applied sciences Article
Advanced Hysteretic Model of a Prototype Seismic Isolation System Made of Polymeric Bearings Tomasz Falborski *
ID
and Robert Jankowski
ID
Faculty of Civil and Environmental Engineering, Gdansk University of Technology, 11/12 Narutowicza St., 80-233 Gdansk, Poland;
[email protected] * Correspondence:
[email protected]; Tel.: +48-58-347-21-17 Received: 12 January 2018; Accepted: 6 March 2018; Published: 9 March 2018
Featured Application: An advanced mathematical model to simulate complex hysteretic behaviour of a prototype seismic isolation system made of Polymeric Bearings in numerical simulations of base-isolated structures. Abstract: The present paper reports the results of acomprehensive study designed to verify the effectiveness of an advanced mathematical model in simulating the complex mechanical behaviour of a prototype seismic isolation system made of polymeric bearings (PBs). Firstly, in order to construct the seismic bearings considered in this research, a specially prepared flexible polymeric material with increased damping properties was employed. High effectiveness of PBs in reducing structural vibrations due to seismic excitations was already confirmed during a previously conducted shaking table investigation. In order to accurately capture the complex mechanical behaviour of PBs in numerical analysis, the proposed mathematical model defines the lateral force as a nonlinear function of shear displacement and deformation velocity. Function parameters were evaluated by fitting the general form of the mathematical model into the experimentally obtained hysteresis loops, using the least squares optimisation method. The effectiveness of the mathematical model was verified by comparing the experimental data (i.e., seismic response of a 1.20 m high single-storey and a 2.30 m high two-storey structure models under various ground motions) with the results obtained from the detailed numerical analysis, where the experimental models were idealized as multi-degree-of-freedom systems. The results obtained from this investigation explicitly confirmed that the proposed mathematical model can be successfully adopted to accurately capture complex mechanical behaviour of PBs in numerical studies. Keywords: hysteretic behaviour; polymeric bearings; base isolation; seismic performance; dynamic excitations
1. Introduction and Motivation Earthquake-induced ground motions are identified as one of the most unpredictable, and more importantly, destructive threats to civil engineering structures. Strong seismic events may cause loss of life and property damage through soft-storey and weak-storey failures, pounding-induced damage (see, for example, [1,2]), brittle fracture and collapse of masonry and concrete structures, or failures resulting from excessive plastic deformations in steel frame structures. Therefore, developing earthquake protection systems in order to improve safety and reliability of structures is an issue of the highest priority in many seismically active regions (see, for example, [3,4]). In general, the family of earthquake protective systems consists of passive, active, and semi- active (hybrid) control systems. Base isolation, which is an example of passive control methods, is recognized as one of the most effective and widely adopted approaches of protecting structures during strong ground motions (see, for example, [5,6]). This solution is also considered to be the safest and most reliable method, Appl. Sci. 2018, 8, 400; doi:10.3390/app8030400
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because, in contrast to active and hybrid (semi-active) devices (see, for example, [7–9]), its activation requires no external source of energy, that may not necessarily be accessible due to earthquake-induced devastation. Broadly, base isolation system works by decoupling the building from the horizontal components of an earthquake ground motion by incorporating a layer with relatively low horizontal stiffness between the structure and its foundation (see, for example, [10–12]). Thus, the primary concept of base isolation systems is to modify structural dynamic properties by elongating the fundamental period of vibration and introducing additional damping so that base-isolated structures may respond more favourably to earthquake ground motions (see, for example, [13,14]). The philosophy of base isolation is well established and commonly accepted in many seismically active regions including California, Japan, and New Zealand. Base isolators, such as lead-rubber bearings (LRBs), high-damping rubber bearings (HDRBs), and friction pendulum bearings (FPBs), are commonly used in practice in order to improve structural safety, and thus, minimize property damage. It should also be mentioned, however, that base isolation devices made with the use of rubbery materials may be susceptible to aging resulting in degradation of stiffness and damping properties. Therefore, every type of rubber bearing requires routine maintenance to ensure the structural safety and reliability. Both numerical and experimental results of research focused on the effectiveness of various types of base isolation system have been widely published (see, for example, [15–18]). More importantly, many of the completed base-isolated buildings have already experienced strong ground motions, and their dynamic performance has been just as predicted, which explicitly confirms high effectiveness of base isolation systems in suppressing structural vibrations (see, for example, [19,20]). However, in order get an accurate insight into the seismic performance of base-isolated structures in numerical analysis, it is absolutely necessary to accurately capture the mechanical behaviour of base isolation system. This issue is particularly important at the design stage of new structures or retrofitting existing ones. Moreover, with the increasing number of base isolators exhibiting different mechanical behaviour, which may also be triggered due to new materials employed, this task is very challenging and often requires and individual approach to a specific type of seismic isolation device. In nonlinear structural dynamics, there are generally a few approaches, which can be incorporated to simulate the nonlinear behaviour of base isolation devices. These mathematical models, however, are