Influence of Soil-structure Interaction on Response of ...

The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

Influence of Soil-structure Interaction on Response of Adjacent SDOF Structures Connected by Viscous Damper Chirag C Patel, R S Jangid Dept. of Civil Engineering, Indian Institute of Technology Bombay, Mumbai, 400076 India. Keywords: Viscous damper, optimum damper damping, Soil-Structure Interaction ABSTRACT: When two adjacent single degree of freedom (SDOF) structures dynamically well separated, connected by appropriate viscous damper (Passive Control), their response to the horizontal loads reduced significantly. The soil-structure interaction change the performance of connected structure system. In this paper the influence of soil properties and damping characteristics on connected structure system is analyzed. The introduction of soil-structure interaction requires a mathematical model for the foundation and surrounding soil. The governing equation of equilibrium have been formulated. A range of soil properties and soil damping characteristics are chosen which gives broad picture of connected structure system behavior on influence of soil-structure interaction. The soil damping characteristics have major influence, but poisson’s ratio has a minor influence. It is concluded that the response of connected structure system on soft soil are more critical than those supported on very stiff soils. Various other energy dissipating systems to connect the adjacent structures to control the response are suggested.

1

Introduction

Vibration control by connecting adjacent structures is very effective to mitigate the dynamic responses and also minimize the chances of pounding. By providing energy dissipation devices of appropriate capacity and at proper position between two adjacent structures, the passive control techniques increase the energy dissipation capacity of the structural system. The control force by control device is function of velocity. There should be a relative velocity between the two ends of the damper, connect the two adjacent structures for damper to be effective and two adjacent structures should be dynamically dissimilar (i.e. One soft structure and other stiff structure). This concept is to allow two dynamically dissimilar structures to exert control force upon the other to reduce over all response of the system. It also overcomes the problem of pounding which is more sever load condition than the case of vibration without pounding. But it alters the dynamic characteristics of the unconnected structures. It enhances undesirable torsional response, if building has asym metric geometry and increase base shear of stiffer structure. The structural control criteria depend on the nature of dynamic loads and the response quantities of interest. Minimizing the relative displacement or absolute acceleration of the system has always been considered as the control objective. In case of flexible structures, displacements are predominant that need to be controlled. Where as, in case of stiff structures, accelerations are of more concern generating higher inertial forces in structu res, which should be mitigated.

2 Adjacent structures connected by viscous damper Considering two adjacent structures connected with a viscous damper idealized as SDOF system, assumed to be symmetric with their symmetric places in alignment, also assuming the ground motion to occur in one direction in symmetric planes of the structures. The viscous damper is modeled in which the force is proportional to the relative velocity of its both ends acting in parallel. Let m1 , c1 ,k1 and m2 , c2 , k2 be the mass, damping coefficient and stiffness of the structure 1 and 2 , respectively. Let ? 1 = k1 /m1 and ?2 =

k2 /m2

be the natural frequencies and

?1 = c1 /2m1 ? 1 and ?2 = c 2 /2m2 ? 2

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damping ratios of

structure 1 and 2 , respectively. Let µ and ß be the mass ratio and frequency ratios of two structures; and ? c be the combined frequency of connected structures defined as µ=

ß=

m1 m2

(1a)

?2

(1b)

?1 ?c =

k1 + k 2 m1 + m2

(1c)

Let cd be the damping coefficient of the damper. The governing equation of motion for the viscous damper connected system can be written as

m1 0 

  x&&1  c1 + c d   +  -c m  x&& 2  2  d 0

x& k d   1 +  1   0 c + c  x& 2 d   2  -c

  x1   -m1&&xg    =  -m x&&  k  x 2  2  2 g  0

(2)

where x1 and x2 are the displacement responses, relative to the ground of structure 1 and 2 , respectively,

&&g is the ground acceleration. Bhaskararao and Jangid 2007, have analyzed the connected structure for and x harmonic base acceleration i? t && xg = a e (3) 0 where a0 and ? are the amplitude and excitation frequency, respectively, of the harmonic ground motion and derived the close form solution for optimum damper damping coefficient of undamped structures . The contol force by damper is function of relative velocity of its both ends, hence damper is more effective if two structures are dynamically well separated. For ß = 2 damper is more effective. Effect of structural dmaping is neligible on reduction of optimum response. For 5% dmaped structures with frequency ratio ß = 2 and different mass ratio values, the reduction in peak responses are given in Table 1. Table 1. Reduction in normalized peak response of viscous damper connected structures for various value of µ µ && && x1 x2 x1 x2 Unconnected peak Unconnected peak Unconnected peak Unconnected peak response = 10.001 response = 10.001 response=10.05 response=10.05 peak % peak % peak % peak % response reduction respons reduction respons reduction respons reduction e e e 1.25 1.3385 86.61 3.7688 62.32 2.5042 75.08 3.1756 68.40 1.00 1.2399 87.61 4.0067 59.90 2.5016 75.11 3.3418 66.75 0.75 1.1741 88.26 4.3701 56.30 2.597 74.16 3.4758 62.73 The above results leads to conclude that the viscous damper is quite effective in response control of the connected structures and higher reductions in response can be achieved if the frequencies of the connected structures are well seperated.

