- Email: [email protected]
A generalized analytical model for sloped rolling-type seismic isolators
Engineering Structures 138 (2017) 434–446
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
A generalized analytical model for sloped rolling-type seismic isolators Shiang-Jung Wang, Chung-Han Yu, Wang-Chuen Lin, Jenn-Shin Hwang, Kuo-Chun Chang ⇑ Department of Civil Engineering, National Taiwan University, Taiwan
a r t i c l e
i n f o
Article history: Received 29 December 2015 Revised 13 December 2016 Accepted 15 December 2016
Keywords: Sloped rolling-type seismic isolator Generalized analytical model Equipment Seismic performance Vertical excitation
a b s t r a c t The exact and simplified generalized analytical models for the sloped rolling-type seismic isolator in which two V-shaped surfaces in contact with cylindrical rollers are designed with arbitrary sloping angles are proposed. Numerical predictions are compared with horizontal seismic simulation test results to demonstrate the accuracy and practicability of the proposed simplified model. Furthermore, the influences arising from two simplification assumptions on the horizontal acceleration prediction are numerically examined. The results indicate that the effect of vertical acceleration excitation plays a more crucial role in prediction accuracy and conservative design compared to that of higher order terms of sloping angles. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Rolling-based metallic seismic isolators [1] in which either a ball rolls on flat [2], concave [3–5], or conical [6] surfaces or a rod rolls on flat [7,8], curved [9–11], or sloped [12–15] surfaces (or rails) have been numerically and experimentally demonstrated for their excellent seismic isolation performance. Since the rolling friction force and the restoring force contributed by additional springs or due to gravity are very limited and much smaller than the input seismic force, the rolling motion can be activated immediately once an earthquake occurs, and the horizontal acceleration transmitted to the protected object can be significantly reduced during an earthquake. The seismic isolators, except those designed with flat surfaces in contact with balls or rods, have an inherent gravity-based self-centering capability after excitation. Among various types of rolling-based metallic seismic isolators, the sloped rolling-type seismic isolators discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies attracted sustained attention. In addition to possessing the advantages aforementioned, the design of constant sloping surfaces in contact with cylindrical rollers can ensure that the horizontal acceleration transmitted to the protected object remains essentially constant regardless of any intensity and frequency content of excitation. Thus, the seismic isolators do not have a fixed vibration natural period, and can offer maximum horizontal decoupling between the protected object and input excitation. It is noteworthy that the zero post-elastic stiffness performance, i.e. the constant transmitted horizontal acceleration ⇑ Corresponding author. E-mail address: [email protected] (K.-C. Chang). http://dx.doi.org/10.1016/j.engstruct.2016.12.027 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
performance, is the most attractive feature. This feature can make the seismic isolators easily meet the rigorous performance-based design requirements for structural and non-structural systems if the maximum transmitted acceleration response is selected as the seismic performance criterion. Furthermore, to have a better displacement control and more effectively stop rolling motion respectively during and after excitation, the built-in damping mechanism provided by additional sliding friction was employed in these two studies. The normal force for the sliding friction was provided by screws in Lee et al.’s study, while that was provided by springs in Wang et al.’s study. As shown in the schematic drawing of Fig. 1, the sloped rollingtype seismic isolators discussed in Lee et al.’s and Wang et al.’s studies were basically composed of three bearing plates (denoted as upper, intermediate, and lower bearing plates hereafter) and cylindrical rollers. In Lee et al.’s study, the upper and lower bearing plates were designed with flat surfaces in contact with the rollers, while the intermediate bearing plate was designed with V-shaped surfaces. In Wang et al.’s study, the three bearing plates were designed with V-shaped surfaces in contact with the rollers, and the sloping angles of the V-shaped surfaces were identical. Without considering the effects arising from higher order terms of sloping angles and vertical excitation, the dynamic behavior of a cylinder moving between a flat surface and a V-shaped surface along each principle horizontal direction was focused in Lee et al.’s study, while that between two V-shaped surfaces designed with the same sloping angle was concentrated in Wang et al.’s study. Because of different designs of rolling surfaces, the compatibility conditions in these two studies were diverse. Hence, the simplified equations of motion derived in these two studies were
435
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Upper bearing plate (Flat surface)
Upper bearing plate (V-shaped sloping surface) Cylindrical roller
Lower bearing plate (V-shaped sloping surface)
Intermediate bearing plate (V-shaped sloping surface)
Lower bearing plate (Flat surface)
Fig. 1. Schematic drawings of sloped rolling-type seismic isolators discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies.
essentially different and cannot be exchanged. In addition, other refinements such as the multi-roller and pounding prevention mechanisms were developed in Wang et al.’s study. The multiroller design can have a better stability performance compared with the conventional single-roller design. The design of an arc rolling range with a fixed curvature radius at the intersection of two inclines of V-shaped surfaces of bearing plates can effectively prevent pounding between rollers and inclined surfaces when rollers are passing through here. In Wang et al.’s study, a simplified analytical model was provided to fully describe the hysteretic behavior of the sloped rolling-type seismic isolator based on a combination of multi-linear elastic and plastic models. It is also mentioned here that the application purposes of Lee et al.’s and Wang et al.’s studies were bridge structures and equipment or facilities, respectively. Therefore, this study first aims to derive generalized equations of motion for the sloped rolling-type seismic isolator in which the cylindrical rollers move between two V-shaped surfaces designed with arbitrary sloping angles. The design is different from and more generalized than Lee et al.’s and Wang et al.’s. The proposed simplified generalized analytical model, of course, is capable of describing the dynamic behavior of the two specific designs respectively discussed in Lee et al.’s and Wang et al.’s studies. The accuracy and practicability of the simplified model are verified by comparing with horizontal shaking table test results. Finally, by further comparing the numerical results predicted by the exact generalized analytical model with those by the simplified one, the influences caused by two simplification assumptions—neglect of higher order terms of sloping angles and vertical excitation— on the transmitted horizontal acceleration responses of the seismic isolator are thoroughly examined. It has never been discussed in detail before. 2. Derivation of generalized equations of motion A simplified model, a cylindrical roller sandwiched between two V-shaped surfaces designed with different sloping angles h1 and h2 , to represent the dynamic behavior of the sloped rollingtype seismic isolator in one principle horizontal direction as well as in the vertical direction is illustrated in Fig. 2(a), in which M, m1 , and m2 are the seismic reactive masses of the protected object, superior bearing plate, and roller, respectively; r is the radius of the roller; and h1 and h2 are the sloping angles of the superior and inferior bearing plates, respectively. The following basic assumptions are made first for deriving the generalized equations of motion of the sloped rolling-type seismic isolator: (1) the roller and bearing plates are ideally contact and in pure rolling motion without any undesired sliding and overturning motions; (2) the rolling motions of the rollers between two bearing plates along two principle horizontal directions are identical; (3) an appropriate rigid-plastic hysteretic model (i.e. the Coulomb friction law) is employed to rep-
resent the force-displacement relationship of built-in damping behavior; (4) the inferior bearing plate is fixed to a rigid base; and (5) the payload of a to-be-protected object is applied on the superior bearing plate. Two opposite rolling motion directions are identified by sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1 and sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1 corresponding to the rightward (sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1) and leftward (sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1) movements of the superior bearing plate (or the roller) relative to the inferior bearing plate. The free body diagrams for one of the four situations, sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1 and xg ð€zg Þ is sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1, are shown in Fig. 2(b), in which € the horizontal (vertical) acceleration excitation; x1 ðz1 Þ, x_ 1 ðz_ 1 Þ, and € x1 ð€z1 Þ are the horizontal (vertical) displacement, velocity, and acceleration responses of the protected object and superior bearing plate relative to the origin O, respectively; x2 ðz2 Þ, x_ 2 ðz_ 2 Þ, and € x2 ð€z2 Þ are the horizontal (vertical) displacement, velocity, and acceleration responses of the roller relative to the origin O, respectively; the positive directions of x and z are correspondingly defined to be rightward and upward in the figure; g is the acceleration of gravity; I is the moment of inertia of the roller; a is the angular acceleration of the roller (the positive rotation is defined to be clockwise in the figure); f 1 ðf 2 Þ and N 1 ðN 2 Þ are the rolling friction force and normal force acting between the superior bearing plate and roller (between the roller and inferior bearing plate), respectively; and F D is the built-in friction damping force acting parallel to the slope of the bearing plates. By considering the free body diagrams of the superior bearing plate and protected object, two dynamic force equilibrium equations are obtained as
F D cos h1 sgnðx_ 1 Þ N1 sin h1 sgnðx1 Þ f 1 cos h1 sgnðx_ 1 Þ ¼ ðM þ m1 Þð€x1 þ €xg Þ
ð1Þ
F D sin h1 sgnðx1 Þsgnðx_ 1 Þ þ N1 cosh1 Mg f 1 sinh1 sgnðx1 Þsgnðx_ 1 Þ m1 g ¼ ðM þ m1 Þð€z1 þ €zg Þ ð2Þ
Moreover, by considering the free body diagram of the roller, two dynamic force equilibrium and one dynamic moment equations are obtained as F D cos h1 sgnðx_ 1 Þ F D cos h2 sgnðx_ 1 Þ þ N1 sin h1 sgnðx1 Þ N2 sin h2 sgnðx1 Þ þ f 1 cos h1 sgnðx_ 1 Þ f 2 cos h2 sgnðx_ 1 Þ ¼ m2 ð€x2 þ €xg Þ ð3Þ F D sin h1 sgnðx1 Þsgnðx_ 1 Þ F D sin h2 sgnðx1 Þsgnðx_ 1 Þ N1 cos h1 þ N2 cos h2 þ f 1 sin h1 sgnðx1 Þsgnðx_ 1 Þ f 2 sin h2 sgnðx1 Þsgnðx_ 1 Þ m2 g ¼ m2 ð€z2 þ €zg Þ ð4Þ
f 1 rsgnðx_ 1 Þ þ f 2 rsgnðx_ 1 Þ ¼ Ia where I ¼ ð1=2Þm2 r . 2
ð5Þ
436
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Protected object Superior bearing plate Cylindrical roller Inferior bearing plate
Ő
Fig. 2. Simplified model for sloped rolling-type seismic isolators in this study.
