Optimal frictional coefficient of structural isolation system

Abstract

An isolation system is not very effective when an inappropriate level of damping is used. This paper proposes a theoretical method which can be used to determine the optimal frictional coefficient of an isolation system. Only a one-dimensional isolation system and ground motion are considered. The frictional coefficient is optimized by minimizing the sum of squares of structural absolute accelerations, with the optimization results being validated graphically. Sensitivity studies were used to verify the feasibility of the optimal frictional coefficient, coupled with a practical example in Taipei under the conditions of the Hualien and El Centro earthquakes. Consequently, the feasibility and reliability of the proposed optimal design were verified.

Keywords

Optimal design isolation friction seismic sensitivity

1. Introduction

Seismic isolation can mitigate structural responses under earthquake. The isolation procedure here involves the installation of an isolator which can elongate the fundamental period between the superstructure and the ground. Many isolation system types have been implemented for the purposes of structural and equipment isolation within the civil and mechanical engineering disciplines. An isolation system with a low frequency can reduce the acceleration transmitted to the superstructure (Chung et al., 2009a; Yang et al., 2011). However, a well-designed isolation system should include damping for energy dissipation. The effectiveness of an isolation system is reduced when the incorporated damping is either too large or too small. Thus, an appropriate level of damping may result in better performance with regard to both the acceleration of superstructure and the base displacement. Therefore, this study proposes a theory for the determination of an optimum frictional damping for an isolation system with a linear restoring force.

In passive structural control, the optimal design of a tuned mass damper (TMD) has been well developed (Chung et al., 2011). However, an optimal design for a structural isolation system is still under development. Optimal design formulas of the linear viscous TMD for an undamped single degree-of-freedom (SDOF) structure have been proposed (Den Hartog, 1956; Warburton and Ayorinde, 1980; Warburton, 1982). If damping is present within the primary structure, it is difficult to obtain the closed-form solutions for the optimal parameters of a TMD (Fujino and Abe, 1993; Sadek et al., 1997; Bakre and Jangid, 2005; Wang et al., 2009). A nonlinear viscous damper may be cost effective in fabrication and maintenance. Hence, Chung et al. (2009b) proposed optimization procedures in the time domain for the design of a TMD with nonlinear viscous damping. Optimal design parameters for the nonlinear TMD were sought using an optimization method to minimize the structural responses. Inaudi and Kelly (1995) proposed a friction-dissipating TMD where the restoring force was provided by linear springs and the dissipating force was provided by friction dampers acting transversely to the direction of motion. Subjected to white-noise support excitation, the mean square structural responses, estimated by statistical linearization, were minimized to obtain the optimum parameters for the friction-dissipating TMD. The results were confirmed by comparing them with those obtained using Monte Carlo simulation.

