Turk J Engin Environ Sci25 (2001) , 617 – 626.c© TUBITAK

Non-Linear Seismic Soil-Structure Interaction Analysis Based onthe Substructure Method in the Time Domain

Mustafa KUTANIS, Muzaffer ELMASSAU Muhendislik Fakultesi, Insaat Muh. Bolumu, Yapı ABD

Esentepe Kampusu, Adapazarı-TURKEY

Received 02.08.2000

Abstract

This paper presents an idealized 2-dimensional plain strain finite element seismic soil-structure interaction(SSI) analysis based on a substructure method by using original software developed by the authors. Toinvestigate the effects of SSI the following types of analysis were performed: linear SSI analysis and non-linearSSI analysis. For the same structure, analysis was carried out by the procedure without the consideration ofsoil-structure interaction. These computations were achieved for different peak accelerations: 0.15g, 0.30gand 0.45g. In another case for a different site soil with a shear wave velocity of 200, 300 and 500 m/s,a linear SSI analysis was performed. In the analysis, the radiation condition was fully accounted for, thesoil plasticity was modeled with the Von Mises failure criterion, basemat uplift was not considered, and theaction of gravity was not taken into consideration.

Key Words: Seismic soil-structure interaction, substructure formulation, finite element method

Sismik Yukler Altında Yapı-Zemin Etkilesiminin Zaman Tanım AlanındaAltsistem Yaklasımı Cercevesinde Analizi

Ozet

Bu calısmada 2-boyutlu duzlem sekil degistirme elemanı olarak sonlu elemanlarla modellenen bir sisteminaltsistem yaklasımı cercevesinde, sismik yukler altında yapı-zemin dinamik etkilesimi incelenmistir. Yapılanhesaplamalarda yazarlar tarafından gelistirilen program kullanılmıstır. Etkilesimi izlemek amacıyla sonsuzrijit zemin, lineer yapı-zemin etkilesimi ve lineer olmayan yapı-zemin etkilesimi ile ilgili durumlar, farklıduzeydeki ivme degerleri ve farklı zemin rijitlikleri icin (Vs=200, 300 ve 500 m/s) hesaplamalar yapılmıstır.Burada radyasyon kosulunun tam olarak saglandıgı, zemin plastik davranısının Von Mises akma kriteri ilebelirlendigi, zemin ayrılma etkisinin ve yapının kendi agırlıgının ihmal edildigi bir cozumleme yapılmıstır.

Anahtar Sozcukler: Dinamik yapı-zemin etkilesimi, altsistem formulasyonu, sonlu elemanlar yontemi

Introduction

During the last quarter of the 20th century, theimportance of dynamic soil-structure interaction forseveral structures founded on soft soils was well rec-ognized. If not accounted for in analysis, the accu-

racy in assessing structural safety in the face of earth-quakes cannot be accounted for adequately. For thisreason, seismic soil-structure interaction analysis hasbecome a major topic in earthquake engineering.

In dealing with the analysis of dynamic soil-structure interaction, one of the most difficult tasks

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is the modeling of unbounded media. Many numer-ical methods or techniques have been developed tosolve this problem, such as transmitting boundariesof different kinds, boundary elements, and infiniteelements and their coupling procedures.

There are two main approaches for analyzing soil-structure interaction, namely the direct method andthe substructure method. These methods are welldocumented in two textbooks by Wolf (1985, 1988).Both methods are still being developed to achieve thedesired results in recent years. Among them, a com-mon formulation equally applicable to both methodsis presented by Aydınoglu (1993a, 1993b). This isachieved by changing the size of an irregular soil zoneand the definition of dynamic boundary conditionsalong the interaction horizon. To determine the in-teraction force-displacement relationships of the de-grees of freedom in the nodes along the soil-structureinterface for use in the consistent formulation of di-rect and substructure methods, rigorous formulationis applied. It is based on the similarity and finite ele-ment method, and was originally developed by (Wolfand Song, 1996). Recent studies have proven that itis very effective and accurate.

Recent research results in the field of soil-structure interaction indicate that SSI has an impor-tant effect on the dynamic response of the structureswhen the soil is soft. In general, there are threemajor influences: (1) It will change the dynamiccharacteristics of the soil-structure system, such asmodal frequencies and vibrating shapes. In particu-lar, the fundamental frequency will have significantdrops and the rigid body motion of the structure

will be produced or enhanced. (2) It will increasethe modal damping as some vibrating energy in thestructure will be transferred to the soil. This type ofdamping is called radiation damping. (3) It will in-fluence free-field ground motion (Menglin and Jingn-ing, 1998).