evaluated for different base isolators, which are characterized by their own specific load-displacement behaviour, and thus, may not be fully appropriate for detailed numerical analysis of structures isolated with different types of base isolation systems. Moreover, most of these models refer to either elastomeric bearings (see, for example, [21–24]), or HDRBs (see, for example, [25,26]), which are also susceptible to vertical loads. Motivated by the preceding discussion, the present paper was designed to verify the effectiveness of an advanced hysteretic model in simulating complex force-deformation behaviour of a prototype seismic isolation system made of PBs, which are constructed with the use of a specially prepared for this purpose polymeric material with improved damping properties [27]. High effectiveness of PBs in reducing structural vibrations due to seismic excitations was confirmed during previously conducted extensive shaking table investigation and the results have already been published (see [28] for details). In order to achieve the above-mentioned aim of the present investigation, the following specific objectives were established: (1) (2)
(3) (4)
Perform dynamic oscillatory test in order to determine the energy dissipation properties of the base isolation system made of PBs based on the experimentally obtained hysteresis loops. Propose a general form of an advanced mathematical model, which defines the lateral force as a nonlinear function of shear displacement and deformation velocity, in order to accurately capture the nonlinear mechanical behaviour of PBs in numerical analysis of base-isolated structures. Evaluate the function parameters by fitting the general form of the proposed mathematical model into the experimentally obtained hysteresis loops using the least squares optimisation method. Implement the proposed mathematical model into the computational analysis in order to perform numerical evaluation of the seismic response of the previously examined experimental models
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(4) Implement the proposed mathematical model the computational analysis order to and with dynamic characteristics typical for low-rise andinto medium-rise buildings, bothin fixed-base perform numerical evaluation of the seismic response of the previously examined experimental base-isolated with the use of PBs, to a number of different ground motions and a mining tremor. models with dynamic characteristics typical for low-rise and medium-rise buildings, both fixedVerifybase the effectiveness of the proposed modelofindifferent simulating nonlinear behaviour of and base-isolated with the use mathematical of PBs, to a number ground motions and a PBs bymining comparing the results of the detailed lumped-parameter analysis with the experimentally tremor. (5) Verify the effectiveness obtained shaking table data. of the proposed mathematical model in simulating nonlinear behaviour of PBs by comparing the results of the detailed lumped-parameter analysis with the experimentally
2. Polymeric Bearings andtable Previous obtained shaking data. Shaking Table Investigation Precise description of and PBs,Previous detailed material testing (i.e., static tension and compression tests, 2. Polymeric Bearings Shaking Table Investigation Dynamic Mechanical Analysis, etc.), and finally the results of the extensive shaking table investigation, Precise description of PBs, detailed material testing (i.e., static tension and compression tests, which was conducted in order to verify the effectiveness of this prototype base isolation system Dynamic Mechanical Analysis, etc.), and finally the results of the extensive shaking table investigation, in reducing vibrations to ground motions,ofhas been (see [27,28]). which structural was conducted in orderdue to verify the effectiveness thisalready prototype basepublished isolation system in To briefly outline the keyvibrations results it due should be mentioned were constructed the use of a reducing structural to ground motions,that has PBs already been published with (see [27,28]). specially for the thiskey purpose polymeric material with increased energy dissipation To prepared briefly outline results it should be mentioned that PBs were constructed with the usepotential. of a specially prepared for this purpose polymeric material with increased energy dissipation potential. Energy dissipation potential of PBs was confirmed by a relatively high value of the loss factor obtained Energy dissipation potential of PBstest. wasMoreover, confirmed static by a tension relativelyand high value of theresults loss factor from the Dynamic Mechanical Analysis compression explicitly obtained from the Dynamic Mechanical Analysis test. Moreover, static tension and compression demonstrated that the tested polymer is considerably nonlinear and its mechanical behaviour is results explicitly demonstrated that the tested polymer is considerably nonlinear and its mechanical strongly dependent on the strain rate. behaviour is strongly dependent on the strain rate.
(a)
(b)
Figure 1. Shaking table investigation: fixed-base two storey experimental model (a) close-up view of
Figure PB 1. Shaking table investigation: used in base-isolated model (b). fixed-base two storey experimental model (a) close-up view of PB used in base-isolated model (b).
In the final step, an extensive shaking table investigation was conducted. This experimental research program was performed with the use of a middle-sized one-directional shaking table platform, which was employed to simulate the lateral forces and displacements caused by seismic excitations.
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In the Appl. Sci. 2018,final 8, 400 step,
