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Dynamic Shear Modulus of Soils, Foundation Stiffness and Damping for Seismic Analysis of Jack-Ups V (Karthi) Karthigeyan, Offshore Division, HSE, UK

Abstract This paper presents methods for calculating the foundation stiffness to be used in the model for seismic analysis of jack-up platforms. It also includes methods for assessing dynamic shear modulus and internal damping of soils to be used in these calculations. The information presented is intended to be used with the International Standard, ISO 19905, Site-specific assessment of mobile offshore units part 1: Jack-Ups, which is in preparation. Guidance on calculating and modelling springs and dampers in different directions or degrees of freedom or formulating the stiffness and damping matrices are presented.

Key Words- Jack-Up, soil dynamics, dynamic shear modulus, shear wave velocity, foundation stiffness, radiation damping, seismic analysis.

Introduction Seismic analyses of jack-ups require the foundation for spud cans to be modelled as discrete springs and dampers or as stiffness and damping matrices. These are calculated using the shear wave velocity vs or dynamic shear modulus G, poisons ratio and mass density of soils below and around the spud cans as well as the natural frequencies in various modes.

The first part of this paper contains different methods for evaluating shear wave velocity of soils. Guidance on preferred methods are provided in the form of a flow chart. It also provides correction factors to be applied depending on the accuracy of the method used. Second part contains methods for calculating stiffness and radiation damping for vertical, horizontal and rocking degrees of freedom. Damping is due to energy radiating away from the foundation in the form of shear, Raleigh and P waves as well as due to hysteresis energy loss or internal damping within the soil mass. Evaluating internal damping is explained in the first part along with other soil properties.

Simplified expressions are provided where possible, together with cautions about their limitations. Expressions or equations are provided in non dimensional form.

Shear Wave Velocity and Shear Modulus Stiffness and damping are calculated using dynamic shear modulus G, poisons ratio and mass density of soil layers below and around the spud-can. Some software uses shear wave velocities (S-wave) vs directly. G can be expressed as a function of Vs using the expression.

G = VS2 -(1) Where is the mass density of soil. If the compressional wave velocity vp is available, it can be converted to vs using eq. (2).

Vs = Vp

2221

-(2)

where is the Poissons ratio. Vs can also be measured directly, indirectly in the laboratory. G can be measured or calculated from Vs or other available soil properties.

Shear modulus G varies with strain. Most in-situ and laboratory methods measure s-waves or modulus at small strains, typically between 1x10-6 to 1x10-5. Both S-wave velocity and shear modulus are highest at these small strains as the soil remains in the elastic region. Shear strains during strong motion earthquakes range from 1x10-4 to 1x10-3. Hence, different subscripts are used for them at different strain or stages:

Symbols, Vsmax and Gmax are used for shear wave velocity and shear modulus of soil: measured in-situ, in the laboratory or calculated using empirical formulas provided.

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Symbol Gmax is also referred to as the tangent modulus. Normally in-situ measurements are carried out before the jack-up is installed. Empirical formulae can be applied to the state of soil with or without jack-up load.

Symbols Vsd and Gd are used for the corrected values of shear wave velocity and shear modulus of soil. These are used in the calculation of stiffness. Corrections applied to Vsmax and Gmax to convert them to Vsd and Gd are:

Higher earthquake strains Increased mean effective stress m in the soil and Possible increased voids ratio e.

Increased m and e are caused by the jack-up weight and pre load. Caution Some in-situ and laboratory measurements are carried out at strains higher than 1x10-5. The geotechnical report can be expected to provide these details.

Methods Fig 1 provides an outline of the method:

Measured Insitu(Preferred)

S-Wave Velocity Vs(or calculated fromP-Wave velocity Vp)

orCone Penetration Test

CPT

Calculate GdGd = vsd2

or

Calculate GmaxGmax = vs2

&Adjust to suit insitu soil pressures +

earthquake strain

Alternative -(3rd option)

Use known soilparameters

to calculate Gmaxor

Measure Gmaxinsample

Using UndisturbedSample

(2nd Choice)Measure S-Wave vsdat earthquake strain

orMeasure S-Wave vs

at small strain

OR

Calculate foundationstiffness'kz, kh & k

&Radiation damping

Dv, Dh & Dusing software utilising soil

layers -Preferred option

AlternativeCalculate foundation

stiffness'kz, kh & k

&Radiation damping

Dv, Dh & Dusing equations in

Apply reduction factors to Dv,h&add internal damping

&Use in Structural model

Fig1

In-situ Measurements Most reliable measurements are obtained by in-situ measurements of s-wave. In the absence of S-wave measurements, Cone Penetration Test (CPT) values may be used to calculate Gmax .

