˆ
f
∂x≈1ˆ
f
1x.(12)
Now using (12) and (11) in (10), the change in the energy dissi-
pation due to a change in a design variable can be approximated
as
1Ep≈lim
n→∞ "1
2
n
X
i=1
(uT
i−uT
i−1)1ˆ
fi+1ˆ
fi−1#.(13)
From (5), the change in the internal force due to removing or
adding an element can be calculated as
1ˆ
f=1xeCT
epe(14)
which can be substituted into (13) to yield
1Ep≈1xelim
n→∞ "1
2
n
X
i=1
(uT
i−uT
i−1)CT
e((pe)i+(pe)i−1)#.(15)
Using the trapezoidal numerical integration method and noting the
definition of the dissipated energy in (2), the above equation can be
simplified as
1Ep≈1xelim
n→∞ [(Ee)i−(Ee)i−1]=1xeEe,(16)
where (Ee)iis the total strain energy of the e-th element after
iiteration through solving the nonlinear equilibrium; and Eeis
the final strain energy of the e-th element upon completion of
the loading cycle. Noting the definition of design variables from
Section 2.2, one can observe that for removing an element 1xe=
−1 and for introducing an element 1xe= +1.
Based on (16) we define the following sensitivity number for an
element e
αe=1EP
1xe
=Ee(17)
which is a measure of efficiency of the e-th element. Note
that the sensitivity numbers defined in (17) are always positive.
Remembering that the maximum value of EPis desirable and
noting that 1EP=αe1xe, for removing an element (1xe=
−1), the element with the lowest sensitivity is the most suitable
candidate for removal. On the other hand, introducing a new
element strengthens the adjacent elements and results in 1xe>
0. Hence, in this case, the new element should be added in the
neighborhood of the elements with higher sensitivity numbers.
K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267 2261
2.4. Shape restrictions
BESO is naturally a topology optimization method which can
introduce new holes and fill the current holes in the domain. To
prevent the algorithm from changing the topology of the domain
and restrict it to shape optimization, it is necessary to restrict the
designable domain to the elements at the boundary of the slits. In
other words, the elements are only allowed to be removed from
and added to the boundary line. The designable domain in each
iteration is thus redefined as
D= {e|∃i,j∈B:i,j∈e∧i6= j},(18)
where Bis the set of boundary nodes defined as
B= {j|∃em∈M,ev∈V:j∈em∩ev}(19)
with Mand Vrepresenting the sets of solid and void elements.
2.4.1. Periodicity
In order to enhance the manufacturability of the solutions,
a periodic cellular design is considered with four identical
cells similar to the initial SSD design. The BESO method has
been previously proved useful in producing optimal periodic
structures [22]. To deal with periodic design problems, the design
domain should be divided into a number of identical cells. The
sensitivity numbers of corresponding elements in all of these cells
are then averaged and this averaged value is used as the sensitivity
number for all of these elements. This procedure can be illustrated
as
αi=1
Ncell
Ncell
X
j=1
αi,j,(20)
where αiis the averaged sensitivity number of the i-th element in
all cells; αi,jis the (original) sensitivity number of the i-th element
in the j-th cell; and Ncell is the number of cells. In this manner, the
BESO algorithm treats all the cells identically and maintains the
periodicity of the design.
2.4.2. Mirroring
Because of the nonlinear nature of the problem, the loading
sequence will affect the mechanical responses. This will generally
result in an unsymmetrical optimal shape. Hence one would get
mirrored shape results if once considers a displacement cycle
starting with an upward moving (↑↓↑) and once with an initial
downward moving (↓↑↓). In real case, however, it is uncertain
which direction is more likely to happen. It is thus reasonable to
consider both of these loading cases. To do so, one should consider
the mechanical responses of the two load cases and add them up
together to obtain the correct sensitivity number. However, as the
loading sequences are just mirror reflections of each other, we just
need to add the sensitivities of mirrored elements together. This
can be mathematically expressed as
¯
ai=ai+a↔
i(21)
where ¯
aiis the corrected (mirrored) sensitivity number of the i-th
element; and ↔
iis the element at the same location as the i-th
element in the mirrored structure.
