Figure 7: Equivalent linear stiffness hysteresis.
The equivalent energy dissipation concept is illustrated in Fig. 7. The red triangle area represents energy
301
dissipation from the linear stiffness element and the gray shaded area represents the energy dissipation from
302
the impact rubber, where both areas are equal. Eq. 10 (b) is modified to include the dynamics of the impact
303
rubber using the equivalent stiffness concept:
304
m¨uc(t) + c ˙uc(t) + (k + keq)uc(t) =−fc (27) where t > trubber. The rubber deformation follows the cladding displacement from the point where uc =
305
uc,max. The analytical solution for the rubber deformation is similar to Eq. 16, but with knew = keq+ kc,
306
where ξr and ωdrare the modified damping ratio and damped frequency.
307
ur(t) =e−ξrωdr(t−trubber)
−fc+ kcuc(trubber) knew
cos ωdr(t− trubber)
+ e−ξrωdr(t−trubber)
˙uc(trubber)knew− (fc+ kcuc(trubber)) ξrωr
knewωdr
sin ωdr(t− trubber)
+fc+ kcuc(trubber) knew
(28)
where uc(trubber) = lc− lr is the space between the cladding element and the impact rubber. Using the
308
analytical solution from Eq. 28 and the time of maximum deformation ur,max, a third transfer function H3
309
is obtained:
310
t2= tan−1
u˙c(trubber)knew√
1−ξr2
−fcωr−kcuc(trubber)ωr+ξrknewu˙c(trubber)
ωdr
+ trubber (29)
ur,max= e−ξrωdr(t2−trubber)
√(uc(trubber)kc+fc)2ω2dr+( ˙uc(trubber)knew−(uc(trubber)kc+fc)ξrωr)2
ωdrknew + fc+kcuknewc(trubber) (30)
H = ur,max
(31)
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where ust = Fkmc is the static deformation. Transfer function H3 provides the deformation of the impact
311
rubber based on the design parameters selected under Steps 1 and 2, with the objective to obtain a rubber
312
deformation smaller than its design thickness lr. Otherwise, other design parameters need to be selected
313
and the PBD procedure be iterated, as explained earlier.
314
5. Numerical Simulations
315
In this section, numerical simulations are conducted to verify the proposed PBD methodology.
Simu-316
lations are based on a 1:4 scale six-story three-bay structure equipped with cladding panels spanning each
317
floor. This structure was selected due to the availability of parameters and results found in the literature
318
[3, 57], in which the authors studied an advanced passive energy-dissipating cladding connector for
seis-319
mic design. Here, we used the cladding panel material consistent with [57], and replaced the connector by
320
the proposed semi-active connection. Each cladding panel is attached to the structure using two tie back
321
connectors and two bearing supports. The semi-active connection is installed as a replacement to tie back
322
connectors, and only the bearing connectors are assumed to provide lateral stiffness and inherent damping
323
between the cladding and the structure (kc and cc, respectively).
324
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5.1. Model assumption
325
5.1.1. 18DOF system
326
Figure 8: Simulated structure with cladding panels.
The 18DOF representation is illustrated in Fig. 8, which contains one translational DOF per floor, and
327
two translational DOF per connection between the cladding and structure. Each cladding panel is modeled
328
as a rigid bar of mass mc and has two degree of freedoms at either end connected to the structure. Again,
329
the nonlinear response of the cladding will be considered in future work, and the assumption of a rigid bar
330
is consistent with current approaches for conservatively calculating the base shear and cladding-to-structure
331
load transfer for buildings under blast loading [45, 46]. The first cladding element spanning between ground
332
level and the first floor has its lower semi-active connection directly attached to the ground. Details of the
333
cladding connections are schematized in Fig. 3 (a). The dynamic properties of the structure and cladding
334
elements are listed in Table 1, based on the properties provided in Reference [57].
335
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Table 1: Dynamic properties of cladding structure [57]
floor mass stiffness damping cladding mass (kg) (kN/m) (kN·s/m) (kg)
6 3175 3000 16.32 450
5 3175 3000 16.32 450
4 3175 3000 16.32 450
3 3175 3000 16.32 450
2 3175 3000 16.32 450
1 3175 3000 16.32 450
The equation of motion for the 18DOF system has the form
336
M¨u + C ˙u + Ku = EpP + EfF (32)
where u∈ R18×1 is the displacement vector, P∈ R12×1is the blast loading input vector, F∈ R12×1 is the
337
control force vector including the friction force and rubber damping force, M ∈ R18×18, C∈ R18×18, K∈
338
R18×18 are the mass, damping, and stiffness matrices, respectively, and Ep∈ R18×12, and Ef ∈ R18×12are
339
the blast loading and control input location matrices, respectively.