From the relative motion between the superior bearing plate and roller, four compatibility conditions can be obtained as
€x2 ¼ r a cos h2
ð6Þ
€z2 ¼ r a sin h2 sgnðx1 Þ
ð7Þ
€x1 ¼ €x2 þ ra cos h1 ¼ ra cos h1 þ r a cos h2
ð8Þ
€z1 ¼ €z2 þ r a sin h1 sgnðx1 Þ ¼ r a sin h1 sgnðx1 Þ þ r a sin h2 sgnðx1 Þ
ð9Þ
A total of nine variables, a, € x1 , € x2 , €z1 , €z2 , N 1 , N 2 , f 1 , and f 2 , can be solved by using Eqs. (1)–(9). With reasonable neglect of the term of m2 =ðM þ m1 Þ due to the fact that m2 is in general much smaller than ðM þ m1 Þ, the exact mathematical forms of the solutions of the nine variables are given in Eqs. (10)–(18)
1 f2F D sgnðx_ 1 Þ 2rðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
a¼
ð10Þ ðcosh1 þ cosh2 Þ €x1 ¼ f2F D sgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
sinh2 f2F D sgnðx1 Þsgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cosh2 Þsgnðx1 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þ
€z2 ¼
ð14Þ
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ 2 ðg þ €zg Þðcos h1 þ cos h2 Þ
ð15Þ
h1 h2 N2 ¼ F D tan sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ 2 ðg þ €zg Þðcos h1 þ cos h2 Þ
ð16Þ
N1 ¼ F D tan
f1 ¼
ð11Þ cos h2 f2F D sgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
€x2 ¼
ð12Þ ðsinh1 þ sinh2 Þ f2F D sgnðx1 Þsgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcosh1 þ cosh2 Þsgnðx1 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þ
€z1 ¼
ð13Þ
f2 ¼
2ðM þ m1 Þsgnðx_ 1 Þ h1 h2 h1 h2 cos2 sin 1 þ cosðh1 h2 Þ 2 2 h1 þ h2 h1 þ h2 ðg þ €zg Þ cos sgnðx1 Þ €xg sin 2 2
ð17Þ
2ðM þ m1 Þsgnðx_ 1 Þ h1 h2 h1 h2 cos2 sin 1 þ cosðh1 h2 Þ 2 2 h1 þ h2 h1 þ h2 ðg þ €zg Þ cos sgnðx1 Þ €xg sin 2 2
ð18Þ
Hereafter, Eqs. (10)–(18) are generally called the exact generalized solutions in which Eqs. (10)–(14) are the exact generalized equations of motion. It can be seen that the insightful information might not be easily observed from the exact generalized solutions. Therefore, the following two assumptions for further simplicity
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
are taken into consideration: (1) Based on the assumption that the sloping angles h1 and h2 are small enough, the higher order terms of h1 and h2 are negligible, i.e. cos2 h1 1, cos2 h2 1, cos h1 cos h2 1, 2
2
sin h1 0, sin h2 0, and sin h1 sin h2 0; and (2) assuming the term of the vertical acceleration excitation €zg is neglected. Thus, the nine variables can be approximated by
a¼
€x1 ¼
F D sgnðx_ 1 Þ 2rðM þ m1 Þ 1 €xg ðcos h1 þ cos h2 Þ þ gðsin h1 þ sin h2 Þsgnðx1 Þ 4r
ð19Þ
Note that during two or more consecutive time steps in numerical integration, it is possible to have no relative motion between the superior and inferior bearing plates of the simplified model shown in Fig. 2. This phenomenon is defined as the relative static state hereafter. To simulate this phenomenon in the analysis program, the critical characteristic strengths due to the built-in friction damping force, F D;C , are defined and calculated by setting € x1 in Eqs. (11) and (20) equal to zero, which are respectively given by
1
F D;C ¼
ðM þ m1 Þ €xg ðcosh1 þ cosh2 Þ þ sgnðx1 Þðg þ €zg Þðsinh1 þ sinh2 Þ
2
ð28Þ
ðcos h1 þ cosh2 Þ ½2F D sgnðx_ 1 Þ þ ðM þ m1 Þgðsin h1 þ sin h2 Þsgnðx1 Þ 4ðM þ m1 Þ x€g
F D;C
1
1 þ cos h1 cos h2 ¼
ðM þ m1 Þ 2€xg þ sgnðx1 Þgðsin h1 þ sin h2 Þ
2 cos h1 þ cos h2 ð29Þ
ð20Þ
€x2 ¼
€z1 ¼
€z2 ¼
cos h2 f2F D sgnðx_ 1 Þ 4ðM þ m1 Þ ðM þ m1 Þ gðsin h1 þ sin h2 Þsgnðx1 Þ þ 2€xg g
ð21Þ
ðsin h1 þ sin h2 Þsgnðx1 Þ 2F D sgnðx_ 1 Þ þ ðM þ m1 Þ€xg ðcos h1 þ cos h2 Þ 4ðM þ m1 Þ ð22Þ
sin h2 sgnðx1 Þ 2F D sgnðx_ 1 Þ þ ðM þ m1 Þ€xg ðcos h1 þ cos h2 Þ 4ðM þ m1 Þ ð23Þ
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ ðg þ €zg Þðcos h1 þ cos h2 Þ 2 ð24Þ
N 1 ¼ F D csc
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ ðg þ €zg Þðcosh1 þ cos h2 Þ 2 ð25Þ
N2 ¼ F D csc
f1 ¼
f2 ¼
437
ðM þ m1 Þsgnðx_ 1 Þ gðsinh1 sinh2 Þsgnðx1 Þ þ €xg ðcos h1 cosh2 Þ 2 ð26Þ ðM þ m1 Þsgnðx_ 1 Þ gðsinh1 sinh2 Þsgnðx1 Þ þ €xg ðcos h1 cos h2 Þ 2 ð27Þ
Hereafter, Eqs. (19)–(27) are generally called the simplified generalized solutions in which Eqs. (19)–(23) are the simplified generalized equations of motion. By using Eqs. (11) and (20), the horizontal dynamic behavior of the sloped rolling-type seismic isolator in which two V-shaped surfaces in contact with cylindrical rollers are designed with arbitrary sloping angles can be analyzed precisely and approximately, respectively. 3. Analysis program The Visual Basic of Application (VBA), a widely used programming language in engineering practice, is adopted to develop the analysis program for the sloped rolling-type seismic isolator in this study. The Newmark-b method with constant average acceleration is used for solving the exact and simplified generalized equations of motion respectively given in Eqs. (11) and (20) at each time step.
It can be seen that F D;C is dependent on € xg and €zg (only € xg for the simplified generalized equation of motion). If F D;C is calculated to be smaller than the design built-in friction damping force F D at a specific time step (e.g. at ti ), the relative static state occurs. Under x1 and x_ 1 should be modified this circumstance, at the time step t i , € to be zero, and x1 should remain the same as that at the previous time step t i1 . In order to have a better acceleration control performance for the sloped rolling-type seismic isolator, Wang et al.’s study [15] suggested employing a pounding prevention mechanism—an arc rolling range with a fixed curvature radius which is larger than the roller radius. The theoretical equations of motion when balls or rods roll on concave or curved surfaces have been thoroughly studied in many past researches [3–5,9–11]. For the sloped rolling-type seismic isolator, its equations of motion when rollers move in the arc range designed with a fixed curvature radius R have also been studied by Wang et al. [15]. R, in detail, is the curvature radius designed for the round surface between two inclines of V-shaped surfaces of bearing plates. Methodologically, in addition to the theoretical equations of motion, the dynamic response when rollers move in the arc range can be numerically analyzed based on a reasonable assumption of the arc range composed of many infinitesimal segments designed with a fixed and continuously varied sloping angle. In this analysis program, the derived generalized equations of motion when rollers move in the sloped range, as given in Eqs. (11) and (20), based on the assumption above are adopted to numerically analyze the dynamic response of the sloped rolling-type seismic isolator when rollers move in the arc range. 4. Verification of simplified generalized analytical model The following will theoretically and experimentally demonstrate that the simplified generalized solutions given in Eqs. (19)–(27) can represent the dynamic behavior of the two different sloped rolling-type seismic isolators respectively discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies, as shown in Fig. 3. 4.1. Theoretical verification When the sloping angles of the superior and inferior bearing plates of the simplified model shown in Fig. 2(a) (i.e. h1 and h2 ) are respectively equal to zero and h, as illustrated in Fig. 3(a), the same solutions as those provided in Lee et al.’s study can be obtained. For example, the transmitted acceleration responses along the horizontal and vertical directions are
€x1 þ €xg ¼
g sin h FD sgnðx1 Þ sgnðx_ 1 Þ 2 ðM þ m1 Þ
ð30Þ
438
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
M m1
(a) Lee et al.’s study
m2
r
(b) Wang et al.’s study
Fig. 3. Simplified models discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies.
Fig. 4. Test installation photo.
Table 1 Acceleration input program for experimental study. Acceleration input
Input earthquake information or response spectrum condition
Excitation direction
Test input PA (g)
Test input PA 100% Original input PA
El Centro Kobe
IMPVALL/I-ELC180, Imperial Valley, U.S., 1940/05/19 KOBE/KJM000, Kobe, Japan, 1995/01/16
Unilateral Unilateral
X X
0.31 0.41
100% 50%
Artificial acceleration history
AC156-1
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8.75 m in elevation) of a 7-story building (24 m in height) at Taipei City RRS specified in IEEE Std 693TM-2005 for seismic qualification test
Unilateral
X
0.50
100%
Unilateral
X
0.50
100%
€z1 þ €zg ¼ €zg
IEEE
€xg sin h F D sin h sgnðx_ 1 Þ 2 2ðM þ m1 Þ
ð31Þ
When the sloping angles of both the superior and inferior bearing plates of the simplified model shown in Fig. 2(a) (i.e. h1 and h2 ) are equal to h, as illustrated in Fig. 3(b), the same solutions as those provided in Wang et al.’s study can be obtained. For example, the transmitted acceleration responses along the horizontal and vertical directions are
€x1 þ €xg ¼
g sin 2h F D cos h sgnðx1 Þ sgnðx_ 1 Þ 2 ðM þ m1 Þ €xg sin 2h F D sin h sgnðx_ 1 Þ 2 ðM þ m1 Þ
ð32Þ
Spectral Acceleratrion (g)
Recorded earthquake history
4
Damping Ratio = 5%
ELCentro Kobe AC156-1 IEEE
3 2 1 0
0
1
2
3
4
5
Period (sec)
ð33Þ
Fig. 5. Acceleration response spectra of acceleration inputs (PA = 1 g) for experimental study.
The tested sloped rolling-type seismic isolator consists of a set of upper, intermediate, and lower bearing plates together with two pairs of mutually orthogonal cylindrical rollers (between the upper and intermediate bearing plates as well as between the intermediate and lower bearing plates). The surfaces of the upper
(or lower) and intermediate bearing plates in contact with the rollers are flat and dual V-shaped with a sloping angle of 6.25°, respectively. The dimension of the seismic isolator without horizontal displacement is 600 mm (in length) 600 mm (in width) 123.9 mm (in height). The seismic reactive masses of the upper, intermediate, and lower bearing plates are 19.44 N-s2/m, 29.44 N-s2/m, and 19.44 N-s2/m, respectively. The sectional radius
€z1 þ €zg ¼ €zg
4.2. Experimental verification
439
0.2
Unilateral-50%-Kobe
0.1 0.1
0 -0.1 -0.2 10
20
30
40
Time (sec)
(a) Acceleration response history Displacement (mm)
0.15
R2 = 0.84
100
Acceleration (g)
Acceleration (g)
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
R2 = 0.96
50
0
0.05
0
-0.05
-0.1
Numerical Prediction Test Result
-50 -100 10
20
30
-0.15 -100
40
-50
0
50
Time (sec)
Displacement (mm)
(b) Displacement response history
(c) Hysteresis loop
100
0.2
Unilateral-100%-AC156-1
0.1 0.1
0 -0.1 -0.2 15
25
35
45
Time (sec)
(a) Acceleration response history Displacement (mm)
0.15
R2 = 0.83
100
R2 = 0.95
50
0
Acceleration (g)
Acceleration (g)
Fig. 6. Comparisons of experimental results and numerical predictions under unilateral-50%-Kobe.
0.05
0
-0.05
-0.1
Numerical Prediction Test Result
-50 -100 15
25
35
45
-0.15 -100
-50
0
50
100
Displacement (mm)
Time (sec)
(b) Displacement response history
(c) Hysteresis loop
Fig. 7. Comparisons of experimental results and numerical predictions under unilateral-100%-AC156-1.
Table 2 Maximum transmitted horizontal acceleration responses of twenty-five design cases predicted by using Eq. (20) (unit:g). Design case (h1 h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(2°, 2°) 0.063
(2°, 3°) 0.072
(2°, 4°) 0.081
(2°, 5°) 0.090
(2°, 6°) 0.098
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(3°, 2°) 0.072
(3°, 3°) 0.081
(3°, 4°) 0.090
(3°, 5°) 0.098
(3°, 6°) 0.107
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(4°, 2°) 0.081
(4°, 3°) 0.090
(4°, 4°) 0.098
(4°, 5°) 0.107
(4°, 6°) 0.115
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(5°, 2°) 0.090
(5°, 3°) 0.098
(5°, 4°) 0.107
(5°, 5°) 0.115
(5°, 6°) 0.124
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(6°, 2°) 0.098
(6°, 3°) 0.107
(6°, 4°) 0.115
(6°, 5°) 0.124
(6°, 6°) 0.133
(r), longitudinal length, and seismic reactive mass of each cylindrical roller are 17.5 mm, 600 mm, and 4.58 N-s2/m, respectively. The built-in friction damping mechanism is composed of a rubber pad
with a thickness of 2 mm vulcanized and attached to the surfaces of the upper, intermediate, and lower bearing plates sliding against the stainless steel surfaces of the side plates. The required normal
440
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Table 3 Acceleration input program for numerical study. Acceleration input El Centro
IMPVALL/I-ELC180, IMPVALL/I-ELC-UP, Imperial Valley, U.S., 1940/05/19
Unilateral Biaxial
Kobe
KOBE/KJM000, KOBE/KJM-UP, Kobe, Japan, 1995/01/16
Unilateral Biaxial
ChiChi
CHICHI/CHY028-N, CHICHI/CHY028-V, Chi-Chi, Taiwan, 1999/09/21
Unilateral Biaxial
AC156-1
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8.75 m in elevation) of a 7-story building (24 m in height) at Taipei City
Unilateral Biaxial
AC156-2
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8 m in elevation) of a 3-story building (12 m in height) at Nantou County
Unilateral Biaxial
IEEE
RRS specified in IEEE Std 693TM-2005 for high performance level
Unilateral Biaxial
Spectral Acceleratrion (g)
Artificial acceleration history
Excitation direction
Spectral Acceleratrion (g)
Recorded earthquake history
Input earthquake information or response spectrum condition
4
Kobe I-ELC ChiChi AC156-1 AC156-2 IEEE
Damping Ratio = 5% 3
2
1
0
0
1
2
3
4
Period (sec)
(a) Horizontal (X) direction
5
Different input PA scales (g)
X X Z X X Z X X Z X X Z X X Z X X Z
25%
50%
75%
100%
0.08 0.08 0.05 0.21 0.21 0.09 0.19 0.19 0.09 0.13 0.13 0.06 0.25 0.25 0.13 0.25 0.25 0.20
0.16 0.16 0.11 0.41 0.41 0.17 0.38 0.38 0.17 0.25 0.25 0.13 0.50 0.50 0.25 0.50 0.50 0.40
0.23 0.23 0.16 0.62 0.62 0.26 0.57 0.57 0.26 0.38 0.38 0.19 0.75 0.75 0.38 0.75 0.75 0.60
0.31 0.31 0.21 0.82 0.82 0.34 0.76 0.76 0.34 0.5 0.5 0.25 1.00 1.00 0.5 1.00 1.00 0.80
4
Kobe I-ELC ChiChi AC156-1 AC156-2 IEEE
Damping Ratio = 5% 3
2
1
0
0
1
2
3
4
5
Period (sec)
(b) Vertical (Z) direction
Fig. 8. Acceleration response spectra of acceleration inputs (PA = 1 g) for numerical study.