2.5. BESO procedure
As already mentioned in Section 2.3, the elements with the
lowest sensitivity numbers are the least efficient and should be
removed while the ones with highest sensitivity numbers are the
most efficient and should be strengthened. In the BESO procedure
strengthening the elements is via introducing new elements in
their vicinity. In the new BESO algorithm [14], a filtering scheme
is used to assign a sensitivity number to the void elements in
the vicinity of solid elements. The filtering scheme is a wighted
Table 1
The specifications of test cases.
Case
name
Slot opening w(Fig. 3)
(mm)
Volume
fraction (%)
Number of elements
N=4Ncell
V84 5 85.6 7360
V76 7.5 78.4 6720
V68 10 71.2 6080
V60 12.5 64.0 5440
averaging which can be expressed as
ˆαi=
N
P
j=1
¯αjwij
N
P
j=1
wij
,(22)
where ˆαiis the filtered sensitivity number of the i-th element and
wij is a linear weighting factor defined as
wij =min{0,R−dij}.(23)
Here Ris a positive scalar value known as filtering radius and dij is
the distance between the centroids of the i-th and j-th elements.
Using this filtering scheme, the void elements in the neighbor-
hood of the elements with higher sensitivity number will obtain
higher filtered sensitivity numbers. Hence, in the BESO procedure,
the void elements with the highest filtered sensitivity numbers can
be assumed as the most efficient choices for introduction to the
system. This filtering scheme also smooths the jagged boundary
lines and overcomes some numerical instabilities such as checker-
board formation [23].
2.5.1. Adding and removing elements
In the optimization problem (1), the volume is fixed to a
predefined value, ¯
V. Starting from a feasible design with V=
¯
V, one needs to add and remove a same amount of material to
keep the volume unchanged. Using identical elements to discretize
the designable domain, the number of adding and removing
elements should be equal. If a solid element has a lower sensitivity
number than a void element, the two elements should be switched.
However, in order to prevent sudden changes, the maximum
number of changes should be restricted. The limiting number of
changes is referred to as ‘move limit’ hereafter and is denoted by
m. Comparing with the BESO algorithm proposed by Huang and
Xie [14], the move limit used here is equivalent to the maximum
adding ratio, i.e. m=ARmax.
The optimization loop continues until the following condition
is met
l
P
i=1
E(k−i+1)
P−E(k−i)
P
l
P
i=1
E(k−i+1)
P
< τ , (24)
where E(i)
Pdenotes the value of the objective function at i-th
iteration; kis the current iteration number; τis the convergence
tolerance selected as 10−4here; and lis chosen as 5 in the
numerical tests.
2.6. Numerical results
The initial design is depicted in Fig. 3. A series of tests are
conducted with different material volumes. Table 1 summarizes
the test cases and their specifications.
In all cases the volume is kept constant so that the objective
function values for different iterations could be compared. The
move limit is chosen as m=0.005N. The number of elements,
N, for each case is reported in Table 1. This relatively small value is
adopted due to the small number of elements in the design domain
2262 K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267
Fig. 3. The initial design.
Fig. 4. The results of case V84.
which is limited to the boundaries of the holes. The filtering radius
is selected as 5 mm through all the tests. The initial and final
cell designs and the evolution of the objective functions (energy
dissipation) for the test cases are illustrated in Figs. 4–7.
The increasing trend of the energy dissipation through opti-
mization iterations, which can be observed in all cases, verifies the
proposed approach. It can be seen that in all cases, the energy dis-
sipation increases significantly. In these four cases, the energy ab-
sorption capacity of the optimum results are improved 58%–96%
compared to the energy dissipation capability of the initial designs.
Fig. 5. The results of case V76.
Table 2
The improvement of the energy dissipation of the test cases after optimization.
Case name EP(J) Improvement (%)
Initial Final
V84 1124 2203. 96
V76 863 1440. 67
V68 631 1000. 58
V60 436 717. 64
Table 2 summarizes the improvements in the energy absorption
capacity of the test cases.
In order to achieve a simple manufacturable shape, some shape
restrictions have been enforced to the shape optimization problem.
Restricting the designable domain to the small set Ddefined in
(18) to maintain the topology, forcing the design to be periodic, and
mirroring will all impose limitations to the optimization process
by making its feasible space smaller. Furthermore because of the
binary nature of the BESO algorithm, the feasible space in not con-
tinuous. This restricted, discrete feasible space causes some oscil-
lations in the objective function values observable at the end of the
optimization procedures, when the solution is going to converge.