an extensive shaking table investigation was conducted. This experimental 4 of 18 research program was performed with the use of a middle-sized one-directional shaking table platform, which was employed to simulate the lateral forces and displacements caused by seismic excitations. Dynamic Dynamic response response of of two two experimental experimental structure structure models models with with dynamic dynamic characteristics characteristics typical typical for for low-rise and medium-rise buildings (i.e., a 1.20 m high single-storey, and a 2.30 m high two-storey low-rise and medium-rise buildings (i.e., a 1.20 m high single-storey, and a 2.30 m high two-storey structures frequencies of of 3.34 3.34 Hz structures with with fundamental fundamental frequencies Hz and and 1.94 1.94 Hz, Hz, respectively), respectively), both both fixed-base fixed-base and and base-isolated with the use of PBs, to a number of different ground motions and a mining tremor base-isolated with the use of PBs, to a number of different ground motions and a mining tremor (as (as an example of so-called induced seismicity) was deeply analyzed. As an example, the 2.30 m high an example of so-called induced seismicity) was deeply analyzed. As an example, the 2.30 m high two-storey model mounted mounted on on the the shaking with the two-storey experimental experimental model shaking table table platform, platform, together together with the close-up close-up view of PB, is presented in Figure 1. The results of this experimental research explicitly confirmed view of PB, is presented in Figure 1. The results of this experimental research explicitly confirmed the the effectiveness of seismic isolation system made of PBs in mitigating seismic-induced structural effectiveness of seismic isolation system made of PBs in mitigating seismic-induced structural vibrations. accelerations recorded the tested tested structures structures subjected subjected to number vibrations. Peak Peak lateral lateral accelerations recorded at at the the top top of of the to aa number of different ground motions were decreased by approximately 11–39% and 12–57% for the single-storey of different ground motions were decreased by approximately 11–39% and 12–57% for the singleand two-storey structure models,models, respectively. Also, it isAlso, worth that a full that scaleaversion of storey and two-storey structure respectively. it mentioning is worth mentioning full scale the PB to be used for real structures was developed, submitted to the Polish Patent Officeand patented version of the PB to be used for real structures was developed, submitted to the Polish Patent in 2017. patented in 2017. Officeand 3. Dynamic Oscillatory Test 3. Dynamic Oscillatory Test The first step in the present research was to perform dynamic oscillatory tests in order to analyze The first step in the present research was to perform dynamic oscillatory tests in order to analyze the load-deformation behaviour of PBs under different horizontal excitation frequencies. In order the load-deformation behaviour of PBs under different horizontal excitation frequencies. In order to to obtain hysteresis loops, four PBs supporting a concrete slab (dimensions of the concrete base obtain hysteresis loops, four PBs supporting a concrete slab (dimensions of the concrete base slab are slab are 50 cm × 50 cm × 7 cm), were mounted on the shaking table platform and subjected to 50 cm × 50 cm × 7 cm), were mounted on the shaking table platform and subjected to harmonic harmonic excitations with gradually increasing amplitudes resulting in different shear strain levels. excitations with gradually increasing amplitudes resulting in different shear strain levels. The The experimental testing was carried out at the excitation frequencies of 1 Hz, 2 Hz, and 5 Hz, experimental testing was carried out at the excitation frequencies of 1 Hz, 2 Hz, and 5 Hz, which which correspond to the range of excitation frequencies typical for earthquakes and mining tremors. correspond to the range of excitation frequencies typical for earthquakes and mining tremors. The The response of the system to cyclic horizontal loading was measured using a laser displacement response of the system to cyclic horizontal loading was measured using a laser displacement sensor sensor and an uniaxial accelerometer, as indicated by Figure 2. and an uniaxial accelerometer, as indicated by Figure 2.
Figure 2. Base-isolated concrete slab during dynamic oscillatory test. Figure 2. Base-isolated concrete slab during dynamic oscillatory test.
Experimentally obtained hysteresis loops for three different excitation frequencies are presented Experimentally obtained hysteresis three different excitation frequencies presented in Figure 3 (it should be mentioned thatloops due for to technical restrictions of the shaking are table, limited in Figureof3hysteresis (it shouldloops be mentioned that due to technical restrictions of the shaking number at the excitation frequency of 1 Hz could be obtained). Basedtable, on thelimited loops, number of hysteresis loops at the excitation frequency of 1 Hz could be obtained). Based on the loops, the equivalent damping ratio, ξ , of the base isolation system was determined using the following the equivalent damping ratio, ξ , of the base isolation system was determined using the following formula (see [26]): formula (see [26]): ∆W ΔW ξξ== (1) (1) 2π ⋅·W W
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where the hysteresis hysteresis loop) loop) where ∆W theenergy energydissipated dissipatedin in one one cycle cycle of of motion motion (the (the area area enclosed enclosed by by the ΔW isisthe and W is a strain energy induced in a seismic isolation bearing given by: and W is a strain energy induced in a seismic isolation bearing given by: WW==FF(u) (u) ⋅·uu
(2) (2)
where u is a maximum shear deflection and F(u) is a lateral force recorded at u. Average Average values values of of equivalent damping damping ratios ratios for for hysteresis hysteresis loops, loops, recorded recorded at at three three excitation excitation frequencies, frequencies, are are briefly briefly summarized in Table Table 1. 1. 200
f = 1 Hz
150
Lateral force (N)
100 50 0 -50 -100 -150 -200 -4
-3
-2
-1 0 1 Shear displacement (m)
2
3
4 -3
x 10
(a) 800
f = 2 Hz
600
Lateral force (N)
400 200 0 -200 -400 -600 -800 -0.015
-0.01
-0.005 0 0.005 Shear displacement (m)
(b) Figure 3. Cont.
0.01
0.015
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800
f = 5 Hz
600
Lateral force (N)
400 200 0 -200 -400 -600 -800 -0.015
-0.01
-0.005 0 0.005 Shear displacement (m)
0.01
0.015
(c) Figure 3. Experimentally obtained hysteresis loops at excitation frequencies of 1 Hz (a); 2 Figure 3. Experimentally obtained hysteresis loops at excitation frequencies of 1 Hz (a); 2 Hz (b); Hz (b); and 5 Hz (c). and 5 Hz (c).
Table 1. Average equivalent damping ratios. Table 1. Average equivalent damping ratios.
