1

FOUNDATIONS FOR DYNAMIC LOADS

VIJAY K. PURI SHAMSHER PRAKASH

Professor Professor Emeritus

Civil Engineering Civil Engineering

SIU Carbondale Missouri S&T, Rolla

IL 62901 MO 65409

puri@engr.siu.edu

praskash@mst.edu

ABSTRACT

Design of foundations in earthquake prone areas needs special considerations.

Shallow foundations may experience a reduction in bearing capacity and increase in

settlement and tilt due to seismic loading. The reduction in bearing capacity depends

on the nature and type of soil and ground acceleration parameters. In the case of

piles, the soil-pile behavior under earthquake loading is generally non-linear. The

nonlinearity must be accounted for by defining soil- pile stiffness in terms of strain

dependent soil modulus. A comparison of observed and predicted pile behavior under

dynamic loads has attracted the attention of several investigators. The lateral dynamic

pile response of single piles predicted by analytical models often yields higher natural

frequencies and lower resonant amplitudes compared to those determined from field

tests in horizontal vibrations only. This has been found to be due to overestimated

shear modulus and radiation damping of the soil. The authors made an investigation

to determine a simple method to improve the theoretical predictions of piles

embedded in fine soils. Based upon this investigation shear strain dependent

reduction factors are proposed for determining the shear modulus and damping for

pile response calculations.

INTRODUCTION

Structures subjected to earthquakes may be supported on shallow foundations or on

piles. The foundation must be safe both for the usual static loads as well for the

dynamic loads imposed by the earthquakes and therefore the design of either type of

foundation needs special considerations compared to the static case. Shallow

foundations in seismic areas are commonly designed by the equivalent static

approach. The observations during 1985 Michoacan-Guerero earthquake (Pecker,

1996) and the Kocaeli 1999 earthquake have shown that initial static pressure and

load eccentricity have a pronounced effect on seismic behavior of foundations. The

soil strength may undergo degradation under seismic loading depending on the type

of soil. Pore pressure buildup and drainage conditions may result in decrease in

strength and an increase in settlement. Most foundation failures due to earthquake

occur due to increased settlement. However, failures due to reduction in bearing

capacity have also been observed during Niigata earthquake (1964) in Japan and

Izmit earthquake (1999) in Turkey (Day, 2002).

Prasad et al (2004) made an experimental investigation of the seismic bearing

capacity of sand. A practical method to account for reduction in bearing capacity

308

2

due to earthquake loading was presented by Richard et al, (1993). The settlement and

tilt of the foundation must also be considered. Gazetas and Anastasopoulos (2008)

have studied the interaction effects of two adjacent buildings, founded on shallow

footings.

For analysis and design of piles and pile groups under seismic loads a simple

approach to account for nonlinear soil-pile interaction using strain dependent soil

modulus to define soil-pile stiffness and radiation damping can be used and is

discussed in the paper. Some studies are available comparing the predicted and

observed response of piles also. In some of these studies the soil pile stiffness and

damping were arbitrarily modified to match the observed and predicted response of

the soil pile system. A study was conducted under the supervision of the authors in

which the data reported in literature was reanalyzed and reduction factors have been

proposed for the stiffness and radiation damping obtained by using the commonly

approach of Novak and El-Sharnouby (1983). Using the proposed reduction factors

the reported test data of Gle (1981) was reanalyzed. The predicted values of natural

frequency and amplitude of vibration at various frequencies showed an excellent

geotechnical comparison with the observed data. The case of shallow foundations is

discussed first followed by several aspects of pile behavior under dynamic loads.

SHALLOW FOUNDATIONS

The problem of static bearing capacity of shallow foundations has been extensively

studied in the past (Terzaghi (1943), Meyerhoff (1951), Vesic (1973) and many

others. Basavanna et al. (1974) obtained analytical solutions for the bearing capacity

under transient loading conditions. The design of foundations subjected to seismic

loading has generally been performed by using psedo-static approach (Prakash;

1981). The foundations are considered as eccentrically loaded and the ultimate

bearing capacity is accordingly estimated. Prakash and Saran (1971) proposed a

method to determine the settlement and tilt of a foundation subjected to vertical load

and moment. The response of a footing to dynamic loads is affected by the (1) nature

and magnitude of dynamic loads, (2) number of pulses and (3) the strain rate response

of soil. To account for the effect of dynamic nature of the load, the bearing capacity

factors are determined by using dynamic angle of internal friction which is taken as

2-degrees less than its static value (Das, 1992). Building codes such as International

Building Code (2006) generally permit an increase of 33 % in allowable bearing

capacity when earthquake loads in addition to static loads are used in design of the

foundation. This recommendation is based on the assumption that the allowable

bearing pressure has adequate factor of safety for the static loads and a lower factor

of safety may be accepted for earthquake loads. This recommendation may be

reasonable for dense granular soils, stiff to very stiff clays or hard bedrocks but is not

applicable for friable rock, loose soils susceptible to liquefaction or pore water

pressure increase, sensitive clays or clays likely to undergo plastic flow (Day, 2006).