In all cases the optimum cell design is tapered in the middle. As
these four cases had different material volume, it can be concluded
that for this sort of dampers the diamond shaped holes provide
the best energy dissipation capacity irrespective of the material
volume.
2.7. Post-processing
In order to remove the jagged boundaries of the resulted shapes
and reduce the stress concentration it is necessary to smooth the
K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267 2263
Fig. 6. The results of case V68.
boundaries of the resulted shapes. A post-processor is written to
automatically smooth the boundaries of the optimal shapes using
Bézier curves. Fig. 8 shows the smoothed results. It can be seen that
irrespective of the volume fraction, all the optimum shapes include
diamond shaped holes.
To check the effect of the shape optimization on stress
distribution, the smoothed optimal shape of the case V60 is tested
against its smoothed initial shape. The force–displacement curve
and the stress distribution of the initial and optimal shapes are
depicted in Figs. 9 and 10 respectively.
It can be seen in Fig. 9 that the optimum shape provides a stiffer
design compared to the initial shape. After undergoing 10 mm
of displacement, the initial shape produces a reaction of 16.4 kN
while the reaction generated in the optimal shape is 24.3 kN. Also
a comparison between the stress levels in Fig. 10 reveals that
the optimal shape provide a much evener stress distribution. The
stress concentration zones in the corner of the holes visible in
the initial design are eliminated in the optimal shape. The stress
concentration zones are prone to fatigue and undesirable brittle
failure under cyclic loads. The fatigue failure of the SSD devices at
these zones have been reported in [18].
2.8. Optimal design
Based on the shape optimization results and Fig. 8, three
specimens with the shape design depicted in Fig. 11 are fabricated.
These specimens are used for experimental tests as discussed in
the following sections. Because of the tapered shape we refer to
this design as TSSD.
Fig. 7. The results of case V60.
3. Experimental verification
The objective of the experiments is to verify the cyclic
characteristics of the optimized shape. Furthermore, strength-
degradation and low-cycle fatigue characteristics of the device are
not predicted by the current finite element model. They must be
investigated by physical experiment.
3.1. Test setup, instrumentation and loading history
Identical setup with previous tests [18] was used such that
comparable results could be obtained. The test setup is shown
in Fig. 12. The specimens were installed between a ground
beam and an L-beam, securely fastened by four M16 bolts (snug
tight) on each side. Forced displacement was applied by an MTS
100 kN capacity computer-controlled actuator quasi-statically to
the specimen via the L-beam. To ensure the verticality of the
applied load, a pantograph system was welded to the right hand
side of the L-beam. The pantograph system also prevented the
L-beam from deflecting out-of-plane. The complete test setup
rested on a reaction frame which was significantly stiffer. The
centerline of the actuator implied an eccentricity to the specimen,
measured 162 mm to the centerline of the specimen. A free-run
of the setup (i.e. without the specimen installed) was performed
and the result showed that friction and the effect of gravity were
considered negligible. The setup was robust and repeatable, no
visible damage occurred after all tests were carried out. Fig. 13
shows a photograph of the setup with specimen installed.
Displacements of the specimens were measured independently
by a pair of LVDTs, marked as 1 and 2 in Fig. 12. While LVDT 1
2264 K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267
(a) Case V84. (b) Case V76.
(c) Case V68. (d) Case V60.
Fig. 8. The smoothed results.
(a) Initial design. (b) Optimal design.
Fig. 9. Comparing the force–displacement curves of the initial and optimal designs for case V60. The stress values are in MPa.
498.40
436.83
375.25
313.68
252.11
190.54
128.97
67.40
5.83
486.51
426.83
367.14
307.46
247.78
188.10
128.42
68.73
9.05
(a) Initial design. (b) Optimal design.
Fig. 10. Comparing the stress distribution of the initial and optimal designs for case V60. The stress values are in MPa.
K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267 2265
Table 3
Specimens and test results.
Specimen Test history Py(kN) δy(mm) kdkN
mm Pmax (kN) Pmin (kN) NcEd(kJ)
TSSD-1 0.5, 1.0, 1.5, 2.5, 5.0, 7.5, 10.0, 12.5, 15.0, 20.0 and 25 mm 9.1 1.8 4.22 15.9 −11.5 33a8.23
TSSD-2 Constant amplitude at 20 mm 9.8 2.7 4.18 15.5 −11.1 15 7.74
TSSD-3 Constant amplitude at 12.5 mm 9.9 2.8 4.55 15.1 −12.0 37 12.31
aNote: TSSD-1 did not break after 33 cycles were completed.