Excitation Frequency Excitation Frequency 1 Hz 1 Hz 2 Hz 2 Hz 5 Hz 5 Hz
Damping Ratio, ξ Damping Ratio, ξ 49.5% 49.5% 30.5% 30.5% 20.4% 20.4%
The results obtained from the dynamic oscillatory tests exhibit pronounced hysteretic behaviour The results obtained from the dynamic oscillatory tests of exhibit pronounced behaviour of PBs, which confirms high energy dissipation properties this base isolationhysteretic system. Moreover, of PBs, which confirms energyloops dissipation properties this base isolation system. Moreover, experimentally obtainedhigh hysteresis indicate that PBs of exhibit higher stiffness at lower shear experimentally obtained hysteresis loops indicate that PBs exhibitshear higher stiffnessthe at lower shear strain levels. Close inspection of Figure 3 shows that with increasing deflection, slope of the strain levels. Close inspection of Figure 3 shows that with increasing shear deflection, the slope of hysteresis loop decreases. The same effect, which is often referred to as the stiffness degradation, can the hysteresis loopfor decreases. The same effect, devises, which is such often as referred to as the Additionally, stiffness degradation, also be observed other seismic isolation HDRBs [12]. results can also be observed for other seismic isolation devises, such as HDRBs [12]. Additionally, results obtained from this investigation clearly demonstrate that the higher the excitation frequency, the obtained from this investigation clearly demonstrate that the higher the excitation frequency, the lower the value of equivalent damping ratio, ξ , as indicated by Table 1. The cyclic loading lower tests the value of the equivalent damping ratio, ξ,by asPBs indicated bythe Table 1. The cyclic20–49%, loadingwhile tests indicate that indicate that damping ratio provided is within range of about for the most the damping ratio provided by PBs is within the range of about 20–49%, while for the most commonly commonly used base isolators (LRBs, HDRBs, FPBs), the damping ratio is approximately 10–20%. used base isolators (LRBs, HDRBs, FPBs), the damping ratio is approximately 10–20%. 4. Mathematical Model 4. Mathematical Model In order to accurately simulate the complex mechanical behaviour of PBs under lateral dynamic In order to accurately simulate the complex mechanical behaviour of PBs under lateral dynamic excitations in numerical analysis, an advanced hysteretic model was firstly developed. The general excitations in numerical analysis, an advanced hysteretic model was firstly developed. The general form of this mathematical model is presented in Equations (3)–(5). It defines the lateral force, Fb , as form of this mathematical model is presented in Equations (3)–(5). It defines the lateral force, Fb , ub. b(t) u b u(t)b (,t)and aasnonlinear function of the shear displacement, the deformation velocity, a nonlinear function of the shear displacement, , and the deformation velocity,u In the the (t). . In proposed mathematical kb ,kand cb represent the stiffness and damping of the PBs,of respectively. c b represent the stiffness and damping the PBs, proposed mathematicalapproach approach b , and Function parameter a defines the initial stiffness of PBs, parameter a depicts stiffness degradation 2 a 2 depicts stiffness respectively. Function 1parameter a1 defines the initial stiffness of PBs, parameter for higher strain rates (i.e., deformation velocity), parameters a and a are responsible broadening 3 4 a 4 are degradation for higher strain rates (i.e., deformation velocity), parameters a 3 for and and rounding hysteresis loops. Finally, parameter a5 defines damping properties of PBs. The function responsible for broadening and rounding hysteresis loops. Finally, parameter a 5 defines damping properties of PBs. The function parameters a1 – a 5 can be estimated by curve fitting the general form
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parameters a1 –a8,5 xcan estimated Appl. Sci. 2018, FORbe PEER REVIEW by curve fitting the general form of the mathematical model 7 of 18to the experimentally obtained hysteresis loops using the least squares optimization method. of the mathematical model to the experimentally obtained hysteresis loops using the least squares . . . optimization method.Fb (u (t), ub (t)) = kb (u (t), ub (t)) · ub (t) + cb · ub (t) b
b
Fb (u.b (t),u b (t)) = k b (u b (t),u b (t)) ⋅ u b (t)a+ cb ⋅ u b (t)
kb (u b (t), ub (t)) = a1 +
.
k b (u b (t),u b (t)) = a1 +
(3)
(3)
2
cos h[a 3 ub (at2)] · cos h[a 4 ub (t)]
3 u b (t)] ⋅ cosh[a 4 u b (t)] ccosh[a b = a5
(4) (4)
(5)
cb =a 5
(5) with In the present study, the curve fitting procedure was conducted for hysteresis loops different In excitation frequencies exemplary results for the frequency of 2 loops Hz are shown the present study, the and curvethe fitting procedure was conducted for hysteresis with different excitation frequencies and the exemplary results for the frequency of 2 Hz are shown below. below. The evaluated set of function parameters is summarized in Table 2. Comparison between The evaluated of functionobtained parameters is summarized Table 2. in Comparison experimentally andset numerically hysteresis loops isinpresented Figure 4. between It should be experimentally and numerically obtained hysteresis loops is presented in Figure It should be underlined that the normalized mean square error calculated for the fit is equal to4.4.44%. underlined that the normalized mean square error calculated for the fit is equal to 4.44%.