In a seismic soil-structure interaction analysis, itis necessary to consider the infinite and layer charac-teristics of soil strata, and the nonlinear behaviors ofsoft soil. The objective of this study is to perform arigorous seismic non-linear soil-structure interactionanalysis in the time domain to satisfy the above re-quirements while the results are compared with thoseof fixed based structural analysis.

Equations of Motion

Basic Equations of Motion for Fixed BaseStructures

If the soil-structure interaction is not considered, theequation of the motion for the structure under theseismic excitation in the time domain can be writtenin the well-known form as follows:

[M ]{r (t)}+ [C]{r (t)}+ [K]{r (t)} = −[M ]{ug (t)}(1)

in which [M], [C] and [K] are nxn mass, dampingand stiffness matrices, respectively, n is the numberof degrees of freedom of the structure, {r} is the totaldisplacement vector of the system, and {ug} is theacceleration vector of the free-field ground motion.

Seismic input

StructureNear-field/Far-field(Interaction Horizon)

Bounded-irregularsoil zone (near-field)

Unbounded-regularsoil zone (far-field)motion

base

Hypothetical distant boundary

Figure 1. Common model for direct and generalized substructure methods (Aydınoglu, 1993a)

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KUTANIS, ELMAS

Governing Equations of the Substructure For-mulation

A systematic formulation and discussion of nonlinearsoil structure interaction is presented by Aydınoglu

(1993a). Referring to the soil structure model givenin Figure 1 together with corresponding indicesshown in Figure 2, the basic equations of the soil-structure system can be expressed in the time do-main as:

s

b

n

h

r

(i)c

d

bn

h

r

(i)f

e

dsbenhrd(i)c

(i)f

: structure: base: soil portion to be excavated: near field: interaction horizon: far field: hypothetical distant boundary: complete system including structure: complete system of virgin soil

Figure 2. Identification of (a) soil structure system, (b) unexcavated free field (Aydınoglu, 1993a)

[M

(c)ii Mih

Mhi M ihh

]{rti(t)rth(t)

}+{Qi(t)Qih(t)

}+{

0Rrh(t)

}={

0P ih(t)

}(2)

where [M], {Q}, {R} and {P } are mass matrix, non-linear internal forces, interaction forces and effectiveforce vector, respectively. The response vector, {r}of Eq. (2) is represented by total displacement in-dicated by superscript t. The first term on the left-hand side represents the inertial forces in respectiveparts of the system with the last component, Qih (t),being the nonlinear internal forces acting on the in-ner face of the interaction horizon.

For the generalized substructure method, theinteraction force-displacement relationships in thetime domain can be expressed in terms of the relativeinteraction displacements calculated along the inter-action horizon, namely, the difference between totaland free-field displacements, which is formulated as:

Rrh(t) =

t∫0

Srhh(t− τ )rth(τ )dτ−P rh (t) (3)

with Srhh (t) representing the far-field dynamic stiff-ness matrix in the time domain. The second term onthe right-hand side is time effective forces, and can

be expressed as:

P rh(t) =

t∫0

Srhh(t− τ )vfh(τ )dτ (4)

where vfh(τ ) is obtained from nonlinear analysis ofthe unexcavated free-field. The relative interactiondisplacements are defined as:

r∆h (t) = rth (t)− vfh (t) (5)

Thus from Eqs. (3)-(5):

Rrh(t) =

t∫0

Srhh(t − τ )r∆h (τ )dτ (6)

Finally, the non-zero effective force vector com-ponent of Eq. (3) can be expressed as:

P ih (t) =[Mhi Mhi

]{ vfivfh

}+ P ih (t) (7)

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KUTANIS, ELMAS

where the second term represents the internal forcesacting on the inner face of the interaction horizonas obtained from one- or two-dimensional nonlinearanalysis of unexcavated free-field system (Figure 2)incident seismic waves.

To overcome the numerical difficulties and sim-plify the formulation and derivation, the interactionforces are expressed as a convolution integral of theaccelerations (Wolf and Song, 1996), as follows:

Rrh(t) =

t∫0

M∞ (t− τ ) r∆h (τ )dτ (8)

where M∞(t) is the acceleration unit impulse re-sponse matrix in the time domain. It can be deter-mined directly with the consistent infinitesimal finiteelement cell method which is addressed in (Wolf andSong 1995). Therefore, it will not be repeated in thispaper.