340
The state-space representation of Eq. (32) is given by
341
U = AU + B˙ pP + BfF (33)
where U = [ u ˙u ]T ∈ R36×1is state vector and
342
A =
0 I
−M−1K −M−1C
36×36
Bp=
0
M−1Ep
36×12
Bf =
0
M−1Ef
36×12
where I is an 18× 18 identity matrix.
343
Table 2: Comparison of modal properties
mode reported [57] (Hz) model (Hz) difference (%) model Γ (%)
first 1.17 1.17 0.00 86.96
second 3.88 3.47 −10.57 8.91
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Table 2 compares modal parameters between the values reported in [57] and the ones from the 18DOF
344
model. The first mode of the model matches the first mode of the six-story building, and the modal
345
participation factor Γ shows that the first mode largely dominates the response of the 18DOF representation.
346
Two design blast loads are considered for this study. The first is based on a 500-kg mass of TNT (approximate
347
explosive mass in a closed van delivery method [52]) at a standoff distance of 25 m. This load will be used in
348
Sections 5.2 and 5.3 to compare the proposed analytical solution with the numerical results and demonstrate
349
the PBD procedure. A smaller charge weight of 100-kg of TNT (approximate explosive mass in a compact
350
car trunk delivery method [52]) at the same 25-m standoff will be introduced in Section 5.4 to evaluate the
351
performance of the proposed model and procedure for a more moderate blast load. There is a greater potential
352
for nonlinear cladding response due to the 500-kg charge than the 100-kg charge, and comparing their effects
353
will evaluate the effectiveness of the proposed connections for a range of blast reaction magnitudes.
354
The amplitude of blast load Fmfor both charge weights are calculated using Eq.7 based on the maximum
355
blast pressure σmax and a total cladding panel area Ac of 3.33 m2 at each floor. Parameter values of the
356
500-kg design blast load for each nodes are listed in Table 3 (values for the 100-kg charge will be provided
357
later in Section 5.4). The resulting values σmax are within a typical range σcap as discussed in Section 3.2.
358
The pressure is taken as a constant between two half floors, resulting in the blast load equal for adjacent
359
connections (p4= p5 in Fig. 8, for instance).
360
Table 3: Design blast load (500-kg TNT) for 18DOF system
node R (m) Z (m/kg1/3) σmax (kPa) Fm (kN) tblast(ms)
u7 25.00 3.15 224.00 370.67 24
u8 25.27 3.18 218.05 359.78 24
u9 25.27 3.18 218.05 359.78 24
u10 26.05 3.28 200.25 330.42 25
u11 26.05 3.28 200.25 330.42 25
u12 27.30 3.44 176.03 290.45 25
u13 27.30 3.44 176.03 290.45 25
u14 28.97 3.65 150.29 247.98 25
u15 28.97 3.65 150.29 247.98 25
u16 30.98 3.90 126.37 208.50 26
u17 30.98 3.90 126.37 208.50 26
u18 33.27 4.18 105.79 174.56 26
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5.1.2. 2DOF system
361
The six story structure is also modeled as 2DOF system by lumping the six structural mass elements ms
362
into a single mass (1DOF) and the six cladding mass elements mc into a single cladding element to obtain
363
a representation similar to the one schematized in Fig. 3 (b). A structural damping ratio of 2% is assumed.
364
A simplified triangular blast load is used for the simulations (p(t)) using blast load parameters calculated
365
following the approach described in Section 4.1.1. The peak design blast load Fm used in 2DOF system
366
is summation of blast peak load in 18DOF system. Note that this 2DOF system assumption is only used
367
for verifying the SDOF approximation and it does not necessarily represent the dynamic of 18DOF system.
368
The resulting parameters used for the numerical simulation are listed in Table 4. Remark that a different
369
value for kc than the one used in Reference [57] to obtain a more compliant connection to improve energy
370
mitigation.
