Dec 28, 2018 - proposed a novel variable friction cladding connection (VFCC), which enables ... elements and supplemental damping systems to achieve a .... The device consists of two sets of .... the friction element Ff is assumed to be under a constant maximum .... by the first mode when subjected to wind excitations [42].
Motion-Based Design Approach for a Novel Variable Friction Cladding Connection used in Wind Hazard Mitigation Yongqiang Gong1 , Liang Cao2 , Simon Laflamme1,3 , James Ricles2 , Spencer Quiel2 , Douglas Taylor4
Abstract Cladding systems typically serve architectural purposes and protect occupants against the external environment. It is possible to leverage these systems to enhance structural resiliency. A common application is the use of blast resistant panels for enhanced protection against man-made hazards, whereas energy is dissipated through sacrificial elements. However, because these protection systems are passive, their mitigation capabilities are bandwidth limited, therefore targeting single types of hazards. The authors have recently proposed a novel variable friction cladding connection (VFCC), which enables the leveraging of cladding inertia for mitigating blast, wind, and seismic hazards. The variation in the friction force is generated by an actuator applying pressure onto sliding friction plates via a toggle system. Previous work has characterized the dynamic behavior of a VFCC prototype, and established design procedures for blast mitigation applications. Here, work is extended for applications to wind mitigation. An analytical model is developed to characterize the dynamic behavior of the VFCC for wind-induced vibrations. A motion-based design framework is developed to enable an holistic integration of the device within the structural design phase. Numerical simulations are conducted on a 24-story building example to demonstrate the motion-based design methodology. Results show that the semi-active cladding system provides significant reduction in the wind-induced inter-story drift and floor acceleration, therefore demonstrating the promise of the VFCC for field applications. Keywords: Motion-based design, cladding connection, wind mitigation, semi-active control, variable friction, high performance control system
1
1. Introduction
2
Motion-based design (MBD) of civil structures is a design methodology that consists of sizing stiffness
3
elements and supplemental damping systems to achieve a desirable structural performance [1]. However,
4
when multiple excitation inputs are considered either individually or combined, termed multi-hazards, the 1 Department
of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA 50011, U.S.A of Civil and Environmental Engineering, Lehigh University, Bethlehem, PA 18015, U.S.A 3 Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, U.S.A 4 Taylor Devices, North Tonawanda, NY 14120, U.S.A 2 Department
Preprint submitted to Journal of Engineering Structures
December 28, 2018
5
MBD approach becomes difficult to implement due to potentially conflicting objectives that may not be
6
simultaneously attained with a traditional structural system. For instance, passive energy dissipation systems
7
have bandwidth-limited mitigation capabilities [2, 3]. A solution is the utilization of high performance control
8
systems (HPCSs) that include active [4, 5, 6], semi-active [7, 8, 9] and hybrid control systems [10, 11, 12].
9
These systems have the particularity of being alterable for adapting to different dynamics, which makes them
10
ideal for multi-hazard mitigation [13, 14].
11
Several types of HPCSs have been proposed of the last decades, with the common objective to provide
12
high mitigation capability on limited power input. Of interest to this paper is the concept of multifunctional
13
cladding systems, where cladding inertia is leveraged to provide vibration mitigation functions. Early work
14
studied blast mitigation capabilities by sacrificing the cladding panel itself. Strategies included double-
15
layer foam cladding [15], sandwich cladding [16, 17], tube-core cladding [18] and metal layer cladding [19].
16
Advanced cladding connections have also been proposed for blast mitigation. For example, Amadio and
17
Bedon [20] developed a viscoelastic spider connector for cable-supported glazing facades. Chen and Hao [21,
18
22] introduced a rotational friction hinge for sandwich panels. Wang et al. [23] experimentally investigated a
19
blast-absorbing cladding connector fabricated from aluminum foam and curved plates. Others have studied
20
wind and seismic mitigation applications. Baird et al. [24] have proposed a U-shape flexural plate dissipator
21
formed by bending a mild steel plate and experimentally investigated its performance at seismic mitigation.
22
Ferrara et al. [25] and Biondini et al. [26] tested bolted friction connectors joining individual cladding
23
elements for energy dissipation. Maneetes et al. [27] studied a spandrel-type precast concrete cladding using
24
supplementary friction devices to establish a lateral force resisting system.
25
It follows that the vast majority, if not all, of the proposed energy dissipation systems leveraging cladding
26
systems are passive strategies. In order to further the applicability to multi-hazard mitigation, the authors
27
have proposed a semi-active cladding connector, termed Variable Friction Cladding Connection (VFCC)
28
[28, 29]. The VFCC is a variable friction device that is engineered to laterally connect cladding elements to
29
the structural system. The variation in the friction force is generated by an actuator applying pressure onto
30
sliding friction plates via a toggle system. Various damping devices based on variable friction mechanism
31
have been studied for structural control applications, including electromechanical- [30, 31], electromagnetic-
32
[32, 33], magnetorheological- [34, 35], hydraulic- [36, 37], and piezoelectric- [38, 39] based technologies. The
33
proposed device is novel by being engineered to leverage cladding panels for vibration mitigation.
34
In previous work, a prototype of the VFCC has been fabricated, experimentally tested, and its dynamic
35
behavior characterized [28]. Afterwards, its passive application for blast energy absorption was studied and
36
a MBD developed [29]. In this paper, the authors extend work to wind-induced vibrations. An analytical
37
model is developed based on cladding-structure transfer functions. It is followed by the development of MBD
38
procedures, which leads to the selection of cladding connection properties, including connection stiffness and
2
39
device damping capacity. While the paper does not consider multi-hazards, results are to be incorporated in
40
future work on multi-hazard applications of the VFCC. Work presented in this paper is different from the
41
investigation on blast loads, because the device is used here in semi-active mode.
42
The paper is organized as follows. Section 2 provides the background on the VFCC device and its
43
dynamics. Section 3 derives a structure-cladding model along with its associated transfer functions to
44
facilitate the MBD procedure. Section 4 develops the MBD procedure used to determine the dynamic
45
parameters of the VFCC. Section 5 demonstrates the VFCC and the MBD procedure through numerical
46
simulations conducted on a 24-story building. Section 6 concludes the paper.
47
2. Variable Friction Cladding Connection
48
The VFCC is a variable friction device engineered to laterally connect cladding elements to the structural
49
system. The device is schematized in Fig. 1 (a) and shown in Fig.1 (b). The device consists of two sets of
50
sliding friction plates upon which a variable pressure is applied by an actuator through toggles. Piezoelectric-
51
based [39] or electromagnetic-based [40] technologies could be good candidates to actuate the device. Figure
52
1 (b) shows a spacer in lieu of the actuator. These spacers were used to emulate actuation during the dynamic
53
characterization process in [28] by maintaining a constant displacement on the toggle. The figure also shows
54
the aluminum frame that was used to mount the device in the testing machine, where the applied force is
55
acting vertically. The blocks are used to prevent the toggles from pushing beyond a vertical alignment.
(a)
(b)
Figure 1: (a) Schematic representation of the VFCC; and (b) annotated picture of a prototype.
3
Figure 2: Example configuration of the VFCC installed in a floor slab (top view) with diagram of forces.