The interaction forces of the soil medium at the

soil-structure interface given by Eq. (8) are dis-cretized at time station n for a piecewise constantacceleration unit impulse response matrix (Wolf andSong 1995) as:

{Rrh}n = γ∆t[M∞]1{rth}n − γ∆t[M∞]1{vfh

}n

+ (1− γ) ∆t[M∞]1{r∆h

}n−1

+n−1∑j=1

[M∞]n−j+1

({r∆h

}j−{r∆h

}j−1

)(9)

rewriting the Eq. (9) as:

{Rrh}n = γ∆t[M∞]1{rth}n

+{Rrh

}n

(10)

Substituting Eq. (9) in Eq. (2) leads to the non-linear seismic soil-structure interaction formulationin the time domain which is expressed as:

[M

(c)ii Mih

Mhi M ihh + γ∆t[M∞]1

]n

{rti(t)rth(t)

}n

+{Qi(t)Qih(t)

}n

+{

0Rrh(t)

}n

={

0P ih(t) − Rrh

}n

(11)

With this formulation, true nonlinearity of soilcan be consistently taken into account within thenear field by properly defined constitutive models(Aydınoglu, 1993a). To solve Eq. (11), Wilson-θmethod is applied to treat the time variable, andthe initial stiffness method is used to treat materialplasticity.

Material Nonlinearity

With SSI analysis there are two kinds of nonlineari-ties. The first one which has received most attentionfrom researchers and practicing engineers, and is as-sociated with the nonlinear behavior of the soil. Thesecond is associated with the partial separation (up-lift) of the foundation from the soil mass, resultingfrom the inability of the soil to resist tension (Jian-guo et al., 1998). Soil is the most complicated en-gineering material, especially when considering theeffects of seismic and dynamic loading.

In this study the authors will follow the mate-rial nonlinearity of both soil and structure withinthe framework of plasticity theory. For this purpose,the well established Von Mises model is employed

to model the failure of the materials. For the plainstrain application the failure surface of the model isexpressed as:

F = σ − σY = 0 (12)

where σY is yield stress and σis deviator stress whichis obtained as:

σ = 1√2

[(σx − σy)2 + (σy − σz)2

+ (σz − σx)2 + 6τ2xy

]1/2 (13)

in which {σxσyσzτxy} is the Cartesian stress tensor.An outline of the elastic-plastic algorithm, which

comes after the stiffness matrix formation, is givenin the flowchart represented in Figure 7.

Numerical Application and Discussions

The proposed analysis model is applied to study thedynamic responses of structures to earthquake exci-tation in the time domain. The computational modelemployed in this section is shown in Figure 3. The

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KUTANIS, ELMAS

parameters of the model are given in Figure 3 andTable 1. The accelerogram (E-W component) for theErzincan earthquake of 1992 (Figure 5) is employedas the horizontal ground motion applied to the anal-

ysis model and scaled to have different peak accel-erations: 0.15g, 0.3g and 0.45g. To investigate theeffects of soil-structure interaction with each inputmotion level, the following cases are studied:

Far field, linear regular soilunbounded medium

Rr

h

r∆

irregular soil medium

B

H

bz

hz

interaction horizon

H=50 mB=20 mhz=20 mbz=90 m

h

Figure 3. Geometry and discretization of the SSI system

1

2

3

45 6

7

8

9

1011

12

13

1415 16

17

18

19

2021

22

2324 25 26 27

28

29

3031

32

33 34 35 36 37 38

39

40

41 42 43 44 45 46

47 48 49 50 51 52

53 54 55 56 57 58

59 60

61 62

63 64

65 66

67 68

69 70

71 72

73 74

75 76

77 78

Figure 4. Element number of the SSI system

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KUTANIS, ELMAS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20-0.5-0.4-0.3-0.2-0.1

00.10.20.30.4

Acc

eler

atio

n, g

Time (s)

Erzincan 1992 E-W

Figure 5. Recorded acceleration time history at the ground surface (max. 0.44g)

Table 1. Material properties of the system considered

Young Modulus Shear Wave Mass Density Yield Stress Poisson’s(kgf/m2) Velocity (m/s) (kg/m3) (kgf/m2) Ratio

Superstructure 2.75 E+09 - 2400 1.0E+08 0.25Near Field - 150 1700 2.0E+05 0.35Far Field - 200 1800 - 0.35

1E-5 1E-4 1E-3 0.01 0.1 1 10

CYCLIC SHEAR STRAIN, PERCENT

0.0

0.2

0.4

0.6

0.8

1.0

1E-5 1E-4 1E-3 0.01 0.1 1 100.0

5.0

10.0

15.0

20.0

DA

MP

ING

RAT

IO P

ER

CE

NT

G/G

max

Soil with

Soil with

OCR=1-15

Roc

Soil with

Soil with

OCR=1-15

Roc

Figure 6. Modulus reduction curves and damping ratio with cyclic shear strain (Vucetic & Dobry, 1991)

(a) Neglecting the effect of soil-structure interac-tion, i.e. assuming the structure being fixed atits base, the soil is assumed to be completelyrigid and only the superstructure is consideredfor analysis.