force for built-in sliding friction is provided by two sets of linear spring modules with a constant compression length installed in the side plates. Considering the pounding prevention mechanism suggested in Wang et al.’s study, an arc rolling range of 21.6 mm with a curvature radius (R) of 100 mm is provided at the intersection of two inclines of V-shaped surfaces. The protected equipment above the seismic isolator is simulated by lead blocks with a total seismic reactive mass of 500 N-s2/m. The test installation on the shaking table is shown in Fig. 4. As detailed in Table 1, two recorded earthquake histories obtained from the Pacific Earthquake Engineering Research (PEER) Ground Motion Database [16], denoted as ElCentro and Kobe hereafter, and two generated acceleration histories compatible with the required response spectra (RRS) specified in AC156 [17,18] and IEEE std 693TM-2005 [19], denoted as AC156-1 and IEEE hereafter, are adopted as the unilateral acceleration inputs. The 5% damped acceleration response spectra of these acceleration inputs corresponding to a peak acceleration (PA) value of 1 g are shown in Fig. 5. The comparisons of experimental results and numerical predictions, including horizontal acceleration and displacement response histories together with hysteresis loops, subjected to unilateral50%-Kobe and unilateral-100%-AC156-1 are presented in Figs. 6 and 7, respectively. The values of coefficient of determination (R2 ), which are regarded as an expedient representation to quantitatively evaluate the correlation between experimental results and numerical predictions, are also shown in the figures. It should be noted that in this study, for simplicity and practicable feasibility, the force-displacement relationship of the built-in friction damp-
ing behavior is numerically represented by a rigid-plastic hysteretic model, rather than a sophisticated model which can fully take velocity- and temperature-dependent prosperities together with other uncertainties into consideration [20]. In addition, the nominal design values, rather than the actual ones identified by further measurement and component tests, including the seismic reactive masses of each component of the tested isolator and lead blocks, the sloping angle of the bearing plates, the curvature radius of the arc surfaces, and the built-in sliding friction damping force, are used for numerical analysis in this study. Therefore, the manufacture tolerances and modeling uncertainties are included in the numerical prediction results. Even so, it can be seen from the figures that the numerical predictions by using the derived simplified generalized analytical model have an acceptable agreement with the unilateral test results.
5. Influences of simplification assumptions on response prediction accuracy By ignoring the effects arising from higher order terms of sloping angles and vertical acceleration excitation, the exact generalized solutions given in Eqs. (10)–(18) can be simplified to Eqs. (19)–(27). However, if the sloping angle or vertical acceleration excitation becomes larger, the difference between the numerical results obtained from the exact and simplified generalized analytical models might be noticeable. The following will numerically discuss the influences of the simplification assumptions on the horizontal acceleration predictions.
441
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
5.1. Design cases A simplified model composed of a cylindrical roller with a sectional radius (r) of 15 mm, a superior bearing plate with a sloping angle of h1 , and an inferior bearing plate with a sloping angle of h2 , as shown in Fig. 2(a), is employed. Twenty-five design cases of sloped rolling-type seismic isolators are numerically studied, in which a commonly used combination of h1 and h2 respectively varying from 2 to 6° with an increment of 1° is designed, i.e. (h1 , h2 ) = (2–6°, 2–6°). To control the maximum transmitted horizontal acceleration response not greater than 0.15 g according to the simplified generalized equation of motion given in Eq. (20), as well as to have an acceptable self-centering performance, the built-in friction damping forces F D are designed to be the same for the twenty-five cases and are dominated by the design case of (h1 , h2 ) = (6°, 6°), i.e. F D = 301 N. The protected object above the seismic isolators is assumed to be an important facility, and the total seismic reactive mass of the protected facility and superior bearing plate, (M þ m1 ), is designed to be 1000 N-s2/m. The protected facility is ideally regarded as a single degree-of-freedom system in this numerical study for simplicity. Considering the pounding prevention mechanism suggested in Wang et al.’s study [15], an arc rolling range of 21.6 mm is designed at the intersection of two inclines of Vshaped surfaces. Therefore, the curvature radiuses R are designed to be 618 mm, 206 mm, 155 mm, 124 mm, and 104 mm corresponding to the sloping angles varying from 2° to 6°, respectively. By using the simplified generalized equation of motion given in Eq. (20), the maximum transmitted horizontal acceleration responses of the twentyfive design cases can be calculated in a preliminary manner and are summarized in Table 2. Since the built-in friction damping forces F D designed for the twenty-five cases are the same, a tendency can be easily observed from Eq. (20) and the calculated results listed in Table 2—the larger the total sloping angles (i.e. h1 þ h2 ), the higher the transmitted horizontal acceleration response obtained. As detailed in Table 3, three recorded earthquake histories obtained from the PEER Ground Motion Database, denoted as ElCentro, Kobe, and ChiChi hereafter, together with three generated acceleration histories compatible with the required response spectra (RRS) specified in AC156 and IEEE Std 693TM-2005, denoted as AC156-1, AC156-2, and IEEE hereafter, are adopted
0.12
as the unilateral and biaxial acceleration inputs with different PA scales (i.e. 25%, 50%, 75%, and 100% of the original PA value). The 5% damped acceleration response spectra of these acceleration inputs corresponding to a PA value of 1 g are shown in Fig. 8.
5.2. Numerical comparison The hysteresis loops of different design cases (h1 and h2 respectively varying from 2° to 6° with an increment of 2°) obtained from the exact and simplified generalized equations of motion under unilateral and biaxial inputs of 25%-AC156-1, 25%-IEEE, 100%Kobe, and 100%-ChiChi are presented in Figs. 9–12, respectively. In these figures and the following discussions, BE and UE denote the numerical predictions by using the exact generalized equation of motion given in Eq. (11) under biaxial and unilateral excitation, respectively, and US denotes those by using the simplified one given in Eq. (20) under unilateral excitation. It can be found from these figures that under some acceleration inputs with smaller PA scales, e.g. all design cases under 25%-AC156-1 and some design cases under 25%-IEEE, the roller still moves within the fixed curvature range (or the arc rolling range). Under this circumstance, the acceleration differences caused by the simplification assumptions are not very significant. When the roller moves between two sloped surfaces, e.g. most design cases under 25%-IEEE together with all design cases under 100%-Kobe and 100%-ChiChi, the horizontal acceleration predictions by using the exact generalized equation of motion reflect many obvious fluctuations rather than a perfect constant. It is mainly attributed to the effect of vertical acceleration excitation, which can be observed from the more noticeable differences between the horizontal acceleration predictions under the conditions of BE and UE compared to those under the conditions of UE and US. The differences of the maximum displacement responses caused by the simplification assumptions are very tiny visually and can be rationally neglected. To separately discuss the extent of influences arising from neglect of higher order terms of sloping angles and vertical acceleration excitation on the horizontal acceleration predictions in a clearer and quantitative manner, four indices, R21 , R22 , ER1 , and ER2 , are defined as
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
(i) θ1/ θ2=6o/6o
0.04 -0.04
Acceleration (g)
-0.12 0.12 0.04 -0.04 -0.12 0.12 0.04 BE UE US
-0.04 -0.12 -20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
Displacement (mm) Fig. 9. Comparison of hysteresis loops of different design cases under 25%-AC156-1.
10
20
442
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
0.16
(a) θ1/ θ2=2o/2o
(c) θ1/ θ2=6o/2o
(b) θ1/ θ2=4o/2o
0.08 0 -0.08
Acceleration (g)
-0.16 0.16
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
(i) θ1/ θ2=6o/6o
0.08 0 -0.08 -0.16 0.16 0.08 0
BE UE US
-0.08 -0.16 -50
-25
0
25
50
-50
-25
0
25
50
-50
-25
0
25
50
Displacement (mm) Fig. 10. Comparison of hysteresis loops of different cases under 25%-IEEE.
0.18 0.09
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
0 -0.09
Acceleration (g)
-0.18 0.18 0.09
(d) θ1/ θ2=2o/4o
(f) θ1/ θ2=6o/4o
(e) θ1/ θ2=4o/4o
0 -0.09 -0.18 0.18 0.09
(g) θ1/ θ2=2o/6o
(i) θ1/ θ2=6o/6o
(h) θ1/ θ2=4o/6o
0
BE UE US
-0.09 -0.18 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450
Displacement (mm) Fig. 11. Comparison of hysteresis loops of different design cases under 100%-Kobe.
2 Pm ðAUE Þi ðAUS Þi R21 ¼ 1 Pm i¼1 2 i¼1 ðAUE Þi ðAUE Þmean
ð34Þ
2 Pm ðABE Þi ðAUE Þi R22 ¼ 1 Pm i¼1 2 i¼1 ðABE Þi ðABE Þmean
ð35Þ
ER1 ¼
maxðjAUE jÞ maxðjAUS jÞ 100% max ðjAUS jÞ
ð36Þ
ER2 ¼
max ðjABE jÞ max ðjAUE jÞ 100% max ðjAUE jÞ
ð37Þ
where R21 is the coefficient of determination to quantitatively evaluate the difference between the horizontal acceleration predictions
under the conditions of UE and US (i.e. with and without considering the higher order terms of sloping angles); R22 is the coefficient of determination to quantitatively evaluate the difference between the horizontal acceleration predictions under the conditions of BE and UE (i.e. with and without considering the vertical acceleration excitation); ER1 and ER2 are the error ratios to quantitatively evaluate the maximum underestimate in horizontal acceleration prediction attributed to neglect of higher order terms of sloping angles and vertical acceleration excitation, respectively; AUS , AUE , and ABE are the horizontal acceleration predictions under the conditions of US, UE, and BE, respectively; the subscript i represents the analysis data point at time t i ; m represents the total number of analysis data points; and the subscript mean represents the mean value of total analysis data points. Through the further discussion on R21 (ER1 ) and R22 (ER2 ), the influences caused by neglect of higher order terms
443
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
0.18 0.09
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
0 -0.09 -0.18
Acceleration (g)
0.18 0.09 0 -0.09 -0.18 0.18
(i) θ1/ θ2=6o/6o
0.09 0
BE UE US
-0.09 -0.18 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450
Displacement (mm) Fig. 12. Comparison of hysteresis loops of different design cases under 100%-ChiChi.
1.00 6% 5%
0.99
ER1
4% 0.98
3% 2%
0.97
2
3
4
5
6
θ1(or θ2) (degrees)
θ2 (or θ1)
o
o
o
o
1% 0%
o
Fig. 13. Variations of R21 values with various h1 and h2 under all unilateral acceleration inputs.
of sloping angles and vertical acceleration excitation on the horizontal acceleration predictions can be clarified, respectively.