Table 2. Estimated values for function parameters for loops at excitation frequency of 2 Hz. Table 2. Estimated values for function parameters for loops at excitation frequency of 2 Hz. Function Parameter Function Parameter a1 a1 a2 a 2 a3 a 3 a4 a 4 a5 a5
Estimated EstimatedValue Value UnitUnit 38.657 38.657
N·m−1
36.301 36.301
N·m N−1·m−1
23.253 23.253
s·ms−1·m
331.626 331.626 2.1932 2.1932
N ·m−1 −1
−1 m−1m kg · s−1 −1
kg·s
800 600
Lateral force (N)
400 200 0 -200 -400 -600 -800 -0.015
-0.01
-0.005 0 0.005 Shear displacement (m)
simulation test data 0.01 0.015
Figure 4. Experimental and numerical hysteresis loops at excitation frequency of 2 Hz. Figure 4. Experimental and numerical hysteresis loops at excitation frequency of 2 Hz. 5. Numerical Analysis
5. Numerical Analysis
The final stage of the current investigation was focused on implementation the proposed mathematical model PBs current into detailed lumped-parameter numerical (see, for example, The final stage ofofthe investigation was focused onanalysis implementation the [29]). proposed Therefore,model in orderoftoPBs verify the effectiveness of the proposed numerical hysteretic model, the(see, dynamic response [29]). mathematical into detailed lumped-parameter analysis for example, of a single-storey two-storey models, base-isolated with the PBs, dynamic to a Therefore, in order and to verify the structure effectiveness ofboth the fixed-base proposedand hysteretic model, number of ground motions was evaluated and compared to the previously obtained experimental response of a single-storey and two-storey structure models, both fixed-base and base-isolated data. The fixed-base and base-isolated single-storey structure models were idealized as a single-degreewith PBs, to a number of ground motions was evaluated and compared to the previously of-freedom (1-DOF), and two-degree-of-freedom (2-DOF) systems, respectively. Consequently, the obtained experimental data. The fixed-base and base-isolated single-storey structure models fixed-base and base-isolated two-storey structures were modelled as a two-degree-of-freedom (2- were
idealized as a single-degree-of-freedom (1-DOF), and two-degree-of-freedom (2-DOF) systems, respectively. Consequently, the fixed-base and base-isolated two-storey structures were modelled
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as a two-degree-of-freedom (2-DOF), and three-degree-of-freedom (3-DOF) systems, respectively. Two numerical models, each with and without base-isolation system, were subjected to the El Centro earthquake of 1940, the San Fernando earthquake of 1971, the Loma Prieta earthquake of 1989, the Northridge earthquake of 1994, and finally the Polkowice (Poland) tremor of 2002 as an example of mining-induced seismicity, which is also a major concern of professional and research communities (see, for example, [30,31]). For numerical evaluation of differential equation of motion, the unconditionally stable Newmark’s average acceleration method was applied [29], as it is the most frequently adopted integration procedure is seismic analysis of structures. 5.1. Single-Storey Structure Model In order to evaluate dynamic response of the fixed-based single-storey model in numerical analysis, the structure was idealized as 1-DOF system, for which the differential equation of motion is given in Equation (6). This simple numerical model is characterized by three parameters: lumped mass m1 lumped at the roof level (i.e., first storey level), lateral stiffness k1 , and viscous damping c1 . The exact values of these parameters, which are also given in Equations (7)–(9), were already determined during the previously conducted shaking table investigation (see [28] for details): ..
.
..
m1 · u(t) + c · u(t) + k1 · u(t) = −m1 · ug (t)
(6)
ω = 2·π·f
(7)
k1 = ω2 · m1
(8)
c1 = 2 · m1 · ω · ξ
(9)
Employing parameters determined in the previous study ([28]), i.e., dominant frequency of the fixed-base single-storey structure, f, equal to 3.34 Hz (and, consequently, angular frequency, ω, equal to 20.98 rad−1 according to Equation (7)), lumped-mass concentrated at the top level, m1 , equal to 46.76 kg, and damping ratio, ξ, equal to 0.5%, into the Equations (8) and (9), stiffness and damping kg N was obtained as k1 =20,571 m and c1 = 10.48 s , respectively. The single-storey structure model mounted on the foundation base plate supported with PBs (as shown in Figure 2) was considered as 2-DOF system, for which the differential equation of motion can be presented in a general matrix form given in Equation (10). ..
..
.
M · u(t) + C · u(t) + K · u(t) = −MI · ug (t)
(10) ..
In this formula, M, C, and K denote mass, damping, and stiffness matrices, respectively, u(t), .. . u(t), u(t) are the acceleration, velocity and displacement vectors, respectively, and ug (t) stands for the acceleration input ground motion. Additional matrix I is the influence coefficient matrix, having 1 for elements corresponding to degrees of freedom in the direction of the applied ground motion and 0 for other degrees of freedom. The structure was mounted on a base plate of mass mb which is supported . on a base isolation system made of PBs characterized by lateral stiffness kb (u b (t), ub (t)), and viscous damping cb , as defined in Equations (4) and (5). Accordingly, for the base-isolated single-storey model .. . idealized as 2-DOF system, matrices M, C, K, u(t), u(t), u(t) and I are defined in Equations (11)–(15). " M= " C=
mb 0
cb + c1 − c1
0 m1
#
− c1 c1
(11) # (12)
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# . kb ub (t), ub (t) + k1 −k1 K= − k1 k1 k (u (t), u (t)) + k 1 - k 1 K= b b( b ) - k11 k 1 I= 11 ) ( ) (I =. 1 ) ( .. . .. ub (t) ub (t) ub (t) , u(t) = , u(t) = u(t) = . .. u1u(bt(t) u1 (t) u b (t) ) u b (t)u1 (t)
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(13)
(13)
(14) (14)
(15)
(t) = u (15) , u (t) = , u(t) = u 1 (t) u1 (t) history u 1 (t) Numerically obtained acceleration-time plots for the single-storey structure model,
both fixed-base and base-isolated considered as 1-DOF systems, respectively, are compared Numerically obtained acceleration-time history and plots2-DOF for the single-storey structure model, both and base-isolated considered as 1-DOF systems, respectively, compared with with thefixed-base experimental data (see [28]) in Figures 5 andand 6. 2-DOF Moreover, computed peakare lateral accelerations the experimental data (see [28]) in Figures 5 and 6. Moreover, computed peak lateral accelerations recorded at the top of the analyzed single-storey structure model, with and without the PBs, recorded at the of the single-storey structure model, with and without theare PBs,also are briefly are briefly reported intop Table 3. analyzed Additionally, reduction levels in lateral accelerations presented reported in Table 2. Additionally, reduction levels in lateral accelerations are also presented and and compared with the results of the previously performed shaking table investigation [28]. compared with the results of the previously performed shaking table investigation [28]. 15
15 fixed-base numerical model (1-DOF)
fixed-base experimental model
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(g)
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2
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Time (s)
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Figure 5. Experimentally determined (left) and numerically obtained (right) (i) (j) time-acceleration history Figure 5.plots Experimentally determined (left)model and numerically obtained (right) time-acceleration history for the fixed-base single-storey during different ground motions: the 1940 El Centro Figure 5. Experimentally determined (left) and numerically obtained (right) time-acceleration history plots forearthquake the fixed-base single-storey modelearthquake during different motions: the 1940(e,f), El Centro (a,b), the 1971 San Fernando (c,d), the ground 1989 Loma Prieta earthquake the plots1994 for (a,b), the fixed-base single-storey during different motions: the 1940 earthquake El Centro Northridge earthquake (g,h),model and the 2002 Polkowice mining tremor (i,j). earthquake the 1971 San Fernando earthquake (c,d),ground the 1989 Loma Prieta (e,f), earthquake (a,b), the 1971 San Fernando earthquake (c,d), the 1989 Loma Prieta earthquake (e,f), the the 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining tremor (i,j). 15 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining15tremor (i,j). base-isolated experimental model 10
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15 10 15
5 10
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5 0 -5 -10
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Figure 6. Cont.