371
Table 4: Model parameters for 2DOF representation
parameter value unit
system
ms 19051 kg
ks 1029 kN/m
cs 56 kN·s/m
mc 2700 kg
blast
W 500 kg
R 25 m
Z 2.5 m/kg1/3
σmax 218 kPa
Fm 3419 kN
tblast 24 ms
5.2. Verification of the SDOF approximation
372
Before conducting simulations of the 18DOF system, the PBD methodology is first verified for the 2DOF
373
representation of the building to demonstrate the assumption that cladding can be designed based on an
374
SDOF approximation. The validity of the assumption is investigated through the comparison of transfer
375
function plots obtained from the PBD procedure and from numerical simulations. In what follows, the PBD
376
transfer functions Hi (i = 1, 2, 3) are compared against the transfer functions of the 2DOF Hi∗ obtained
377
numerically, using a performance metric ˜Hi:
378
H˜i= Hi− Hi∗
Hi∗ for i = 1, 2, 3 (34)
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First, H1 and H1∗ transfer functions are plotted as functions of the blast duration ratio tblast/Tn for
379
different friction capacity ratios fc/Fm in Fig. 9 (a), over the range 0.1% to 3.0%. The performance metric
380
H˜1 is plotted in Fig. 9 (b). The error in H1 is larger for small ratios tblast/Tn, and converges to 0 with
381
increasing tblast/Tn. The magnitude of the error increases with increasing friction ratio. Generally, the error
382
is positive, which results in an over-estimation of the cladding displacement. The error is negative for a zero
383
friction ratio (fc/Fm = 0%), because the blast energy will be transmitted to the structure (no dissipation)
384
and the model will underestimate cladding displacement due to the unmodeled displacement of the structure.
385
Second, transfer function H2is verified through the investigation of its fitting performance index ˜H2over
386
different values of relative rubber thickness (lc− lr)/ust. Plots are shown in Figs. 10 and 11 for the ratio
387
range 1% to 4%, which represents a large range of spacing lc− lr as ustis relatively large under blast load. A
388
null value for H2signifies that the cladding does not collide with the structure. The error ˜H2is high for H2
389
close to 0, and increases with increasing relative rubber thicknesses. The error is also higher for increasing
390
friction ratio. In both cases, this high level of error is due to increasing nonlinearities that are not captured
391
by the analytical solutions. However, the error is positive in all cases, which signifies that the cladding will
392
be over-designed therefore provide additional safety. Also, the error converges to zero with increasing blast
393
duration ratio.
394
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transfer function H2H∗
2
Third, the H3function is plotted under different friction capacity ratios fc/Fm, relative rubber thickness
395
ratios (lc− lr)/ust and impact rubber stiffness ratios kc/kr in Figs. 12 to 15. The error ˜H3 is high when
396
rubber has smaller deformations (H3 close to 0). As tblast/Tn increases, the error ˜H3 quickly decreases to
397
reach a negative value for a higher relative rubber stiffness or lower ratio kc/kr.
398
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5.3. Demonstration of PBD procedure
399
In this section, the proposed PBD procedure is demonstrated using the six-story structure shown in Fig.
400
8. Recall that, similar to Section 5.2, the 500-kg charge size is used to make this comparison. The dynamic
401
parameters of the cladding system and impact rubber, at each floor, are designed based on the PBD transfer
402
function results Hi (i = 1, 2, 3) obtained in the previous section for an SDOF system.
403
Step 1: Performance Criteria
404
The design blast load parameters Fm and tblastfor each node are taken from Table 3. The spacing lc at
405
each floor is set to 0.4 m for the preliminary design, based on the minimum requirements reviewed in Section
406
4.1.2.
407
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Step 2: Dynamic Parameters of Cladding System
408
The mass of each cladding panel mc is taken as fixed, as provided in Reference [57]. Using the H1 plot
409
(Fig. 9 (a))with the assumption of a cladding damping ratio ξ = 2%, a blast duration ratio tblast/Tn= 3%
410
with a friction capacity ratio fc/Fm= 1% yields H1= 0.081. This H1value is used to compute uc,maxat each
411
floor using Equation 19. Table 5 lists the results for each node, as well as kcand cc to obtain tblast/Tn= 3%
412
and ξ = 2%. With these design parameters, all of the nodes exceed the allowable deformation of 0.4 m,
413
necessitating design step 3 for appropriate sizing of the impact rubber.