3 Soil – structure interaction The results in Table 1. are based on assumption that the structures are on firm ground. During the earthquake, waves propagated and move through the ground and finally vibrate the structures which develop a force that acts on the soil as an intertail force. A building above the rock, during earthquake, the rock movement directly transferred to the bottom of the building, The input acceleration acts on the structure as a horizontal inertial force,but since the rock is extremely rigid, the input seismic wave is not affected regardless of whether a structure is present or not. Hence, the response of a structure depends only on its own dynamic properties. The response of building on firm ground is different than on soft ground because movements of structures built on soft ground differ from that of one on rocky ground due to vibrations of the nearby ground. The dynamic rsponse of buildings is modified depending on the structural and soil properties by the

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translational and rotationa of the foundation relative to the soil during dynamic structure-soil interaction.The free field motion is the ground vibration, as the site before construction a buildings. When earthquake wave approaches the ground, the vibrations are generally amplified depending on the presence of overlaying soft ground. There is also kinematics mutual interaction that occurs because the incident seismic wave is itself modified from the one under free field as the built-up structure alters the rigidity of the foundation. There is mutual interaction due to inertia. When structure vibrates develop inertia force and this force is fed back to the ground causing its deformation. The earthquake time history of natural ground and the time history at the foundation thus differ due to vibration of the ground and the structure which are interrelated (Seto 2000; Dowrick 2001). Gupta and Trifunac 1991, have explain the soil-structure interaction effects in the analysis of multistoried building response via response spectrum superposition method, by incorporating a few modificaitons in the input excitation. The complete soil-structure interaction analysis is much complicated.

3.1 Adjacent connected structures on soft ground The two adjacent structures on rock, connected with damper of optimum damping coefficient subjected to harmonic base excitation performs well. When structures are on soft soil, their performance much affected by the soil condition. For soft soil there are rotational motion as well as translational motion, let us assume free field motion and consider the effect of translational motion only. Consider two adjacent structures connected with a viscous damper idealized as SDOF system, assumed to be symmetric with their symmetric place in alignment. Also assuming the ground motion to occur in the one direction in symmetric plane of the structure. Also neglecting radiation damping.The structure model of two SDOF adjacent structures on soft soil, connected with viscous damper and its mathematical model is as shown in Figure 1. Let ms be the effective mass of foundation and soil and ks be the effective stiffness of soil mass, ß1 be the frequency ratio of soil and soft structure defined as ? ß1 = s (4) ?1 x2

x1 Viscous damper Structure 1

cd m1

Structure 2

m2 c1

k1

c2

k2 xg2

xg1 m s1 ks

m s2 cs

ks

cs

(a) Connected structure on soft soil.

(b) Mathematical model. Figure 1 Connected structures on soft ground and its mathematical model. Considering the equilibrium of each mass, the equilibrium equation for each mass can be written as m1&&x1 +(cd + c1 )x& 1 - c1 x& g1 - cd x&2 + k1x1 - k1xg1 = -m1x&&g (5a) ms1&&xg1 + ( cs + c1 )x& g1 - c1x&1 +(k s +k1 )x g1 - k1x1 = - ms1 x&&g

(5b)

m2 && x2 + ( cd + c2 )x&2 - c 2 x&g2 - c d x&1 + k2 x2 - k 2 xg2 = - m2 && xg

(5c)

ms2 &&xg2 + ( cs + c2 )x& g2 - c2 x& 2 + ( ks + k2 )x g2 - k2 x2 = - ms2 x&&g The governing equation of motion for the given system can be written in matrix form as && + C X& + K X = - M 1 x&& M X g

[ ]{ } [ ]{ } [ ]{ } [ ]{ } 2776

(5d)

(6)

where

 m1 0 [M ] =  0   0

 k1 -k [K ] =  01   0

0

0

ms1

0

0

m2

0

0

-k1 k1 + k s

0

0

k2 -k2

0

0

 0   0  ms2   0

 0   -k2  k2 + ks   0

c1 + cd  -c [C] =  -c1  d  0

-c1 c1 + cs

-c d

0

c2 + cd -c2

0

0

 x&1   x&  { X& } =  x&g1   2  x&   g2 

 &&x1   &&x  && { X} =  &&xg1   2  x&&g2 

   -c2  c2 + cs   0

0

 x1   x  { X} =  xg1   2  xg2 

(7 a,b)