56
Figure 2 illustrates a possible configuration of the device, where it is embedded into a floor slab with the
57
inner plates extended outward to connect to the cladding panel. The actuator force Fa produces an axial
58
force Ft onto the toggles, which in turn generates a normal force N onto the friction plates. A compressive
59
pressure pc is consequently applied and is assumed to be uniformly distributed between both toggles pc =
N
(1)
Ac,max
60
where Ac,max = bp (lp − 2d) is the maximum contact area of the friction plates under the normal pressure, bp
61
and lp are the width and length of the friction plate, respectively, and d is the distance between the toggle
62
and the end of the friction plate, as illustrated in Fig. 2. The effective contact area Ac between friction
63
64
65
66
67
plates varies with their relative displacement y and the generated Coulomb friction force Fc is written 2µc N if 0≤y 1 and Cσ are constants. Table 1 lists parameters of the VFCC prototype characterized in [28].
(a)
(b)
Figure 3: Dynamics of the VFCC under a harmonic excitation of amplitude 13 mm at 0.05 Hz under various levels of actuation capacity: (a) force-displacement loop; and (b) force-velocity loop.
5
Table 1: Parameters of the VFCC prototype [28].
parameter
units
value
lp
mm
165
bp
mm
60
d
mm
45
Cs
−
1.052
Cσ
mm
2.185
kN·mm−1
1.147
σ1
N·s·mm
−1
0.200
σ2
N·s·mm−1
0.200
σ0 |Fc0 =0
79
−1
3. Analytical Transfer Functions
80
This section derives the analytical transfer functions of a linear time-invariant structure-cladding system
81
used for the MBD procedure. The equations of motion are first derived, followed by the non-dimensional
82
transfer functions.
83
3.1. Equations of motion
84
Fig. 4 is the diagram of a n-story structure equipped with a cladding system and VFCCs. The primary
85
structure is represented by a lumped-mass shear system and the cladding panel at each floor is modeled as a
86
uniform rigid bar connected to adjacent floors. The lateral cladding connection is represented by a stiffness
87
element kc , a viscous damping element cc0 , and a variable friction element Ff . In this configuration, the
88
VFCC is assumed to provide only lateral resistance. The gravity resistance is considered as decoupled and
89
provided by conventional gravity connections. To develop a mathematically trackable analytical solution,
90
the friction element Ff is assumed to be under a constant maximum capacity (i.e., passive-on configuration)
91
and the corresponding passive friction element is transformed into an equivalent viscous damping element
92
cv [1], yielding a general viscous damping element for the connection cc cc = cc0 +
4Ff πΩˆ x
(9)
93
where Ω is the excitation frequency and x ˆ is the amplitude of structure-cladding displacement due to the
94
dynamic load. Note that this assumption does not hold for the intended semi-active application, and only
95
provides a rough approximation of the friction behavior. The intent is to develop a design procedure that
96
would provide starting points in the sizing of the device. As it will be discussed later, this technique will
97
also require the approximation of the wind load into a harmonic load.
6
(a) Figure 4: Diagram of a n-story structure equipped with a cladding system and VFCCs.
The equations of motion of the linear time-invariant n-story structure-cladding system with equivalent
98
99
viscous damping are given by Ms x ¨s + Cs x˙ s + Ks xs = Ec Cc x˙ c + Ec Kc xc
(10a)
Mc x ¨c + Cc x˙ c + Kc xc = Ep p − Mc ET ¨s cx
(10b)
100
where xs ∈ Rn×1 and xc ∈ R2n×1 are the displacement vectors of the primary structure and of the cladding
101
relative to the primary structure, respectively, p ∈ Rn×1 is the external load input vector, Ep ∈ R2n×n and
102
Ec ∈ Rn×2n are the external loading and cladding location matrices, respectively, and Ms ∈ Rn×n , Cs ∈
103
Rn×n , Ks ∈ Rn×n are the mass, damping, and stiffness matrices of the primary structure and Mc ∈ R2n×2n ,
104
Cc ∈ R2n×2n , Kc ∈ R2n×2n are the mass, damping, and stiffness matrices associated with the cladding
105
system.
106
The mass of the cladding panel at each floor mc is taken to be identical at the preliminary design stage for
107
simplicity. The cladding connection, which includes the lateral stiffness kc and the equivalent viscous damping
108
cc , is assumed to be identical at each floor for simplicity of the design process and field implementation. The
109
mass, damping, and stiffness matrices of the cladding elements, Mc ∈ R2n×2n , Cc ∈ R2n×2n , Kc ∈ R2n×2n ,
110
are element diagonal matrices composed with the cladding mass, damping and stiffness matrices mc , cc and
111
kc :
7
mc =
mc 3 mc 6
mc 6 mc 3
cc cc = 0
0 cc
kc and kc = 0
(a)
0 kc
(b)
Figure 5: Diagram of the ith cladding element connected to adjacent floors: (a) two DOFs representation; and (b) SDOF representation.
112
113
Fig. 5(a) illustrates the two degrees-of-freedom (DOFs) representation of the ith cladding element connected to adjacent floors. Its equation of motion is written mc x ¨c,i + cc x˙ c,i + kc xc,i = ep pi − mc x ¨s,i xs,i ]T and xc,i = [xc,2i−1
(11)
xc,2i ]T are the displacement vectors of the primary structure
114
where xs,i = [xs,i−1
115
and of the ith cladding element relative to its connected floors, respectively, pi is the external load acting on
116
the ith cladding, and ep = [ 12
117
is connected to the ground and to the first floor, as illustrated in Fig. 4 and xs,0 corresponds to ground
118
motion, which is taken as xs,0 = 0 for wind loading.
1 T 2]
is the load location vector. Note that the first cladding panel (i.e., i = 1)
119
To obtain a mathematically trackable displacement vector xc,i , its dynamics is expressed in terms of the
120
modal vectors Φcr and modal coordinates qcr,i (r = 1, 2) for DOFs r = 1, 2, and is assumed to be governed
121
by the first mode xc,i ≈ Φc1 qc1,i
(12) φc,12 ]T = [1
1]T , obtained by
122
where qc1,i is the modal coordinate of the first mode and Φc1 = [φc,11
123
solving the algebraic equation [kc − ωc2 mc ]Φc1 = 0 [41]. Pre-multiplying Eq. 11 by ΦT c1 and substituting
124
xc,i from Eq.12 yields a linear time-invariant single DOF (SDOF) expression governing the dynamics of the
125
cladding element mce q¨c1,i + cce q˙c1,i + kce qc1,i = pi − mce
126
x ¨s,i−1 + x ¨s,i 2
with the equivalent cladding mass mce , damping cce , and stiffness kce 8
(13)
mce = ΦT c1 mc Φc1 = mc
cce = ΦT c1 cc Φc1 = 2cc
and kce = ΦT c1 kc Φc1 = 2kc
127
where the displacement qc1,i ≈ xc,2i−1 ≈ xc,2i . The SDOF schematic representation is illustrated in Fig.
128
5(b).