Richards et al (1993) observed seismic settlements of foundations on partially

saturated dense or compacted soils. These settlements were not associated with

liquefaction or densification and could be easily explained in terms of seismic bearing

3

capacity reduction. They have proposed a simplified approach to estimate the

dynamic bearing capacity q

ue

and seismic settlement S

Eq

of a strip footing

Figure 1. Failure surface in soil for seismic bearing capacity (After Richards et

al, 1993; See also Prakash and Puri, 2008)

Figure 1 shows the assumed failure surfaces. The seismic bearing capacity

(q

uE

) is given by Eq. 1:

q

uE

= cN

cE

+ qN

qE

+ ½ γ BN

γE

(1)

where, γ = Unit weight of soil

D

f

= Depth of the foundation and q= γD

f

N

cE

, N

qE

, and N

γE

= Seismic bearing capacity factors which are functions of φ and

tanψ = k

h

/ (1-k

v

)

k

h

and k

v

are the horizontal and vertical coefficients of acceleration due to

earthquake.

For static case, k

h

= k

v

, = 0 and Eq. (1) becomes

q

u

= cN

c

+ qN

q

+ ½ γBN

γ

(2)

in which N

c

, N

q

and N

γ

are the static bearing capacity factors. Figure 2 shows plots of

N

γE

/N

γ

, N

qE

/N

q

and N

cE

/N

c

with tan ψ

and φ.

tanψ = k

h

/ (1-k

v

) tan ψ = k

h

/ (1-k

v

), tanψ = k

h

/ (1-k

v

),

Figure 2. Variation of N

qE

/N

q

, N

γE

/N

γ

and N

cE

/N

c

with φ and tan ψ (After

Richard et al 1993; See also Prakash and Puri, 2008)

SEISMIC SETTLEMENT OF FOUNDATION

Bearing capacity-related seismic settlement takes place when the ratio k

h

/(1 - k

v

)

reaches a critical value (k

h

/1 – k

v

)*. If k

v

= 0, then (k

h

/1 - k

v

)* becomes equal to k

h

*

.

Figure 3 shows the variation of k

h

*

(for k

v

= c = 0; granular soil) with the static factor

4

of safety (FS) applied to the ultimate bearing capacity [Eq, (2)], for φ = 10°, 20°, 30°,

and 40° and D

f

/B of 0, 0.25, 0.5 and 1.0.

Figure 3. Critical acceleration

*

h

k

(After Richards et al, 1993; See also Prakash

and Puri, 2008)

The settlement (S

Eq

) of a strip foundation due to an earthquake can be estimated as

4

2

*

( ) 0.1 7 4 t an

k

V

h

S m

E q A E

A g A

α

−

=

(3)

where V = peak velocity for the design earthquake (m/sec),A = acceleration

coefficient for the design earthquake, g = acceleration due to gravity (9.81 m/sec

2

).

tan α

AE

depends on φ and k

h

*, (Richards et al, (1993). In Figure 4, variation of tan

α

AE

with k

h

* for φ

of 15° - 40° is shown.

Suppose a typical strip foundation is supported on a sandy soil with B = 2 m, and D

f

= 0.5 m, and γ = 18 KN/m

3

, φ = 34°, and c = 0. The value of k

h

= 0.3 and k

v

= 0 and

the velocity V induced by the design earthquake is 0.4 m/sec. The static ultimate

bearing capacity for this footing for vertical load will be 1,000 KN/m

2

(Eq. 2). The

reduced ultimate bearing capacity for vertical load is calculated as 290 KN/m

2

(Eq 1).

If the footing is designed using a FS = 3 on the static ultimate bearing capacity (i.e.,

for an allowable soil pressure of 333 kN/m

2

), the additional settlement due to

earthquake will be 20.5 mm (Eq 3, Richards et al, 1993). This settlement reduces to

7.0 mm if FS of 4 is used. Besides ensuring that the footing soil system does not

experience a bearing capacity failure or undergo excessive settlement, the

foundations should be tied together using interconnecting beams.

