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Seismic Pounding between Reinforced Concrete Buildings: A Study using two recently proposed Contact Element Models

Deepak Raj Pant & Anil C. Wijeyewickrema Department of Civil Engineering, Tokyo Institute of Technology, Japan

Tastsuo Ohmachi Department of Built Environment, Tokyo Institute of Technology, Japan

ABSTRACT: In this paper, three-dimensional (3D) simulation of seismic pounding between reinforced concrete (RC) moment-resisting frame buildings is presented considering material as well as geometric nonlinearities. The building configuration considered consists of 8-story and 10-story buildings designed according to the 2006 International Building Code (IBC). Two recently proposed variations of the linear contact element model namely, modified Kelvin-Voigt (MKV) model and modified Kelvin (MK) model are compared. The relative performance of the adjacent buildings is evaluated based on maximum inter-story drift ratios for four earthquakes and different gap sizes. It is concluded that the MKV model is more rational than the MK model for seismic pounding simulation. In general the response of the 8-story building is amplified due to pounding. The pounding response is found to be more dependent on earthquake characteristics than the gap between buildings.

Keywords: contact elements; finite elements; nonlinear analysis; pounding; reinforced concrete buildings.

1. INTRODUCTION

Seismic pounding between reinforced concrete (RC) buildings has been reported as the cause of severe damage in several past earthquakes. Therefore, the analysis of seismic pounding has gained much attention in recent years (see for example Karayannis and Favvata 2005, Jankowski 2006, 2009, 2010 and Dimitrakopoulos et al. 2009).

Numerical simulation of seismic pounding between adjacent buildings involves many complexities due to the inherent nature of nonlinearity of the problem, crushing at the contact surfaces and vibrations induced in the structures. Hence, the previous efforts to simulate the pounding phenomenon have included many assumptions. Previous studies show diversity in the methods of modeling colliding buildings and simulating impact. While single-degree-of-freedom systems or multi-degree-of-freedom lumped mass systems had been a choice for the modeling of structures in the early days, the more recent research efforts include a more rigorous analysis based on finite element method (FEM), see for example Mouzakis and Papadrakakis (2004), Karayannis and Favvata (2005), Anagnostopoulos and Karamaneas (2008), Jankowski (2009) and Shakya and Wijeyewickrema (2009). However, only limited research has been reported using three-dimensional (3D) models considering material as well as geometric nonlinearities. On the other hand, for the simulation of impact, many researches have used a contact element approach. The most widely used type of contact element model is the Kelvin-Voigt model.

Pant et al. (2010) among others identified the drawbacks associated with the Kelvin-Voigt model and proposed the modified Kelvin-Voigt (MKV) model for the simulation of seismic pounding. Komodromos et al. (2007) had proposed a variation of Kelvin-Voigt model namely, modified linear viscoelastic model, in which a permanent deformation is allowed at the contact surface. Pant et al. (2010) compared MKV model with the modified linear viscoelastic model (Komodromos et al. 2007)

and concluded that the MKV model is more rational and appropriate for the simulation of seismic pounding.

In the present study, 3D simulation of seismic pounding between RC buildings designed according to the 2006 International Building Code (ICC 2006) is presented. The contact element approach is used to simulate the impact. The MKV model of contact element is compared with the modified Kelvin (MK) model proposed by Kun et al. (2009a). The concept used to develop MK model had also been used by Kun et al. (2009b) to propose a modification to nonlinear contact element model. Effect of earthquake characteristics and gap between the buildings on the relative performance of adjacent buildings is also investigated in the present study.

2. LINEAR CONTACT ELEMENT MODELS

A linear contact element model is best suited for the simulation of seismic pounding between multi-story buildings (Pant et al. 2010). The schematic representation of a contact element is shown in Fig. 2.1. The most fundamental linear contact element model, which can take into account the energy dissipated during impact is the Kelvin-Voigt model. There have been two important modifications to the model as outlined below.

ContactElement

Figure 2.1. Schematic representation of contact element.