129
Similarly, the displacement vector of the primary structure xs is written in terms of its modal vectors
130
Φsk and modal coordinates qsk (k = 1, 2, ..., n), with the assumption that its dynamic behavior is governed
131
by the first mode when subjected to wind excitations [42] xs =
n X
Φsk qsk ≈ Φs1 qs1
(14)
k=1
φs,12
...
φs,1n ]T with φs,1n normalized to unity. Pre-multiplying Eq. 10a by ΦT s1
132
where Φs1 = [φs,11
133
and substituting xs from Eq.14 yields an uncoupled equation of motion for qs1 , mse q¨s1 + cse q˙s1 + kse qs1 = ΦT ˙ c + ΦT s1 Ec Cc x s1 Ec Kc xc
134
(15)
with the equivalent structural mass mse , damping cse , and stiffness kse equal to mse = ΦT s1 Ms Φs1
cse = ΦT s1 Cs Φs1
and kse = ΦT s1 Ks Φs1
135
where qs1 ≈ xsn is the displacement of the SDOF structure and the damping matrix Cs is taken proportional
136
to the stiffness matrix Ks .
137
138
It follows that the governing equations of motion of the linear time-invariant structure-cladding system are reduced to mse q¨s1 + cse q˙s1 + kse qs1 =
n X
αi (kce qc1,i + cce q˙c1,i )
(16a)
i=1
mce q¨c1,i + cce q˙c1,i + kce qc1,i = pi − αi mce q¨s1 139
where αi = 12 (φs1,i−1 + φs1,i ) for i = 2, 3, ..., n and α1 = 12 φs1,1 .
140
3.2. Non-dimensional transfer functions
(16b)
141
To derive the non-dimensional transfer functions, the wind load pi is approximated as a harmonic exci-
142
tation pi (t) = pbi ejΩt . The corresponding steady state response of the structure-cladding system is written
143
as [1] qs1 = qbs1 ej(Ωt+δs ) qc1,i = qbc1,i ej(Ωt+δci )
144
(17)
where the hat denotes an amplitude, j is the imaginary unit, Ω the excitation frequency, and δ the phase.
9
145
Substituting Eq. 17 into Eqs. 16a and 16b yields n X kse − mse Ω2 + jcse Ω qbs1 ejδs = (kce + jcce Ω) αi qbc1,i ejδci
(18a)
i=1
kce − mce Ω2 + jcce Ω qbc1,i ejδci = pbi + αi mce Ω2 qbs1 ejδs 146
Multiplying Eq. 18b by αi and summing from i = 1 to n yields n X
αi qbc1,i ejδci =
i=1
147
148
where pbe =
n P i=1
pbe + Γmce Ω2 qbs1 ejδs kce − mce Ω2 + jcce Ω
αi pbi is the amplitude of the equivalent load and Γ =
n P i=1
(19)
αi2 . Substituting Eq. 19 into Eqs.
18a and rearranging Eq.18b yields
kce + jcce Ω qbs1 ejδs = 2 pbe (kse − mse Ω + jcse Ω) (kce − mce Ω2 + jcce Ω) − (kce + jcce Ω) Γmce Ω2 qbc1,i ejδci pbi + αi mce Ω2 qbs1 ejδs = pbe pbe (kce − mce Ω2 + jcce Ω) 149
(20)
Defining the mass ratio µ, tuning frequency ratio f , and excitation frequency ratio λ as µ=
150
(18b)
mce mse
f=
ωc ωs
and λ =
Ω ωs
with r ωs =
kse mse
cse ξs = 2mse ωs
r ωc =
kce mce
and ξc =
cce 2mce ωc
151
yields non-dimensional transfer functions Hs (λ) and Hc,i (λ) that represent the dynamic amplification of the
152
displacement of the SDOF structure and of the displacement of the ith cladding relative to its connected
153
floor, respectively, where qbs1 ejδs f 2 + j2ξc f λ = 2 2 2 2 pbe /kse (1 − λ )(f − λ ) − 4ξc ξs f λ − µΓf 2 λ2 + j [2ξc f λ(1 − (1 + µΓ)λ2 ) + 2ξs λ(f 2 − λ2 )] qbc1,i ejδci pbi /b pe + αi µλ2 Hs (λ) = Hc,i (λ) = 2 pbe /kse µ(f − λ2 ) + j2ξc µf λ Hs (λ) =
154
155
An additional transfer function Ha (λ) representing the dynamic amplification of the structural acceleration is given by Ha (λ) =
156
(21)
b as1 ejδs = −λ2 Hs (λ) pbe /mse
where the amplitude of the acceleration b as1 = Ω2 qbs1 .
10
(22)
157
4. Motion-Based Design Procedure
158
This section describes the MBD procedure associated with the semi-active cladding connection used for
159
wind hazard mitigation. This procedure is conducted assuming a passive behavior of the device to provide
160
preliminary design values, including a preliminary value for the VFCC’s friction capacity. To optimize the
161
design parameters, one would be required to conduct simulations with semi-active control rules. Such a
162
design strategy is commonly used in the design of semi-active or hybrid control systems in order to provide
163
a target damping capacity [7, 43, 44].
164
The procedure, diagrammed in Fig. 6, starts by quantifying the design wind load (e.g., return period).
165
After the allowable structural acceleration ap and inter-story drift ratio ∆p are defined, as well as the
166
allowable structure-cladding spacing lc , both based on the design wind load, the dynamic parameters of
167
the cladding connection consisting of stiffness kce and equivalent viscous damping ratio ξc , are selected.
168
Subsequently, three non-dimensional analytical solutions Ra , Rs , and Rc associated with the maximum
169
structural acceleration amax , the maximum drift ratio ∆max , and the maximum structure-cladding spacing
170
lmax , respectively, are used to verify the satisfaction of the performance metrics. If the structural motion
171
is unsatisfactory, the process is iterated by either re-designing dynamic parameters kce and ξc (option 1) or
172
updating the allowable structure-cladding spacing lc (option 2). These MBD steps are described in what
173
follows.
Figure 6: Motion-based design procedure.