414
Table 5: Cladding connection design parameters from Step 2
node mc (kg) Tn (s) kc (kN/m) cc (kN·s/m) fc (kN) ust(m) uc,max (m)
u7 225 0.76 15.11 7.38×10−2 3.71 27.27 2.21
u8 225 0.76 15.11 7.38×10−2 3.60 26.62 2.16
u9 225 0.76 15.11 7.38×10−2 3.60 26.62 2.16
u10 225 0.76 15.11 7.38×10−2 3.30 24.85 2.01
u11 225 0.76 15.11 7.38×10−2 3.30 24.85 2.01
u12 225 0.76 15.11 7.38×10−2 2.90 22.40 1.81
u13 225 0.76 15.11 7.38×10−2 2.90 22.40 1.81
u14 225 0.8 15.11 7.38×10−2 2.48 19.74 1.60
u15 225 0.8 13.88 7.07×10−2 2.48 19.74 1.60
u16 225 0.8 13.88 7.07×10−2 2.09 17.21 1.40
u17 225 0.8 13.88 7.07×10−2 2.09 17.21 1.40
u18 225 0.8 13.88 7.07×10−2 1.75 14.97 1.21
Step 3: Parameters of Impact Rubber
415
For simplicity, both the cladding-structure spacing lc= 0.4 m and the rubber thickness lr are assumed
416
constant throughout the height of the structure. The design of lris based on the blast load at the first floor,
417
which represents the worst case scenario. An initial rubber thickness is selected taking (lc− lr)/ust = 1%
418
and yielding lr= 0.12m. The ultimate compression capacity is ur,ult= 0.8lr= 0.096m. The value H2= 0.96
419
is obtained from the H2 plot (Fig. 10 (a)) using the design parameters from Step 2 (tblast/Tn = 3%,
420
fc/Fm = 1%), which results in uc,max = 2.21m. This value is much higher than the cladding-structure
421
spacing lc. Therefore, H3must be obtained such that H3≤ ur,ult/ust= 0.053. Using H3from Fig. 12 (a), a
422
value of kc/kr = 0.001% would satisfy this requirement, with H3= 0.0025. The resulting maximum rubber
423
deflection is ur,max= 0.068 m. Since the maximum deformation of rubber ur,maxis smaller than the rubber’s
424
design thickness lr, the design is completed. A detailed design of the impact rubbers can be conducted using
425
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5.4. Simulation results
427
The 18DOF model is first verified under a design blast excitation of a 500-kg charge. The discrete form
428
of a Duhamel integral is used to numerically simulate Eq. (33) [2]:
429
U(t + 1) = eA∆tU(t) + A−1(eA∆t− I)[BfF(t) + BpP(t)] (35) where ∆t is the time interval used in the simulation, taken as 0.0001s. Fig. 16 is a plot of the maximum
430
cladding and rubber displacement results. The maximum rubber and cladding deformations at each node
431
are compared with design values lc=0.4m and lr=0.12m. Maximum deformation of rubber reduce from first
432
floor (node 8) to top floor (node 18) caused from decreasing peak blast load. All deformations are smaller
433
than design values, demonstrating that the proposed PBD procedure from an SDOF system provides an
434
adequate design methodology for an MDOF system.
435
7 8 9 10 11 12 13 14 15 16 17 18 node
0 0.1 0.2 0.3 0.4
deformation (m)
rubber cladding
Figure 16: Maximum deformation of cladding and rubber.
The performance of proposed connection (“controlled” case) at mitigating inter-story drift and
accel-436
eration is also evaluated versus a cladding system attached with conventional connections (“uncontrolled”
437
case), where kc is assumed to be infinitely stiff and no lateral stiffness is provided by the cladding [57]. A
438
comparison of the time series inter-story displacements at the first and top floors under design blast load
439
are plotted in Figs. 17 and 18, as well as the corresponding rubber peak deformations. Results demonstrate
440
that proposed cladding system provides great mitigation performance under significant blast excitation. The
441
peak deformation of the rubber impact bumpers is larger at the first impact, as expected.
442
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0 1 2 3 4 5
time (s) -0.1
-0.05 0 0.05 0.1
inter-story displacement (m)
controlled uncontrolled
(a)
0 0.1 0.2 0.3 0.4 0.5
time (s) 0
0.01 0.02 0.03 0.04 0.05 0.06
rubber deformation (m)
(b)
Figure 17: (a) Comparison of displacement response at first floor; (b) peak rubber deformation at node 8 (zoom on 0.5s).
0 1 2 3 4 5
time (s) -0.1
-0.05 0 0.05 0.1
inter-story displacement (m)
controlled uncontrolled
(a)
0 0.1 0.2 0.3 0.4 0.5
time (s) 0
0.01 0.02 0.03 0.04
rubber deformation (m)
(b)
Figure 18: (a) Comparison of displacement response at top floor; (b) peak rubber deformation at node 18 (zoom on 0.5s).
To evaluate the relative effectiveness of the proposed connection under more moderate blast loads (for
443
which nonlinear cladding response may realistically be less likely), additional simulations of the 18DOF
444
numerical model are conducted using the blast loads due to the 100-kg charge mass of TNT. Parameter
445
values of this excitation are calculated based on Section 4.1.1 and are listed in Table 6.
