8CUEE CONFERENCE PROCEEDINGS 8th International Conference on Urban Earthquake Engineering March 7-8, 2011, Tokyo Institute of Technology, Tokyo, Japan
NONLINEAR SEISMIC RESPONSE OF BASE-ISOLATED BUILDINGS CONSIDERING POUNDING
Deepak Raj Pant1) and Anil C. Wijeyewickrema2) 1) Graduate Student, Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Japan 2) Associate Professor, Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Japan
[email protected],
[email protected]
Abstract: Previous studies have shown that pounding of seismically isolated buildings during earthquakes could significantly affect the performance of these buildings. In this paper analysis of seismic pounding of a typical 4-story base-isolated reinforced concrete (RC) building with retaining walls at the isolation level is presented. A modified Kelvin-Voigt impact force model, which does not have the limitations of the conventional model, is used to simulate impact. Geometric and material nonlinearities are considered in the analysis. The effect of seismic pounding on the performance of the building is evaluated based on drift demands and damage indices. The findings of the study are expected to assist design and evaluation of typical mid-rise base-isolated RC buildings.
1. INTRODUCTION Seismic pounding refers to collision between adjacent structures during earthquakes. It occurs when structures with different dynamic characteristics, having insufficient separation between them vibrate out of phase. Earthquake reconnaissance surveys have reported that seismic pounding of fixed-base buildings may lead to severe damage and even result in complete collapse (Rosenblueth and Meli 1986, Kasai and Maison 1997). At least one case of seismic pounding of base-isolated buildings has been reported viz. the Fire Command and Control (FCC) building in Los Angeles experienced one-sided pounding with the entry bridge during the 1994 Northridge earthquake. Increased story shear, drift and acceleration demands on the steel structure were evident due to pounding (Nagarajaiah and Sun 2001). Although seismic pounding of fixed-base buildings has been studied extensively (Maison and Kasai 1992, Pant et al. 2010) for two decades, it is only recently that the seismic pounding of base-isolated buildings has drawn the attention of researchers (see for example Matsagar and Jangid 2003, Polycarpou and Komodromos 2010). All previous studies related to seismic pounding of base-isolated buildings have been carried out where the buildings were modeled as elastic or elastoplastic shear beams or elastic multi-degree-of-freedom lumped mass systems. However, these simplified models cannot incorporate characteristics of the realistic building materials such as reinforced concrete (RC). Therefore, the response of a base-isolated RC building considering seismic pounding is not well understood. Owing to the increasing use of base isolation technology for seismic protection of mid-rise RC buildings around the world, it is important to study the seismic
pounding response of such buildings. In this study, seismic pounding of a typical mid-rise base-isolated RC building with retaining walls at the base is studied. The effect of seismic pounding on the performance of the building is assessed using three-dimensional nonlinear finite element (FE) analysis. 2. IMPACT SIMULATION The contact element approach is widely used for the simulation of impact between structures, due to its clear physical meaning and simple algorithm. The appropriate force-deformation relationship of the contact element is one of the important issues in pounding simulations. Hence, there have been numerous efforts to develop various impact force models for these contact elements. Although nonlinear impact force models are also available, the linear impact force models provide a good balance between simplicity and accuracy. The Kelvin-Voigt (KV) model, which can be represented by the combination of a spring and a dashpot in parallel, is the most widely used impact force model. In this model, the impact force F between the impacting bodies is,
k δ + cδ δ > 0, F = l δ ≤ 0, 0
(1)
where kl is stiffness of spring element, c is damping coefficient and indentation δ and relative velocity of impact δ are given by,
δ = ui − u j − d g ,
δ = ui − u j ,
(2)
where ui and u j are displacements and ui and u j are velocities of impacting nodes i and j, respectively and d g is at-rest gap distance between the impacting nodes. The damping coefficient c in Eq. (1) can be expressed in terms of the coefficient of restitution r , which is defined as the ratio of the relative velocity after impact to the relative velocity before impact. The expression for the damping coefficient in the KV model is (Anagnostopoulos 2004),
c = 2ξ
(
)
kl meff ,
(3)
where the stiffness kl in the absence of experimental studies is determined based on the axial stiffness of colliding bodies and the effective mass meff and damping ratio ξ are given by,
meff =