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base-isolated experimental model 3
3
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30
(f) base-isolated experimental model
8
1 0 -1
1 0 -1
-2
-2
-3
-3
-4
25
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10
Acceleration at top (m/s 2)
20 Time (s)
0
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4
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8
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14
-4
0
2
4
6
8
10
12
14
Time (s)
Time (s)
(i)
(j)
Figure 6. Experimentally determined (left) and numerically obtained (right) time-acceleration history
Figure 6. Experimentally determined (left) andduring numerically obtained (right) time-acceleration plots for the base-isolated single-storey model different ground motions: the 1940 El Centro history plotsearthquake for the base-isolated during ground motions: the 1940 El Centro (a,b), the 1971single-storey San Fernandomodel earthquake (c,d),different the 1989 Loma Prieta earthquake (e,f), the 1994 Northridge earthquake (g,h), and the 2002 Polkowice(c,d), mining earthquake (a,b), the 1971 San Fernando earthquake thetremor 1989 (i,j). Loma Prieta earthquake (e,f), the 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining tremor (i,j). Table 2. Results of the seismic tests for the single-storey structure model.
Table 3. Results of the Computed seismic tests the single-storey structure model. Numerically Peak for Lateral Dynamic Excitation
Acceleration for the Single-Storey Structure Model (m/s2) Peak Lateral Numerically Computed Fixed-Base (1-DOF Base-Isolated (2Acceleration for the Single-Storey 2) System) Model (m/sDOF System) Structure
Dynamic Excitation 1940 El Centro earthquake Fixed-Base 11.39 System) NS component, original record (1-DOF with PGA = 3.070 m/s2 1940 El Centro earthquake 1971 San Fernando earthquake 15.64 NS component, original record 11.39 Pacoima Dam station, N74°E with PGA = 3.070 m/s2 1971 San Fernando earthquake Pacoima Dam station, N74◦ E component, scaled record with PGA = 5.688 m/s2
15.64
Base-Isolated 7.21 (2-DOF System)
Reduction Based on Reduction Based on Numerical Analysis Shaking Table (%) Investigation (%) [28] Reduction Based on Reduction Based on Shaking Table Numerical Analysis (%) Investigation (%) [28] 36.70 38.7
9.66 7.21
38.24 36.70
9.66
38.24
32.8
38.7
32.8
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Table 3. Cont. Numerically Computed Peak Lateral Acceleration for the Single-Storey Structure Model (m/s2 )
Dynamic Excitation
Reduction Based on Numerical Analysis (%)
Reduction Based on Shaking Table Investigation (%) [28]
Fixed-Base (1-DOF System)
Base-Isolated (2-DOF System)
1989 Loma Prieta earthquake Corralitos station, NS component, scaled record with PGA = 3.158 m/s2
13.86
8.16
41.16
40.2
1994 Northridge earthquake Santa Monica station, EW component, scaled record with PGA = 4.332 m/s2
8.73
7.85
10.08
11.3
2002 Polkowice mining tremor NS component, original record with PGA = 1.634 m/s2
3.50
2.17
38.00
38.2
Where: PGA: Peak Ground Acceleration, NS: North-South, EW: East-West.
5.2. Two-Storey Structure Model In the next step of the numerical investigation, 2-DOF and 3-DOF systems were applied to simulate the dynamic behaviour of the fixed-base and base-isolated two-storey structure models, respectively. The general matrix form of differential equation of motion has already been presented in Equation (10). Similarly to the single-storey structure model, for the fixed-based two-storey structure .. . model, matrices M, C, K, u(t), u(t), u(t) and I are defined as given in Equations (16)–(20), whereas for the base-isolated case, these matrices are presented in Equations (21)–(25): " M= " C=
m1 0
c1 + c2 − c2
0 m2
#
−c2 c2
(16) #
# k1 + k2 −k2 K= − k2 k2 ( ) 1 I= 1 ( .. ) ( . ) ( ) .. . u1 ( t ) u1 (t) u1 ( t ) u(t) = , u(t) = , u(t) = .. . u2 ( t ) u2 (t) u2 ( t ) mb 0 0 M = 0 m1 0 0 0 m2 cb + c1 −c1 0 C = −c1 c1 + c2 −c2 0 −c2 c2 . −k1 0 kb (u b (t), ub (t)) + k1 K= −k1 k1 + k2 −k2 0 −k2 k2
(17)
"
(18)
(19)
(20)
(21)
(22)
(23)
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I=
1 1 1
(24)
.. Appl. Sci. 2018, 8, x FOR PEER REVIEW
. ub (t) u. b (t) .. . .. u(t) = u(t) = uu1(t)(t) u1 (t) u , (t) b u.. (t) .b (t) u (t) (t) = 1 , u (t) = uu 2u 2 , 1 (t)
ub ( t ) , u(ut)(t)= u1 (t) b u (t) u (t) 2 u(t) = 1
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(25)
(25) u 2 (t) u 2 (t) Numerically obtained acceleration-time history plots for the two-storey structure model, both fixedu 2 (t)
base and base-isolated considered as 2-DOF and 3-DOF systems, respectively, are compared with the Numerically obtained acceleration-time history plots for the two-storey structure model, both fixed-basedata and (see base-isolated considered as 2-DOF and 3-DOF systems, respectively, are compared experimental [28]) in Figures 7 and 8. Computed peak lateral accelerations recorded at the experimental data (seestructure [28]) in Figures 7 and 8. Computed peak lateral accelerations recorded top ofwith thethe analyzed two-storey model, with and without the PBs, are briefly reported in at the top of the analyzed two-storey structure model, with and without the PBs, are briefly reported Table 4. Additionally, reduction levels in lateral accelerations are also presented and compared with in Table 3. Additionally, reduction levels in lateral accelerations are also presented and compared the results of the previously performed shaking table investigation [28]. with the results of the previously performed shaking table investigation [28]. 6
6
fixed-base experimental model
fixed-base numerical model (2-DOF) 4
Acceleration at top (m/s 2)
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Figure 7. Cont.