Seismic bearing capacity factors for a strip footing resting on cohesionless soil were

determined by Dormieux and Pecker (1995) using the upper bound theorem of yield.

5

Figure 4. Variation of tan α

AE

with k

h

* and φ (After Richards et al 1993; See

also Prakash and Puri, 2008)

Using the classical Prandtl like mechanism, it was established that the reduction in

bearing capacity was mainly caused by load inclination (Dormieux and

Pecker;1995).Choudhury and Subba Rao (2005) determined seismic bearing capacity

factors for shallow strip footing using the limit equilibrium approach and pseudo-

static method of analysis. The reduction in bearing capacity under the combined

effect of vertical and horizontal forces was explained by using smaller failure surface

compared to case when only static vertical loads are applied.

It is thus seen that presently (2009), the pseudo-static approach is being used to

determine bearing capacity and settlement and tilt of the foundations subjected to

seismic loads in non-liquefying soils. Dynamic nature of the load and other factors

which affect the dynamic response are not being accounted for. Also, no guidelines

are available design of footings in liquefying soil.

PILES FOUNDATIONS

Piles may often be the preferred choice of foundations in seismic areas. The seismic

loading induces large displacements or strains in the soil. The shear modulus of the

soil degrades and damping (material) increases with increasing strain. The stiffness of

piles should be determined for these strain effects. The elastic solutions for

determining response of piles subjected to dynamic loads have been presented by

several investigators in the past in several modes of vibrations, (Novak, 1974; Novak

and El-Sharnouby, 1983; Novak and Howell, 1977; Poulos, 1971; Prakash and

Puri,1988; and Prakash and Sharma, 1991). Displacement dependent spring and

damping factors for piles for vertical, horizontal and rotational vibrations have been

presented by Munaf and Prakash (2002), Munaf et al. (2003) and Prakash and Puri

(2008). The stiffness of the pile group is estimated from that of the single piles by

6

using group interaction factors. The contribution of the pile cap, if any, is also

included. The response of the single pile or pile groups may then be determined using

principles of structural dynamics. The design of pile foundations subjected to

earthquakes requires a reliable method of calculating the effects of earthquake

shaking and post-liquefaction displacements on pile foundations. Keys to good design

include reliable estimates of environmental loads, realistic assessments of pile head

fixity, and a mathematical model which can adequately account for all significant

factors that affect the response of the pile-soil-structure system to ground shaking

and/or lateral spreading in a given situation.

The equivalent spring stiffness and damping of the soil-pile system are a function of

Young’s modulus of pile material (E

p

), shear modulus of soil (G

s

), and geometry of

the piles in the group. Shear modulus and hence spring and damping factors are strain

or displacement dependent. There are six independent spring factors for a piles-cap

system; ie; k

x

, k

y

,

k

z

, in translation in x, y and z directions, respectively and k

φ

, k

θ

, k

ψ

rotational-springs about x, y and z directions respectively. There are two rotational

cross-coupled springs; i.e.; k

xφ

and k

yθ

which include 2-components of displacement;

i.e., translation and rotation about the appropriate axis. Also there are corresponding

eight damping factors; i.e., c

x

, c

y,

c

z

, c

φ

, c

θ

, c

ψ

, and. c

xφ

and c

yθ

. To develop

displacement dependent relationships for the spring and damping factors, appropriate

relationships between strain and displacement are needed. Also, modulus

degradation with strain needs be built into these relationships. Stiffness and damping

in all the modes; i.e., vertical, horizontal, rocking and torsion and cross coupling in

both the x and y direction have been evaluated (Munaf and Prakash, 2002). The sign

convention is explained in Fig. 5.

Figure 5: Sign Convention

For the case of earthquake loads, the response of piles under horizontal loads are

important and are discussed here.

Sliding and Rocking Stiffness and Damping Factors

Because, the pile is assumed to be cylindrical with a radius r

o,

its stiffness and

damping factors in any horizontal direction are the same. However, in the pile group,

the number of piles in the x and y directions may be different. Therefore the stiffness

and damping factors of a pile group are dependent on the number of piles and their

spacing in each direction, (Figure 6).