m1m2 ln(r ) , ξ= − , 2 m1 + m2 π + (ln(r )) 2
(4)
where m1 and m2 are masses of the colliding bodies. Although the model is attractive due to its simplicity, the limitations are: (i) Due to a constant damping coefficient, it shows a sudden jump in the impact force at the beginning of impact, which is not reasonable for concrete-to-concrete impact (Fig. 1(a)). This sudden jump in the force induces unrealistic high accelerations in the impacting bodies. (ii) Due to the dashpot being activated even in the restitution phase, the colliding bodies exert tension on each other just before the separation (Fig. 1(a)), which has no physical meaning. Hence, a modification to the KV model referred to as the Modified Kelvin-Voigt (MKV) model was proposed by Pant et al. 2010. The MKV model is based on the assumption that most of the energy during impact is dissipated in the compression phase. Hence, the energy dissipated in the restitution phase can be neglected. This assumption allows for the removal of the dashpot from the spring-dashpot assembly in the restitution phase. The impact force F between the impacting bodies can be expressed as,
beginning of impact, the damping coefficient c is taken to be directly proportional to the indentation δ as,
c = ξδ . This leads to a new relationship between ξ and r as,
ξ=
(
(7)
0
where δ0 is relative velocity before impact. The relationship between impact force and indentation for the MKV model is shown schematically in Fig. 1(b). It is clear from Fig. 1(b) that the sudden jump in impact force at the beginning of impact and the tensile impact force just before the separation of the KV model are no longer present. Thus, all the major limitations of the KV model are eliminated in the MKV model, while maintaining the simplicity and clear physical meaning of the KV model. Furthermore, the model is implemented in FE program OpenSees 2010 in the form of a uniaxial material and is used to simulate seismic pounding of a base-isolated building in subsequent sections. It is noted that other researchers have also proposed different modifications to the KV model. However, a comparison of these models with the MKV model has shown that the MKV model is more rational for seismic pounding simulation of multi-story buildings (Pant 2010). To verify the correctness of the derived formula (Eq. (7)) and its proper implementation in OpenSees, numerical verification is conducted by simulating the first instance of impact between a spherical ball and a stationary rigid surface. A ball of mass m = 1.0 kg is dropped freely from a height h = 0.5 m on a stationary rigid surface (Fig. 2(a)). Different values of coefficient of restitution rpre are specified, pounding simulations are performed and the coefficient of restitution from the simulation rpost are obtained and compared with the pre-specified values. In Op enSees, the si mu latio n is co nd ucted usin g two-dimensional FE method. The FE model of the problem is shown in Fig. 2(b). The stationary rigid surface and the spherical ball are modeled as node 1 and node 2, respectively. Node 1 is fixed and node 2 is free to move F
F
kl
δ
(5)
In order to avoid a sudden jump in the impact force at the
)
3kl 1 − r 2 , 2r 2δ
kl
kl δ + cδ δ > 0 and δ > 0, = δ > 0 and δ ≤ 0, F kl δ 0 δ ≤ 0.
(6)
δ
(a)
(b)
Fig. 1. Schematic force-indentation curve for one instance of impact for: (a) KV model; (b) MKV model.
spherical ball
uniform acceleration g
node 2
m
contact element
h
node 1 stationary rigid surface
(a)
(b)
Fig. 2. Drop weight impact simulation: (a) schematic diagram; (b) finite element idealization. Table 1. Comparison of pre-defined and simulated values of coefficient of restitution using MKV model.
rpre
rpost
Relative error (%)
0.10
0.113
13.4
0.20
0.223
11.5
0.30
0.328
9.2
0.40
0.428
7.0
0.50
0.525
5.0
0.60
0.620
3.3
0.70
0.713
1.9
0.80
0.807
0.9
0.90
0.902
0.2
1.00
1.000
0.0
only in the vertical direction. A contact element needs to be placed between the nodes to simulate the pounding phenomenon. Here, a so called zero-length element available in OpenSees is chosen. With the zero-length element, end nodes of the element can have same coordinates. Particular choice of zero-length element as contact element becomes very advantageous while performing a large number of numerical simulations of seismic pounding between structures with various at-rest gap distances. The nodal coordinates of the adjacent structures need not to be revised for each gap case; rather the gap parameter in the associated uniaxial material only needs to be changed. For the present simulation, the MKV model is assigned to the aforementioned contact element. The stiffness of spring element kl is set to 2.0 × 107 N/m. In the free fall condition, node 2 has a constant acceleration of g in downward direction. Hence, a uniform acceleration loading is applied at node 2. To be consistent with the seismic pounding simulations going to be discussed in forthcoming sections, the applied acceleration is treated from the viewpoint of uniform ground acceleration. The resulting system of nonlinear equations is solved using Newmark’s method of constant acceleration ( β = 0.25, γ =0.5), where Modified Newton-Raphson