11
174
175
4.1. Wind load The time-varying wind load acting on cladding panels at height z is given by [45] Pd (t) =
1 ρCd AVd (t)2 2
(23)
176
where Cd is the drag coefficient, ρ is the air density, A is the cladding area exposed to the wind pressure,
177
and Vd (t) is the time-varying wind speed [45] Vd (t) = V + v(t)
(24)
178
where v(t) is a zero mean fluctuation of the wind flow Vd (t) around its mean wind speed V . The design
179
value for V is computed from a 3-second wind gust speed V0 that can be obtained from wind hazard maps
180
[45, 46] V = 0.13V0
v∗ ln(z/z∗ ) v∗0
(25)
181
where z∗ is the surface roughness length of the building’s terrain, and v∗ and v∗0 are the shear velocities
182
of wind flow of the building and open terrain, respectively. Example values for the ratio v∗ /v∗0 are listed
183
in Table 2. Substituting Eq. 24 into Eq. 23, the wind load is expressed as the sum of a steady wind force
184
component P and a fluctuating wind force component p(t) [45] Pd (t) =
1 ρCd A V 2 + v 2 (t) + ρCd AV v(t) 2
(26)
≈ P + p(t) 185
where v 2 (t) may be neglected for high-rising buildings or, as done in this paper, approximated into a steady
186
component using its mean value v 2 (t) [45]. The mean square value of the wind fluctuation v 2 (t) can be
187
written [45] v 2 (t) = βv∗ = β
0.4V ln(z/z∗ )
2 (27)
188
where the constant β depends on surface roughness length at the building terrain, also listed in Table 2. The
189
wind fluctuation v(t) at height z can be modeled as a stochastic process characterized by a two-sided power
190
spectral density function (PSD) at excitation frequency Ω [47], where Sv (z, Ω) =
191
192
1 200 2 z h Ωz i−5/3 v∗ 1 + 50 2 2π V 2πV
(28)
The fluctuating wind forces can be modeled as a zero-mean Gaussian stationary and spatiotemporal field that is characterized by its cross-spectral density function Sp (zi , zl , Ω) [48] p Sp (zi , zl , Ω) = (ρCd Ai Vi )(ρCd Al Vl ) Sv (zi , Ω)Sv (zl , Ω)Coh(zi , zl , Ω)
12
(29)
193
where subscripts i and l refer to different heights, and the coherence function Coh is given by [46] h 10Ω|z − z | i i l Coh(zi , zl , Ω) = exp − π(Vi + Vl )
(30)
Table 2: Terrain exposure constants[45]
194
ocean
open terrain
suburb
city center
downtown
z∗ (m)
0.005
0.07
0.30
1.00
2.50
v∗ /v∗0
0.83
1.00
1.15
1.33
1.46
β
6.5
6.0
5.25
4.85
4.00
4.2. Performance objectives
195
Motion criteria associated with wind-induced vibrations are linked to the average return period of the
196
wind excitation under consideration (e.g., 1, 10, 50 and 475 years). The drift ratio ∆p can be selected on
197
the basis of structural damage minimization. Typical values are within the range 1/750 ≤ ∆p ≤ 1/250
198
corresponding to a serviceability limit state [49].
199
The acceleration criterion ap is typically associated with building serviceability and its performance
200
threshold is left to the designer. Here, the selection of ap is conducted based on Refs. [50, 51] to ensure
201
occupancy comfort. The acceleration threshold ap (m·s−2 ) is function of the average return period Q (yr)
202
and the fundamental frequency of the structure ωs (Hz) p ap = (0.68 + 0.2 ln Q) 2 ln(ωs T )e−3.65−0.41 ln ωs
(31)
203
where T is the observation time of the wind event. A typical value for T is T = 3600 s (1 hr) [50]. Fig. 7
204
plots different values for ap as a function of various ωs and Q.
Figure 7: Acceleration criterion ap as a function of ωs and Q.
13
205
Conventional cladding panels are connected to the structural system using vertical bearing connections
206
to support gravity loads and push-pull (tieback) connections to resist lateral loads [52]. A common push-
207
pull connection is a flexible threaded rod designed to either slide inside a slot (sliding connection) or to
208
bend (flexing connection) [53]. A minimum cladding-structure spacing lmin is required for the installation
209
and maintenance of the lateral connections, and this value can be as much as 15 cm [54]. Here, allowable
210
structure-cladding spacing lc is selected such that lc ≥ lmin .
211
4.3. Connection design
212
A first step in the design of the cladding connection is the selection of the dynamic parameters. This is
213
done through initially selecting connection parameters and obtaining the peak building responses based on
214
three non-dimensional analytical solutions Ra , Rs and Rc , yielding amax , ∆max , and lmax , respectively, to
215
be compared with the performance objectives. These three analytical solutions are derived in what follows,
216
and their solutions verified in section 5.4.
217
4.3.1. Maximum structural response
218
219
Consider the response of a structure exposed to wind where the wind velocity fluctuation is taken as a stochastic stationary Gaussian process. The structural response q(t) is given by [55] q(t) = q¯ + qs (t)
(32)
220
where qs (t) is the zero mean fluctuation of structural displacement q(t) around its mean value q¯. The mean
221
static displacement q¯ can be obtained by applying the equivalent mean wind load component Pe to the
222
structure [56], with Pe =
n X
αi Pi
(33)
i=1 223
yielding q¯ = Pe /kse . Following Davenport’s method [57], the expected maximum displacement qmax over
224
observation time T and the corresponding peak response factor η are given by qmax = q¯ + ησq η=
p
2 ln(ωs T ) + p
0.5772 2 ln(ωs T )
(34)
225
where σq is the standard deviation of qs (t). The maximum structural acceleration amax = ησa [55], where
226
σa is the standard deviation of the wind-induced structural acceleration as (t).
227
The mean square values of the structural displacement and acceleration, σq2 and σa2 , are computed by
228
integrating the PSD based on the product of the fluctuating wind load component and the transfer functions
229
of the structure-cladding model [51]. For a lightly damped structure, the response power spectrum is dom-
230
inated by the contribution of the excitation power spectrum around the natural frequency of the structure 14
231
[58, 59]. It follows that a constant power spectrum of the excitation at the first natural frequency of the
232
structure Spe (ωs ) is used to compute σq2 and σa2 [58]. σq2 = σa2 =
233
ωs 2 kse
ωs m2se
+∞
Z
Spe (Ω)|Hs (λ)|2 dλ ≈
−∞ Z +∞
Spe (Ω)|Ha (λ)|2 dλ ≈
−∞
ωs Spe (ωs ) 2 kse
ωs Sp (ωs ) m2se e
where the PSD of the equivalent fluctuating wind force Spe (Ω) =
+∞
Z
|Hs (λ)|2 dλ
−∞ Z +∞
|Ha (λ)|2 dλ
(35a) (35b)
−∞
n P n P
αi αl Sp (zi , zl , Ω) [51]. A solution for
i=1 l=1 234
the integration of transfer functions,Js and Ja , can be obtained using the integral formula from Gradshteyn
235
and Ryzhik [60] Z
+∞
Js = −∞ Z +∞
Ja =
|Hs (λ)|2 dλ = π
b20 /a0 (a2 a3 − a1 a4 ) − a3 b21 a1 (a2 a3 − a1 a4 ) − a0 a23
(36a)
|Ha (λ)|2 dλ = π
a1 b22 + b23 /a4 (a0 a3 − a1 a2 ) a1 (a2 a3 − a1 a4 ) − a0 a23
(36b)
−∞ 236
where a0 = f 2 , a1 = 2ξc f + 2ξs f 2 , a2 = −4ξc ξs f − [1 + (1 + µΓ)f 2 ], a3 = −2ξc f (1 + µΓ) − 2ξs , a4 = 1,
237
b0 = f 2 , b1 = 2ξc f , b2 = f 2 , and b3 = 2ξc f . Two non-dimensional analytical solutions Ra and Rs are used
238
to compute the maximum acceleration and the maximum interstory drift ratio of the structure, respectively, p η ωs Spe (ωs )Ja amax Ra = = (37a) Pe /mse Pe p η ωs Spe (ωs )Js ∆max =1+ (37b) Rs = φ −φ Pe maxi | s,1i s,1i−1 |Pe /kse hi
239
φ −φ where the maximum inter-story drift ratio ∆max = maxi s,1i his,1i−1 qmax based on the assumption that the
240
structure vibrates at its first modal shape and hi is the inter-story height of the ith floor.
