Performance Evaluation of a Variable Friction Cladding System for Seismic Hazard Mitigation Y. Gong 1 , L. Cao 2 , S. Laflamme 3 , S. Quiel4 , J. Ricles5 , D. Taylor6 1. 2.
3. 4. 5. 6.
Ph.D. Student, Dept. of Civil, Construction, and Environmental Engineering, Iowa State University, United States. E-mail:
[email protected] Postdoctoral Research Associate, Dept. of Civil, Construction, and Environmental Engineering, Iowa State University, United States. E-mail:
[email protected] Associate Professor, Dept. of Civil, Construction, and Environmental Engineering, Iowa State University, United States. E-mail:
[email protected] Assistant Professor, Dept. of Civil and Environmental Engineering, Lehigh University, United States. E-mail:
[email protected] Professor, Dept. of Civil and Environmental Engineering, Lehigh University, United States. E-mail:
[email protected] Chief Executive Officer, Taylor Devices, Inc. United States. E-mail:
[email protected]
ABSTRACT A novel semi-active frict ion device, termed Variable Frict ion Cladding Connection (VFCC), has been proposed by the authors. The device is a variable frict ion damper that links cladding elements to the structural system. The friction force is generated by sliding plates and controlled by a varying normal force applied fro m an adjustable toggle system. Prev iously, its performance for blast mitigation has been validated in its passive in -situ configuration (constant friction force). In this paper, we investigate the applicability of VFCC for seismic hazard mit igation. First, dynamic transfer functions for structural responses under harmonic ground motion are derived fro m a 2DOF structure-cladding model. Second, the effect of vary ing cladding damping is investigated in the frequency domain. Third, the VFCC is simulated on a 4-story building. Simulat ion results show that the VFCC offers significant performance improvement compared with conventional cladding connection under seismic excitation. KEYWORDS: Semi-active damper, Variable friction, Cladding connection, Multi-hazard mitigation, Structural Control, High performance control system.
1. INTRODUCTION Cladding systems are nonstructural elements serving architectural purposes and protecting occupants against the environment (e.g., wind, rain). A typical cladding is mounted to the structure via rigid tie -back connections that transfer the cladding reactions directly to the structure with very limited energy dissipation. It also contributes to additional inertia due to its non-negligible load. Early work on engaging cladding systems for energy dissipation includes the development of advanced cladding panels for blast mitigation [1] [2] [3] and the imp lementation of passive energy dissipating connections for seismic mitigation. Examp les of passive connectors include a ductile cladding connection that dissipates energy via plastic deformation, where the device was shown to be capable of reducing inter-story drift in a structure under earthquakes [4], a U-shaped flexu ral plates connector also for seis mic mitigation [5], and a v iscoelastic spider connector for blast resilience [6]. To the best knowledge of the authors, all of the proposed energy absorption or dissipation strategies design ed for cladding systems are passive, restricting mit igation performance within a given frequency bandwidth, i.e. single hazard [7]. The authors have recently proposed a new semi-active cladding connection, with the objective to leverage cladding systems for enhancing protection against mult iple hazards. The device is a variab le friction damper, termed variable friction cladding connection (VFCC). It consists of two sets of sliding plates that produce a frict ion force, and an actuated toggle system that varies the normal force applied to the slid ing plates. This actuation could either be provided, for instance, by a pneumatic [8] [9] or hydraulic [10] [11] systems. A possible imp lementation of the VFCC is its embed ment into a floor slab laterally connecting the cladding panel to the structural system. The VFCC is designed to act as a passive damper during daily operations with its toggle system locked to provide a high static
friction force, therefore adding resistance against blast loads. The toggle system is to be actuated during high wind and earthquake events. Previously, the performance of the VFCC under blast excitation has been theoretically validated [12]. The objective of this paper is to investigate the performance of the VFCC at seismic mitigation. The paper is organized as fo llows. Section 2 introduces theoretical background on the VFCC and its dynamic friction model. Section 3 studies the dynamic response of a t wo-degree-of-freedo m (2DOF) structure-cladding system equipped with a VFCC under harmonic ground motion. Section 4 simulates the VFCC on a 4-story structure and evaluates its performance under a realistic earthquake event. Section 5 concludes the paper.
2. VARIABLE FRICTION CLADDING CONNECTION A configuration of the VFCC frict ion mechanism is shown in Fig 2.1 (a). It is designed to dissipate energy via friction using two sets of sliding friction plates generated by an actuator varying the geometry of the toggles . A possible implementation of the VFCC is shown in Fig 2.1 (b ) with the device embedded in a floor slab laterally connecting the cladding panel to the structural frame.
(a) (b) Figure 2.1 The VFCC configuration: (a) friction mechanism; and (b) implementation in a floor slab (top view). Fig 2.2 (a) shows the force diagram of the VFCC. The actuation force acts on toggles and generates a normal force acting on the friction plates. The relationship between normal fo rce and actuator stroke length (Fig. 2.2 (b)) is derived based on the geometric deformation of the toggles: ( ) where
is the toggle stiffness,
( )
⁄
[((
the initial toggle length, and
)
)
]
(2.1)
the initial toggle-plate angle.