(b) Linear soil-structure interaction analysis.

(c) Nonlinear soil-structure interaction analysis.

For this purpose, at first, seismic free-field inputmotion along the interaction horizon is determined.

This is achieved by the analysis of unexcavated vir-gin soil in the absence of the structure. The free fieldmotion is calculated by assuming an upwards prop-agating shear wave. To carry out this step, usingthe data given in Table 2 and Figure 6, a well-knowncomputer program, SHAKE, is employed. Then, as-suming the far-field to be linear, dynamic boundaryconditions along the interaction horizon are definedby calculating the unit-impulse response matrix ofthe far-field in the time domain. In the third step,

622

KUTANIS, ELMAS

the analysis of the soil structure system under theaction of free-field input motion determined in thefirst step, subject to the dynamic boundary condi-tions determined in the second step, is carried outby using the finite element software developed by the

authors. The flowchart of the computational proce-dure is given in Figure 7. In this analysis, basematuplift is not considered and the action of gravity isneglected.

Calculatestrains/stress

Yield violated

Use ElasticStress-Strain

Matrices

Use PlasticStress-Strain

Matrices

Yes No

Update GlobalStiffnesses

SolveEquations of

Motion

Calculate Displacements,Velocities, Accelerations

Assemblage the StartingGlobal Stiffness

Input andInitialization

For all time steps

Figure 7. Flowchart for proposed procedure

Table 2. Soil profile data for free field response analysis

Layer No Soil Type Thickness Damping Unit Weight Shear Wave Velocity(m) (%) (kgf/m3) (m/s)

1 PI=15 3 0.05 1600 1502 PI=200 4 0.05 1700 1803 PI=200 6 0.05 1700 1804 PI=200 4 0.05 1700 1805 PI=200 3 0.05 1800 2006 PI=200 40 0.05 1900 3007 Rock - 0.01 2400 1600

The calculated soil-structure interaction re-sponses for different cases are shown in Figures 8, 9,10 and 11. As shown in Figure 8, at the 0.15g acceler-ation, the level linear and nonlinear response are co-

incident, but, as the acceleration level increases non-linear response becomes significant (Figures 9 and10). From these figures, it is seen that fixed baseanalysis gives somewhat greater displacements. As

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KUTANIS, ELMAS

some of the elements have exceeded the defined yieldstress at the 0.45g acceleration level, the others re-main within the elastic limit (Figure 12). In Figure11, the linear response that is calculated for the dif-

ferent soil properties is shown. From the numericalresults, it is clear that as the shear wave velocity ofthe soil increases the response decreases.

0 1 2 3 4 5 6 7 8 9 10Time (sn)

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Top

disp

lace

men

t (m

)

8

6

4

2

Fixed base

Linear SSI

Nonlinear SSI

Fixed base

Linear SSI

Nonlinear SSI

Figure 8. At 0.15g acceleration level top displacement of the structure vs time

4

2

1

8

6

4

2

0 1 2 3 4 5 6 7 8 9 10Time (sn)

-0.020

0.020.040.060.08

0.10.12

Top

disp

lace

men

t (m

)

-0.04-0.06-0.08-0.1

-0.12-0.14

Fixed base

Linear SSI

Nonlinear SSI

Figure 9. At 0.30g acceleration level top displacement of the structure vs time

2864218642

Fixed base

Linear SSI

Nonlinear SSI

0 1 2 3 4 5 6 7 8 9 10Time (sn)

Top

disp

lace

men

t (m

)

Fixed base

Linear SSI

Nonlinear SSI

0.180.160.140.120.1

0.080.060.040.02

0-0.02-0.04-0.06-0.08-0.1

-0.12-0.14-0.16-0.18-0.2

Figure 10. At 0.45g acceleration level top displacement of the structure vs time

624

KUTANIS, ELMAS

864218642

Vs=200 m/s

Vs=300 m/s

Vs=500 m/s

0 1 2 3 4 5 6 7 8 9 10Time (sn)

Top

disp

lace

men

t (m

)