7

Sliding (k

x,

c

x

)

1

3

0

x

pp

x

f

r

IE

K

⎥

⎦

⎤

⎢

⎣

⎡

=

(4a)

2

2

0

x

s

pp

x

f

Vr

IE

C

⎥

⎦

⎤

⎢

⎣

⎡

=

(4b)

Rocking (k

φ

, c

φ

) and (k

θ

, c

θ

)

1

2

φθϕ

f

r

IE

KK

o

pp

⎥

⎦

⎤

⎢

⎣

⎡

==

(5a)

2

2

φθφ

f

Vr

IE

CC

so

pp

⎥

⎦

⎤

⎢

⎣

⎡

==

(5b)

Cross-coupling (k

xφ

, c

xφ

) and (k

yθ

,

c

yθ

)

1

2

0

θθϕ

x

pp

yx

f

r

IE

KK

⎥

⎦

⎤

⎢

⎣

⎡

==

(6a)

2

φθφ

x

so

pp

yx

f

Vr

IE

CC

⎥

⎦

⎤

⎢

⎣

⎡

==

(6b)

Where;

I

p

= moment of inertia of single pile about x or y axis

r

o

= pile radius

E

p

= modulus of elasticity of pile material

V

s

= shear wave velocity of soil along the floating pile

f

x1,

f

x2,

f

xφ1,

f

xφ

2

,

f

xφ1,

f

xφ2

are Novak’s coefficient obtained from Table 1 for

homogeneous and parabolic soil profiles, with appropriate interpolation for ν between

0.25 and 0.4.

Group interaction factor

For group effect in lateral directions (Poulos, 1971), obtained a solution for α

L

for

each pile in the horizontal x-direction, considering departure angle β (degrees). α

L

’s

are a function of L, r

o

and flexibility K

R

as defined in figure 7 and departure angle β.

This procedure will also apply for horizontal y- direction. The group interaction

factor (∑α

L

) is the summation of α

L

for all the piles. Note that the group

interactionfactor in x-direction and y-direction may be different depending on number

and spacing of piles in each direction.

Group Stiffness and Damping

Figure 6 shows schematically the plan and cross sections of an arbitrary pile group.

This figure will be used to explain the procedure for obtaining the stiffness and

damping for a group of pile for all modes of vibration.

8

Figure 6: Plan and Cross Section of Pile Group

Figure 7: Graphical solution of α

L

(Poulus, 1971)

Sliding and Rocking and Cross Coupled Group Stiffness and Damping Factors

Translation along X-axis

∑

∑

=

Lx

x

g

x

k

k

α

(7a)

∑

∑

=

Lx

x

g

x

c

c

α

(7b)

9

Table 1. Stiffness and Damping Parameters of Horizontal Response for Pile with

L/R

o

>25 for Homogeneous Soil Profile and L/R

o

>30 for Parabolic Soil Profile

( Novak and El-Sharnouby, 1983)

Translation along Y Axis (k

y

g

,

c

y

g

)

∑

∑

=

LA

y

g

y

k

k

α

(8a)

∑

∑

=

Ly

y

g

y

c

c

α

(8b)

Rocking About Y- Axis (k

φ

g

,

c

φ

g

)

[ ]

φφφ

α

xccxrz

Lx

g

kzzkxkkk 2

1

22

−++=

∑

(9a)

[ ]

φφφ

α

xccxrz

Lx

g

czzcxccc 2

1

22

−++=

∑

(9b)

Rocking About X- Axis (k

θ

g

,

c

θ

g

)

[ ]

θθθ

α

yccyrz

Ly

g

kzzkykkk 2

1

22

−++=

∑

(10a)

10

[ ]

θθθ

α

yccxrz

Ly

g

czzcyccc 2

1

22

−++=

∑

(10b)

Cross-Coupling: Translation along X Axis and Rotation about Y Axis. (k

xφ

g

,

c

xφ

g

)

( )

∑

−=

cxx

Lx

g

x

zkkk

φφ

α

1

(

11a)

( )

∑

−=

cxx

Lx

g

x

zccc

φφ

α

1

(

11b)

Cross-Coupling: Translation along Y-Axis and Rotation about X Axis. (k

yθ

g

,

c

yθ

g

)

( )

∑

−=

cyy

Ly

g

y

zkkk

θθ

α

1

(

12a)

( )

∑

−=

cyy

Ly

g

y

zccc

θθ

α

1

(

12b)

COMPARISION OF COMPUTED AND PREDICTED PILE RESPONSE

The present methods for design of pile foundations subjected to dynamic loads are

generally based on the models developed by Novak (1974) and Novak and

El_Sharnouby (1984) and also presented by Prakash and Puri (1988) and Prakash and