The sliding type isolator with frictional energy dissipation has been well researched. Most of the research found that sliding supports can be quite effective in controlling the level of acceleration response of structures under earthquake excitation. Some of the research also found that for low coefficients of friction, the acceleration response does not vary with the frequency content of the ground motion. This implies that sliding supports can be effectively used for all kinds of sites, whether hard or soft soil, and whether close or far from causative faults (Mostaghel and Tanbakuchi, 1983; Mostaghel et al., 1983; Zayas et al., 1990; Mokha et al., 1991). Derby and Calcaterra (1970) proposed an optimal design formula for an undamped SDOF structure with a massless isolation system. In this system, the restoring force was provided by linear springs and the dissipating force was provided by elastic-viscous dampers. Subjected to white-noise ground excitation by minimizing the root mean square of the structural absolute acceleration, the optimal damping ratio of the isolation system can then be found. Inaudi and Kelly (1992) presented a procedure for defining optimal damping for a linear isolation system. For the statistical approach used, the ground motion was modeled as a stationary Gaussian random process and optimal damping was defined as that which renders the minimum peak floor acceleration. Within the deterministic approach presented, the maximum floor acceleration of the base isolated structure subjected to recorded ground motions was evaluated. This was performed as a function of the isolation energy dissipation capacity using numerical simulation methods. It was shown that optimal isolation damping increases with an increase in the damping of the superstructure. Thakkar and Jain (2004) studied the role of isolation damping for base-isolated buildings of moderate height by controlling their seismic response under different earthquake motion types. Two reinforced concrete frame buildings and eight real earthquake motions were considered. It was observed that the optimal value of isolation damping is dependent upon the characteristics of the input ground motion and has a low value for ground motions with only high frequencies. Jangid (2007) investigated the seismic response of multi-story buildings isolated using leader-rubber bearing (LRB) under near-fault motions. It was found that an optimum value of yield strength of the LRB exists for which the absolute acceleration of the building’s top floor attains the minimum value. The optimum value of yield strength of the LRB was found to be in the range of 10%–15% of the total weight of the building under near-fault motions. Furthermore, the energy-dissipating mechanism can be replaced with friction dampers. Jangid (1996, 2000, 2005) first considered the resilient-friction base isolator (R-FBI) as an isolation system and investigated optimal isolation damping to minimize the top floor root-mean-square acceleration when subjected to earthquake ground excitation. It was shown that the optimal isolation damping decreases with an increasing frictional coefficient of the R-FBI system, the number of floors in the superstructure, and the period of the superstructure. Further, the optimal friction coefficient of a sliding system with a restoring force to minimize the root mean square of the top-floor acceleration response of a base-isolated structure under earthquake ground motion was investigated. It was also shown that the optimal friction coefficient decreases with the increase of the period and damping ratio of base isolation system, and depends upon the intensity of earthquake excitation. Therefore, the optimal friction coefficient increases with the increase of the intensity of earthquakes. Furthermore, the analytical seismic response of multi-story buildings isolated by a friction pendulum system (FPS) was investigated under near-fault motions. It was also found that an optimal frictional coefficient of FPS exists for minimizing both the absolute acceleration of the building’s top floor and the sliding displacement. The optimal frictional coefficient of the FPS was found to be in the range of 0.05 to 0.15 under near-fault motions. It was also found to decrease with increasing flexibility of the FPS.

In a passive isolation system, it is always a problem that the system is not adaptive when it is excited by a non-design earthquake including different waveform or intensity. Thus, the optimal friction coefficient from the proposed design procedure may be suitable under an earthquake similar to the design one. However, passive isolation systems (e.g., FPS, LRB) are generally implemented in practical application (Naeim and Kelly, 1999) because their reliability has been verified and they are easy to design.

This paper investigates the optimal design of the frictional coefficient of an isolation system. The dynamic equation of a structure implemented on a frictional isolation system is expressed in discrete-time state space. A single equation was developed to determine the frictional force in both stick and slip states (Lu et al., 2006) so that time-history analysis can be conducted iteratively. The design parameter is optimized such that the sum of squares of the chosen structural response is minimized. If the friction coefficient becomes lower and lower, the structural acceleration and isolation displacement are larger and larger. On the other hand, if the frictional coefficient becomes higher and higher, the structural acceleration is higher and higher but the isolation displacement is still smaller and smaller. Therefore, there exists an optimal frictional coefficient to minimize the structural acceleration but not the isolation displacement. Hence, only structural acceleration is considered in the performance index in this study. As a result of the equations derived from the optimization procedure being nonlinear, the steepest descent method was adopted to solve the necessary, and sufficient, conditions of optimization. The design parameters were updated with the optimal step using the golden section search method. The feasibility of the optimization method is illustrated numerically using an isolated structure attached to a frictional isolation system subjected to Hualien and El Centro ground accelerations. The accuracy of the optimization result was also confirmed graphically. Furthermore, a sensitivity analysis was performed to study the influence of the system parameters. Finally, the isolated structure subjected to the Hualien and El Centro ground accelerations was taken as an example to verify the effectiveness of the frictional isolation system with an optimal frictional coefficient.

2. Dynamics of frictional isolation system

A superstructure was modeled as an SDOF structure and was mounted on a frictional isolation base, transforming it into a two-degree-of-freedom (2DOF) system. By utilizing state space, the discrete-time first order difference equation is transformed from the equation of motion. In computing the frictional force, a simplified equation which includes two kinds of behavior of motion (stick and slip states) is proposed. Through each iteration of the discrete-time state equation, all the time-histories of structural responses can then be obtained.