241
4.3.2. Structure-cladding spacing
242
243
The maximum relative displacement between the cladding and the primary structure lmax is estimated using the mean and fluctuating wind force components lmax = maxi |li + qbc1,i |
(38)
244
where li = Pi /kce is the displacement of the ith cladding relative to that of the connected floor due to the mean
245
static wind force component, and qbc1,i is the maximum relative displacement resulting from the fluctuating
246
wind force component p(t). Unlike for the computation of Ra and Rs , the cladding response analysis using
247
PSD of a stochastic wind load excitation could require numerical simulations unless a simplification for
248
the term pbi /b pe is conducted in Hc,i (λ) (Eq. 21). Instead, the forcing fluctuation of the wind flow pi (t) is
249
directly simplified as a harmonic process to obtain a mathematically trackable solution for qbc1,i . Although
250
numerical solutions are more exact, the mathematically trackable solutions enables a quick estimation of the 15
251
structure-cladding spacing and therefore facilitates a selection of design parameters during the structural
252
design phase. The quality of the analytical solution will be discussed in Section 5.4. The approximate
253
harmonic wind velocity fluctuation vi (t) is then written vi (t) = vbi sin(ωt)
254
where vbi is the amplitude of the harmonic excitation. The mean square value of vi (t) is written [61] 1 T0 →∞ T0
vi2 (t) = lim 255
256
0
vbi2 2
(40) √
2βv∗ and the fluctuating
wind load component is taken as p 2βv∗ ρCd Ai Vi sin(ωt)
(41)
giving the amplitude of the equivalent fluctuating wind force component pbe n X i=1
259
2
[b vi sin(ωt)] dt =
Combining Eqs. 27 and 40 yields an amplitude for the wind fluctuation vbi =
pbe = 258
T0
Z
pi (t) = pbi sin(ωt) = 257
(39)
αi pbi =
n X
αi
p
2βv∗ ρCd Ai Vi
(42)
i=1
It follows that the maximum cladding-structure displacement qbc1,i attributed to the fluctuating wind loads is written qbc1,i =
pbe b Hc,i kse
(43)
260
b c,i = max|Hc,i (λ)| represents the maximum value of the amplitude of the transfer function Hc,i (λ) where H
261
over the excitation frequency ratio λ. The non-dimensional analytical solution Rc is used to compute the
262
maximum structure-cladding spacing Rc =
263
P b c,i | maxi |Pi /kce + pbe /kse H pbe b lmax i = = maxi 2 + Hc,i Pe /kse Pe /kse µf Pe Pe
(44)
4.3.3. Dynamic parameters for cladding connection
264
The dynamic parameters for the cladding connection are selected on the basis of all three analytical
265
solutions Ra , Rs , and Rc . First, the structure-cladding displacement ln (i.e., ln ≤ lc ) due to the mean static
266
wind force component Pn at the top floor is assigned, yielding kce = Pn /ln
267
Second, an initial stiffness value kce is selected, and the tuning frequency ratio f =
(45) q
kce µkse
obtained.
268
Third, the damping ratio of the connection ξc is selected through a minimization of the structural acceleration,
269
because acceleration is mostly related to serviceability under wind hazards. Fourth, using Eq. 35b, an
16
270
estimated value for ξc is obtained by setting ∂Ja /∂ξc = 0. For simplicity, structural damping is taken as
271
ξs = 0 and Ja (Eq. 36b) reduces to Ja =
272
π µΓ
f3 + 2ξc f 2ξc
and applying ∂Ja /∂ξc = 0 yields ξc = 0.5f
(46)
273
where note that structural damping is not considered to be negligible (ξs 6= 0) in the computation of the
274
peak building responses. The resulting peak structural responses are then compared against the performance
275
objectives. If amax ≤ ap , ∆max ≤ ∆p , and lmax ≤ lc , the design is completed. Otherwise, an additional
276
iteration is required, where the design parameters kce , ξc , and/or lc are altered until the process is completed.
277
The friction damping capacity Fcp at each connection is then obtained using Eq. 9 with the VFCC arbitrarily
278
designed for a harmonic excitation acting on the first natural frequency of the structure ωs and the amplitude
279
of structure-cladding displacement due to dynamic load taken as x ˆ = lmax − ln , Fcp =
1 πmce ωc ωs (ξc − ξc0 )(lmax − ln ) 4
(47)
cc0 2mce ωc .
280
with ξc0 =
281
5. Numerical Simulations
282
Numerical simulations are conducted on a 24-story office occupancy building located in Los Angeles, CA
283
[62]. The office tower is a steel moment-resisting frame structure with six bays in the North-South direction
284
and eight bays in the East-West direction. Each bay is 9 m wide and each floor is 3.9 m high, except for
285
the first floor that is 4.5 m high. The building is simulated as a lumped-mass shear system in the East-West
286
direction. The dynamic properties of the primary structure are listed in Table 3. The structural damping
287
ratio is assumed to ξs = 1%. The cladding is taken as concrete panels with 30% window opening area and
288
the thicknesses of the concrete and glass panels are 15 cm and 0.6 cm, respectively [53]. The densities of
289
the concrete and glass are taken as 2400 kg/m3 and 2800 kg/m3 , respectively, giving a cladding mass of
290
5.51 × 104 kg at each floor, except at the first floor where its mass is 6.36 × 104 kg.
17
Table 3: Dynamic properties of building.
floor
291
mass
stiffness
damping
mass
stiffness
damping
3
(10 kg)
(kN/m)
(kN·s/m)
12
2011
622953
9879
1993
11
2011
668616
10603
177431
2814
10
2011
713378
11313
2011
224212
3556
9
2011
761873
12082
20
2011
271092
4299
8
2011
818123
12974
19
2011
315938
5010
7
2011
913068
14480
18
2011
359101
5695
6
2011
102413
16241
17
2011
398817
6325
5
2011
111175
17631
16
2011
436525
6923
4
2011
119869
19010
15
2011
473719
7513
3
2011
131956
20927
14
2011
512098
8121
2
2011
156541
24826
13
2011
560090
8882
1
2041
172255
27318
3
(10 kg)
(kN/m)
(kN·s/m)
24
1423
67355
1068
23
2011
125646
22
2011
21
floor
The equation of motion for the 24-story building has the form M¨ x + Cx˙ + Kx = Ep + Ef F
(48)
292
where x ∈ R72×1 is the displacement vector, p ∈ R24×1 is the wind force input vector, F ∈ R48×1 is the
293
control input vector, E ∈ R72×24 and Ef ∈ R72×48 are the wind loading and control input location matrices,
294
respectively, and M ∈ R72×72 , C ∈ R72×72 , K ∈ R72×72 are the mass, damping and stiffness matrices of the
295
building, respectively.