(7 c,d,e,f)

and x1 and x2 are the displacement responses, relative to the ground of structure 1 and 2 , respectively, and && xg is the ground acceleration. The effect of structural damping on optimum response by optimum damper damping coefficient have negligible effect, hence for simplicity we are considering the undamped system. For simplicity the mass ratio µ = m1 m2 is considered to be one. The effective mass of soil is so small as to be of little consequence ( Nandakumar et al. 1977; Claude and Singhal 1968); here we modeled the soil mass as 2m where m is the mass of structure 1 . Let the harmonic base acceleration is given by equation 3. The displacement response of two structure can be derived by solving equation 6. First let us consider higher value for ß1 say ß1 =10, means the ground is 10 times stiffer than soft structure, indicates very stiff soil (rock). The variations of the structural responses (displacement and acceleration) of two structures against the excitation frequency, for different damper damping coefficient are shown in Figure 2. It indicates the viscous damper is quite effective or vibration control of connected structures on stiff soil.

6

15

ξ1=ξ2=0.00

ξ1=ξ2=0.00 12

2

ξd=0.00

ξd=0.00

0.1 0.4252 0.4436 1.0

x 2(a 0/ω22)

2

x 1(a 0/ω1 )

4

9

0.1 0.312 0.2653 1.0

6 3

0

0 12

8 ξ1=ξ2=0.00

4

9

ξd=0.00

0.1 0.3402 0.3257 1.0

6

.. x 2(a0)

.. x 1(a0)

6

ξ1=ξ2=0.00

3

2

0 0.0

0.5

1.0

ω/ ω1

1.5

2.0

2.5

0 0.0

ξ d=0.00

0.1 0.3547 0.3785 1.0

0.5

ω/ω2

1.0

1.5

Figure 2. Displacement andAbsolute acceleration variation against excitation frequency (β =2, µ =1 and β1 =10) for different damper damping coefficient.

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Now considering ß1 =1, means connected structures on soft soil. The variation of displacement response of structure 1 and 2 and acceleration response of structure 1 and 2 for different soil damping are shown in Figure 3,4,5 and 6 respectively. It shows that connected structures on soft soil gives extremely different behavior, optimum damper damping coefficient to control structural response no longer remain optimum. Soil damping affect the structural response of connected structures. Figure 7 shows increase in soil damping decrease the optimum response of the structures.

6

ξs = 0.0

ξ1=ξ2=0.00

β1 = 1.0 ξs = 0.05

24

ξd=0.0

2

18

0.1 0.4252 0.4436 1.0

2

2

0.1 0.4252 0.4436 1.0

ξd=0.0

x 1(a0/ ω1 )

4

x 1(a0/ ω1 )

30

ξ1=ξ2=0.00

β1 = 10.0

12 6

0 20

0 15

ξ1=ξ2=0.00

β1 = 1.0

= 0.07

ξd=0.0

5

9

2

x1(a0/ω1 )

10

0 0.0

β1 = 1.0 ξs = 0.1

ξd=0.0

0.1 0.4252 0.4436 1.0

2

x1(a0/ω 1 )

15 ξs

ξ1=ξ2=0.00 12

6

0.1 0.4252 0.4436 1.0

3

0.5

1.0

ω/ ω1 1.5

2.0

2.5

0 0.0

0.5

ω/ω 1

1.0

1.5

Figure 3. Displacement varation of structure 1 considering SSI against excitation frequency (β =2 and µ =1) for different damper damping coefficient.

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ξd = 0.0

6

0.1 0.312 0.2653 1.0

β1=10.0

120

ξs = 0.00

β1=1.0

100

ξs = 0.05

ξ1=ξ2=0.00 ξd = 0.0

80

0.1 0.312 0.2653 1.0

2

x2(a0/ω2 )

9 2

x2(a0/ω2 )

12 ξ1=ξ2=0.00

60 40

3 20 0 100

0

ξ1=ξ2=0.00

β1=1.0

= 0.07

60

ξd = 0.0

60 40

β1=1.0

ξ1=ξ2=0.00

ξs = 0.1

ξd = 0.0

40

0.1 0.312 0.2653 1.0

2

x 2(a0/ω 2 )

0.1 0.312 0.2653 1.0

2

x 2(a0/ω 2 )

80 ξ s

20

20 0 0.0

0

0.5

1.0 1.5 0.0 0.5 1.0 1.5 ω/ ω2 ω/ω 2 Figure 4. Displacement varation of structure 2 considering SSI against excitation frequency (β =2 and µ = 1)

for different damper damping coefficeint.