2.1. Equation of motion of structure with frictional isolation system

A superstructure is modeled as an SDOF structure with mass $m s$ , damping coefficient $c s$ , and stiffness $k s$ . The superstructure is mounted on a frictional isolation system with frictional coefficient μ. The mass and stiffness of the frictional isolation system are $m i$ and $k i$ , respectively. After the superstructure is isolated, it becomes a 2DOF system (Figure 1). When subjected to a ground disturbance $w (t)$ , the equation of motion of the system can be expressed as:

M \overset{··}{x} (t) + C \overset{\cdot}{x} (t) + Kx (t) = b f r (t) - M 1 w (t)

(1)

where

x (t) = [\binom{x s (t)}{x i (t)}]

is the displacement vector,

x i (t)

and

x s (t)

are the displacements of the isolation base and the structure, respectively;

M = [m s 00 m i] = m [R m 00 1 - R m]

is the mass matrix,

m = m s + m i

is the total mass of the system,

R m = m s / m

is the mass ratio;

K = [k s - k s - k s k s + k i] = m (2 π f i) 2 [R m R_{f}^{2} - R m R_{f}^{2} - R m R_{f}^{2} R m R_{f}^{2} + 1]

is the stiffness matrix,

R f = f s / f i

is the frequency ratio of the superstructure and the isolation system,

f s = \sqrt{k s / m s} / 2 π

and

f i = \sqrt{k i / m} / 2 π

are the structural frequency and isolation frequency, respectively;

C = [c s - c s - c s c s] = 2 m (2 π f i) [R m R f ζ s - R m R f ζ s - R m R f ζ s R m R f ζ s]

is the damping matrix,

ζ s

is the damping ratio of the superstructure;

f r (t)

is the frictional force of the isolation system, and

b = [\binom{0}{1}]

and

1 = [\binom{1}{1}]

are the location vectors of the frictional force and the ground disturbance, respectively.

Figure 1.

Schematic diagram of structural isolation with frictional damping.

The equation of motion (equation (1)) in configuration space is conveniently expressed as a first-order differential equation in state space,

\overset{\cdot}{z} (t) = Az (t) + B f r (t) + E w (t)

(2)

where

z (t) = [\binom{x (t)}{\overset{\cdot}{x} (t)}]

is the state vector,

A = [0 I - M - 1 K - M - 1 C]

is the system matrix, and

B = [0 M - 1 b] and E = [0 - 1]

are the location vectors of the frictional force and the ground disturbance in state space, respectively. The exact solution of equation (2) is

z (t 2) = e A (t 2 - t 1) z (t 1) + \int_{t 1}^{t 2} e A (t 2 - τ) [B f r (τ) + E w (τ)] d τ

(3)

In the interval between two consecutive time steps, $t 1 = k Δ t$ and $t 2 = (k + 1) Δ t$ , it is assumed that the frictional force and the base disturbance are piecewise constant,

f r (τ) = f r (k Δ t), k Δ t \leq τ < (k + 1) Δ t w (τ) = w (k Δ t), k Δ t \leq τ < (k + 1) Δ t

(4)

where

Δ t

is the sampling period. Under the assumption mentioned above, the solution of equation (3) can be expressed in discrete-time fashion as a first-order difference equation,

z [k + 1] = A d z [k] + B d f r [k] + E d w [k]

(5)

where

A d = e A Δ t

is the discrete-time system matrix, and

B d = A - 1 (A d - I) B

and

E d = A - 1 (A d - I) E

are the location vectors of the frictional force and the ground disturbance in discrete-time state space, respectively.

2.2. Frictional force of structural isolation system

Under disturbance, sliding of the isolation interface has two states: the slip state and the stick state. In the slip state, the velocity of the isolation base relative to the ground is unknown but the magnitude of the frictional force is known to be $| f r [k] | = μ N$ , where $N = mg$ is the normal force. In the stick state, the frictional force is unknown and bounded by $| f r [k] | \leq μ N$ , but the velocity of the isolation base relative to the ground is known to be zero. It is assumed that the isolation system is in the stick state at the ( $k + 1$ )^st step. Therefore, the relative velocity of the isolation base is zero,

\overset{\cdot}{x} i [k + 1] = d T z [k + 1] = 0

(6)

where

d T = [0 T b T]

is the output vector of the isolation base relative velocity.

After equation (6) is substituted into equation (5), the frictional force at the k^th step $f r [k]$ can be estimated as

\overset{\land}{f} r [k] = - (d T B d) - 1 d T (A d z [k] + E d w [k]) .