Fig. 11. TSSD shape design used for experiments.
700
Ground beam
L-Beam
12
Actuator Pantograph
system
Fig. 12. Test setup.
measures the elastic deformation of the support, the difference
across LVDT 1 and 2 measured the absolute distortion of the test
specimen.
Specimen TSSD-1 was tested under identical displacement
history with previously study on SSD [18]. The load history
comprised three repeated cycles at amplitudes of 0.5, 1.0, 1.5, 2.5,
5.0, 7.5, 10.0, 12.5, 15.0, 20.0 and 25 mm. TSSD-2 and TSSD-3 were
tested under a constant displacement until complete breakage
of the specimens. Table 3 summarizes the test histories and key
results.
3.2. Specimens
Based on the result of optimization, three specimens were
fabricated. Dimensions are shown in Fig. 11. All specimens (each
100 mm long) in this study were cut from the same structural
wide-flange section 152 ×152 ×37 Universal Column to BS4449
(depth ×flange width ×web thickness ×flange thickness is
161.8×152.2×8×11.5 mm respectively). Consequently, the
web thickness tis identical and material strengths of all specimens
may be assumed equal. Four 16 mm diameter holes were drilled
on each flange for connection to the test rig. Two standard test
coupons were taken from the web of the section. Coupon tests gave
Fig. 13. Overview of test setup.
an average tensile yield stress of 316.5 N/mm2and an average
Modulus of Elasticity of 206.1 kN/mm2.
3.3. Test results and discussion
All three specimens deformed in a stable manner under the
cyclic tests. The strips deformed in double curvature as expected.
Fig. 14(a)–(c) present the force–displacement hysteresis obtained
from the cyclic tests. A positive sign refers to downward force and
displacement. Shear strain γ(i.e. distortion divided by total width
of device) of the specimens are also shown. Positive yield strength
Py, its corresponding yield displacement δy, elastic stiffness kd,
positive peak strength Pmax, negative peak strength Pmin , the
number of cycles to failure Ncand energy dissipation Edare
tabulated in Table 3.
It is clear that all specimens have yielded at small displace-
ment and exhibited very stable hysteretic behavior with a gradual
transition between the elastic and inelastic regime. The specimen
response magnitude was slightly lesser than the input displace-
ment history due to elastic deformation of the support. Absolute
displacements (difference across LVDT 1 and 2) are used to deter-
mine the mechanical properties of the specimens. The connection
of the specimens by four structural bolts on each flange performed
satisfactory; no significant distortion was observed after the tests.
Fig. 15 shows the damaged specimen TSSD-1 after testing.
3.4. Specimen TSSD-1
TSSD-1 completed all cycles without breakage. At the end of
the 25 mm cycle, force was released such that the specimen
was left deformed. A photograph of the specimen after all cycles
2266 K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267
Shear Strain ( )
γ
15
10
5
–5
–10
–15
-25 -20 -15 -10
-1.0 -0.05
-5 0 5 10 15 20 25
Displacement (mm)
0
0
0.05 0.1 0.15
20
–20
Force (kN)
Shear Strain ( )
γ
15
10
5
–5
–10
–15
-20 -15 -10
-1.0 -0.05
-5 0 5 10 15 20
Displacement (mm)
0
0
0.05 0.1 0.15
20
Force (kN)
(a) TSSD-1. (b) TSSD-2.
Shear Strain ( )
γ
15
10
5
–5
–10
–15
-25 -20 -15 -10
-1.0 -0.05
-5 0 5 10 15 20 25
Displacement (mm)
0
0
0.05 0.1 0.15
20
–20
Force (kN)
(c) TSSD-3.
Fig. 14. Force–displacement hysteresis of TSSD specimens.
is shown in Fig. 15. Its force–displacement hysteresis is shown
in Fig. 14(a). Very stable hysteresis without noticeable sign of
strength deterioration was observed. Cracks have developed on
the bar surfaces due to repeated loading, but they did not cause
breakage. Initial yield strength and displacement were recorded at
9.1 kN and 1.8 mm respectively. Bauschinger effect was apparent,
with negative peak strength only 72% of the positive peak.