method is used for iterative solution. The analyses are performed for 0.35 sec with a time step of 1× 10−7 sec. The values of rpost obtained from pounding simulation are compared with rpre and relative error is computed as rpre − rpost rpre . The comparison is shown in Table 1. The The relative error reduces with increasing r. coefficient of restitution used to simulate structural pounding ranges from 0.5 to 0.75 (Jankowski 2005). It is clear from Table 1 that the relative error is small for the values of coefficient of restitution in this range. This shows that the relationship between ξ and r given by Eq. (7) is correct and hence can be used for rational simulation of structural pounding. 3. SEISMIC POUNDING CASES A 4-story, 2-bay by 2-bay base-isolated RC building is considered to investigate the effect of pounding on the response of a typical mid-rise structure. The bay widths and story heights are 6 m and 3.6 m, respectively. The building is supposed to be used as an elementary school such that substantial hazard to human life is expected in case of failure. The building is assumed to be located at a stiff soil site. Two cases of seismic pounding of this building are examined: (i) Pounding with the retaining wall on one side (Fig. 3(a)), (ii) Pounding with the retaining walls on both sides (Fig. 3(b)). The retaining wall extends from ground level up to the isolation level (Fig. 3). The building was designed following the 2009 International Building Code (ICC 2009). According to the code, an equivalent lateral force procedure was used for the design of this building. ACI 318-08 (ACI 2008) was followed for the design of the structural concrete. Compressive strength, unit weight and modulus of elasticity of concrete are taken as 28 MPa, 22.76 kN/m3 and 2.5 × 104 MPa, respectively. The yield strength of main steel reinforcement bars and ties are taken as 420 MPa and 300 MPa, respectively. The unit weight and modulus of elasticity of steel are assumed to be (a)
base-isolated building
(b)
retaining wall
Fig. 3. The base isolated building with: (a) retaining wall on one side; (b) retaining walls on both sides.
Pseudo acceleration (g)
Table 2. Details of earthquake ground motions. 76.97 kN/m3 and 1.99 × 105 MPa, respectively. Live loads on floors and roofs are assumed to be 4.79 kN/m 2 and 0.96 kN/m 2 , respectively. Total wall thickness is Earthquake Date Station PGA (g) taken as 113 mm with a unit weight of approximately Hachinohe 1968/05/16 Hachinohe city 0.239 15.0 kN/m3 . The slab thickness is 200 mm. For the El Centro 1940/05/19 El Centro Array #9 0.313 floor finish, a 10 mm thick porcelain tile with a unit weight 3 of 23.57 kN/m is taken. Seismic force-resisting system Kobe 1995/01/17 JMA Kobe 0.821 of the building is chosen as special moment-resisting frame Rinaldi Receiving 0.825 Northridge 1994/01/17 (SMRF). The lead rubber bearing (LRB) isolation system Station is selected for the building. Bearings are designed for gravity and earthquake loads using SAP2000 (2009) and 1 Microsoft Excel spreadsheets. Identical circular bearings Design El Centro of 750 mm diameter and 572 mm height are provided under 0.8 Hachinohe each of the 9 column bases. A total of 30 layers of rubber Northridge Kobe with a layer thickness of 15 mm are used. Steel shim 0.6 thickness is taken as 2 mm. The design displacement DD and total maximum displacement DTM of the isolation 0.4 system are 225 mm and 462 mm, respectively. Figure 4 0.2 shows the cross-section of the bearing. In order to investigate the influence of separation, the 0 structures are assumed to be separated by different at-rest 0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6 gap distances d g in both of the pounding cases. It is Natural period (sec) considered that the gap between buildings can be as small as half of the design displacement DD . Therefore, ratio of Fig. 5. Comparison of elastic pseudo acceleration response at-rest gap distance to the design displacement is taken as 1.0, spectra of matched ground motions with design 0.9, 0.8, 0.7, 0.6 and 0.5. acceleration response spectrum. For performance evaluation, four earthquake ground motions are selected from 1968 Hachinohe, 1940 El Centro, modeling issues to be addressed viz. modeling of structural 1995 Kobe and 1994 Northridge earthquakes (see Table 2 members which are beams, columns, lead rubber bearings for details of ground motions). The earthquake ground and slabs, modeling of retaining walls and modeling of motions are matched to the design response spectrum at 5% impact. Reinforced concrete beams and columns are damping using wavelet adjustments. The program modeled using force-based, fiber beam-column elements, RspMatch2005 (Hancock et al. 2006) is used to match the which are considered as