296
The state-space model of Eq. 48 for simulation is then given by ˙ = AX + Bp p + Bf F X
297
where X = [x
(49)
x] ˙ T ∈ R144×1 is the state vector and the constant coefficient matrices are defined as follows 0 I A= (50) −M−1 K −M−1 C 144×144
0 Bp = M−1
(51)
144×24
0 Bf = M−1 Ef
144×48
18
(52)
298
The numerical algorithm follows the discrete form of the Duhamel integral [1]: X(t + ∆t ) = eA∆t X(t) + A−1 (eA∆t − I)[Bp p(t) + Bf F(t)]
(53)
299
where ∆t = 0.001 s is the simulation time interval and I ∈ R144×144 is the identity matrix. This discrete
300
state-space linear formulation has been used in literature to simulate the dynamic response of a linear
301
structural system with nonlinear damping devices [63, 64]. Of interest to this paper, the authors used the
302
same method in previous work to simulate a different variable friction device [7]. Three simulations cases
303
are conducted to investigate the performance of the VFCC device, namely the semi-active, passive-on and
304
uncontrolled cases. The semi-active scheme is the VFCC used with a full-state feedback control rule based
305
on a linear quadratic regulator (LQR) computing the required control force Freq for each device Freq = −Gf X
306
(54)
where Gf ∈ R48×144 is the control gain matrix, tuned to minimize a performance objective index JLQR Z 1 ∞ T JLQR = (X Rx X + FT Rf F)dt (55) 2 0
307
where Rx ∈ R144×144 is the regulatory weight matrix and Rf ∈ R48×48 is the actuation weight matrix. Note
308
that the design and optimization of the controller is out-of-the-scope of this work. In the later numerical
309
simulation, the regulatory and actuation weight matrices Rx and Rf are pre-tuned as Rx = diag[I24×24
0.5I10×10
10I10×10
15I28×28
50I15×15
200I5×5
1000I4×4
I48×48 ]
310
and Rf = 10−10 I48×48 . The performance of the semi-active VFCC is compared against that of a passive-on
311
friction case where the VFCC is used under a constant maximum capacity, and of an uncontrolled case with
312
a conventional stiffness connection. In the uncontrolled case, a set of conventional stiffness connectors are
313
used to laterally connect the cladding elements to the structural system with 24 tie-back connectors at the
314
top and 24 bearing connectors at the bottom of cladding panels for each floor. The lateral stiffnesses of each
315
tie-back connector and bearing connector are taken as 39 kN/mm and 2335 kN/mm, respectively, based on
316
Ref. [53]. The stiffness element of the lateral connection for the lumped model is taken as the sum of the 24
317
connectors.
318
5.1. Verification of SDOF simplification for cladding system
319
In the derivation of the transfer functions in Section 3, an assumption was made that a linear time-
320
invariant 2DOF cladding system could be reduced to a linear time-invariant SDOF representation. In what
321
follows, the quality of this assumption is verified on a linear and time-invariant 4DOF representation of the
322
building. To create a realistic system, the 4DOF is created by lumping the 24 structural mass elements into
323
two floors of identical masses and the 24 cladding mass elements into a single cladding mass, as illustrated 19
324
in Fig. 8(a). The dynamic properties of the 4DOF system are listed in Table 8. The dynamic parameters
325
at each floor are taken as identical and such that the 4DOF’s first structural frequency is equal to the first
326
natural frequency of the building (0.2 Hz), with the structural damping ratio set at ξs = 1%. Note that this
327
4DOF system is only used to verify the SDOF simplification for a 2DOF cladding system and it does not
328
necessarily represent the dynamics of the 24-story building system. Following the methodology discussed in
329
Section 3, the 4DOF system is reduced into its equivalent 2DOF representation, illustrated in Fig.8(b) with
330
its dynamic properties listed in Table 4. The amplitude of the equivalent harmonic loading pˆe acting on the
331
2DOF system is taken as pˆe =
332
φs,11 + φs,12 pˆ 2φs,12
where pˆ is the amplitude of the harmonic load acting on the 4DOF system.
(a)
(b)
Figure 8: Representations of the building for verifications of the assumptions: (a) 4DOF model; and (b) 2DOF model.
Table 4: Dynamic properties of the 4DOF system and its equivalent 2DOF system.
4DOF
2DOF
parameter
value
unit
ms
23848
103 kg
ks
98595
kN/m
cs
1569
kN·s/m
mc
1332
103 kg
mse
32958
103 kg
kse
52045
kN/m
cse
828
kN·s/m
mce
1332
103 kg
20
333
The transfer functions Hs (λ), Ha (λ) and Hc (λ) for the equivalent 2DOF system are written f 2 + j2ξc f λ (1 − λ2 )(f 2 − λ2 ) − 4ξc ξs f λ2 − µΓs f 2 λ2 + j [2ξc f λ(1 − (1 + µΓs )λ2 ) + 2ξs λ(f 2 − λ2 )] 1 1 − λ2 + j2ξs λ Hc (λ) = √ 2 2 2 2 µ Γs (1 − λ )(f − λ ) − 4ξc ξs f λ − µΓs f 2 λ2 + j [2ξc f λ(1 − (1 + µΓs )λ2 ) + 2ξs λ(f 2 − λ2 )]
Hs (λ) =
(56)
Ha (λ) = −λ2 Hs (λ) φs,11 +φs,12 2 ) . 2φs,12
The verification of the model reduction assumption is conducted through the
334
where Γs = (
335
comparison of transfer functions derived from the equivalent 2DOF system with those computed numerically
336
on the 4DOF system. In the numerical simulation, each friction damping element is simulated as passive
337
Coulomb friction force Fc of capacity Fc =
338
element of damping ratio ξv under harmonic loading. The initial viscous damping ratio of the lateral
339
connection is taken as ξc0 =
340
Hc (λ).
√ cc0 2 kce mce
π 4 µf λξv |Hc (λ)|,
which is equivalent to a linear viscous damping
= 1% and |Hc (λ)| denotes the magnitude of the transfer function
341
The magnitudes of transfer functions are plotted in Figs. 9 to 11, with tuning frequency ratio f = 3 which
342
exhibits a typical behavior, and two representative damping cases ξv = 10% and ξv = 150% to investigate the
343
effect of the connection’s damping. Note that ξv = 150% represents an overdamped case; it is investigated
344
because the design procedure is likely to yield overdamped strategies based on Eq. 46. In general, the
345
analytical solutions (2DOF) can track the numerical solution (4DOF). There is a disagreement between
346
both solutions for Hs and Ha that occurs around the excitation frequency ratio λ = 2.5, in particular under
347
lower damping (ξv = 10%). The reason is that the second mode of the primary structure, which corresponds
348
to the excitation frequency ratio λ = 2.5, is not part of the analytical solution that is based on an SDOF
349
representation. For Hc , there is a good agreement between the analytical and numerical solutions at low
350
damping (ξv = 10%), but a significant disagreement for the overdamped connection (ξv = 150%), observable
351
in Fig. 11(b). This is likely attributed to the discontinuous motion from the stick-slip phenomenon that is
352
amplified with large friction force.
21
(a)
(b)
Figure 9: Transfer function Hs at f = 3: (a) ξv = 10%; and (b) ξv = 150%.
(a)
(b)
Figure 10: Transfer function Ha at f = 3: (a) ξv = 10%; and (b) ξv = 150%.
22
(a)
(b)
Figure 11: Transfer function Hc at f = 3: (a) ξv = 10%; and (b) ξv = 150%.