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fixed-base experimental model
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4 6
2 Acceleration at top (m/s ) (m/s 2) Acceleration at top
2 Acceleration at top (m/s ) (m/s 2) Acceleration at top
6 8
2 4 0 2 -2 0 -4 -2 -6 -4 -8 -6 0 -8
5
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2 Acceleration at top (m/s ) (m/s 2) Acceleration at top
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0
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(g)
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0 2
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40
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25
fixed-base numerical model (2-DOF)
4 6
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(j)
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Time (s)
Time (s)
Figure 7. Experimentally history (i) determined (left) and numerically obtained (right) time-acceleration (j)
Figureplots 7. Experimentally determined and numerically (right) time-acceleration for the fixed-base two-storey(left) model during different obtained ground motions: the 1940 El Centrohistory Figure 7. Experimentally determined (left) and numerically obtained (right) time-acceleration history plots earthquake for the fixed-base two-storey model during (c,d), different ground motions: the 1940 (a,b), the 1971 San Fernando earthquake the 1989 Loma Prieta earthquake (e,f),El theCentro plots for the fixed-base two-storey model during different ground motions: the 1940 El Centro 1994 Northridge and theearthquake 2002 Polkowice mining (i,j). Prieta earthquake (e,f), earthquake (a,b), theearthquake 1971 San(g,h), Fernando (c,d), thetremor 1989 Loma earthquake (a,b), the 1971 San Fernando earthquake (c,d), the 1989 Loma Prieta earthquake (e,f), the the 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining tremor (i,j). 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining tremor (i,j). 6
6
base-isolated numerical model (3-DOF)
base-isolated experimental model
64
64
base-isolated numerical model (3-DOF) 2 Acceleration at top (m/s Acceleration at )top (m/s 2)
Acceleration at top (m/sat2)top (m/s2) Acceleration
base-isolated experimental model 42
20
0-2
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4 0 2 -2 0 -4 -2 -6 -4 -8 -6 -10
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Figure 8. Cont.
30
35
(b)
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40
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base-isolated numerical model (3-DOF)
8 4
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10 6
4 0
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base-isolated model
6 2
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2
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6 base-isolated experimental model
base-isolated numerical model (3-DOF) 4
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4 Acceleration at top (m/s 2)
Acceleration at top (m/s 2)
40
base-isolated numerical model (3-DOF)
base-isolated experimental model
2 0 -2
2 0 -2
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base-isolated experimental model
base-isolated numerical model (3-DOF)
2
2
Acceleration at top (m/s 2)
Acceleration at top (m/s 2)
35
(f)
6
1
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8 Time (s)
(i)
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(j)
Figure 8. Experimentally determined (left) and numerically obtained (right) time-acceleration Figure 8. Experimentally determinedtwo-storey (left) andmodel numerically (right)motions: time-acceleration history plots for the base-isolated during obtained different ground the 1940 Elhistory plots Centro for theearthquake base-isolated during different theearthquake 1940 El Centro (a,b),two-storey the 1971 Sanmodel Fernando earthquake (c,d),ground the 1989motions: Loma Prieta (e,f), the(a,b), 1994 Northridge earthquake (g,h),earthquake and the 2002 (c,d), Polkowice mining tremorPrieta (i,j). earthquake (e,f), earthquake the 1971 San Fernando the 1989 Loma the 1994 Northridge earthquake (g,h), and the 2002 Polkowice mining tremor (i,j). As indicated by Figures 5–8 and Tables 2 and 3, the results of the numerical analysis explicitly confirm high effectiveness of the developed mathematical model in simulating complex hysteretic As indicated by prototype Figures 5–8 Tables 3 andmade 4, theofresults of the numerical analysis explicitly behaviour of the baseand isolation system PBs. Close inspection of Tables 2 and 3 confirm high effectiveness of the developed mathematical modelwhich in simulating complex demonstrates that the reduction levels in peak lateral accelerations, were calculated basedhysteretic on both numerical results and previously obtained dataClose (see [28]), are veryofsimilar behaviour of the prototype base isolation systemexperimental made of PBs. inspection Tablesfor3 and 4 every ground motion analyzed. In the case of the base-isolated single-storey model, the largest based demonstrates that the reduction levels in peak lateral accelerations, which were calculated difference between numerical and experimental peak lateral acceleration has been obtained for thesimilar on both numerical results and previously obtained experimental data (see [28]), are very 1971 San Fernando earthquake (difference of 7.74%), while the lowest one for the 2002 Polkowice for every ground motion analyzed. In the case of the base-isolated single-storey model, the largest mining tremor (difference of 0.46%). On the other hand, in the case of the base-isolated two-storey difference between numerical and experimental peak lateral acceleration has been obtained for the model, the largest difference between numerical and experimental peak lateral acceleration has been
1971 San Fernando earthquake (difference of 7.74%), while the lowest one for the 2002 Polkowice mining tremor (difference of 0.46%). On the other hand, in the case of the base-isolated two-storey model, the largest difference between numerical and experimental peak lateral acceleration has been obtained for the 1971 San Fernando earthquake (difference of 3.86%), while the lowest one for the 1994 Northridge earthquake (difference of 0.78%).