most accurate and robust distributed ground motions simultaneously to the design acceleration plasticity elements. In fiber beam-column elements, the response spectrum (Fig. 5) and the design displacement element response is evaluated at certain number of response spectrum for natural periods up to 6 sec. integration points i.e. sections along the length of the element and each section is discretized into unconfined 4. NUMERICAL MODELING AND ANALYSIS concrete, confined concrete and steel fibers (Fig. 6, 7). Uniaxial materials with nonlinear constitutive relationship A macromodel-based approach is adopted for are assigned to these fibers. Section constitutive three-dimensional modeling of seismic pounding of the relationship is derived from the integration of constitutive base-isolated building in OpenSees. There are three relationship of fibers. There are several material models to describe nonlinear constitutive relationship of concrete and steel under monotonic as well as cyclic loading. In this study the modified Kent and Park model (Park et al. 1982) is 32 used for the response of concrete in compression. In tension, a linear elastic branch is followed by a linear 572 softening branch up to zero stress (Fig. 8(a)). For reinforcing steel, the constitutive model of Menegotto and Pinto (Menegotto and Pinto 1973) is used (Fig. 8(b)). Lead rubber bearings are modeled using elastomeric bearing 32 65 elements. A bilinear hysteretic model (Fig. 8(c)) is used to describe shear force-deformation relationship of these 750 elements. Floor and roof slabs are not modeled explicitly; rather their restraining effect is accounted for by assuming Fig. 4. Cross-section of the bearing (all dimensions are in an in-plane rigid diaphragm response. Retaining walls are mm) modeled as rigid objects. Backfill soil-structure
integration points
Fig. 6. A fiber beam-column element.
= RC section
+ unconfined concrete fibers
+ confined steel fibers concrete fibers
Fig. 7. Discretization of RC section into unconfined concrete, confined concrete and steel fibers. σ
σ
ε
F
ε δ
(a)
(b)
(c)
Fig. 8. (a) Stress-strain relationship of concrete; (b) stress-strain relationship of reinforcing steel; (c) bilinear hysteretic model of elastomeric bearing elements.
interaction is considered outside the scope of this study. Impact is modeled using zero length elements, which are used as contact elements between structures at floor levels (Fig. 3). The material property of the contact elements is based on the MKV model. The coefficient of restitution r is taken as 0.65 and stiffness of spring element kl is assessed based on the axial stiffness of a slab. Nonlinear time-history analyses are performed. Time integration of equations of motion is accomplished using Newmark’s method of constant acceleration. A time step of 0.005 sec is used for all the cases. The nonlinearity of the problem demands an iterative solution procedure. At the beginning of each time step modified Newton-Raphson method is employed for the iterative solution and the method is changed if the convergence is not achieved in the time step. There are many alternatives to modified Newton-Raphson method in OpenSees such as Newton with
line search or Broyden method. The convergence of the solution is based on energy increment. For damping in the building, stiffness proportional damping is applied to the superstructure only. A damping ratio of 5% is used for the first mode of the isolated structure. The accuracies of impact force model and FE modeling have been validated using an available impact experiment and a shake table test of a base-isolated RC building, respectively. Details of the validation are shown elsewhere (Pant 2010). 5. RESULTS AND DISCUSSION Inter-story drift ratio and overall damage index (ODI) based on Park and Ang damage index (Park et al. 1985) are presented to illustrate the influence of pounding on the structural performance of the base-isolated building.
Table 3. Values of inter-story drift ratios and overall damage indices for performance evaluation of RC buildings. Degree of damage
Physical appearance
Inter-story drift ratio (%)
No damage
Minor localized cracking
drift ratio < 0.5
ODI < 0.1
Minor to moderate damage
Light cracking throghout to extensive cracking and localized spalling of concrete
0.5 ≤ drift ratio < 1.5
0.1 ≤ ODI < 0.4
Severe damage
Extensive crushing of concrete and exposure of buckled reinforcement
1.5 ≤ drift ratio < 3.0
0.4 ≤ ODI < 1.0
Total collapse
Complete loss of load carrying capacity
drift ratio ≥ 3.0
ODI ≥ 1.0
R
1.4
No pounding dg/DD = 1.0 dg/DD = 0.9 dg/DD = 0.8 dg/DD = 0.7 dg/DD = 0.6 dg/DD = 0.5
3F 2F 1F
1.2 1
ODI
Floor level
ODI
0.6 0.4
B 0
0.8
0.2
1 2 3 Inter-story drift ratio (%)
0 No pounding dg/DD = 1.0 dg/DD = 0.9 dg/DD = 0.8 dg/DD = 0.7 dg/DD = 0.6 dg/DD = 0.5
(a)
(b)
Fig. 9. Response of the base-isolated building considering pounding with the retaining wall on one side: (a) peak inter-story drifts; (b) overall damage indices.