Sharma (1990). Several researchers have attempted to make a comparison of the

observed and predicted pile response. Small scale pile tests, centrifuge and full scale

pile tests have been used for this purpose (Gle, 1981; Novak and ElSharnouby, 1984;

Woods, 1984; and Poulos, 2007). Woods (1984) reported results of 55 horizontal

vibration tests on 11 end bearing piles 15 - 48 m long. The outer diameter of piles was

35.56 cm and the wall thickness varied from 0.47 cm to 0.94 cm. Typical amplitude –

frequency plot for one of the piles in soft clay is shown in Fig. 8. It may be seen from

this plot that the observed natural frequency decreases with an increase in the value of

‘ө’ (increase in ‘ө’ means an increase in dynamic force at the same frequency of

vibrations) indicating non-linear behavior. Woods (1984) also compared the

observed and computed response of the piles. The stiffness and damping values were

obtained using computer program PILAY which uses continuum model

accommodating soil layers and assumes homogeneous soil in the layer with elastic

behavior. A typical comparison of the pile response so computed with the observed

response is shown in Fig. 9. It may be observed from Fig.9 that the calculated and

computed responses do not match.

Efforts were made to obtain a match between observed and predicted response by

using reduced values of stiffness obtained from PILAY, which did not help much. A

better match could, however, be obtained when a considerably softened or weakened

zone was assumed surrounding the piles (program PILAY 2) simulating disturbance

to soil during pile installation. A loss of contact of the soil with the pile for a short

length close to the ground surface also improved the predicted response.

11

Fig. 8. Response curves; a decrease Fig 9. Typical response curves

in resonant frequency with increasing predicted by PILAY superimposed

amplitudes. (Woods, 1984) on measured pile response.

(Woods, 1984)

Novak and El-Sharnouby (1984) performed tests on 102 model pile groups using

steel pipe piles. A typical comparison of the theoretical and experimental horizontal

response is shown in Fig.10. Plot ‘a’ shows the theoretical group response without

interaction effects. Response shown in plot ‘b’ was obtained by applying static

interaction factors to stiffness only. Plot ‘c’ was obtained with arbitrary interaction

factor of 2.85 applied to stiffness only. Plot‘d’ was obtained by using an arbitrary

interaction factor of 2.85 on stiffness and 1.8 on damping respectively. Plot ‘e’ shows

the experimental data. The plot which shows an excellent match with experimental

data was obtained by

arbitrarily

increasing the damping factor by 45%.

Fig. 10. Experiment horizontal response curves and theoretical curves calculated

with static interaction factors. (Novak and El-Sharnouby, 1984)

The comparison of theoretical response obtained by using dynamic interaction

factors of Kaynia and Kausel (1982) is shown in plot ‘a’ in Fig.11. Plot ‘b’ in Fig. 11

shows the calculated data based on dynamic analysis of Wass and Hartmann(1981).

12

Fig. 11. Experimental horizontal response curve and theoretical curves.

(a) Calculated with Kaynia and Kausel dynamic interaction factors

(b) Calculated with Wass and Hartmann impedances

(c ) Experimental (Novak and El-Sharnouby, 1984).

The experimental data of Novak and EL-Sharnouby (1984) is shown by plot ‘c’.

Novak and El-Sharnouby, (1984) also compared the observed response for vertical

and torsional vibrations with the predicted response.

Jadi (1999) and Prakash and Jadi (2001) reanalyzed the reported pile test data of

Gle (1981) for the lateral dynamic tests on single piles and proposed reduction factors

for the stiffness and radiation damping obtained by using the approach of Novak and

El-Sharnouby (1983). The suggested equations for the reduction factors are:

λ

G

= -353500 γ

2

– 0.00775 γ + 0.3244 (16)

λ

c

= 217600 γ

2

– 1905.56 γ + 0.6 (17)

where, λ

G

and λ

c

are the reduction factors for shear modulus and damping and γ is

shear strain at computed peak amplitude without correction factors.

Typical

comparison of the computed pile response using the modified values of stiffness and

damping values and the observed response is shown in figures 12-20.

Fig 12 shows prediction and performance of Gle’s pile in Fig 9. Fig 13, 14 and 15

are for the cases where prediction and performance matched well. However in Fig16-

18, the prediction and performance does not match well. Fig 19 and 20 shows

prediction and performance from other sites.