(7)

When the magnitude of the estimated frictional force is less than the maximum frictional force $| \overset{\land}{f} r [k] | < μ N$ , the isolation system is in the stick state and the frictional force is the same as the estimated one $f r [k] = \overset{\land}{f} r [k]$ . On the other hand, when the magnitude of the estimated frictional force is larger than the maximum frictional force $| \overset{\land}{f} r [k] | > μ N$ , the isolation system is in the slip state and the magnitude of the frictional force is the same as the maximum frictional force, and the sign is the same as that of the estimated frictional force $f r [k] = μ N sgn (\overset{\land}{f} r [k])$ . A unified equation to compute the frictional force for both the slip and stick states is given as:

f r [k] = 0.5 \overset{\land}{f} r [k] [1 + sgn (μ N - | \overset{\land}{f} r [k] |)] + 0.5 μ N sgn (\overset{\land}{f} r [k]]) [1 + sgn (| \overset{\land}{f} r [k] | - μ N)]

(8)

After the frictional force $f r [k]$ is calculated, the state vector at the ( $k + 1$ )^th step, $z [k + 1]$ can be calculated from equation (5). The time history of the state vector is obtained iteratively. The structural absolute acceleration $y [k]$ can be further obtained from the equation of motion in configuration space and defined as:

y [k] = d_{a}^{T} \overset{··}{x} a [k] = d_{a}^{T} (D T z [k] + S f r [k])

(9)

where

\overset{··}{x} a [k]

is the absolute acceleration vector,

d_{a}^{T} = [10]

is the output location vector of the structural absolute acceleration,

D T = [- M - 1 K - M - 1 C]

is the lower part of the system matrix A, and

S = M - 1 b

. Hence, all the time-histories of the structural responses can be obtained.

3. Optimization of frictional coefficient

In order to achieve the maximum vibration isolation of the structure, the design of the frictional isolation system must be optimized. The optimal design parameter for the frictional isolation system can be obtained such that a performance index is minimized. The most appropriate performance index J is the sum of squares of the structural absolute accelerations $y [k]$ ,

J = \sum_{k = 0}^{k 1} y T [k] y [k]

(10)

where k₁ is the instant at which ground disturbance is terminated. When equation (9) is substituted into equation (10), the performance index J can be rewritten as

J = \sum_{k = 0}^{k 1} y T [k] y [k] = \sum_{k = 0}^{k 1} {z [k] T Dd a d_{a}^{T} D T z [k] + z [k] T Dd a d_{a}^{T} S f r [k] + S T d a f r [k] d_{a}^{T} D T z [k] + S T d a d_{a}^{T} S f_{r}^{2} [k]}

(11)

In order to solve the optimization problem, the state vector $z [k]$ in the above equation should satisfy the state in equation (5), which becomes the equality constraint to the performance index J. After incorporating the equality constraint with a time-varying Lagrange multiplier vector $λ [k + 1]$ , the augmented performance index $J'$ is obtained as:

J' = \sum_{k = 0}^{k 1} {z [k] T Dd a d_{a}^{T} D T z [k] + z [k] T Dd a d_{a}^{T} S f r [k] + S T d a f r [k] d_{a}^{T} D T z [k] + S T d a d_{a}^{T} S f_{r}^{2} [k] + λ T [k + 1] (A d z [k] + B d f r [k] + E d w [k] - z [k + 1])}

(12)

According to calculus of variations, the first variation of the augmented performance index $J'$ is

δ J' = \sum_{k = 0}^{k 1} {δ λ T [k + 1] \frac{\partial J'}{\partial λ [k + 1]} + δ z T [k] \frac{\partial J'}{\partial z [k]} + δ μ \frac{\partial J'}{\partial μ}} = 0 .

(13)

The augmented performance index

J'

is a function of the state vector

z [k]

, the co-state vector

λ [k + 1]

, and the frictional coefficient μ. The necessary conditions for the minimization of the augmented performance index

J'

are derived such that the first variation

δ J'

is zero. Therefore, the first imposed condition is

z [k + 1] = A d z [k] + B d f r [k] + E d w [k], z [0] = z 0

(14)

where

z [0] = z 0

is the initial condition for the state equation. The second condition is

λ [k] = 2 Dd a d_{a}^{T} D T z [k] + 2 Dd a d_{a}^{T} S f r [k] + 2 (\frac{\partial f r [k]}{\partial z [k]}) T S T d a d_{a}^{T} D T z [k] + 2 (\frac{\partial f r [k]}{\partial z [k]}) T f r [k] S T d a d_{a}^{T} S + A_{d}^{T} λ [k + 1] + (\frac{\partial f r [k]}{\partial z [k]}) T B_{d}^{T} λ [k + 1], λ [k 1 + 1] = 0