2
3
4
5
6
θ1(or θ2) (degrees)
θ2 (or θ1)
o
o
o
o
o
Fig. 14. Variations of ER1 values with various h1 and h2 under all unilateral acceleration inputs.
The variations of R21 and ER1 values with respect to h1 and h2 under all the unilateral acceleration inputs are shown in Figs. 13 and 14, respectively. The variations of ER1 values with respect to h1 and h2 under each 100% unilateral acceleration input are shown in Fig. 15. As observed from Eqs. (11) and (20) together with Figs. 9–12, it is evident that an exchange of design values for h1 and h2 will not affect the horizontal responses of the protected
R21 values shown in Fig. 13, a large number of R22 values shown in Fig. 16 are farther from 1. In this numerical study, the minimum
object and superior bearing plate. The variations of R22 and ER2 values with various vertical PA values under all the biaxial acceleration inputs are shown in Figs. 16 and 17, respectively. The variations of ER2 values with various vertical PA values for the design cases of (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°) under all the biaxial acceleration inputs are shown in Fig. 18. To focus on the dynamic behavior of the sloped rolling-type seismic isolator when the roller moves between two sloped surfaces, as well as to avoid erroneous judgment due to small excitation and/or responses, the calculation results of the roller moving within the fixed curvature range (or the arc rolling range) are excluded in the figures and the following discussion.
R22 value is 0.675 for the design case of (h1 , h2 ) = (5°, 3°) under biaxial-100%-Kobe, which might be an unacceptable value in engineering practice. It is found from Figs. 14 and 15 that the larger the sloping angle is, the more apparent effect revealed from neglect of higher order terms of sloping angles on the ER1 value exerts. Nevertheless, as observed from Figs. 13–15, even if the design sloping angle becomes larger, the difference between the acceleration predictions by using the simplified and exact generalized equations of motion under horizontal excitation is very limited. In this numerical study, the maximum ER1 value is only 5.10% for the design case of (h1 , h2 ) = (6°, 6°) under unilateral-100%-ChiChi. Therefore, it can
It can be seen from Fig. 13 that the R21 values are in general very close to 1. In this numerical study, the minimum R21 value is 0.973 for the design case of (h1 , h2 ) = (6°, 6°) under unilateral-50%-Kobe, which is very acceptable in engineering practice. Compared to the
444
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
2%
3%
unilateral-100%-ElCentro
unilateral-100%-Kobe
ER 1
ER 1
2% 1%
1%
0%
0% 2
3
4
5
2
6
5
6
unilateral-100%-AC156-1
unilateral-100%-ChiChi
2%
ER 1
4%
ER 1
4
3%
6% 5%
3
θ 1 (or θ 2) (degrees)
θ 1 (or θ 2) (degrees)
3% 2%
1%
1% 0%
0% 2
3
4
5
6
2
θ 1 (or θ 2) (degrees)
3
4
5
6
θ1 (or θ 2) (degrees)
4%
6% unilateral-100%-AC156-2
5%
unilateral-100%-IEEE
3%
ER 1
ER 1
4% 2%
3% 2%
1% 1% 0%
θ 2 (or θ 1)
0% 2
3
4
5
6
2
θ1 (or θ2) (degrees)
3
4
2
o
o
3
5
o
4
o
5
o
6
6
θ 1 (or θ 2) (degrees)
Fig. 15. Variations of ER1 values with various h1 and h2 under 100% unilateral acceleration inputs.
50% 40%
ER2
30% Kobe EL ChiChi AC156-1 AC156-2 IEEE
20% 10% 0%
0
0.2
0.4
0.6
0.8
1
Vertical PA (g) Fig. 16. Variations of R22 values with various vertical PA scales under all biaxial acceleration inputs.
Fig. 17. Variations of ER2 values with various vertical PA scales under all biaxial acceleration inputs.
be concluded that the effect of higher order terms of sloping angles on the acceleration prediction accuracy is very limited and can be rationally negligible in engineering practice. By comparing Figs. 17 and 14, it can be seen that the difference between the maximum horizontal acceleration predictions by using the exact generalized analytical model under unilateral and biaxial excitation, i.e. the ER2 value, might be somewhat noticeable. In addition, it is apparent that a larger vertical PA value will lead to a more significant effect arising from neglect of vertical acceleration excitation on the ER2 value. As observed from Eq. (11), the term of €zg is multiplied by a parameter which is related to the
sloping angles h1 and h2 , i.e. ðcos h1 þ cos h2 Þðsin h1 þ sin h2 Þ= ½1 þ cosðh1 h2 Þ. Therefore, to more clearly understand the tendency of variations of ER2 values with various vertical PA values, the calculation results of five design cases, (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°), under all the biaxial acceleration inputs are marked with different symbols in Fig. 18. As observed from Fig. 18, it is very evident that the larger the vertical PA value is, the more significant effect without considering vertical acceleration excitation on the ER2 value produces. In this numerical study, the maximum ER2 value is 36.59% for the design case of (h1 , h2 ) = (2°, 2°) under biaxial-100%-IEEE, which might not be very
445
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
15%
25%
biaxial-ElCentro
biaxial-Kobe
10%
15%
ER2
ER2
20%
10%
5%
5% 0%
0
0.1
0.2
0.3
0%
0.4
0
0.1
Vertical PA (g)
0.2
0.3
Vertical PA (g)
30%
15% biaxial-ChiChi
biaxial-AC156-1
20%
ER2
ER2
10%
10%
0%
5%
0
0.1
0.2
0.3
0%
0.4
0
0.1
Vertical PA (g)
0.3
40%
30% biaxial-AC156-2
biaxial-IEEE
30%
ER2
ER2
20%
10%
20% 10% 2/2
0%
0.2
Vertical PA (g)
0
0.2
0.4
0.6
0%
0
Vertical PA (g)
0.3
3/3
4/4
0.6
5/5
6/6
0.9
Vertical PA (g)
Fig. 18. Variations of ER2 values with various vertical PA scales for the design cases of (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°) under all biaxial acceleration inputs.
acceptable compared to the maximum ER1 value aforementioned, i.e. 5.10%. As observed from Figs. 16–18, it is obvious that if the sloped rolling-type seismic isolator is subjected to an acceleration excitation with a considerable vertical component, its real dynamic responses might not be precisely predicted by using the simplified generalized analytical model with neglect of vertical acceleration excitation. More importantly, the significant underestimate in the maximum transmitted horizontal acceleration response under biaxial excitation due to neglect of vertical acceleration excitation will result in a non-conservative design result. Therefore, more attention should be paid to the effect of vertical acceleration excitation on the maximum transmitted horizontal acceleration response of the seismic isolator, especially when the vertical excitation possesses a larger PA value. 6. Conclusions Lee et al.’s [13,14] and Wang et al.’s [15] studies respectively proposed the simplified analytical models for different designs of sloped rolling-type seismic isolators. Lee et al.’s study focused on the dynamic behavior of a cylinder moving between a flat surface and a V-shaped surface along each principle horizontal direction, while Wang et al.’s study concentrated on that between two V-shaped surfaces designed with an identical sloping angle. To make the design of the sloped rolling-type seismic isolator more flexible, the exact and simplified generalized analytical models which can represent the dynamic behavior of a cylindrical roller moving between two V-shaped surfaces designed with arbitrary sloping angles are proposed in this study. Under the specific design
conditions, the proposed simplified model can make a valid generalization in Lee et al.’s and Wang et al.’s models by theoretical verification. In addition, the accuracy and practicability of the proposed simplified model are experimentally demonstrated. A series of numerical study are performed to quantitatively discuss the influences arising from neglect of higher order terms of sloping angles and vertical excitation on the transmitted horizontal acceleration responses of the seismic isolator, which has never been discussed in detail before. The numerical results reveal that the effect of vertical acceleration excitation plays a more crucial role in prediction accuracy compared to that of higher order terms of sloping angles. When the acceleration excitation possesses a considerable vertical component, adopting the exact generalized analytical model, rather than the simplified one, can achieve more precise and conservative analysis and design results for the sloped rolling-type seismic isolator. Acknowledgements The study was supported by the National Center for Research on Earthquake Engineering (NCREE), National Applied Research Laboratories (NARL) of Taiwan. The support is greatly acknowledged. References [1] Harvey Jr PS, Kelly KC. A review of rolling-type seismic isolation: historical development and future directions. Eng Struct 2016;125:521–31. [2] Guerreiro L, Azevedo J, Muhr AH. Seismic tests and numerical modeling of a rolling-ball isolation system. J Earthq Eng 2007;11(1):49–66. [3] Zhou Q, Lu X, Wang Q, Feng D, Yao Q. Dynamic analysis on structures baseisolated by a ball system with restoring property. Earthq Eng Struct Dyn 1998;27(8):773–91.
446
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
[4] Harvey Jr PS, Gavin HP. Double rolling isolation systems: a mathematical model and experimental validation. Int J Non-Linear Mech 2014;61:80–92. [5] ISO-BaseTM seismic isolation platform, from the World Wide Web: http:// www.worksafetech.com/products/iso-base/. [6] Kasalanati A, Reinhorn AM, Constantinou MC, Sanders D. Experimental study of ball-in-cone isolation system. In: Proceedings of the ASCE structures congress XV, Portland, Oregon, April 13–16. p. 1191–5. [7] Lin TW, Chern CC, Hone CC. Experimental study of base isolation by free rolling rods. Earthq Eng Struct Dyn 1995;24(12):1645–50. [8] Jangid RS. Stochastic seismic response of structures isolated by rolling rods. Eng Struct 2000;22(8):937–46. [9] Mahmood H, Amirhossein S. Using orthogonal pairs of rollers on concave beds (OPRCB) as a base isolation system - Part I: analytical, experimental and numerical studies of OPRCB isolators. Struct Desi Tall Spec Build 2011;20 (8):928–50. [10] Jangid RS, Londhe YB. Effectiveness of elliptical rolling rods for base isolation. J Struct Eng ASCE 1998;124(4):469–72. [11] CRS, cosine curved rail system, from the World Wide Website: http://www. oiles.co.jp/en/menshin/building/menshin/kikimenshin/. [12] Tsai MH, Wu SY, Chang KC, Lee GC. Shaking table tests of a scaled bridge model with rolling type seismic isolation bearings. Eng Struct 2007;29(9):694–702.
[13] Lee GC, Ou YC, Niu T, Song J, Liang Z. Characterization of a roller seismic isolation bearing with supplemental energy dissipation for highway bridges. J Struct Eng ASCE 2010;136(5):502–10. [14] Ou YC, Song J, Lee GC. A parametric study of seismic behavior of roller seismic isolation bearings for highway bridges. Earthq Eng Struct Dyn 2010;39 (5):541–59. [15] Wang SJ, Hwang JS, Chang KC, Shiau CY, Lin WC, Tsai MS, et al. Sloped multiroller isolation devices for seismic protection of equipment and facilities. Earthq Eng Struct Dyn 2014;43(10):1443–61. [16] Pacific Earthquake Engineering Research (PEER) Ground Motion Database, from World Wide Website: http://peer.berkeley.edu/peer_ground_motion_ database/. [17] AC156. Acceptance criteria for seismic qualification by shake-table testing of nonstructural components and systems. ICC Evaluation Service Inc.; 2007. [18] Seismic design code for buildings. Taipei (Taiwan): Ministry of Interior; 2011. [19] IEEE Std 693TM-2005. IEEE recommended practice for seismic design of substations. NY: Institute of Electrical and Electronics Engineers (IEEE) Power Engineering Society; 2006. [20] Tsopelas P, Constantinou MC, Okamoto S, Fujii S, Ozaki D. Experimental study of bridge seismic sliding isolation systems. Eng Struct 1996;18(4): 301–10.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
A generalized analytical model for sloped rolling-type seismic isolators Shiang-Jung Wang, Chung-Han Yu, Wang-Chuen Lin, Jenn-Shin Hwang, Kuo-Chun Chang ⇑ Department of Civil Engineering, National Taiwan University, Taiwan
a r t i c l e
i n f o
Article history: Received 29 December 2015 Revised 13 December 2016 Accepted 15 December 2016
Keywords: Sloped rolling-type seismic isolator Generalized analytical model Equipment Seismic performance Vertical excitation
a b s t r a c t The exact and simplified generalized analytical models for the sloped rolling-type seismic isolator in which two V-shaped surfaces in contact with cylindrical rollers are designed with arbitrary sloping angles are proposed. Numerical predictions are compared with horizontal seismic simulation test results to demonstrate the accuracy and practicability of the proposed simplified model. Furthermore, the influences arising from two simplification assumptions on the horizontal acceleration prediction are numerically examined. The results indicate that the effect of vertical acceleration excitation plays a more crucial role in prediction accuracy and conservative design compared to that of higher order terms of sloping angles. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Rolling-based metallic seismic isolators [1] in which either a ball rolls on flat [2], concave [3–5], or conical [6] surfaces or a rod rolls on flat [7,8], curved [9–11], or sloped [12–15] surfaces (or rails) have been numerically and experimentally demonstrated for their excellent seismic isolation performance. Since the rolling friction force and the restoring force contributed by additional springs or due to gravity are very limited and much smaller than the input seismic force, the rolling motion can be activated immediately once an earthquake occurs, and the horizontal acceleration transmitted to the protected object can be significantly reduced during an earthquake. The seismic isolators, except those designed with flat surfaces in contact with balls or rods, have an inherent gravity-based self-centering capability after excitation. Among various types of rolling-based metallic seismic isolators, the sloped rolling-type seismic isolators discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies attracted sustained attention. In addition to possessing the advantages aforementioned, the design of constant sloping surfaces in contact with cylindrical rollers can ensure that the horizontal acceleration transmitted to the protected object remains essentially constant regardless of any intensity and frequency content of excitation. Thus, the seismic isolators do not have a fixed vibration natural period, and can offer maximum horizontal decoupling between the protected object and input excitation. It is noteworthy that the zero post-elastic stiffness performance, i.e. the constant transmitted horizontal acceleration ⇑ Corresponding author. E-mail address: [email protected] (K.-C. Chang). http://dx.doi.org/10.1016/j.engstruct.2016.12.027 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.