446
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Table 6: Simulated blast load (100-kg TNT) for 18DOF system
node R (m) Z (m/kg1/3) σmax (kPa) Fm (kN) tblast(ms)
u7 25.00 5.39 59.49 98.16 28
u8 25.27 5.44 58.20 96.03 28
u9 25.27 5.44 58.20 96.03 28
u10 26.05 5.61 54.67 90.21 28
u11 26.05 5.61 54.67 90.21 28
u12 27.30 5.88 49.75 82.09 29
u13 27.30 5.88 49.75 82.09 29
u14 28.97 6.23 44.35 73.18 29
u15 28.97 6.23 44.35 73.18 29
u16 30.98 6.66 39.11 64.54 30
u17 30.98 6.66 39.11 64.54 30
u18 33.27 7.15 34.40 56.75 30
The maximum inter-story drift and acceleration reductions at each floor resulting from the controlled case
447
under various blast excitations are compared and listed in Table 7. The cladding system under 100-kg TNT
448
provides higher inter-story reduction values than the 500-kg case below the third floor due to lower peak blast
449
load values. Above the third floor, the 500-kg case demonstrates better inter-story drift mitigation since no
450
rubber-cladding collision and energy dissipation from impact rubber occur for the 100-kg case. Under both
451
blast load scenarios, the decrease of inter-story drift reduction from the first to the third floor are caused by
452
a reduced cladding stiffness and friction capacity (Table 5). Above the third floor, the performance increases
453
as the peak blast load magnitude rapidly decreases with increasing diagonal standoff. Comparing the overall
454
acceleration mitigation shows that the designed cladding system can provide significant improvement for
455
different severities of blast excitations. The high acceleration reduction is caused by the absorption of the
456
blast reaction at the cladding level.
457
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Table 7: Maximum response reduction
floor 500-kg TNT 100-kg TNT
inter-story drift (%) acceleration (%) inter-story drift (%) acceleration (%)
1 28.32 90.93 30.20 89.93
2 26.91 87.19 32.83 86.43
3 31.22 88.54 45.93 88.06
4 47.27 78.42 40.65 82.64
5 58.79 84.81 43.38 85.31
6 67.67 71.68 38.93 79.80
6. Conclusions
458
Explosive materials deliver large shock-wave pressures to nearby structures and can cause significant
459
damage in a very short duration. Typically, cladding systems for buildings are connected to the structure
460
using connections with very high stiffness, which provide direct transfer of blast-induced reactions to the
461
structural system. In this paper, a novel semi-active cladding connection is proposed for mitigating this
462
high-rate load transfer. Instead of simply providing rigid lateral support for cladding, this new mechanism
463
is an active participant in energy dissipation under blast load. A semi-active friction mechanism is used to
464
provide a variable damping force, and an impact rubber bumper is utilized to absorb pounding energy for
465
large connection displacements.
466
A performance-based procedure has been proposed for the design of this new cladding connection. It
467
contains three steps: (1) blast load design, (2) cladding and friction device design, and (3) impact rubber
468
design. Three dimensionless transfer functions were derived using an equivalent SDOF system. These
func-469
tions allow rapid preliminary design of the cladding system and have been verified by numerical simulation
470
of the 2DOF system. Results show that the PBD procedure offers adequate design values with positive
471
performance metrics in most cases. It is computationally convenient and reasonably accurate to implement
472
the design value from the SDOF system for the MDOF system. Negative error values may arise in the design
473
of the rubber bumper for a limited range of scenarios, but this can be accommodated by recommending a
474
slightly larger design thickness value to provide additional safety.
475
Furthermore, the proposed cladding connection was designed and simulated in a six story structure as
476
an example for the proposed PBD procedure. A 18DOF system is used as a prototype, and the
blast-477
induced performance of the structure with the proposed friction-based connections is compared with that
478
using conventional cladding connections. Numerical simulation results show that the proposed cladding
479
connection offers significant reductions of blast-induced story displacements and accelerations. Moreover,
480
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the simulation results show that the largest rubber deformation occurs in the first cladding impact cycle,
481
which supports the assumptions inherent in models associated with the PBD approach. The results of this
482
study indicate that this new semi-active cladding system shows promise for practical considerations of blast
483
mitigation for buildings.
484
Acknowledgments
485
This material is based upon the work supported by the National Science Foundation under Grant No.
486
1463252 and No. 1463497. Their support is gratefully acknowledged. Any opinions, findings and conclusions
487
or recommendations expressed in this material do not necessarily reflect the views of the National Science
488
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