(15)

where

λ [k 1 + 1] = 0

is the terminal condition for the co-state equation (15),

(\frac{\partial f r [k]}{\partial z [k]}) T = 0.5 (\frac{\partial \overset{\land}{f} r [k]}{\partial z [k]}) T [1 + sgn (μ N - | \overset{\land}{f} r [k])]

(15a)

and

(\frac{\partial \overset{\land}{f} r [k]}{\partial z [k]}) T = - A_{d}^{T} d [(d T B d) - 1] T

(15b)

Further, the third condition is

\frac{\partial J'}{\partial μ} = \sum_{k = 0}^{k 1} {N sgn (\overset{\land}{f} r [k]) [1 + sgn (| \overset{\land}{f} r [k] - μ N)] z [k] T Dd a d_{a}^{T} S + f r [k] N sgn (\overset{\land}{f} r [k]) [1 + sgn (| \overset{\land}{f} r [k] - μ N)] S T d a d_{a}^{T} S + 0.5 N sgn (\overset{\land}{f} r [k]) [1 + sgn (| \overset{\land}{f} r [k] - μ N)] λ T [k + 1] B d} .

(16)

In this paper, the steepest gradient method of optimization is adopted. The initial guess for the frictional coefficient μ is assigned as $μ (0) = 0$ . The state vector $z [k]$ and the co-state vector $λ [k]$ are solved from equations (14) and (15) in sequence. The gradients of the frictional coefficient μ in equation (16) are typically non-zero and hence, it is used to update the frictional coefficient,

μ (i + 1) = μ (i) - s \frac{\partial J' (i)}{\partial μ (i)}

(17)

where s is the step increment for the frictional coefficient. If the step s is too large, the solution may not converge. Conversely, if the step s is too small, convergence is guaranteed but the number of iterations may be very large. Therefore, the golden section search method was adopted to find the optimal step increment (Lee et al., 2006). The iterative procedure is repeated until an acceptable tolerance between two consecutive iterations has been achieved (Figure 2).

Figure 2.

Flowchart of iterative procedure.

4. Numerical verification and sensitivity analysis

The feasibility of the optimization method was verified numerically using an isolated structure mounted on a frictional isolation system subjected to the 2002 Hualien, Taiwan, and 1940 El Centro, USA, earthquakes. Two-dimensional graphs of the objective function and the searching parameter were used to validate the optimization results. Following the optimization procedure, the performances of the superstructure with and without the frictional isolation system under the two specific earthquakes were compared. In addition, the influence of the optimal design parameters of the frictional isolation system was also investigated using sensitivity analysis.

4.1. Introduction of structural isolation system

In this research, a reinforced concrete building with an isolation system assumed to be located at the National Taiwan University (NTU), Taipei, Taiwan was considered and simplified as a SDOF model. When the superstructure is constructed on a frictional isolation system, it becomes a 2DOF system. As a simplified SDOF superstructure for the first mode, the mass $m s$ , the natural frequency $f s$ and the damping ratio $ζ s$ of the superstructure were 560.57 mt (metric ton), 1.1389 Hz, and 1.7%, respectively. For the isolation system, the mass of the base $m i$ was 131.58 mt, the effective period was 2.93 s, and the allowable base displacement was 0.3 m. Based on these parameters, the mass ratio $R m$ was 0.87, and the frequency ratio $R f$ was 3.337, with an isolation frequency $f i$ equal to 0.3413 Hz.