353
In order to further investigate the effect of dynamic parameters, the analytical solutions of the transfer
354
functions (Eq. 56) are used to conduct a parametric study of the mass ratio µ (Fig. 12), tuning frequency
355
ratio f (Fig. 13), and damping ratio ξc (Fig. 14, showing ξce ). A study of the transfer functions shows that
356
adding mass to the cladding (i.e., increasing µ) improves the mitigation performance for both the inter-story
357
displacement (Hs ), absolute acceleration (Ha ), and structure-cladding displacement (Hc ). Variation in the
358
cladding mass can generally be attained through using different materials (e.g. glass, masonry or precast
359
concrete). A typical cladding-structural system mass ratio ranges from 0.01 to 0.1 [65, 66]. Augmenting
360
the connection’s stiffness (i.e., increasing f ) has the converse effect on Hs and Ha , yet still decreases Hc .
361
Also, using a more flexible connection leads to a behavior analogous to that of a tuned mass damper,
362
observable under f = 1.5 and f = 1.8. However, this added performance comes at the cost of a much
363
larger structure-cladding displacement which will unlikely meet the design target. A similar performance is
364
observed by decreasing ξc , although one must be careful in evaluating the behavior under high damping given
365
the lower accuracy of the transfer functions to represent structural behavior. Once again, higher mitigation
366
(i.e., lowering ξc ) comes at the cost of higher structure-cladding displacement that may not be possible to
367
achieved. It follows that the VFCC is expected to be used under low stiffness and high damping.
23
(a)
(b)
(c) Figure 12: Plot of transfer functions vs various mass ratios µ at f = 0.9 and ξc = 20%: (a) Hs ; (b) Ha ; and (c) Hc .
24
(a)
(b)
(c) Figure 13: Plot of transfer functions vs various tuning frequency ratios f at µ = 0.05 and ξc = 10%: (a) Hs ; (b) Ha ; and (c) Hc .
25
(a)
(b)
(c) Figure 14: Plot of transfer functions vs various connection damping ratios ξc at µ = 0.05 and f = 0.8: (a) Hs ; (b) Ha ; and (c) Hc .
368
5.2. Demonstration of MBD procedure
369
The proposed MBD procedure is demonstrated on the 24-story office tower building. A basic design
370
wind speed with a return period of Q = 50 years is selected with a 3-second wind gust speed V0 at reference
371
height 10 m of 38 m/s based on the wind hazard map in ASCE 7-10 (2010) [67]. The cladding connection
372
parameters are designed based on the MBD non-dimensional analytical solutions Ra , Rs and Rc .
373
Wind Hazard Quantification
374
The wind loading parameters are determined based on the building location’s terrain with the ratio of
375
shear velocity of wind flow v∗ /v∗0 = 1.15, surface roughness length z∗ = 0.3 m and its corresponding value
376
β = 5.25, and the drag coefficient Cd = 1.4 based on building cross-section [45]. The cladding area exposed
377
to wind pressure at each floor is A = 210 m2 , except at the first floor A = 243 m2 . Using Eq. 25 to Eq. 27,
26
378
the mean static wind load at the top floor is obtained as Pn = 1.93 × 105 N and the equivalent mean static
379
wind load is calculated as Pe = 1.82 × 106 N using Eq. 33. The design parameters are listed in Table 5.
380
Performance Objectives
381
The performance objectives are then quantified. Using the average return period Q = 50 years and the
382
first nature frequency of the structure 0.2 Hz, the acceptable peak acceleration for occupancy comfort is
383
calculated as ap = 27 mg based on Eq. 31. The allowable lateral drift ∆p = 1/250 and a design value of the
384
allowable structure-cladding spacing lc is set to 0.5 m for the preliminary design, corresponding to half the
385
allowable distance (1 m) reported in Ref. [66]. The performance objectives are listed in Table 5.
386
Connection Design
387
The equivalent mass and stiffness of the primary structure is mse = 1.48×107 kg and kse = 2.36×107 N/m,
388
yielding the mass ratio between cladding and the primary structure µ = mce /mse = 0.37%. The structure-
389
390
cladding displacement ln due to the mean static wind force is first set to 0.3 m, yielding the equivalent q kce = 2.7, and lateral connection stiffness kce = Pn /ln = 6.43 × 105 N/m, a tuning frequency ratio f = µk se
391
a damping ratio ξc = f /2 = 1.35. However, these values give a maximum structure-cladding displacement
392
lmax = 0.52 m based on Rc = 6.73 (Eq. 44) that exceeds the allowable value lc = 0.5 m. An iteration is
393
conducted by decreasing the ln value to 0.28 m, yielding an equivalent connection stiffness kce = 6.89 × 105
394
N/m, a tuning frequency ratio f = 2.8, and a damping ratio ξc = f /2 = 1.40. This non-dimensional
395
analytical solutions Ra , Rs and Rc are 2.166, 3.060, and 6.278, respectively. The corresponding maximum
396
structural acceleration and the maximum inter-story drift ratio are amax = Pe /mse Ra = 27 mg 6 ap and
397
∆max = Pe /kse Rs = 1/320 6 ∆p , respectively, which meets the performance objectives. The total structure-
398
cladding spacing at each floor is within the allowable spacing with the maximum value lmax = 0.48 m
399
occurring at the top floor. The friction capacity of the cladding connection at each floor is designed for
400
Fcp = 54.2 kN (Eq. 47). The dynamic parameters are listed in Table 5.
27
Table 5: Cladding connection design parameters.
parameter
variable
wind load
motion criteria
cladding connection
401
402
403
value
units
note
Pn
193
kN
Eqs. 25 - 27
Pe
1820
kN
Eq. 33
ap
27
mg
Eq. 31
∆p
1/250
−
−
lc
0.5
m
−
Ra
2.166
−
Eq. 37a
Rs
3.060
−
Eq. 37b
Rc
6.278
−
Eq. 44
kce
689
kN·m−1
Eq. 45
Fcp
54.2
kN
Eq. 47
5.3. Simulated wind model The time series of wind fluctuation vi (t) is simulated as a multivariate stochastic process with crossspectral density matrix S(Ω) with its element expressed as [47] Sv (zi , Ω) if i = l Sil (Ω) = p Sv (zi , Ω)Sv (zl , Ω)Coh(zi , zl , Ω) if i 6= l
(57)
404
with the two-sided PSD Sv (zi , Ω) in Eq. 28 and the coherence function Coh(zi , zl , Ω) from Eq.30. To simulate
405
the stochastic process, the power density matrix S(Ω) is first decomposed into the following product [68]: S(Ω) = Λ(Ω)Λ
406
Λ11 (Ω)
Λ21 (Ω) Λ(Ω) = .. . Λn1 (Ω)
408
(Ω)
(58)
where subscript asterisk denotes the complex conjugate and Λ(Ω) is a lower triangular matrix
407
T∗
0
...
0
Λ22 (Ω) .. .
... .. .
0 .. .
Λn2 (Ω) . . .