performance, is the most attractive feature. This feature can make the seismic isolators easily meet the rigorous performance-based design requirements for structural and non-structural systems if the maximum transmitted acceleration response is selected as the seismic performance criterion. Furthermore, to have a better displacement control and more effectively stop rolling motion respectively during and after excitation, the built-in damping mechanism provided by additional sliding friction was employed in these two studies. The normal force for the sliding friction was provided by screws in Lee et al.’s study, while that was provided by springs in Wang et al.’s study. As shown in the schematic drawing of Fig. 1, the sloped rollingtype seismic isolators discussed in Lee et al.’s and Wang et al.’s studies were basically composed of three bearing plates (denoted as upper, intermediate, and lower bearing plates hereafter) and cylindrical rollers. In Lee et al.’s study, the upper and lower bearing plates were designed with flat surfaces in contact with the rollers, while the intermediate bearing plate was designed with V-shaped surfaces. In Wang et al.’s study, the three bearing plates were designed with V-shaped surfaces in contact with the rollers, and the sloping angles of the V-shaped surfaces were identical. Without considering the effects arising from higher order terms of sloping angles and vertical excitation, the dynamic behavior of a cylinder moving between a flat surface and a V-shaped surface along each principle horizontal direction was focused in Lee et al.’s study, while that between two V-shaped surfaces designed with the same sloping angle was concentrated in Wang et al.’s study. Because of different designs of rolling surfaces, the compatibility conditions in these two studies were diverse. Hence, the simplified equations of motion derived in these two studies were
435
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Upper bearing plate (Flat surface)
Upper bearing plate (V-shaped sloping surface) Cylindrical roller
Lower bearing plate (V-shaped sloping surface)
Intermediate bearing plate (V-shaped sloping surface)
Lower bearing plate (Flat surface)
Fig. 1. Schematic drawings of sloped rolling-type seismic isolators discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies.
essentially different and cannot be exchanged. In addition, other refinements such as the multi-roller and pounding prevention mechanisms were developed in Wang et al.’s study. The multiroller design can have a better stability performance compared with the conventional single-roller design. The design of an arc rolling range with a fixed curvature radius at the intersection of two inclines of V-shaped surfaces of bearing plates can effectively prevent pounding between rollers and inclined surfaces when rollers are passing through here. In Wang et al.’s study, a simplified analytical model was provided to fully describe the hysteretic behavior of the sloped rolling-type seismic isolator based on a combination of multi-linear elastic and plastic models. It is also mentioned here that the application purposes of Lee et al.’s and Wang et al.’s studies were bridge structures and equipment or facilities, respectively. Therefore, this study first aims to derive generalized equations of motion for the sloped rolling-type seismic isolator in which the cylindrical rollers move between two V-shaped surfaces designed with arbitrary sloping angles. The design is different from and more generalized than Lee et al.’s and Wang et al.’s. The proposed simplified generalized analytical model, of course, is capable of describing the dynamic behavior of the two specific designs respectively discussed in Lee et al.’s and Wang et al.’s studies. The accuracy and practicability of the simplified model are verified by comparing with horizontal shaking table test results. Finally, by further comparing the numerical results predicted by the exact generalized analytical model with those by the simplified one, the influences caused by two simplification assumptions—neglect of higher order terms of sloping angles and vertical excitation— on the transmitted horizontal acceleration responses of the seismic isolator are thoroughly examined. It has never been discussed in detail before. 2. Derivation of generalized equations of motion A simplified model, a cylindrical roller sandwiched between two V-shaped surfaces designed with different sloping angles h1 and h2 , to represent the dynamic behavior of the sloped rollingtype seismic isolator in one principle horizontal direction as well as in the vertical direction is illustrated in Fig. 2(a), in which M, m1 , and m2 are the seismic reactive masses of the protected object, superior bearing plate, and roller, respectively; r is the radius of the roller; and h1 and h2 are the sloping angles of the superior and inferior bearing plates, respectively. The following basic assumptions are made first for deriving the generalized equations of motion of the sloped rolling-type seismic isolator: (1) the roller and bearing plates are ideally contact and in pure rolling motion without any undesired sliding and overturning motions; (2) the rolling motions of the rollers between two bearing plates along two principle horizontal directions are identical; (3) an appropriate rigid-plastic hysteretic model (i.e. the Coulomb friction law) is employed to rep-
resent the force-displacement relationship of built-in damping behavior; (4) the inferior bearing plate is fixed to a rigid base; and (5) the payload of a to-be-protected object is applied on the superior bearing plate. Two opposite rolling motion directions are identified by sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1 and sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1 corresponding to the rightward (sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1) and leftward (sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1) movements of the superior bearing plate (or the roller) relative to the inferior bearing plate. The free body diagrams for one of the four situations, sgnðx1 Þ ¼ sgnðx2 Þ ¼ 1 and xg ð€zg Þ is sgnðx_ 1 Þ ¼ sgnðx_ 2 Þ ¼ 1, are shown in Fig. 2(b), in which € the horizontal (vertical) acceleration excitation; x1 ðz1 Þ, x_ 1 ðz_ 1 Þ, and € x1 ð€z1 Þ are the horizontal (vertical) displacement, velocity, and acceleration responses of the protected object and superior bearing plate relative to the origin O, respectively; x2 ðz2 Þ, x_ 2 ðz_ 2 Þ, and € x2 ð€z2 Þ are the horizontal (vertical) displacement, velocity, and acceleration responses of the roller relative to the origin O, respectively; the positive directions of x and z are correspondingly defined to be rightward and upward in the figure; g is the acceleration of gravity; I is the moment of inertia of the roller; a is the angular acceleration of the roller (the positive rotation is defined to be clockwise in the figure); f 1 ðf 2 Þ and N 1 ðN 2 Þ are the rolling friction force and normal force acting between the superior bearing plate and roller (between the roller and inferior bearing plate), respectively; and F D is the built-in friction damping force acting parallel to the slope of the bearing plates. By considering the free body diagrams of the superior bearing plate and protected object, two dynamic force equilibrium equations are obtained as
F D cos h1 sgnðx_ 1 Þ N1 sin h1 sgnðx1 Þ f 1 cos h1 sgnðx_ 1 Þ ¼ ðM þ m1 Þð€x1 þ €xg Þ
ð1Þ
F D sin h1 sgnðx1 Þsgnðx_ 1 Þ þ N1 cosh1 Mg f 1 sinh1 sgnðx1 Þsgnðx_ 1 Þ m1 g ¼ ðM þ m1 Þð€z1 þ €zg Þ ð2Þ
Moreover, by considering the free body diagram of the roller, two dynamic force equilibrium and one dynamic moment equations are obtained as F D cos h1 sgnðx_ 1 Þ F D cos h2 sgnðx_ 1 Þ þ N1 sin h1 sgnðx1 Þ N2 sin h2 sgnðx1 Þ þ f 1 cos h1 sgnðx_ 1 Þ f 2 cos h2 sgnðx_ 1 Þ ¼ m2 ð€x2 þ €xg Þ ð3Þ F D sin h1 sgnðx1 Þsgnðx_ 1 Þ F D sin h2 sgnðx1 Þsgnðx_ 1 Þ N1 cos h1 þ N2 cos h2 þ f 1 sin h1 sgnðx1 Þsgnðx_ 1 Þ f 2 sin h2 sgnðx1 Þsgnðx_ 1 Þ m2 g ¼ m2 ð€z2 þ €zg Þ ð4Þ
f 1 rsgnðx_ 1 Þ þ f 2 rsgnðx_ 1 Þ ¼ Ia where I ¼ ð1=2Þm2 r . 2
ð5Þ
436
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Protected object Superior bearing plate Cylindrical roller Inferior bearing plate
Ő
Fig. 2. Simplified model for sloped rolling-type seismic isolators in this study.