Vs=200 m/s

Vs=300 m/s

Vs=500 m/s

0.180.160.140.120.1

0.080.060.040.02

0-0.02-0.04-0.06-0.08-0.1

-0.12-0.14-0.16-0.18-0.2

Figure 11. At 0.45g acceleration level the investigation of the effect of different soil properties, Linear SSI analysis

-0.2 0 0.2Shear Strain

-1.2E+5

-9E+4

-6E+4

-3E+4

0

3E+4

6E+4

9E+4

1.2E+5

Element No: 20

-0.2 0 0.2Shear Strain

-1.2E+5

-9E+4

-6E+4

-3E+4

0

3E+4

6E+4

9E+4

1.2E+5

Element No: 10

-0.04 0 0.04Shear Strain

-6E+4

-3E+4

0

3E+4

6E+4

Element No: 29

-4E-4 0 4E-4Shear Strain

-1.5E+5

-1.2E+5

-9E+4

-6E+4

-3E+4

0

3E+4

6E+4

9E+4

1.2E+5

1.5E+5

Element No: 60

She

ar S

tres

s kg

f/m2

She

ar S

tres

s kg

f/m2S

hear

Str

ess

kgf/m2

She

ar S

tres

s kg

f/m2

Figure 12. Calculated shear stress-strain response obtained by elastic-plastic procedure at different elements

Conclusions

The authors performed a calculation for the lin-ear and nonlinear soil-structure interaction analy-sis based on the substructure method. It has been

shown that the procedure works well and it can beemployed for the analysis of important structures. Itis desirable to expand it to a three-dimensional case.Research on this subject is continuing. Much hasto be done in investigating the performance of the

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KUTANIS, ELMAS

model, and the numerical procedures, as well as thevarious influence factors on the response of a soil-structure system. Moreover, the material dampingeffect of foundation media in the time domain needsto be improved.

Notation

[C] damping matrixG shear modulusg gravitational acceleration[K] stiffness matrix[M ] mass matrixn time station

OCR over consolidation ratio{P } effective force vectorPI plasticity index{Q} nonlinear internal forces{R} interaction forces{r} displacement vector[S] far field dynamic stiffness matrix∆ t time step{ug} acceleration vector of free fieldVs shear wave velocity{v} free field displacement vectorγ Newmark’s method constantσ deviator stressσY yield stress

References

Aydınoglu, M.N., “Consistent Formulation of Di-rect and Substructure Methods in Nonlinear Soil-Structure Interaction”, Soil Dynamics and Earth-quake Engineering, 12, 403-410, 1993a.

Aydınoglu, M.N., “Development of Analytical Tech-niques in Soil-Structure Interaction”, in Develop-ment in Dynamic Soil-Structure Interaction, ed.P.Gulkan & R.W.Clough, NATO Advanced StudyInstitute, Kemer-Antalya, Turkey, 1992, KluwerAcademic Publishers, 25-42, Dordrecht, 1993b.

Jianguo, X., Danming, W., Tieming, F., and Jun,L., “Nonlinear SSI–Simplified Approach, ModelTest Verification and Parameter Studies for Seis-mic and Air-Blast Environment”, Dynamic Soil-Structure Interaction: Current Research in Chinaand Switzerland, ed. Z. Chuhan & J.P. Wolf, Else-vier Science, Amsterdam, 245-259, 1998.

Menglin, L., and Jingning, W., “Effects of Soil-Structure Interaction on Structural Vibration Con-trol”, Dynamic Soil-Structure Interaction: CurrentResearch in China and Switzerland, ed. Z. Chuhan

& J.P. Wolf, Elsevier Science, Amsterdam, 189-202,1998.

Vucetic, M., and Dobry, R., “Effect of Soil Plas-ticity on Cyclic Response”, Journal of GeotechnicalEngineering, 117(1), 89-107, 1991.

Wolf, J.P., “Dynamic Soil Structure Interaction”,New Jersey: Prentice-Hall, Inc., Englewood Cliffs,1985.

Wolf, J.P., “Soil Structure Interaction Analysis inTime Domain”, New Jersey: Prentice-Hall, Inc., En-glewood Cliffs, 1988.

Wolf, J.P., and C., Song, “Unit-Impulse ResponseMatrix of Unbounded Medium by InfinitesimalFinite-Element Cell Method”, Computer Methodsin Applied Mechanics and Engineering, 122, 251-272, 1995.

Wolf, J.P., and C., Song, “Finite Element Modelingof Unbounded Media”, West Sussex: John Wiley &Sons, 1996.

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