2.1. The Modified Kelvin-Voigt (MKV) Model (Pant et al. 2010)

Here, the impact force F is expressed as,

> 0 and 0 > 0 and 0 ,

0 0

l

l

k cF k

+ >=

$ $$ (2.1.1)

where is stiffness of spring element, is the damping coefficient and indentation at contact surface

lk c and relative velocity of impact $ are given as,

; ,i j i ju u gap u u = = $ $ $ (2.1.2)

where i and u ju are the displacements of nodes and i ,j respectively, gap is at-rest separation between the nodes and i and u$ ju$ are the velocities of nodes and i ,j respectively. The damping coefficient in Eqn. 2.1.1 is expressed as, c

,c = (2.1.3)

where damping ratio is,

2

2

3 (1 ),

2l

o

k rr

= $ (2.1.4)

where is coefficient of restitution and r o$ is the relative velocity just before the impact.

node i jnode

2.2. The Modified Kelvin (MK) Model (Kun et al. 2009a)

Here, the impact force F is given as,

> 0 ,

0 0 lk cF += $ (2.2.1)

where damping coefficient is given by Eqn. 2.1.3, in which damping ratio c is expressed as,

3 (1 ).

2l

o

k rr

= $ (2.2.2)

The main difference between MKV and MK models lies in the use of dashpot in parallel with the spring in the contact element. The dashpot is only activated during the approach period in the MKV model while the dashpot is activated throughout the contact period in the MK model. Furthermore, both the models have been implemented in finite element program OpenSees (2009) in the form of uniaxial material models and are used to simulate seismic pounding of two multi-story RC buildings in subsequent sections.

3. BUILDING DESCRIPTION AND MODELING

Two adjacent 8-story and 10-story RC office buildings designed according to 2006 IBC (ICC 2006) for seismic design category D are considered in this study. The seismic force-resisting system was considered to be special moment-resisting frame. Design details of the buildings can be found in Pant et al. (2010). The fundamental natural periods of the 8-story and 10-story buildings were found to be 1.59 sec and 1.63 sec, respectively.

Three-dimensional frame models of the buildings with an assumed rigid slab response are used in OpenSees (2009). RC beams and columns are modeled as force-based finite elements with fiber-based section discretization. Material nonlinearity and geometric nonlinearity due to P-Delta effect are considered in the analysis. The confining effect of stirrups is implicitly modeled by increasing the core concrete strength based on the model proposed by Mander et al. (1988). Contact between the buildings is modeled using zero length elements as contact elements with uniaxial material properties based on MKV and MK models (Fig. 3.1). A value of l used by Jankowski (2005) for concrete-to-concrete impact is also used in this study. The coefficient of restitution is taken to be 0.65 as used by other researchers for concrete-to-concrete impact (Jankowski 2005 and Anagnostopoulos and Karamaneas 2008).

93,500 kN/m=k

Figure 3.1 Three-dimensional model of buildings with contact elements.

4. ANALYSIS AND RESULTS

Nonlinear dynamic analysis of buildings is carried out using two far-field (Hachinohe and El Centro) and two near-field (Kobe and Northridge) earthquake records. Peak ground acceleration, velocity and displacement (i.e. PGA, PGV and PGD), significant duration ( )sD and other details of the input ground motion records are given in Table 4.1. Different gap sizes between the buildings are considered in the analysis. In the first case, the buildings are assumed to be in contact at rest (i.e.

In subsequent cases, gaps of 50 mm, 75 mm, 100 mm and 125 mm are considered. In order to demonstrate the effect of pounding on structural response, the analysis of buildings is also performed by providing a significantly large gap to avoid pounding. The analyses for all the cases are carried out up to 40 sec, with a time step of 0.005 sec.

0).gap =

Table 4.1. Input ground motion records.