From the relative motion between the superior bearing plate and roller, four compatibility conditions can be obtained as
€x2 ¼ r a cos h2
ð6Þ
€z2 ¼ r a sin h2 sgnðx1 Þ
ð7Þ
€x1 ¼ €x2 þ ra cos h1 ¼ ra cos h1 þ r a cos h2
ð8Þ
€z1 ¼ €z2 þ r a sin h1 sgnðx1 Þ ¼ r a sin h1 sgnðx1 Þ þ r a sin h2 sgnðx1 Þ
ð9Þ
A total of nine variables, a, € x1 , € x2 , €z1 , €z2 , N 1 , N 2 , f 1 , and f 2 , can be solved by using Eqs. (1)–(9). With reasonable neglect of the term of m2 =ðM þ m1 Þ due to the fact that m2 is in general much smaller than ðM þ m1 Þ, the exact mathematical forms of the solutions of the nine variables are given in Eqs. (10)–(18)
1 f2F D sgnðx_ 1 Þ 2rðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
a¼
ð10Þ ðcosh1 þ cosh2 Þ €x1 ¼ f2F D sgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
sinh2 f2F D sgnðx1 Þsgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cosh2 Þsgnðx1 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þ
€z2 ¼
ð14Þ
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ 2 ðg þ €zg Þðcos h1 þ cos h2 Þ
ð15Þ
h1 h2 N2 ¼ F D tan sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ 2 ðg þ €zg Þðcos h1 þ cos h2 Þ
ð16Þ
N1 ¼ F D tan
f1 ¼
ð11Þ cos h2 f2F D sgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcos h1 þ cos h2 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þsgnðx1 Þ
€x2 ¼
ð12Þ ðsinh1 þ sinh2 Þ f2F D sgnðx1 Þsgnðx_ 1 Þ 2ðM þ m1 Þ½1 þ cosðh1 h2 Þ þðM þ m1 Þ €xg ðcosh1 þ cosh2 Þsgnðx1 Þ þ ðg þ €zg Þðsinh1 þ sinh2 Þ
€z1 ¼
ð13Þ
f2 ¼
2ðM þ m1 Þsgnðx_ 1 Þ h1 h2 h1 h2 cos2 sin 1 þ cosðh1 h2 Þ 2 2 h1 þ h2 h1 þ h2 ðg þ €zg Þ cos sgnðx1 Þ €xg sin 2 2
ð17Þ
2ðM þ m1 Þsgnðx_ 1 Þ h1 h2 h1 h2 cos2 sin 1 þ cosðh1 h2 Þ 2 2 h1 þ h2 h1 þ h2 ðg þ €zg Þ cos sgnðx1 Þ €xg sin 2 2
ð18Þ
Hereafter, Eqs. (10)–(18) are generally called the exact generalized solutions in which Eqs. (10)–(14) are the exact generalized equations of motion. It can be seen that the insightful information might not be easily observed from the exact generalized solutions. Therefore, the following two assumptions for further simplicity
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
are taken into consideration: (1) Based on the assumption that the sloping angles h1 and h2 are small enough, the higher order terms of h1 and h2 are negligible, i.e. cos2 h1 1, cos2 h2 1, cos h1 cos h2 1, 2
2
sin h1 0, sin h2 0, and sin h1 sin h2 0; and (2) assuming the term of the vertical acceleration excitation €zg is neglected. Thus, the nine variables can be approximated by
a¼
€x1 ¼
F D sgnðx_ 1 Þ 2rðM þ m1 Þ 1 €xg ðcos h1 þ cos h2 Þ þ gðsin h1 þ sin h2 Þsgnðx1 Þ 4r
ð19Þ
Note that during two or more consecutive time steps in numerical integration, it is possible to have no relative motion between the superior and inferior bearing plates of the simplified model shown in Fig. 2. This phenomenon is defined as the relative static state hereafter. To simulate this phenomenon in the analysis program, the critical characteristic strengths due to the built-in friction damping force, F D;C , are defined and calculated by setting € x1 in Eqs. (11) and (20) equal to zero, which are respectively given by
1
F D;C ¼
ðM þ m1 Þ €xg ðcosh1 þ cosh2 Þ þ sgnðx1 Þðg þ €zg Þðsinh1 þ sinh2 Þ
2
ð28Þ
ðcos h1 þ cosh2 Þ ½2F D sgnðx_ 1 Þ þ ðM þ m1 Þgðsin h1 þ sin h2 Þsgnðx1 Þ 4ðM þ m1 Þ x€g
F D;C
1
1 þ cos h1 cos h2 ¼
ðM þ m1 Þ 2€xg þ sgnðx1 Þgðsin h1 þ sin h2 Þ
2 cos h1 þ cos h2 ð29Þ
ð20Þ
€x2 ¼
€z1 ¼
€z2 ¼
cos h2 f2F D sgnðx_ 1 Þ 4ðM þ m1 Þ ðM þ m1 Þ gðsin h1 þ sin h2 Þsgnðx1 Þ þ 2€xg g
ð21Þ
ðsin h1 þ sin h2 Þsgnðx1 Þ 2F D sgnðx_ 1 Þ þ ðM þ m1 Þ€xg ðcos h1 þ cos h2 Þ 4ðM þ m1 Þ ð22Þ
sin h2 sgnðx1 Þ 2F D sgnðx_ 1 Þ þ ðM þ m1 Þ€xg ðcos h1 þ cos h2 Þ 4ðM þ m1 Þ ð23Þ
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ ðg þ €zg Þðcos h1 þ cos h2 Þ 2 ð24Þ
N 1 ¼ F D csc
h1 h2 sgnðx1 Þsgnðx_ 1 Þ 2 1 ðM þ m1 Þ €xg ðsin h1 þ sin h2 Þsgnðx1 Þ ðg þ €zg Þðcosh1 þ cos h2 Þ 2 ð25Þ
N2 ¼ F D csc
f1 ¼
f2 ¼
437
ðM þ m1 Þsgnðx_ 1 Þ gðsinh1 sinh2 Þsgnðx1 Þ þ €xg ðcos h1 cosh2 Þ 2 ð26Þ ðM þ m1 Þsgnðx_ 1 Þ gðsinh1 sinh2 Þsgnðx1 Þ þ €xg ðcos h1 cos h2 Þ 2 ð27Þ
Hereafter, Eqs. (19)–(27) are generally called the simplified generalized solutions in which Eqs. (19)–(23) are the simplified generalized equations of motion. By using Eqs. (11) and (20), the horizontal dynamic behavior of the sloped rolling-type seismic isolator in which two V-shaped surfaces in contact with cylindrical rollers are designed with arbitrary sloping angles can be analyzed precisely and approximately, respectively. 3. Analysis program The Visual Basic of Application (VBA), a widely used programming language in engineering practice, is adopted to develop the analysis program for the sloped rolling-type seismic isolator in this study. The Newmark-b method with constant average acceleration is used for solving the exact and simplified generalized equations of motion respectively given in Eqs. (11) and (20) at each time step.
It can be seen that F D;C is dependent on € xg and €zg (only € xg for the simplified generalized equation of motion). If F D;C is calculated to be smaller than the design built-in friction damping force F D at a specific time step (e.g. at ti ), the relative static state occurs. Under x1 and x_ 1 should be modified this circumstance, at the time step t i , € to be zero, and x1 should remain the same as that at the previous time step t i1 . In order to have a better acceleration control performance for the sloped rolling-type seismic isolator, Wang et al.’s study [15] suggested employing a pounding prevention mechanism—an arc rolling range with a fixed curvature radius which is larger than the roller radius. The theoretical equations of motion when balls or rods roll on concave or curved surfaces have been thoroughly studied in many past researches [3–5,9–11]. For the sloped rolling-type seismic isolator, its equations of motion when rollers move in the arc range designed with a fixed curvature radius R have also been studied by Wang et al. [15]. R, in detail, is the curvature radius designed for the round surface between two inclines of V-shaped surfaces of bearing plates. Methodologically, in addition to the theoretical equations of motion, the dynamic response when rollers move in the arc range can be numerically analyzed based on a reasonable assumption of the arc range composed of many infinitesimal segments designed with a fixed and continuously varied sloping angle. In this analysis program, the derived generalized equations of motion when rollers move in the sloped range, as given in Eqs. (11) and (20), based on the assumption above are adopted to numerically analyze the dynamic response of the sloped rolling-type seismic isolator when rollers move in the arc range. 4. Verification of simplified generalized analytical model The following will theoretically and experimentally demonstrate that the simplified generalized solutions given in Eqs. (19)–(27) can represent the dynamic behavior of the two different sloped rolling-type seismic isolators respectively discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies, as shown in Fig. 3. 4.1. Theoretical verification When the sloping angles of the superior and inferior bearing plates of the simplified model shown in Fig. 2(a) (i.e. h1 and h2 ) are respectively equal to zero and h, as illustrated in Fig. 3(a), the same solutions as those provided in Lee et al.’s study can be obtained. For example, the transmitted acceleration responses along the horizontal and vertical directions are
€x1 þ €xg ¼
g sin h FD sgnðx1 Þ sgnðx_ 1 Þ 2 ðM þ m1 Þ
ð30Þ
438
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
M m1
(a) Lee et al.’s study
m2
r
(b) Wang et al.’s study
Fig. 3. Simplified models discussed in Lee et al.’s [13,14] and Wang et al.’s [15] studies.
Fig. 4. Test installation photo.
Table 1 Acceleration input program for experimental study. Acceleration input
Input earthquake information or response spectrum condition
Excitation direction
Test input PA (g)
Test input PA 100% Original input PA
El Centro Kobe
IMPVALL/I-ELC180, Imperial Valley, U.S., 1940/05/19 KOBE/KJM000, Kobe, Japan, 1995/01/16
Unilateral Unilateral
X X
0.31 0.41
100% 50%
Artificial acceleration history
AC156-1
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8.75 m in elevation) of a 7-story building (24 m in height) at Taipei City RRS specified in IEEE Std 693TM-2005 for seismic qualification test
Unilateral
X
0.50
100%
Unilateral
X
0.50
100%
€z1 þ €zg ¼ €zg
IEEE
€xg sin h F D sin h sgnðx_ 1 Þ 2 2ðM þ m1 Þ
ð31Þ
When the sloping angles of both the superior and inferior bearing plates of the simplified model shown in Fig. 2(a) (i.e. h1 and h2 ) are equal to h, as illustrated in Fig. 3(b), the same solutions as those provided in Wang et al.’s study can be obtained. For example, the transmitted acceleration responses along the horizontal and vertical directions are
€x1 þ €xg ¼
g sin 2h F D cos h sgnðx1 Þ sgnðx_ 1 Þ 2 ðM þ m1 Þ €xg sin 2h F D sin h sgnðx_ 1 Þ 2 ðM þ m1 Þ
ð32Þ
Spectral Acceleratrion (g)
Recorded earthquake history
4
Damping Ratio = 5%
ELCentro Kobe AC156-1 IEEE
3 2 1 0
0
1
2
3
4
5
Period (sec)
ð33Þ
Fig. 5. Acceleration response spectra of acceleration inputs (PA = 1 g) for experimental study.
The tested sloped rolling-type seismic isolator consists of a set of upper, intermediate, and lower bearing plates together with two pairs of mutually orthogonal cylindrical rollers (between the upper and intermediate bearing plates as well as between the intermediate and lower bearing plates). The surfaces of the upper
(or lower) and intermediate bearing plates in contact with the rollers are flat and dual V-shaped with a sloping angle of 6.25°, respectively. The dimension of the seismic isolator without horizontal displacement is 600 mm (in length) 600 mm (in width) 123.9 mm (in height). The seismic reactive masses of the upper, intermediate, and lower bearing plates are 19.44 N-s2/m, 29.44 N-s2/m, and 19.44 N-s2/m, respectively. The sectional radius
€z1 þ €zg ¼ €zg
4.2. Experimental verification
439
0.2
Unilateral-50%-Kobe
0.1 0.1
0 -0.1 -0.2 10
20
30
40
Time (sec)
(a) Acceleration response history Displacement (mm)
0.15
R2 = 0.84
100
Acceleration (g)
Acceleration (g)
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
R2 = 0.96
50
0
0.05
0
-0.05
-0.1
Numerical Prediction Test Result
-50 -100 10
20
30
-0.15 -100
40
-50
0
50
Time (sec)
Displacement (mm)
(b) Displacement response history
(c) Hysteresis loop
100
0.2
Unilateral-100%-AC156-1
0.1 0.1
0 -0.1 -0.2 15
25
35
45
Time (sec)
(a) Acceleration response history Displacement (mm)
0.15
R2 = 0.83
100
R2 = 0.95
50
0
Acceleration (g)
Acceleration (g)
Fig. 6. Comparisons of experimental results and numerical predictions under unilateral-50%-Kobe.
0.05
0
-0.05
-0.1
Numerical Prediction Test Result
-50 -100 15
25
35
45
-0.15 -100
-50
0
50
100
Displacement (mm)
Time (sec)
(b) Displacement response history
(c) Hysteresis loop
Fig. 7. Comparisons of experimental results and numerical predictions under unilateral-100%-AC156-1.
Table 2 Maximum transmitted horizontal acceleration responses of twenty-five design cases predicted by using Eq. (20) (unit:g). Design case (h1 h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(2°, 2°) 0.063
(2°, 3°) 0.072
(2°, 4°) 0.081
(2°, 5°) 0.090
(2°, 6°) 0.098
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(3°, 2°) 0.072
(3°, 3°) 0.081
(3°, 4°) 0.090
(3°, 5°) 0.098
(3°, 6°) 0.107
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(4°, 2°) 0.081
(4°, 3°) 0.090
(4°, 4°) 0.098
(4°, 5°) 0.107
(4°, 6°) 0.115
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(5°, 2°) 0.090
(5°, 3°) 0.098
(5°, 4°) 0.107
(5°, 5°) 0.115
(5°, 6°) 0.124
Design case (h1 , h2 ) Maximum horizontal acceleration calculated by using Eq. (20)
(6°, 2°) 0.098
(6°, 3°) 0.107
(6°, 4°) 0.115
(6°, 5°) 0.124
(6°, 6°) 0.133
(r), longitudinal length, and seismic reactive mass of each cylindrical roller are 17.5 mm, 600 mm, and 4.58 N-s2/m, respectively. The built-in friction damping mechanism is composed of a rubber pad
with a thickness of 2 mm vulcanized and attached to the surfaces of the upper, intermediate, and lower bearing plates sliding against the stainless steel surfaces of the side plates. The required normal
440
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
Table 3 Acceleration input program for numerical study. Acceleration input El Centro
IMPVALL/I-ELC180, IMPVALL/I-ELC-UP, Imperial Valley, U.S., 1940/05/19
Unilateral Biaxial
Kobe
KOBE/KJM000, KOBE/KJM-UP, Kobe, Japan, 1995/01/16
Unilateral Biaxial
ChiChi
CHICHI/CHY028-N, CHICHI/CHY028-V, Chi-Chi, Taiwan, 1999/09/21
Unilateral Biaxial
AC156-1
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8.75 m in elevation) of a 7-story building (24 m in height) at Taipei City
Unilateral Biaxial
AC156-2
RRS specified in AC156. Isolated equipment is placed at 3rd floor (8 m in elevation) of a 3-story building (12 m in height) at Nantou County
Unilateral Biaxial
IEEE
RRS specified in IEEE Std 693TM-2005 for high performance level
Unilateral Biaxial
Spectral Acceleratrion (g)
Artificial acceleration history
Excitation direction
Spectral Acceleratrion (g)
Recorded earthquake history
Input earthquake information or response spectrum condition
4
Kobe I-ELC ChiChi AC156-1 AC156-2 IEEE
Damping Ratio = 5% 3
2
1
0
0
1
2
3
4
Period (sec)
(a) Horizontal (X) direction
5
Different input PA scales (g)
X X Z X X Z X X Z X X Z X X Z X X Z
25%
50%
75%
100%
0.08 0.08 0.05 0.21 0.21 0.09 0.19 0.19 0.09 0.13 0.13 0.06 0.25 0.25 0.13 0.25 0.25 0.20
0.16 0.16 0.11 0.41 0.41 0.17 0.38 0.38 0.17 0.25 0.25 0.13 0.50 0.50 0.25 0.50 0.50 0.40
0.23 0.23 0.16 0.62 0.62 0.26 0.57 0.57 0.26 0.38 0.38 0.19 0.75 0.75 0.38 0.75 0.75 0.60
0.31 0.31 0.21 0.82 0.82 0.34 0.76 0.76 0.34 0.5 0.5 0.25 1.00 1.00 0.5 1.00 1.00 0.80
4
Kobe I-ELC ChiChi AC156-1 AC156-2 IEEE
Damping Ratio = 5% 3
2
1
0
0
1
2
3
4
5
Period (sec)
(b) Vertical (Z) direction
Fig. 8. Acceleration response spectra of acceleration inputs (PA = 1 g) for numerical study.