4.2. Optimal frictional coefficient

In this paper, the ground accelerations of two design earthquakes, Hualien and El Centro, were both scaled to a peak ground acceleration (PGA) equal to 3.14 m/s², which corresponds to the maximum considered earthquake with a return period of 2500 years in Taipei. The ground accelerations of the Hualien earthquake were taken at the NTU campus. The root mean square of the ground accelerations for the Hualien and El Centro earthquakes were 0.356 m/s² and 0.423 m/s², respectively (Figure 3). From the frequency analysis, the dominant frequency of the Hualien earthquake was in the range of 0.68 to 1.52 Hz. Similarly, the dominant frequency of the El Centro earthquake was in the range of 1.16 to 2.21 Hz (Figure 4). As a result, the Hualien earthquake may cause larger responses for the isolation system due to its low-frequency content which is influenced by the soft soil in the Taipei basin. The optimal design frictional coefficients μ which minimizes the performance index J for the frictional isolation system were computed by substituting $R m = 0.87$ , $R f = 3.337$ , $ζ s = 0.017$ , and $f i = 0.3413$ into the proposed optimization procedures under the two specific design earthquakes.

Figure 3.

Ground accelerations. (a) 2002 Hualien earthquake (PGA = 0.32 g). (b) 1940 El Centro earthquake (PGA = 0.32 g).

Figure 4.

Spectra of ground accelerations. (a) 2002 Hualien earthquake. (b) 1940 El Centro earthquake.

Furthermore, to confirm the reliability of the optimization procedures and to evaluate the effectiveness of the structural isolation system, an acceleration ratio R_A, which is a normalization of the performance index J, was defined as:

R A = \sqrt{J w / iso / J w / o iso}

(18)

where

J w / iso

and

J w / o iso

are the performance indices J with and without the frictional isolation system, respectively. Following the optimization procedures, the optimal frictional coefficients under the Hualien and El Centro earthquakes were

μ opt = 0.0223

and

μ opt = 0.0175

, respectively. The corresponding optimal acceleration ratios were

R_{A}^{opt} = 0.1634

and

R_{A}^{opt} = 0.0837

, respectively. The spectral accelerations of Hualien and El Centro earthquakes at the isolation frequency

f i

are

S a (f i) = 1.66

m/s² and

S a (f i) = 1.114

m/s², respectively. It is found that under different earthquakes, the larger the spectral acceleration, the higher the optimal frictional coefficient.

Figure 5 shows the acceleration ratio R_A versus the design parameter of the frictional isolation system. From this figure, the design parameter of the frictional isolation system with the lowest acceleration ratio R_A has the same value as the optimal design parameter from the optimization procedure under each one of the two earthquakes investigated. Therefore, the proposed method for the optimal design parameter of the frictional isolation system is feasible.

Figure 5.

Variation of acceleration ratios with frictional coefficient. (a) 2002 Hualien earthquake. (b) 1940 El Centro earthquake.

Moreover, the optimal friction coefficient of isolation systems is proportional to the PGA of an earthquake (Jangid 2000). In Jangid (2005), the optimal frictional coefficient ranged from 0.05 to 0.15 under near-fault earthquakes with intensities from 0.36 to 0.87 g. However, in Jangid (2000), the optimal frictional coefficients ranged from 0.01 to 0.03 under excitations with smaller intensities. In this study, the intensity of the far-field earthquakes is 0.32 g and the optimal friction coefficients are about 0.02 which fall into the range proposed by Jangid (2000).

4.3. Effectiveness of structural isolation system with optimal frictional coefficients

Considering the Hualien earthquake for the structure without an isolation system, the peak structural absolute acceleration, root-mean-square structural absolute acceleration, peak structural displacement, and root-mean-square structural displacement were 9.488 m/s², 1.435 m/s², 0.1851 m, and 0.0279 m, respectively (Table 1). For the isolated structure with an optimal frictional coefficient equal to 0.0223, the peak structural absolute acceleration, root-mean-square structural absolute acceleration, peak structural relative displacement $(x s - x i) peak$ , and root mean square structural relative displacement $(x s - x i) rms$ were reduced to 1.528 m/s², 0.234 m/s², 0.0298 m, and 0.0046 m, respectively. These are 16.1%, 16.9%, 16.1%, and 16.5% of the corresponding figures without isolation, respectively (Table 1, Figure 6(a)). The time history of the base displacement is shown in Figure 7(a). The maximum base displacement and its root mean square value were 0.2651 m and 0.0365 m. Hence, the base displacement was not over the allowable isolation displacement.

Figure 6.

Time histories of structural absolute accelerations. (a) 2002 Hualien earthquake. (b) 1940 El Centro earthquake.

Figure 7.

Time histories of base displacements. (a) 2002 Hualien earthquake. (b) 1940 El Centro earthquake.

Table 1.