(a) (b) Figure 2.2 The VFCC configuration: (a) force diagram; and (b) annotated geometric parameters .
2.1. Dynamic Model The dynamics of the VFCC has been characterized on a prototype subjected to various harmonic excitations in laboratory environment [13]. A modified LuGre model was developed to characterize its dynamic behavior.
The friction force
is written as ( ) ̇
̇
(2.2)
where , , and represent the aggregate bristle stiffness, micro-damping, and viscous friction respectively, ̇ are the sliding displacement and velocity of the device, respectively, is an evolutionary variable with ̇ | ̇| , and ( ̇) is a function describing the Stribeck effect:
and ̇
( ̇)
( ̇)
(
)
( ̇⁄ ̇ )
(2.3)
where ̇ is a constant modeling the Stribeck velocity, is the Coulo mb frict ion force and is the magnitude of the Stribeck effect. Force is modeled as , where is a constant. Force is modeled as a function of the ( ) sliding motion y and stroke length : ( ) , where , is the friction coefficient , is the friction p late length and is the d istance between the toggle and the end of the friction plate. Parameter | is taken as a linear function of stroke length . Tab le 2.1 lists identified values for stroke length independent parameters, and Fig. 2.3 are plots of modeled dynamic loops based on the developed dynamic model. Table 2.1 Stroke length independent parameters of the VFCC dynamic model parameters value unit
1.052 -
0.428
0.043
0.200
0.200
| 1.302
(a) (b) Figure 2.3 (a) Force-displacement and (b) force-velocity loops for a harmonic excitation of 13 mm at 0.05 Hz.
3. 2DOF STRUCTURE-CLADDING MODEL In this section, a 2DOF structure-cladding system equipped with the VFCC is investigated analytically under harmonic ground motion. Dynamic transfer functions that represent the ratio of the steady response to the magnitude of the excitation are derived. The analytical work is facilitated by converting the VFCC into an equivalent viscous damper. The developed transfer functions are verified via numerical simu lations considering the dynamic friction instead of equivalent viscous damping, along with linear feedback control.
3.1. Equivalent Viscous Damping The dynamic model o f a 2DOF model o f a structure-cladding system with structure properties of mass , damping , and stiffness , subjected to a harmonic ground motion of amplitude and frequency is diagrammed in Fig 3.1 (a). A cladding system of mass is connected to a primary structure via a stiffness element and the VFCC of capacity . The natural frequency of structure and cladding are defined as √ ⁄ and
√ ⁄ , respectively. The equivalent 2DOF model with a variable v iscous damping is shown in Fig. 3.1 (b). Remark that with this approach, the system resembles a tuned mass damper equipped with a semi-act ive friction element [14]. While the analytical work is derived following this model, the configuration of the VFCC system will differs, because it will link a cladding element spanning two floor, as it will be discussed in the next section.
(a) (b) Figure 3.1 2DOF structure-cladding model: (a) semi-active friction damping; and (b) equivalent viscous damping. A linear feedback control ru le is used to computed the required control force the actual friction force ( ) is given by ()
{
()
(
̇ ( ))
( )
(
()
( ) ̇ () ( ) ̇ ()
̇ ( ) ) and
(3.1)
where and are constant control parameters, and ( ) and ̇ ( ) are the relative displacement and velocity of the cladding to the structure, respectively. The energy dissipated by the semi-active friction force over one period is [ ( where
( ) and
is the amplitude of
. By equating
and
)
(
)]
(3.2)
. The energy dissipated by viscous damping over one period is
, an equivalent viscous damping ratio [ (
)
is obtained )]
(
(3.3)
3.2. Dynamic Transfer Functions The equations of motion of the 2DOF model with equivalent viscous damping are written ̈
̇ ( ̈
̇ ̈ )
(3.4)
̇
(3.5)
( ) ( ) The steady dynamic response of the system has the form and , where and are the response amp litudes , and and are phase angles between the response and the excitation . The dynamic transfer functions that represent the amplitude of the steady response of structure and cladding are defined as ̈ , and , where ̈ is the amplitude of the structure acceleration. Using
non-dimensional parameters [ The dynamic transfer functions
, ] and
and [
into Eq. 3.4 and 3.5 yields
(
)
]
[ ]
can be expressed in terms of these non-dimensional parameters
(3.6)
[(
√[(
)(
)
√[(
)(
)
)
]
[ (
)
]
[
(
(
)
)
(
)]
(
(
)
)
(
)]
( ]
]
) [
(3.7) (3.8)
3.3. Verification Two performance metrics
and
are defined to evaluate the accuracy of equivalent viscous damping model, ;
(3.9)
where and are dynamic transfer functions obtained fro m nu merical simu lations using semi-active friction damping. Results for dimensionless parameters , , , and various values of are plotted in Fig. 3.2. The error increases around and converges to zero when the excitation frequency is far fro m . A comparison of and for various values of shows that the error diminished with the damping demand.