Earthquake Station PGA (g) PGV

(cm/sec)PGD (cm)

sD (sec) Remarks

Hachinohe (1968) Hachinohe City

a 0.221g 67.42 33.92 46.86 Far-field

El Centro (1940)

USGS 117 El Centro Array #9b 0.313g 29.69 13.03 24.10 Far-field

Kobe (1995) KJMA

b 0.821g 81.30 17.69 8.63 Near-field

Northridge (1994) DWP 77 Rinaldi

b 0.825g 160.12 29.62 7.25 Near-field

a http://www.eq.pari.go.jp/kyosin/

b http://peer.berkeley.edu/nga/

Through numerical simulations using MK model, it was found that the model does not always avoid the appearance of tensile force just before the separation. It is recalled that this is one of the inherent disadvantages of the original Kelvin-Voigt model. However, by virtue of Eqn. 2.1.3, the force just at the end of impact is always zero. The existence of tensile force is quite possible due to the activation of dashpot even in the restitution phase of contact. For example, in the case of Hachinohe earthquake, when the buildings are in contact at rest, the MK model does not produce tensile forces at the fourth floor level (Fig. 4.1 a). However, the appearance of tensile force is evident when the gap between the buildings is 50 mm (Fig. 4.1 b). It shall be noted that due to the absence of the dashpot in the restitution phase of impact, the MKV model never produces tensile force just before the separation (Fig. 4.1 a, b). It can be observed from Fig 4.1 that both the models are consistent in terms of time and instances of impact. Nonetheless, the maximum impact forces obtained using MK model are much lower (up to 20% less) than those obtained using MKV model (Fig. 4.1 a, b). In spite of having such a large difference in impact forces, the displacement response of structures was found to be nearly insensitive to different contact element models. For example, maximum inter-story drift ratios for 8-story building in case of Hachinohe and Kobe earthquakes when the buildings are in contact at rest are shown in Fig. 4.2 (a) and (b), respectively, where it can be observed that the maximum difference in drift ratio is about 0.25%. Similar response was also found for other earthquakes and gap cases, but not shown here. Although the difference in the displacement response using both the models is not significant, MKV model, which never produces tensile force just before the separation is found to be more rational for the seismic pounding simulation. Hence, the relative performance of adjacent buildings is evaluated in the following sections using only the MKV model.

Maximum inter-story drift ratios of both the buildings for Hachinohe and Northridge earthquakes for different gap sizes are presented in Fig. 4.3 (a), (d) and 4.4 (a), (d). For 8-story building, the Hachinohe and Northridge earthquakes produce very large inter-story drift ratios (in access of 10%), when the pounding between the buildings is not allowed (Fig. 4.3 a, d). This marks the collapse of 8-story building under these two earthquakes. However, such response is not observed for 10-story building, where the inter-story drift ratio is limited to a maximum of nearly 3.7% (Fig. 4.4 a, d).

Time (sec)

Impa

ct fo

rce

(kN)

0 2 4 6 8 10 12 14 16 18 20-50

0

50

100

150

200

250MKVMK

Time (sec)

Impa

ct fo

rce

(kN)

0 2 4 6 8 10 12 14 16 18 20-50

0

50

100

150

200

250MKVMK

(a) (b)

Figure 4.1. Impact force time histories at fourth floor level for Hachinohe earthquake: (a) In contact; (b) 50 mm gap.

(a) (b) Inter-story drift ratio (%)

Floo

r le

vel

0 0.5 1 1.5 2 2.5 30

2

4

6

8MKVMK

Inter-story drift ratio (%)

Floo

r le

vel

0 0.5 1 1.5 2 2.5 30

2

4

6

8

MKVMK

Figure 4.2. Maximum inter-story drift ratios for 8-story building when buildings are in contact at rest: (a) Hachinohe; (b) Kobe.

Furthermore, the occurrence of pounding avoids the generation of excessively large inter-story drift ratios at the critical floor levels of 8-story building, where the drift ratio gets limited to a maximum of about 5% (Fig. 4.3 a, d). Therefore, pounding can sometimes reduce the excessively large inter-story drifts that occur in the absence of pounding. Fig. 4.3 (b) and 4.4 (b) show the maximum inter-story drift ratios for El Centro earthquake. It was found that for the case of El Centro earthquake pounding only occurs when the buildings are in contact at rest. It is clear from the figures that the response of 8-story building is increased due to pounding; however that of 10-story building is decreased. Maximum inter-story drift ratios for Kobe earthquake are shown in Fig. 4.3 (c) and 4.4 (c). The drift response for 8-story building due to pounding is found to be more than that for 10-story building. Unlike the case of El Centro earthquake, the response due to pounding is found to have increased at some levels and decreased at others. It can be observed from Fig. 4.3 and 4.4 that there is no clear trend in the drift response with respect to the gap between the buildings. Earthquake characteristics were found to be governing factors in the seismic pounding response. Except a few cases, the response of buildings is amplified due to pounding, which is consistent with the results obtained in previous studies.