force for built-in sliding friction is provided by two sets of linear spring modules with a constant compression length installed in the side plates. Considering the pounding prevention mechanism suggested in Wang et al.’s study, an arc rolling range of 21.6 mm with a curvature radius (R) of 100 mm is provided at the intersection of two inclines of V-shaped surfaces. The protected equipment above the seismic isolator is simulated by lead blocks with a total seismic reactive mass of 500 N-s2/m. The test installation on the shaking table is shown in Fig. 4. As detailed in Table 1, two recorded earthquake histories obtained from the Pacific Earthquake Engineering Research (PEER) Ground Motion Database [16], denoted as ElCentro and Kobe hereafter, and two generated acceleration histories compatible with the required response spectra (RRS) specified in AC156 [17,18] and IEEE std 693TM-2005 [19], denoted as AC156-1 and IEEE hereafter, are adopted as the unilateral acceleration inputs. The 5% damped acceleration response spectra of these acceleration inputs corresponding to a peak acceleration (PA) value of 1 g are shown in Fig. 5. The comparisons of experimental results and numerical predictions, including horizontal acceleration and displacement response histories together with hysteresis loops, subjected to unilateral50%-Kobe and unilateral-100%-AC156-1 are presented in Figs. 6 and 7, respectively. The values of coefficient of determination (R2 ), which are regarded as an expedient representation to quantitatively evaluate the correlation between experimental results and numerical predictions, are also shown in the figures. It should be noted that in this study, for simplicity and practicable feasibility, the force-displacement relationship of the built-in friction damp-
ing behavior is numerically represented by a rigid-plastic hysteretic model, rather than a sophisticated model which can fully take velocity- and temperature-dependent prosperities together with other uncertainties into consideration [20]. In addition, the nominal design values, rather than the actual ones identified by further measurement and component tests, including the seismic reactive masses of each component of the tested isolator and lead blocks, the sloping angle of the bearing plates, the curvature radius of the arc surfaces, and the built-in sliding friction damping force, are used for numerical analysis in this study. Therefore, the manufacture tolerances and modeling uncertainties are included in the numerical prediction results. Even so, it can be seen from the figures that the numerical predictions by using the derived simplified generalized analytical model have an acceptable agreement with the unilateral test results.
5. Influences of simplification assumptions on response prediction accuracy By ignoring the effects arising from higher order terms of sloping angles and vertical acceleration excitation, the exact generalized solutions given in Eqs. (10)–(18) can be simplified to Eqs. (19)–(27). However, if the sloping angle or vertical acceleration excitation becomes larger, the difference between the numerical results obtained from the exact and simplified generalized analytical models might be noticeable. The following will numerically discuss the influences of the simplification assumptions on the horizontal acceleration predictions.
441
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
5.1. Design cases A simplified model composed of a cylindrical roller with a sectional radius (r) of 15 mm, a superior bearing plate with a sloping angle of h1 , and an inferior bearing plate with a sloping angle of h2 , as shown in Fig. 2(a), is employed. Twenty-five design cases of sloped rolling-type seismic isolators are numerically studied, in which a commonly used combination of h1 and h2 respectively varying from 2 to 6° with an increment of 1° is designed, i.e. (h1 , h2 ) = (2–6°, 2–6°). To control the maximum transmitted horizontal acceleration response not greater than 0.15 g according to the simplified generalized equation of motion given in Eq. (20), as well as to have an acceptable self-centering performance, the built-in friction damping forces F D are designed to be the same for the twenty-five cases and are dominated by the design case of (h1 , h2 ) = (6°, 6°), i.e. F D = 301 N. The protected object above the seismic isolators is assumed to be an important facility, and the total seismic reactive mass of the protected facility and superior bearing plate, (M þ m1 ), is designed to be 1000 N-s2/m. The protected facility is ideally regarded as a single degree-of-freedom system in this numerical study for simplicity. Considering the pounding prevention mechanism suggested in Wang et al.’s study [15], an arc rolling range of 21.6 mm is designed at the intersection of two inclines of Vshaped surfaces. Therefore, the curvature radiuses R are designed to be 618 mm, 206 mm, 155 mm, 124 mm, and 104 mm corresponding to the sloping angles varying from 2° to 6°, respectively. By using the simplified generalized equation of motion given in Eq. (20), the maximum transmitted horizontal acceleration responses of the twentyfive design cases can be calculated in a preliminary manner and are summarized in Table 2. Since the built-in friction damping forces F D designed for the twenty-five cases are the same, a tendency can be easily observed from Eq. (20) and the calculated results listed in Table 2—the larger the total sloping angles (i.e. h1 þ h2 ), the higher the transmitted horizontal acceleration response obtained. As detailed in Table 3, three recorded earthquake histories obtained from the PEER Ground Motion Database, denoted as ElCentro, Kobe, and ChiChi hereafter, together with three generated acceleration histories compatible with the required response spectra (RRS) specified in AC156 and IEEE Std 693TM-2005, denoted as AC156-1, AC156-2, and IEEE hereafter, are adopted
0.12
as the unilateral and biaxial acceleration inputs with different PA scales (i.e. 25%, 50%, 75%, and 100% of the original PA value). The 5% damped acceleration response spectra of these acceleration inputs corresponding to a PA value of 1 g are shown in Fig. 8.
5.2. Numerical comparison The hysteresis loops of different design cases (h1 and h2 respectively varying from 2° to 6° with an increment of 2°) obtained from the exact and simplified generalized equations of motion under unilateral and biaxial inputs of 25%-AC156-1, 25%-IEEE, 100%Kobe, and 100%-ChiChi are presented in Figs. 9–12, respectively. In these figures and the following discussions, BE and UE denote the numerical predictions by using the exact generalized equation of motion given in Eq. (11) under biaxial and unilateral excitation, respectively, and US denotes those by using the simplified one given in Eq. (20) under unilateral excitation. It can be found from these figures that under some acceleration inputs with smaller PA scales, e.g. all design cases under 25%-AC156-1 and some design cases under 25%-IEEE, the roller still moves within the fixed curvature range (or the arc rolling range). Under this circumstance, the acceleration differences caused by the simplification assumptions are not very significant. When the roller moves between two sloped surfaces, e.g. most design cases under 25%-IEEE together with all design cases under 100%-Kobe and 100%-ChiChi, the horizontal acceleration predictions by using the exact generalized equation of motion reflect many obvious fluctuations rather than a perfect constant. It is mainly attributed to the effect of vertical acceleration excitation, which can be observed from the more noticeable differences between the horizontal acceleration predictions under the conditions of BE and UE compared to those under the conditions of UE and US. The differences of the maximum displacement responses caused by the simplification assumptions are very tiny visually and can be rationally neglected. To separately discuss the extent of influences arising from neglect of higher order terms of sloping angles and vertical acceleration excitation on the horizontal acceleration predictions in a clearer and quantitative manner, four indices, R21 , R22 , ER1 , and ER2 , are defined as
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
(i) θ1/ θ2=6o/6o
0.04 -0.04
Acceleration (g)
-0.12 0.12 0.04 -0.04 -0.12 0.12 0.04 BE UE US
-0.04 -0.12 -20
-10
0
10
20
-20
-10
0
10
20
-20
-10
0
Displacement (mm) Fig. 9. Comparison of hysteresis loops of different design cases under 25%-AC156-1.
10
20
442
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
0.16
(a) θ1/ θ2=2o/2o
(c) θ1/ θ2=6o/2o
(b) θ1/ θ2=4o/2o
0.08 0 -0.08
Acceleration (g)
-0.16 0.16
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
(i) θ1/ θ2=6o/6o
0.08 0 -0.08 -0.16 0.16 0.08 0
BE UE US
-0.08 -0.16 -50
-25
0
25
50
-50
-25
0
25
50
-50
-25
0
25
50
Displacement (mm) Fig. 10. Comparison of hysteresis loops of different cases under 25%-IEEE.
0.18 0.09
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
0 -0.09
Acceleration (g)
-0.18 0.18 0.09
(d) θ1/ θ2=2o/4o
(f) θ1/ θ2=6o/4o
(e) θ1/ θ2=4o/4o
0 -0.09 -0.18 0.18 0.09
(g) θ1/ θ2=2o/6o
(i) θ1/ θ2=6o/6o
(h) θ1/ θ2=4o/6o
0
BE UE US
-0.09 -0.18 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450
Displacement (mm) Fig. 11. Comparison of hysteresis loops of different design cases under 100%-Kobe.
2 Pm ðAUE Þi ðAUS Þi R21 ¼ 1 Pm i¼1 2 i¼1 ðAUE Þi ðAUE Þmean
ð34Þ
2 Pm ðABE Þi ðAUE Þi R22 ¼ 1 Pm i¼1 2 i¼1 ðABE Þi ðABE Þmean
ð35Þ
ER1 ¼
maxðjAUE jÞ maxðjAUS jÞ 100% max ðjAUS jÞ
ð36Þ
ER2 ¼
max ðjABE jÞ max ðjAUE jÞ 100% max ðjAUE jÞ
ð37Þ
where R21 is the coefficient of determination to quantitatively evaluate the difference between the horizontal acceleration predictions
under the conditions of UE and US (i.e. with and without considering the higher order terms of sloping angles); R22 is the coefficient of determination to quantitatively evaluate the difference between the horizontal acceleration predictions under the conditions of BE and UE (i.e. with and without considering the vertical acceleration excitation); ER1 and ER2 are the error ratios to quantitatively evaluate the maximum underestimate in horizontal acceleration prediction attributed to neglect of higher order terms of sloping angles and vertical acceleration excitation, respectively; AUS , AUE , and ABE are the horizontal acceleration predictions under the conditions of US, UE, and BE, respectively; the subscript i represents the analysis data point at time t i ; m represents the total number of analysis data points; and the subscript mean represents the mean value of total analysis data points. Through the further discussion on R21 (ER1 ) and R22 (ER2 ), the influences caused by neglect of higher order terms
443
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
0.18 0.09
(a) θ1/ θ2=2o/2o
(b) θ1/ θ2=4o/2o
(c) θ1/ θ2=6o/2o
(d) θ1/ θ2=2o/4o
(e) θ1/ θ2=4o/4o
(f) θ1/ θ2=6o/4o
(g) θ1/ θ2=2o/6o
(h) θ1/ θ2=4o/6o
0 -0.09 -0.18
Acceleration (g)
0.18 0.09 0 -0.09 -0.18 0.18
(i) θ1/ θ2=6o/6o
0.09 0
BE UE US
-0.09 -0.18 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450 -450 -300 -150
0
150 300 450
Displacement (mm) Fig. 12. Comparison of hysteresis loops of different design cases under 100%-ChiChi.