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Table 4. Results of the seismic tests for the two-storey structure model.
Dynamic Excitation
Numerically Computed Peak Lateral Acceleration for the Two-Storey Structure Model (m/s2 )
Reduction Based on Numerical Analysis (%)
Reduction Based on Shaking Table Investigation (%) [28]
Fixed-Base (2-DOF System)
Base-Isolated (3-DOF System)
1940 El Centro earthquake NS component, original record with PGA = 3.070 m/s2
5.77
3.89
32.58
33.3
1971 San Fernando earthquake Pacoima Dam station, N74◦ E component, scaled record with PGA = 5.688 m/s2
8.45
3.77
55.38
56.8
1989 Loma Prieta earthquake Corralitos station, NS component, scaled record with PGA = 3.158 m/s2
5.54
4.00
27.80
28.1
1994 Northridge earthquake Santa Monica station, EW component, scaled record with PGA = 4.332 m/s2
7.85
5.10
35.03
33.1
2002 Polkowice mining tremor NS component, original record with PGA = 1.634 m/s2
2.67
2.40
10.11
12.4
Where: PGA: Peak Ground Acceleration, NS: North-South, EW: East-West.
In addition to the peak lateral acceleration values, the whole time-acceleration records (Figures 5–8) were also compared and the difference between numerical and experimental results was assessed using the normalized mean square error. The error calculated for the base-isolated single-storey model for the 1940 El Centro, 1971 San Fernando, 1989 Loma Prieta, 1994 Northridge, and 2002 Polkowice ground motions is as low as 4.56%, 5.21%, 5.59%, 2.41% and 6.21%, respectively. On the other hand, the normalized mean square error calculated for the base-isolated two-storey model for the same suite of dynamic excitations is 7.26%, 5.88%, 7.89%, 6.71%, and 7.53%, respectively. The above results concerning the whole time-acceleration records indicate that the accuracy of the developed mathematical model somewhat drops for the two-storey structure model, which is believed to be triggered by more complex modelling related to the increased number of degrees of freedom. It should also be underlined that the set of function parameters a1 –a5 (incorporated in the numerical model) was evaluated to provide the best fit for the frequency range of 1 Hz to 5 Hz, which was taken as the most representative range for all ground motions considered in the present study. That approach has produced satisfactory results as the normalized mean square error has not exceeded 10% for all the earthquakes and the mining tremor analyzed in the research. 6. Conclusions The paper reports the results of the comprehensive investigation designed to verify the effectiveness of an advanced mathematical model in simulating complex hysteretic behaviour of a prototype base isolation system made of PBs in numerical analysis. High effectiveness of PBs in reducing structural vibrations due to seismic excitations has already been experimentally confirmed during previously conducted shaking table investigation [28]. In the present paper, an advanced hysteretic model was developed to capture mechanical behaviour of PBs in numerical analysis. The effectiveness of the mathematical model was verified by comparing the previously obtained shaking table investigation data (i.e., seismic response of a single-storey and two-storey structure models under various ground motions) with the results of the detailed lumped-parameter analysis. The results obtained from the investigation explicitly confirmed that the proposed mathematical model can be successfully adopted to accurately capture complex mechanical behaviour of PBs in numerical studies. They show that the reduction levels in peak lateral accelerations between numerical
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and experimental results are very similar for all ground motions considered in the study. Moreover, relatively low normalized mean square errors were obtained when the whole time-acceleration numerical and experimental records were compared. It has to be also added that the results of the study indicated that the accuracy of the numerical analysis somewhat decreases as the number of DOFs increases. The calibration procedure presented in the current investigation (performed by curve-fitting the general form of the developed mathematical model into the experimentally obtained hysteresis loops) was conducted for the prototype seismic isolation system made of PBs. Nevertheless, the preliminary experimental results obtained from the full-scale PBs, indicate that very similar shape of hysteresis loops can be observed. Therefore, the general form of the proposed mathematical model is expected to be also suitable for different PBs, regardless of their dimensions or scale factors. However, curve-fitting procedure has to be performed to evaluate a new set of function parameters for each specific PB. Further development of the seismic isolation system made of full-scale PBs is to be performed. More studies are necessary to be conducted in order to fully verify the effectives of full-scale multi-dimensional PBs as a base isolation system for real structures. These should include detailed numerical simulations as well as experimental tests conducted on a large six degrees-of-freedom shaking table. Author Contributions: Tomasz Falborski and Robert Jankowski conceived, designed and performed the numerical investigation presented in this paper. Tomasz Falborski analyzed the data and wrote the paper. Both authors discussed the results obtained and commented on the paper at all stages. Conflicts of Interest: The authors declare no conflict of interest.
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