(59)
Λnn (Ω)
Once the matrix S(Ω) is decomposed, the stochastic process of wind fluctuation vi (t) is generated by the following series [68] vi (t) = 2
NΩ i X X
√ |Λiq (Ωqτ )| ∆Ω cos Ωqτ t − θiq (Ωqτ ) + δqτ
q=1 τ =1
28
(60)
409
where δqτ is a random phase uniformly distributed between 0 and 2π; and the phase θiq (Ωqτ ) is given by θiq (Ωqτ ) = tan−1
410
n Im[Λ (Ω )] o iq qτ Re[Λiq (Ωqτ )]
(61)
The double-indexing frequency Ωqτ is defined as: Ωqτ = (τ −
i−q )∆Ω i
(62)
411
with the frequency step ∆Ω = Ωu /NΩ , where Ωu is an upper cutoff frequency, taken as Ωu = 20π, and
412
the total number of frequency points NΩ = 213 [69]. Lastly, the simulated wind load Pd,i (t) acting on the
413
cladding panels at the ith floor is generated Pd,i (t) =
1 2 ρCd Ai [Vi + vi (t)] 2
(63)
414
where Vi is the mean wind speed at building height zi obtained using Eq. 25 and the air density is taken
415
as ρ = 1.225 kg/m3 . A typical 10-minute duration wind excitation at the top floor is plotted in Fig. 15.
416
Fig. 16(a) shows a comparison of the power spectral density function of the wind fluctuation at the top
417
floor between the simulated wind data and the analytical model (Eq. 28). The turbulence intensity of the
418
generated wind velocity fluctuation is verified against the analytical model, illustrated in Fig. 16(b). These
419
comparisons show a good match between the analytical and computational wind model.
(a)
(b)
Figure 15: Typical realization of a wind hazard time series over a 10-minute duration (a) wind speed and (b) wind load.
29
(a)
(b)
Figure 16: Comparison of the analytical wind model and the simulated wind data (a) power spectral density function and (b) turbulence intensity.
420
5.4. Simulation results
421
The maximum response profiles of the building is plotted in Fig. 17, showing the maximum inter-story
422
displacement (Fig. 17(a)), the maximum absolute acceleration (Fig. 17(b)), and the maximum structure-
423
cladding displacement (Fig. 17(c)). Results show that the semi-active controlled VFCC significantly reduces
424
the structural response compared with the passive-on case and the uncontrolled case. The maximum reduc-
425
tion of the inter-story displacement and acceleration reached 16.4 % and 31.4 %, respectively. In particular,
426
the semi-active VFCC brought the acceleration under the limit threshold, while the passive-on and uncon-
427
trolled cases were unsatisfactory. In all cases, the structure-cladding spacing remains under the threshold
428
value of 0.5 m, but the additional mitigation performance provided by the semi-active scheme does come at
429
the cost of higher structure-cladding displacement.
30
(a)
(b)
(c) Figure 17: Maximum building response profiles: (a) inter-story displacement; (b) absolute acceleration; and (c) claddingstructure displacement (uncontrolled case not shown).
430
It can also be observed in Fig. 17 that the passive design indicated that the passive-on strategy was within
431
the acceptable acceleration bounds. This discrepancy is mainly attributed to the non-negligible errors in
432
the analytical solutions arising from the assumptions made to reach a mathematically trackable solution,
433
for instance the negligence of the higher structural modes, the simplification of the wind spectral density
434
function, and the modeling of the friction element through equivalent viscous damping. Nevertheless, the
435
objective is to obtain a connection that would perform appropriately under control, which objective has been
436
attained given the higher teachability of the device under its semi-active mode.
437
To further investigate the quality of the analytical solutions, Fig. 18 plots three non-dimensional analyt-
438
ical solutions (Rs , Ra and Rc ) at the designed connection stiffness under various damping capacities. These
31
439
analytical solutions are compared against the simulation results, using performance metric R∗ : R∗ =
Rmodel − Rsimulation × 100% Rmodel
440
where the R refers to Rs , Ra and Rc and the subscript ‘model’ represents the analytical solutions and
441
‘simulation’ the simulation results of passive-on case or semi-active case. Results show a non-negligible dis-
442
agreement between the passive-on (i.e., numerical solution) and all three analytical solutions. In particular,
443
Rs and Rc are overestimated, while Ra is underestimated by the analytical solutions, consistent with the
444
results from Fig. 17. The large discrepancies are attributed to the assumptions made to develop a mathe-
445
matically trackable solution, whereas 1) higher modal responses of the structure were neglected; 2) the wind
446
spectrum was taken as constant; and 3) friction was simplified into an equivalent viscous system ignoring
447
high nonlinearities. However, in all cases, the analytical solutions overestimate the responses with respect
448
to the semi-active case, demonstrating that the design methodology provides a conservative design for the
449
VFCC used in its semi-active mode. This finding also agrees with results from Fig. 17.
32
(a)
(b)
(c) Figure 18: Comparison of the non-dimensional design factors (a) Rs , (b)Ra , and (c) Rc .
450
6. Conclusions
451
A novel variable friction cladding connection (VFCC) has been previously proposed by the authors to
452
leverage a structure’s cladding system for multiple hazard mitigation. This friction device provides a lateral
453
connection between the cladding element and the structural system. Its variation in the friction force is
454
provided by an actuator that applies a variable pressure onto sliding friction plates via a toggle system. In
455
this paper, a motion-based design (MBD) approach to the design of the VFCC is developed for wind hazard
456
mitigation. While this paper investigates a single hazard, results are to be integrated in future work for
457
multi-hazard applications.
458
A set of analytical transfer functions were derived to enable the MBD approach, based on a simplification
459
of the structural-cladding system into a two degrees-of-freedom (2DOF) system. These transfer functions
33
460
were used to estimate the maximum building responses under a quantified wind load, which responses in-
461
cluded inter-story displacement, absolute acceleration, and structure-cladding displacement. These analytical
462
solutions were numerically verified, and show good agreement between the numerical and analytical solu-
463
tions, except when the connection was overdamped where the equivalent viscous damping representation of
464
the friction element failed at accurately modeling the friction behavior by ignoring important nonlinearities,
465
such as the stick-slip motion.
466
The MBD procedure was demonstrated on a 24-story building, and compared to the performance of the
467
VFCC under semi-active control with the passive-on mode and the uncontrolled structure. Simulation results
468
show that the VFCC was capable of reducing the response of the uncontrolled structure under the prescribed
469
performance objectives. However, while the design of the VFCC under the MBD approach assumes a passive-
470
on mode, the passive-on scenario underestimated the maximum absolute acceleration. This was attributed
471
to an underestimation of the acceleration response in the analytical solutions due to the various assumptions
472
in the model. Nevertheless, it was also found that the analytical solutions always overestimate structural
473
response with respect to the semi-active mode of the VFCC, therefore demonstrating that the MBD approach
474
yields a conservative design of the semi-active VFCC. Overall, this study demonstrated the promise of the
475
VFCC at mitigating wind-induced vibrations in structures by leveraging the cladding system, and provided
476
an MBD approach enabling its holistic integration during the structural design phase.
477
Acknowledgments
478
This material is based upon the work supported by the National Science Foundation under Grant No.
479
1463252 and No. 1463497. Their support is gratefully acknowledged. Any opinions, findings, and conclusions
480
or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the
481
views of the National Science Foundation.
482
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