Effectiveness of optimal structural isolation system.

2002 Hualien		1940 El Centro
w/o isolation	w/isolation	w/o isolation	w/isolation
Peak structural absolute acc.(m/s²)	9.488	1.528	8.00	0.85
RMS structural absolute acc. (m/s²)	1.435	0.234	2.337	0.195
Peak structural relative disp. (m)	0.1851	0.0298	0.156	0.0165
RMS structural relative disp. (m)	0.0279	0.0046	0.0456	0.0038
Peak base disp. (m)	—	0.2651	—	0.1275
RMS base disp. (m)	—	0.0365	—	0.0271
Acceleration ratio R_A	—	0.1634	—	0.0837

Considering the El Centro earthquake for the structure without an isolation system, the peak structural absolute acceleration, root-mean-square structural absolute acceleration, peak structural displacement, and root-mean-square structural displacement were 8.00 m/sec², 2.337 m/sec², 0.156 m, and 0.0456 m, respectively (Table 1). Post isolation with the optimal frictional coefficient equal to 0.0175, the peak structural absolute acceleration, root-mean-square structural absolute acceleration, peak structural relative displacement $(x s - x i) peak$ , and root-mean-square structural relative displacement $(x s - x i) rms$ of the building were reduced to 0.85 m/s², 0.195 m/s², 0.0165 m, and 0.0038 m. These are 10.6%, 8.3%, 10.6%, and 8.3% of the corresponding figures for the case without isolation, respectively (Table 1, Figure 6(b)). The time history of the base displacement is shown in Figure 7(b). The maximum base displacement and its root mean square values were 0.1275 m and 0.0271 m, respectively. Therefore, the maximum base displacement was also within the allowable base displacement. Comparing the maximum base displacements under the Hualien and El Centro earthquakes, the former was twice that of the latter. This may be caused by the low-frequency content of the Hualien earthquake. Thus, the effectiveness of the isolation system designed using the proposed optimization method under the design earthquakes has been verified.

4.4. Sensitivity analyses

This section presents sensitivity analyses to evaluate the influence of system parameters on the optimal frictional coefficient. Four parameters, the mass ratio, the structural damping ratio, the frequency ratio, and the isolation frequency, were analyzed. Firstly, for the Hualien earthquake, in the case when the mass ratio varies from 0.5 to 0.95 and the other parameters are fixed (i.e. $R f = 3.337$ , $ζ s = 1.7 %$ , $f i = 0.3413 Hz$ ), the relationship of the optimal frictional coefficient, acceleration ratio, and the maximum base displacement versus the mass ratio is shown in Figure 8. The optimal frictional coefficient varies from 0.01488 to 0.02235 for this specific range of mass ratios. Similarly for this case, the acceleration ratio and the maximum base displacement varied from 0.16 to 0.2 (Figure 8(a)) and 0.26 m to 0.284 m (Figure 8(b)), respectively. When the seismic excitation was changed from the Hualien to the El Centro earthquake, the optimal friction coefficient varied from 0.01434 to 0.01915 for the same range of mass ratios. The acceleration ratio and the maximum base displacement varied from 0.0805 to 0.106 (Figure 9(a)) and 0.113 m to 0.147 m (Figure 9(b)), respectively. From Figures 8 and 9, it is obvious that the optimal frictional coefficient was not sensitive to the mass ratio $R m$ , and through this range of mass ratios, none of the corresponding maximum base displacements exceeded the allowable base displacement.

Figure 8.

Sensitivity of the mass ratio under 2002 Hualien earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Figure 9.

Sensitivity of mass ratio under 1940 El Centro earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Following the investigation into the sensitivity of the mass ratio, the influence of the structural damping ratio $ζ s$ with values ranging from 0 to 5 % was studied. Considering that the Hualien earthquake produced a range of optimal frictional coefficients from 0.02168 to 0.02355, the corresponding acceleration ratio R_A and the maximum base displacement varied from 0.04 to 0.25 (Figure 10(a)) and 0.26 m to 0.268 m (Figure 10(b)), respectively. When the earthquake under consideration was changed to the El Centro earthquake, the optimal frictional coefficients obtained varied from 0.01747 to 0.01785, and the acceleration ratio and the maximum base displacement varied from 0.023 to 0.161 (Figure 11(a)) and 0.12 m to 0.131 m (Figure 11(b)), respectively. Based on these simulation results under the two specific earthquakes, the optimal frictional coefficient was found to be insensitive to the structural damping ratio $ζ s$ . From the definition of the acceleration ratio R_A, a smaller structural damping ratio $ζ s$ causes a larger denominator ( $J w / o iso$ ), meaning that the acceleration ratio may be smaller. Therefore, the acceleration ratio R_A was sensitive to the structural damping ratio $ζ s$ . Similarly, through the variation of $ζ s$ , no corresponding maximum base displacement exceeded the allowable base displacement.