1.00 6% 5%
0.99
ER1
4% 0.98
3% 2%
0.97
2
3
4
5
6
θ1(or θ2) (degrees)
θ2 (or θ1)
o
o
o
o
1% 0%
o
Fig. 13. Variations of R21 values with various h1 and h2 under all unilateral acceleration inputs.
of sloping angles and vertical acceleration excitation on the horizontal acceleration predictions can be clarified, respectively.
2
3
4
5
6
θ1(or θ2) (degrees)
θ2 (or θ1)
o
o
o
o
o
Fig. 14. Variations of ER1 values with various h1 and h2 under all unilateral acceleration inputs.
The variations of R21 and ER1 values with respect to h1 and h2 under all the unilateral acceleration inputs are shown in Figs. 13 and 14, respectively. The variations of ER1 values with respect to h1 and h2 under each 100% unilateral acceleration input are shown in Fig. 15. As observed from Eqs. (11) and (20) together with Figs. 9–12, it is evident that an exchange of design values for h1 and h2 will not affect the horizontal responses of the protected
R21 values shown in Fig. 13, a large number of R22 values shown in Fig. 16 are farther from 1. In this numerical study, the minimum
object and superior bearing plate. The variations of R22 and ER2 values with various vertical PA values under all the biaxial acceleration inputs are shown in Figs. 16 and 17, respectively. The variations of ER2 values with various vertical PA values for the design cases of (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°) under all the biaxial acceleration inputs are shown in Fig. 18. To focus on the dynamic behavior of the sloped rolling-type seismic isolator when the roller moves between two sloped surfaces, as well as to avoid erroneous judgment due to small excitation and/or responses, the calculation results of the roller moving within the fixed curvature range (or the arc rolling range) are excluded in the figures and the following discussion.
R22 value is 0.675 for the design case of (h1 , h2 ) = (5°, 3°) under biaxial-100%-Kobe, which might be an unacceptable value in engineering practice. It is found from Figs. 14 and 15 that the larger the sloping angle is, the more apparent effect revealed from neglect of higher order terms of sloping angles on the ER1 value exerts. Nevertheless, as observed from Figs. 13–15, even if the design sloping angle becomes larger, the difference between the acceleration predictions by using the simplified and exact generalized equations of motion under horizontal excitation is very limited. In this numerical study, the maximum ER1 value is only 5.10% for the design case of (h1 , h2 ) = (6°, 6°) under unilateral-100%-ChiChi. Therefore, it can
It can be seen from Fig. 13 that the R21 values are in general very close to 1. In this numerical study, the minimum R21 value is 0.973 for the design case of (h1 , h2 ) = (6°, 6°) under unilateral-50%-Kobe, which is very acceptable in engineering practice. Compared to the
444
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
2%
3%
unilateral-100%-ElCentro
unilateral-100%-Kobe
ER 1
ER 1
2% 1%
1%
0%
0% 2
3
4
5
2
6
5
6
unilateral-100%-AC156-1
unilateral-100%-ChiChi
2%
ER 1
4%
ER 1
4
3%
6% 5%
3
θ 1 (or θ 2) (degrees)
θ 1 (or θ 2) (degrees)
3% 2%
1%
1% 0%
0% 2
3
4
5
6
2
θ 1 (or θ 2) (degrees)
3
4
5
6
θ1 (or θ 2) (degrees)
4%
6% unilateral-100%-AC156-2
5%
unilateral-100%-IEEE
3%
ER 1
ER 1
4% 2%
3% 2%
1% 1% 0%
θ 2 (or θ 1)
0% 2
3
4
5
6
2
θ1 (or θ2) (degrees)
3
4
2
o
o
3
5
o
4
o
5
o
6
6
θ 1 (or θ 2) (degrees)
Fig. 15. Variations of ER1 values with various h1 and h2 under 100% unilateral acceleration inputs.
50% 40%
ER2
30% Kobe EL ChiChi AC156-1 AC156-2 IEEE
20% 10% 0%
0
0.2
0.4
0.6
0.8
1
Vertical PA (g) Fig. 16. Variations of R22 values with various vertical PA scales under all biaxial acceleration inputs.
Fig. 17. Variations of ER2 values with various vertical PA scales under all biaxial acceleration inputs.
be concluded that the effect of higher order terms of sloping angles on the acceleration prediction accuracy is very limited and can be rationally negligible in engineering practice. By comparing Figs. 17 and 14, it can be seen that the difference between the maximum horizontal acceleration predictions by using the exact generalized analytical model under unilateral and biaxial excitation, i.e. the ER2 value, might be somewhat noticeable. In addition, it is apparent that a larger vertical PA value will lead to a more significant effect arising from neglect of vertical acceleration excitation on the ER2 value. As observed from Eq. (11), the term of €zg is multiplied by a parameter which is related to the
sloping angles h1 and h2 , i.e. ðcos h1 þ cos h2 Þðsin h1 þ sin h2 Þ= ½1 þ cosðh1 h2 Þ. Therefore, to more clearly understand the tendency of variations of ER2 values with various vertical PA values, the calculation results of five design cases, (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°), under all the biaxial acceleration inputs are marked with different symbols in Fig. 18. As observed from Fig. 18, it is very evident that the larger the vertical PA value is, the more significant effect without considering vertical acceleration excitation on the ER2 value produces. In this numerical study, the maximum ER2 value is 36.59% for the design case of (h1 , h2 ) = (2°, 2°) under biaxial-100%-IEEE, which might not be very
445
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
15%
25%
biaxial-ElCentro
biaxial-Kobe
10%
15%
ER2
ER2
20%
10%
5%
5% 0%
0
0.1
0.2
0.3
0%
0.4
0
0.1
Vertical PA (g)
0.2
0.3
Vertical PA (g)
30%
15% biaxial-ChiChi
biaxial-AC156-1
20%
ER2
ER2
10%
10%
0%
5%
0
0.1
0.2
0.3
0%
0.4
0
0.1
Vertical PA (g)
0.3
40%
30% biaxial-AC156-2
biaxial-IEEE
30%
ER2
ER2
20%
10%
20% 10% 2/2
0%
0.2
Vertical PA (g)
0
0.2
0.4
0.6
0%
0
Vertical PA (g)
0.3
3/3
4/4
0.6
5/5
6/6
0.9
Vertical PA (g)
Fig. 18. Variations of ER2 values with various vertical PA scales for the design cases of (h1 , h2 ) = (2°, 2°), (3°, 3°), (4°, 4°), (5°, 5°), and (6°, 6°) under all biaxial acceleration inputs.
acceptable compared to the maximum ER1 value aforementioned, i.e. 5.10%. As observed from Figs. 16–18, it is obvious that if the sloped rolling-type seismic isolator is subjected to an acceleration excitation with a considerable vertical component, its real dynamic responses might not be precisely predicted by using the simplified generalized analytical model with neglect of vertical acceleration excitation. More importantly, the significant underestimate in the maximum transmitted horizontal acceleration response under biaxial excitation due to neglect of vertical acceleration excitation will result in a non-conservative design result. Therefore, more attention should be paid to the effect of vertical acceleration excitation on the maximum transmitted horizontal acceleration response of the seismic isolator, especially when the vertical excitation possesses a larger PA value. 6. Conclusions Lee et al.’s [13,14] and Wang et al.’s [15] studies respectively proposed the simplified analytical models for different designs of sloped rolling-type seismic isolators. Lee et al.’s study focused on the dynamic behavior of a cylinder moving between a flat surface and a V-shaped surface along each principle horizontal direction, while Wang et al.’s study concentrated on that between two V-shaped surfaces designed with an identical sloping angle. To make the design of the sloped rolling-type seismic isolator more flexible, the exact and simplified generalized analytical models which can represent the dynamic behavior of a cylindrical roller moving between two V-shaped surfaces designed with arbitrary sloping angles are proposed in this study. Under the specific design
conditions, the proposed simplified model can make a valid generalization in Lee et al.’s and Wang et al.’s models by theoretical verification. In addition, the accuracy and practicability of the proposed simplified model are experimentally demonstrated. A series of numerical study are performed to quantitatively discuss the influences arising from neglect of higher order terms of sloping angles and vertical excitation on the transmitted horizontal acceleration responses of the seismic isolator, which has never been discussed in detail before. The numerical results reveal that the effect of vertical acceleration excitation plays a more crucial role in prediction accuracy compared to that of higher order terms of sloping angles. When the acceleration excitation possesses a considerable vertical component, adopting the exact generalized analytical model, rather than the simplified one, can achieve more precise and conservative analysis and design results for the sloped rolling-type seismic isolator. Acknowledgements The study was supported by the National Center for Research on Earthquake Engineering (NCREE), National Applied Research Laboratories (NARL) of Taiwan. The support is greatly acknowledged. References [1] Harvey Jr PS, Kelly KC. A review of rolling-type seismic isolation: historical development and future directions. Eng Struct 2016;125:521–31. [2] Guerreiro L, Azevedo J, Muhr AH. Seismic tests and numerical modeling of a rolling-ball isolation system. J Earthq Eng 2007;11(1):49–66. [3] Zhou Q, Lu X, Wang Q, Feng D, Yao Q. Dynamic analysis on structures baseisolated by a ball system with restoring property. Earthq Eng Struct Dyn 1998;27(8):773–91.
446
S.-J. Wang et al. / Engineering Structures 138 (2017) 434–446
[4] Harvey Jr PS, Gavin HP. Double rolling isolation systems: a mathematical model and experimental validation. Int J Non-Linear Mech 2014;61:80–92. [5] ISO-BaseTM seismic isolation platform, from the World Wide Web: http:// www.worksafetech.com/products/iso-base/. [6] Kasalanati A, Reinhorn AM, Constantinou MC, Sanders D. Experimental study of ball-in-cone isolation system. In: Proceedings of the ASCE structures congress XV, Portland, Oregon, April 13–16. p. 1191–5. [7] Lin TW, Chern CC, Hone CC. Experimental study of base isolation by free rolling rods. Earthq Eng Struct Dyn 1995;24(12):1645–50. [8] Jangid RS. Stochastic seismic response of structures isolated by rolling rods. Eng Struct 2000;22(8):937–46. [9] Mahmood H, Amirhossein S. Using orthogonal pairs of rollers on concave beds (OPRCB) as a base isolation system - Part I: analytical, experimental and numerical studies of OPRCB isolators. Struct Desi Tall Spec Build 2011;20 (8):928–50. [10] Jangid RS, Londhe YB. Effectiveness of elliptical rolling rods for base isolation. J Struct Eng ASCE 1998;124(4):469–72. [11] CRS, cosine curved rail system, from the World Wide Website: http://www. oiles.co.jp/en/menshin/building/menshin/kikimenshin/. [12] Tsai MH, Wu SY, Chang KC, Lee GC. Shaking table tests of a scaled bridge model with rolling type seismic isolation bearings. Eng Struct 2007;29(9):694–702.
[13] Lee GC, Ou YC, Niu T, Song J, Liang Z. Characterization of a roller seismic isolation bearing with supplemental energy dissipation for highway bridges. J Struct Eng ASCE 2010;136(5):502–10. [14] Ou YC, Song J, Lee GC. A parametric study of seismic behavior of roller seismic isolation bearings for highway bridges. Earthq Eng Struct Dyn 2010;39 (5):541–59. [15] Wang SJ, Hwang JS, Chang KC, Shiau CY, Lin WC, Tsai MS, et al. Sloped multiroller isolation devices for seismic protection of equipment and facilities. Earthq Eng Struct Dyn 2014;43(10):1443–61. [16] Pacific Earthquake Engineering Research (PEER) Ground Motion Database, from World Wide Website: http://peer.berkeley.edu/peer_ground_motion_ database/. [17] AC156. Acceptance criteria for seismic qualification by shake-table testing of nonstructural components and systems. ICC Evaluation Service Inc.; 2007. [18] Seismic design code for buildings. Taipei (Taiwan): Ministry of Interior; 2011. [19] IEEE Std 693TM-2005. IEEE recommended practice for seismic design of substations. NY: Institute of Electrical and Electronics Engineers (IEEE) Power Engineering Society; 2006. [20] Tsopelas P, Constantinou MC, Okamoto S, Fujii S, Ozaki D. Experimental study of bridge seismic sliding isolation systems. Eng Struct 1996;18(4): 301–10.