6

12 ξ s

.. x 1(a0)

ξd=0.00

0.1 0.3402 0.3257 1.0

6 3

0

1

0 8

ξ1=ξ2=0.00

= 1.0

ξs = 0.07

6 ξs

ξd=0.00

0.1 0.3402 0.3257 1.0

.. x1(a0)

6 4

ξ1=ξ2=0.00

β1 = 1.0

= 0.1

ξ d=0.00

0.1 0.3402 0.3257 1.0

4

.. x 1(a0)

10 β 8

ξ1=ξ 2=0.00

= 0.05

9

0.1 0.3402 0.3257 1.0

2

β1 = 1.0

ξs = 0.0

ξd=0.00

4

15

β1 = 10.0

ξ1=ξ 2=0.00

.. x1(a0)

8

2

2 0 0.0

0.5

1.0

1.5

2.0

2.5

0 0.0

0.5

1.0

1.5

2.0

ω/ω 1 ω /ω1 Figure 5. Absolute acceleration varation of structure 1 considering SSI against excitation frequency (β =2 and µ =1) for different damper damping coefficeint.

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2.5

12

ξ1=ξ2=0.00

9

ξs = 0.0

ξd = 0.00

15

.. x2(a0)

.. x2(a0)

3

ξ1=ξ2=0.00

β1 = 1.0 ξs = 0.05

ξd = 0.00

12

0.1 0.3547 0.3785 1.0

6

18

β1 = 10.0

0.1 0.3547 0.3785 1.0

9 6 3

0

0

ξ 1=ξ2=0.00

12 β 1

= 1.0

10 ξs

= 0.07

9

0.1 0.3547 0.3785 1.0

.. x 2(a0)

.. x 2(a0)

4

ξd = 0.00

6

0.1 0.3547 0.3785 1.0

6

ξ1=ξ2=0.00

ξs = 0.10

ξd = 0.00

8

β1 = 1.0

3

2 0 0.0

0.5

1.0

0 0.0

1.5

0.5

1.0

ω/ω 2 ω /ω2 Figure 6. Absolute acceleration varation of structure 2 considering SSI against excitation frequency

1.5

(β =2 and µ =1) for different damping damping coefficeint.

100 150

ξ1=ξ2=0.0

0.1 0.4252 0.4436

120 2

x2(a0/ω2 )

ξd = 0.0

2

x1(a0/ω1 )

75

β1=1.0

50

β1=1.0

ξd = 0.0

ξ1=ξ2=0.0

0.1 0.312 0.2653

β1=1.0

ξd = 0.0

ξ1=ξ2=0.0

0.1 0.312 0.2653

90 60

25 30 0 50

0

40

β1=1.0

ξd = 0.0

ξ1=ξ2=0.0

0.1 0.3402 0.3257

.. x1(a0)

30

24

18

.. x2(a0)

12

20

6

10 0 0.02

0.04

0.06

ξs

0.08

0 0.02

0.10

0.04

0.06

0.08

0.10

ξs Figure 7 Soil dmaping effect on displacement and absolute acceleration considering SSI (β =2 and µ =1)

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4

Conclusions

In this paper, influence of soil condition on performance of viscous damper used for vibration control of adjacent connected structures is analyzed. The following conclusions are drawn. 1. Viscous damper is quite effective for vibration control of adjacent connected dynamically dissimilar structures on stiff groun. 2. The natural frequency of the building with foundation is always lower than that obtained for a building in which foundation effects are neglected. 3. The inclusion of soil structure interaction is very critical. The optimum damper damping coefficient will not remain optimum, if the adjacent connected structures are on soft soil. 4. The Poisson ratio of soil effect is negligible to the structural response. 5. Soil damping influence the performance of viscous damper used for vibration control of adjacent connected structures. 6. Semi-active control system may be more effective to control the response of adjacent connected structures on stiff as well as on soft ground.

5

References

Bhaskararao, A. V., and Jangid, R. S., 2007, ”Optimum viscous damper for connecting adjacent SDOF structures for harmonic and stationery white noise random excitation”, Earthquake engineering and structural dynamics, Vol 36 (4), pp 563-571. Claude, T., and Singhal, A., 1968, ” Influence of soil-structure interaction on earthquake response of buildings”, Laval University, Quebec, Canada. Dowrick, D. J., 2001, ” Earthquake resistant design”, 2nd Edition, John Wiley & Sons. Chichester. Seto Tadanobu., 2000, “Dynamic Analysis and earthquake resistant design vol.2”, Japanese Society of Civil Engineers, Oxford & IBH Publishing co. Pvt. Ltd. New Delhi. Gupta, V. K., and Trifunac, M. D., 1991, ”Seismic response of multistoried buildings including the effect of soilstructure interaction”, Soil Dynamics and Earthquake Engineering, Vol 10(8), pp- 414-422. Nandakumaran, P., Paul, D. K., and Jadia, N. N., 1977, “Foundation type and seismic response of buildings”, International symposium on Soil Structure Interaction. Department of Civil Engineering , University of Roorkee. pp 157-164

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