No pounding dg/DD = 1.0 dg/DD = 0.9 dg/DD = 0.8 dg/DD = 0.7 dg/DD = 0.6 dg/DD = 0.5
3F 2F
ODI
Floor level
R
1F B 0
(a)
1.4 1.2 1 0.8 0.6 0.4 0.2
1 2 3 Inter-story drift ratio (%)
0 No pounding dg/DD = 1.0 dg/DD = 0.9 dg/DD = 0.8 dg/DD = 0.7 dg/DD = 0.6 dg/DD = 0.5
(b)
Fig. 10. Response of the base-isolated building considering pounding with retaining walls on both sides: (a) peak inter-story drifts; (b) overall damage indices. The damage can be classified as no damage, minor to moderate damage, severe damage and total collapse based on inter-story drift ratio and ODI as shown in Table 3. Note that to highlight the trends in the response, the results presented represent maximum responses generated due to all four earthquake excitations. When the pounding of the base-isolated building is considered with the retaining wall on one side, the peak inter-story drift ratio, which is less than 0.5% at the first story when there is no pounding, becomes more than 1.2% due to pounding (Fig. 9(a)). This implies that the base-isolated building experiencing no damage without any
interaction with adjacent structures undergoes minor to moderate damage due to pounding. In general, the inter-story drift demand increases with reduction in the gap. In addition, the maximum inter-story drift occurs at the first story and gradually reduces at upper stories. The minor to moderate damage to the building is also evident from Fig. 9(b), where ODI remains less than 0.4. A clear pattern of overall damage is also observed from Fig. 9(b). Overall damage index increases rapidly for the gaps up to 70% of the design displacement and becomes nearly constant for further reduction in the gap. Significant increase in the demands is observed in case
of pounding with the retaining walls on both sides compared to one-sided pounding (Fig. 10(a)). The peak inter-story drift demand at the first story clearly exceeds 1.5% for the gaps smaller than 80% of design displacement, denoting severe damage to the story. Imminent collapse of the story is evident at a gap equal to 60% of the design displacement. The influence of pounding is less severe at upper stories. Fig. 10(b) shows clear trend in the overall damage to the building for various gaps. With decreasing gap, the ODI first increases and then reduces with further reduction of the gap. This implies that there exists a critical at-rest gap distance causing maximum damage to the base-isolated building. 6. CONCLUDING REMARKS This study investigates the performance of a typical mid-rise base-isolated RC building considering seismic pounding. It is found through series of numerical analyses that the seismic pounding has detrimental effects on the response of a base-isolated RC building. While one-sided pounding with a retaining wall at the base causes minor to moderate damage to the base-isolated building, two-sided pounding with retaining walls at the base could even lead to its collapse, depending upon the clear space maintained around the building. The present study only focuses on the evaluation structural damage to the building due to pounding. Evaluation of the non-structural damage, which requires accurate estimation of floor accelerations in the building, could be studied in the future. Furthermore, the present study is only focused on the seismic performance evaluation of base-isolated buildings. The mitigating measures against pounding should also be studied in the future. Acknowledgements: The first author is pleased to acknowledge a Monbukagakusho (Ministry of Education, Culture, Sports, Science and Technology, Japan) scholarship for graduate students. Financial support from the Center for Urban Earthquake Engineering (CUEE) through the GCOE Program “International Urban Earthquake Engineering Center for Mitigating Seismic Mega Risk,” is gratefully acknowledged. References: ACI (2008), “Building Code Requirements for Structural Concrete (ACI 318-08) and Commentary (ACI 318R-08)”, ACI Committee 318, American Concrete Institute, Farmington Hills, MI. Anagnostopoulos S.A. (2004), “Equivalent viscous damping for modeling inelastic impacts in earthquake pounding problems”, Earthquake Engineering and Structural Dynamics, 33, 897-902. Hancock, J., Watson-Lamprey, J., Abrahamson, N.A., Bommer, J.J., Markatis, A., Mccoy, E. and Mendis, R. (2006), “An improved method of matching response spectra of recorded earthquake ground motion using wavelets”, Journal of Earthquake Engineering, 10 (Sp. 1), 67-89. International Code Council (ICC) (2009), “International Building Code”, IBC 2009, Country Club Hills, Illinois. Jankowski, R. (2005), “Non-linear viscoelastic modelling of earthquake-induced structural pounding”, Earthquake
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