COMMENTS ON PREDICTIONS

Novak and El Sharnouby (1984) have attempted to match the observed and the

predicted response by adjusting, arbitrarily, the group stiffness and damping values.

No guidelines were developed to modify these values.

Woods (1984) used PILAY program with modified stiffness to match prediction and

performance.

Jadi (1999) developed rational correction factors to both stiffness and damping to

match the computed and predicted responses. She was successful in her efforts. Her

13

Fig.12 Measured and predicted Fig. 13 Measured and predicted lateral

response of pile of Fig. 11 Dynamic Response of pile GP 13-7 (ө=2.5

˚

)

(Prakash and Jadi 1991) (Prakash and Jadi, 2001)

Fig. 14 Measured Vs arbitrarily reduced Fig. 15 Measured vs. predicted lateral

Dynamic Response of pile K 16-7 (ө=5

˚

) Dynamic Response with proposed

(Prakash and Jadi, 2001) reduction factors for pile K 16-7, (ө=5

˚

)

(Prakash and Jadi, 2001)

Fig.16 Measured and predicted lateral Fig.17 Measured and predicted lateral

Dynamic Response of piles K16-7 Dynamic Response of pile GP 13- 7

(ө=5

˚

)(Prakash and Jadi 2001) (ө=2.5

˚

) (Prakash and Jadi, 2001)

14

Fig.18 Measured and predicted lateral Fig.19 Measured and predicted lateral

Dynamic Response of pile LF 16 Dynamic Response of pile I-WES

(ө=10

˚

)(Prakash and Jadi 2001) vibrator λ

G

= 0.31, λ

c

= 0.5

(Prakash and Jadi, 2001)

Fig.20 Measured vs. Reduced predicted lateral Dynamic Response of the 2.4”

Pile tested by Novak and Grigg, 1976, λ

G

= 0.44, λ

c

= 0.34, (Prakash and Jadi,

2001)

approach, however is more scientific and more efforts needs to be devoted to develop

relationships for correction factors for different modes of vibration.

CONCLUSIONS

Considerable attention has been paid to the design of foundations for earthquake

loads, both for shallow foundations and piles. Efforts have been made to understand

the behavior of the foundations under seismic loading. However the shallow

foundations are mostly designed using the equivalent static approach.

For the case of pile foundations, many efforts have been made for comparing the

predicted and observed response of single piles and pile groups under dynamic

loading and arbitrary modifications to stiffness and damping were made. Jadi’s

(1999) method of modifying shear modulus and damping appears reasonable.

15

However this has been tested against a limited data base. More analyses are needed to

make this tool easily usable in practice.

REFERENCES

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Soils under Transient Loading.” Bull. Indian Soc. Earthquake Tech., Vol.2, No. 3,

pp.67-84.

Choudhury, D. and Subba Rao, K.S. (2005). “Seismic Bearing capacity of Shallow

strip Foundations”, Geotechnical and Geological Engineering, Vol.23, No. 4, pp.

403-418.

Das, B.M.(1992). Principles of Soil Dynamics, PWS Kent.

Day, R. W. (2002). Geotechnical Earthquake Engineering Handbook, McGraw Hill

Dormieux , L. and Pecker, A. (1995). “Seismic Bearing Capacity of Foundations on

Chesionless Soils”,

J. Geot. Engg

. Dn. ASCE, Vol. 121,No 3, pp. 300-303.

Gazetas, D. and Anastasopoulos, I., (2008). “Case Histories of Foundations on top of

a Rupturing Normal Fault during the Kocaeli 1999 Earthquake”,

Sixth

International Conference on Case Histories in Geotechnical Earthquake

Engineering and Symposium in Honor of Professor James K. Mitchell

, CD Rom,

Aug. 11-19, Arlington, Virginia.

Gle, D. R., (1981). “The Dynamic Lateral Response of Deep Foundations”,

PhD

dissertation

, University of Michigan, Ann Arbor.

Jadi, H. (1999). “Prediction of lateral Dynamic Response of single Piles Embedded in

Clay.” MS

Thesis

, UMR, Rolla.

Kaynia, A.M., and Kausel, F., (1982). “Dynamic behavior of Pile Groups.”

Proc.

Second International Conference on numerical Methods in Offshore Piling,

Austin, TX, pp. 509-532.

Meyerhoff, G.G. (1951). “The Ultimate Bearing capacity of Foundations.”

Geotechnique,Vol.2, pp. 301-332.

Munaf, Y. and Prakash, S. [2002],”Displacement Dependent Spring and Damping

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