TSSD-1 dissipated 8.23 kJ at the end of the cycles. Energy
dissipation of TSSD-1 is shown in Fig. 16. Result from specimen
SL-1 [18] is plotted on the same chart for comparison. To account
for different volume of steel involved in specimens, energy is
expressed as energy dissipated per unit volume of steel. Here
the volume of steel between fillets is taken into consideration
as the flange does not contribute to energy dissipation. There
is a 37% increase in energy dissipation and it is evident that
the optimization process has resulted in a more efficient design.
From the same diagram it is clear that, after optimization, TSSD-1
sustained a much larger cumulative displacement compared to
SL-1. The enhanced resistance against low-cycle fatigue is clear.
3.5. Specimens TSSD-2 and TSSD-3
TSSD-2 and TSSD-3 were fabricated to identical dimensions
as TSSD-1, but these two specimens were tested under constant
amplitudes until complete breakage. They enable us to identify
their energy dissipating capacity under different displacement
amplitudes. TSSD-2 was tested at 20 mm (µ ≈11)while TSSD-
3 was tested at 12.5 mm (µ ≈7). Their load–displacement
hysteresis are shown in Fig. 14(b) and (c). Both specimens
exhibited stable behavior without noticeable degradation during
their early cycles. TSSD-2 sustained 15 complete cycles prior to
Fig. 15. Specimen TSSD-1 after the test.
failure, while TSSD-3 sustained 37 cycles. TSSD-2 dissipated 7.74 kJ
of energy, while TSSD-3 dissipated 12.31 kJ. It is interesting that at
relatively low displacements, the device dissipated a much larger
amount of energy.
4. Conclusion
This paper proposed a new steel slit damper design, TSSD,
based on the numerical shape optimization results. A previously
proposed steel slit damper, SSD, with straight uniform slit width
has been taken as the initial design. An optimization procedure
has been proposed based on the well-known BESO method to
find the optimum shape of the slits. Some shape restrictions have
K. Ghabraie et al. / Engineering Structures 32 (2010) 2258–2267 2267
200 400 600 800 1000
SL–1
TSSD–1
×
10–4
1.6
1.4
1.2
1.0
0.8
0.2
0.4
0.6
0 1200
Cumulative Displacement (mm)
1.8
0
Energy/Volume (kJ/mm3)
Fig. 16. Cumulative energy dissipation.
been introduced and imposed in the optimization procedure to
maintain the topology of the design and restrict it to a symmetric
periodic cellular shape with 4 cells. The plastic energy dissipation
of the damper after one cycle of displacement loading with
10 mm amplitude has been taken as the objective function. The
optimization problem has been then stated as maximizing the
objective function while the material volume is kept constant
and the shape is restricted. Four initial models with similar
shape to SSD has been considered as the initial designs each
having a different material volume. The resulted evolution of
the objective function values for all the cases has shown a
significant increase in the energy dissipation capacity verifying the
proposed optimization procedure. Improvements of 58%–96% in
the energy absorption capacity of the designs have been recorded.
The optimum shapes were all include bars tapered in the middle
forming diamond shaped slits irrespective of the material volume.
A post-processor has been used to smooth the optimum results
using Bézier curves. It has been demonstrated that the optimum
tapered slit design provides an even stress distribution and the
stress concentration noticeable in the initial straight slit design
has been eliminated in the optimum design. This even stress
distribution can significantly improve the behavior of the damper
under fatigue.
The finite element model used in the optimization process was
not capable of predicting failure and fatigue of the design. There-
fore, based on the optimization findings, three TSSD specimens
were fabricated and put under cyclic tests. Under identical test
setup and load history, the TSSD specimen dissipated 37% more en-
ergy per unit volume compared to the previously tested SSD, and
significantly delayed low-cycle fatigue. It should be noted that this
figure is not comparable with the improvements reported on Ta-
ble 2 because the volume of the original SSD and the proposed TSSD
are not equal. Experiments confirmed that the optimization pro-
cess is robust and it is suitable for future development of energy
dissipaters.
Acknowledgement
Experimental works described in this paper is carried out
during an academic visit to City University of Hong Kong by the
second author, supported by the Research Grant Council of Hong
Kong City University RGC 115208.
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