Figure 10.

Sensitivity of structural damping ratio under 2002 Hualien earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Figure 11.

Sensitivity of structural damping ratio under 1940 El Centro earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

The sensitivity of a variation in the frequency ratio $R f$ for values ranging from 2 to 10 was also considered. When the system was subjected to the Hualien earthquake, the optimal frictional coefficient $μ opt$ , acceleration ratio, and maximum base displacement varied from 0.0155 to 0.0227, 0.067 to 0.371 (Figure 12(a)), and 0.257 m to 0.288 m (Figure 12(b)), respectively. Similarly, the optimal frictional coefficient, acceleration ratio, and maximum base displacement varied from 0.01563 to 0.01895, 0.068 to 0.279, and 0.121 m to 0.141 m under the El Centro earthquake, respectively (Figure 13(a) and (b)). Therefore, although the acceleration ratio may be more sensitive to the frequency ratio, the optimal frictional coefficient and the maximum base displacement remain insensitive to it.

Figure 12.

Sensitivity of frequency ratio under 2002 Hualien earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Figure 13.

Sensitivity of frequency ratio under 1940 El Centro earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Finally, the influence of the isolation frequency $f i$ over the range of 0.2 to 0.5 Hz was also simulated. For the Hualien earthquake, the optimal frictional coefficient obtained varied from 0.00586 to 0.02854, and the acceleration ratio and maximum base displacement varied from 0.053 to 0.251 (Figure 14(a)) and 0.166 m to 0.335 m (Figure 14(b)), respectively. For the El Centro earthquake, the obtained optimal frictional coefficient varied from 0.004175 to 0.02398, and the acceleration ratio and maximum base displacement varied from 0.07 to 0.168 (Figure 15(a)) and 0.01 m to 0.186 m (Figure 15(b)), respectively. It was found that as a result of the increasing flexibility, a lower isolation frequency $f i$ resulted in a higher base displacement. The results show that the optimal frictional coefficient was insensitive to the mass ratio, the structural damping ratio, and the frequency ratio, but was sensitive to the isolation frequency.

Figure 14.

Sensitivity of isolation frequency under 2002 Hualien earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

Figure 15.

Sensitivity of isolation frequency under 1940 El Centro earthquake. (a) Optimal acceleration ratio. (b) Maxiumum base displacement.

5. Conclusions

An optimization method for the optimal design of a frictional isolation system is proposed in this paper. Only one-dimensional isolation system and ground motion are considered. The frictional coefficient was optimized by minimizing the sum of squares of the structural absolute accelerations. Since the conditions of the optimization are highly nonlinear, a numerical method was proposed to systematically solve for the optimal frictional coefficient, thereby guaranteeing the convergence of the solution. Based on the presented simulation results, several conclusions are summarized as follows:

The feasibility of the proposed optimization method was verified by the implementation of the frictional isolation system for a structure assumed to be located at the National Taiwan University under the influence of two specific design earthquakes (the Hualien and El Centro earthquakes). The isolation system with the optimal frictional coefficient, which was obtained using the proposed optimization method, performed well under the two earthquakes in terms of both the isolation displacement and the structural acceleration.

The proposed optimization method is easily implemented by substituting the three system parameters, the mass ratio, the frequency ratio, and the structural damping ratio, with one design earthquake. The accuracy of the proposed optimization method was also proven diagrammatically.

The sensitivity analysis shows that the optimal frictional coefficient is sensitive to the isolation frequency but not the mass ratio, the frequency ratio, or the structural damping ratio.

As a preliminary design, after the design parameter is suggested from the optimization method, if the isolation displacement of the system with optimal frictional coefficient exceeds the design one, the frictional coefficient may increase until the design isolation displacement is satisfied but the structural acceleration is no longer optimal.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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