Floo

r le

vel

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

ElCentro

(a) (b) Inter-story drift ratio (%)

Floo

r le

vel

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Hachinohe

In contact50 mm gap75 mm gap100 mm gap125 mm gapNo pounding


Floo

r le

vel

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

Northridge


Floo

r le

vel

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

Kobe


(c) (d)

Figure 4.3. Maximum inter-story drift ratios for 8-story building using MKV model: (a) Hachinohe; (b) El Centro; (c) Kobe; (d) Northridge.


Floo

r lev

el

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

Hachinohe


Floo

r lev

el

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

ElCentro


(a) (b)


Floo

r lev

el

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10Northridge

(c) (d) Inter-story drift ratio (%)

Floo

r lev

el

0 0.5 1 1.5 2 2.5 3 3.5 40

2

4

6

8

10

Kobe In contact50 mm gap75 mm gap100 mm gap125 mm gapNo pounding

Figure 4.4. Maximum inter-story drift ratios for 10-story building using MKV model: (a) Hachinohe; (b) El Centro; (c) Kobe; (d) Northridge.

5. CONCLUSIONS

Three-dimensional simulation of seismic pounding between code-designed RC buildings is presented using MKV and MK contact element models. Material and geometric nonlinearities are considered. Two far-field and two near-field earthquake records and different gap sizes between the buildings are used. The MKV model is found to be more rational to simulate seismic pounding, compared to the MK model. The displacement response of the buildings is found to be nearly insensitive to the variation of contact element models. In general, the pounding amplifies the response of buildings significantly. In general the response of 8-story building is amplified due to pounding. The pounding response is found to be more dependent on earthquake characteristics than the gap between buildings.

ACKNOWLEDGEMENTS The first author gratefully acknowledges a Monbukagakhusho scholarship from the Japanese government. 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

Anagnostopoulos, S. A. and Karamaneas, C. E. (2008). Use of collision shear walls to minimize seismic separation and to protect adjacent buildings from collapse due to earthquake-induced pounding. Earthquake Engineering and Structural Dynamics. 37, 1371-1388.

Dimitrakopoulos, E., Nicos, M. and Kappos, A. J. (2009). Dimensional analysis of the earthquake-induced pounding between adjacent structures. Earthquake Engineering and Structural Dynamics. 38, 867-886.

International Code Council (ICC). (2006). International Building Code. Country Club Hills, Illinois. Jankowski, R. (2005). Non-linear viscoelastic modelling of earthquake-induced structural pounding. Earthquake

Engineering and Structural Dynamics. 34, 595-611. Jankowski, R. (2006). Analytical expression between the impact damping ratio and the coefficient of restitution

in the non-linear viscoelastic model of structural pounding. Earthquake Engineering and Structural Dynamics. 35, 517-524.

Jankowski, R. (2009). Non-linear FEM analysis of earthquake-induced pounding between the main building and the stairway tower of the Olive View Hospital. Engineering Structures. 31, 1851-1864.

Jankowski, R. (2010). Experimental study on earthquake-induced pounding between structural elements made of different building materials. Earthquake Engineering and Structural Dynamics. 39, 343-354.

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Kun, Y., Li, L. and Hongping, Z. (2009a). A modified Kelvin impact model for pounding simulation of base-isolated building with adjacent structures. Earthquake Engineering and Engineering Vibration. 8:3, 433-446.

Kun, Y., Li, L. and Hongping, Z. (2009b). A note on the Hertz contact model with nonlinear damping for pounding simulation. Earthquake Engineering and Structural Dynamics. 38, 1135-1142.

Mander, J., B., Priestley, J. N. and Park, R. (1988). Theoretical stress strain model for confined concrete. Journal of Structural Engineering (ASCE). 114, 1804-1826.

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Shakya, K. and Wijeyewickrema, A. C. (2009). Mid-column pounding of multi-story reinforced concrete buildings considering soil effects. Advances in Structural Engineering. 12:1, 71-85.

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