(a)
(b) Figure 3.2 Performance metrics (a)
and (b)
.
3.4. Effect of VFCC Damping In what follows, the effect of damping rat io Assuming , can be written
on the dynamic transfer functions is investigated analytically. (
| | √(
√
⁄
)
⁄
)
(3.10)
( ) ( ) ( )( ) [ where , , , and ( ) ]. Values for and are plotted with various damping ratio in Fig. 3.3. These curves intersect at two fixed points, P and Q, where and are independent of . The values at points P and Q can be obtained by solving | | | | [15] (
)
√[
( (
) )
]
[ (
) ]
(3.11)
[ ] and [ ]. Otherwise, a A largest cladding damp ing ratio can provide a s maller value when smaller is required to minimize . Similarly, an optimal can also be achieved with the optimizat ion of . In summary, the cladding damping can have a significant influence on the structural dynamics by varying the damping ratio .
Q
Q P P
(a) Figure 3.3 Dynamic transfer functions at
,
and
(b) : (a)
and (b)
.
4. NUMERICAL SIMULATIONS 4.1. Building Model A 4-story office occupancy building located in Los Angeles was selected to numerically investigate the performance of the VFCC for seis mic mitigation. The weak direction of the building equipped with the VFCC is modeled as a lu mped mass system, as shown in Fig. 4.1. Each cladding panel is modeled as a rigid bar of mass and has two degree of freedoms each connected to adjacent floors. The dynamic properties of the building model are listed in Table 4.1. In the simu lated model, the connection element between the cladding and the structure includes a stiffness element , a viscous dashpot , and a semi-active friction element . The capacity of the friction element at each floor is designed to meet the maximu m value of required force ( ) , taken as 200 kN, based on Eqs. 3.1 and 3.3 using parameter values , , and m. Note this capacity represents the sum of the maximu m damping force provided by individual devices at each floor. In applications, several VFCC devices could be installed per cladding element as necessary.
connection connection
Figure 4.1 Simulated 4-story building model with cladding panels .
floor 4 3 2 1
mass ( ) 1423000 2011000 2011000 2041000
Table 4.1 Dynamic properties of the building structure cladding stiffness damping mass stiffness ⁄ ) ( ⁄ ) ( ( ) ( ⁄ ) 143900 2664 172300 1485 213900 2472 172300 1485 254300 2079 172300 1485 274000 1399 198800 1485
(
damping ⁄ ) 14.58 14.58 14.58 14.58
4.2. Results and Discussion The north-south component of the Hollister earthquake recorded at the Ho llister City Station is used as the excitation input. In the uncontrolled case, the cladding system is treated as a mass contributor to the structure assuming and . In the semi-act ive control case, the VFCC force is controlled using a Linear Quadratic Regulator (LQR) control law with pre-tuned control parameters. The distribution of peak inter-story displacements and absolute accelerations are plotted in Fig 4.2 (a) and (b). Results show that the VFCC significantly reduces both maximu m displacement and acceleration under earthquake excitation. The maximu m reduction of inter-story drift and acceleration reached 33.1% and 26.8%, respectively. The time histories of the inter-story drift, acceleration, and damping force taken at the top floor are p lotted in Fig. 4.3 (a)-(c). The VFCC reduces the maximu m inter-story drift and acceleration of the top floor by 35.2% and 26.8% , respectively. The maximu m damping force occurred at approximately 10 s and did not exceed the capacity, as expected. Results show that the semi-active control case rapidly mitigates the event.
(a) (b) Figure 4.2 Comparison of (a) maximu m inter-story drift; and (b) maximu m absolute acceleration for each floor.
(a)
(c) Figure 4.3 Comparison of time histories of top floor response: (a)
(b)
; (b) ̈ and (c)
.
5. CONCLUSION A variable frict ion cladding connection (VFCC) has been previously proposed by the authors to leverage cladding motion for mit igating multip le hazards. In this paper, the effectiveness of the VFCC at mit igating seismic loads has been analytically investigated on a 2DOF structure-cladding model under harmonic acceleration. The analytical solution showed that the cladding connection had a significant effect on the structure’s dynamics by varying the friction force. A realistic numerical simu lation was conducted on a 4-story structure equipped with the VFCC subjected to an earthquake. Simu lation results demonstrated that the proposed structure-cladding system with the VFCC provided considerable inter-story displacement and absolute acceleration reduction comparing with an uncontrolled strategy. AKCNOWLEDGEMENT This material is based upon the work supported by the National Science Foundation under Grant No. 1463252 and No. 1463497. Their support is gratefully acknowledged. Any opinions, findings and conclusions or recommendations expressed in this material do not necessarily reflect the views of the Natio nal Science Foundation. REFERENCE 1. 2. 3. 4.
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