Shear wave velocity

Shear wave velocity Vs for different layers can be easily measured by geotechnical contractors using seismic cone penetrometer. These are measured at small strains from 1x10-6 to 1x10-5. If P-wave velocity is available from

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geophysical investigations can be converted to vsmax using the expression (2). In the absence of measured poisons ratios, the values given in table1 may be used.

When both Vsmax and Vp are available from tests or geophysical investigations using the same strain range, can also be used to calculated using equation 2. The values given intable 1 should be used with caution as the Vs to Vp ratio is sensitive to at values close to 0.5 and can produce zero S-wave.

Soil Type Poisons Ratio Saturated Clay 0.45 0.50

Partially Saturated Clay 0.35 0.45 Dense Sand or Gravel 0.40 0.50

Medium Dense Sand or Gravel 0.30 0.40 Silt 0.30 0.40

Table1

Geophysical investigations carried out for hydrocarbon drilling purposes or as preliminary to geotechnical investigations and they do not normally provide the same detail and accuracy with depth as seismic down-hole test. Hence CPT results, if available in preference or the resulting values should be checked against and complemented by those measured in the laboratory or calculated using empirical relationships. Shear Modulus Using CPT Results

In the absence of shear wave measurements, Gmax may be obtained using CPT measurements using the expressions:

Gmax = 290.6(qc)0.25(v)0.375 pa0.375 for quartz sand [1] -(3) Gmax = 99.5(qc)0.695 pa0.305 e-1.13 for clay [2] -(4)

Where qc is cone tip resistance, a is the atmospheric pressure, v is vertical effective stress and e is voids ratio. Laboratory Measurements of Shear Wave velocity and Shear modulus Only brief information is provided to assist in specifying requirements to geotechnical contractors. Geotechnical contractors use Resonant Column Tests in tensional mode or longitudinal mode to calculate and provide Vsmax and Vpmax for small strains of 1x10-6 to 1x10-5. Peizoelectric bender element tests can also be used to provide Vsmax.

High strain tests such as cyclic triaxial tests or cyclic direct simple shear tests are used to measure shear modulus Gd or elastic modulus Ed. If both Gd and Ed are measured at the same strain, Poissons ratio can be calculated using the expression

= GE

2-1 (5)

Geotechnical laboratories will also be able to determine internal or hysteresis damping from the frequency response curve using half power bandwidth or by stopping and measuring the decay of vibration. Laboratory measurements of S-wave or G should be reported along with mean principal effective principal stress m and strains used, in order that they can be corrected to the in-situ values in calculating foundation stiffness.

Alternative Empirical Methods for Gmax Clay

-Using Voids ratio

The value Gmax for clay can be obtained using the following expression by Hardin [3]:

Gmax = 625 Fe (OCR)k pa0.5 m0.5 (6)

Where OCR is over consolidation ratio, m is mean effective principal stress, pa is the atmospheric pressure, and F(e) = 1/(0.3 + 0.7e2) (7)

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F(e) is valid for voids ratios e from 0.4 to 1.2

The parameter k is given in table 2 as a function of Plasticity Index PI.

Plasticity Index PI k

0

20

40

60

80

>100

0

0.18

0.30

0.41

0.48

0.50

Table 2

Values of k

-Using Undrained Shear Strength

The value of Gmax as a function of un-drained shear strength Su is given in table 3[4].

Gmax / Su

Over Consolidation ratio OCR Plasticity Index PI 1 2 5

15-20

20-25

35-45

1100

700

450

900

600

380

600

500

300

Table 3

Sand

Gmax for sand is given by the expression

Gmax = 700 Fe pa0.5m0.5 -(8)

Fe = ee

+

1)17.2( 2

-(9)

Which is valid for round and angular sands for voids ratios e from 0.5 to 1.0 [5].

or Gmax= 22K2max pa0.5m0.5 -(10) K2 is as a function of relative density Dr is given below in table 4 [6].

Dr K2 30

40

45

60

75

90

34

40

43

52

59

70

Table 4

Values of K2

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Gravel

Gravels with the same relative density, exhibit slightly higher shear modulus than sands, hence It is suggested that equation 10 is used for gravels, with values for K2 in table 4 increased by 50%. It can be up to double the values given in table depending on the type of gravel. Seed Provides more detailed information on Gmax of gravels if required.

Corrections for Shear Modulus and S-wave Velocity Whether Gmax or Vs are measured in-situ, in the laboratory, or using empirical relationships, they need to be corrected to account for:

mean effective principal stress under jack-up possible change in voids ratio e due to pre-load and weight of jack-up and anticipated strain during earthquake

Equations 3, 4 and 6 to 10 indicates how the change in the above parameters affect G.

G is proportional to square root of mean effective pressure and directly proportionally to Fe.

Corrected value of G for calculating stiffness and damping is given by:

Gd = Gmax c fc c -(11a) Where c, fc and c are corrections for earthquake strain, and changes in Fe & mean effective stress respectively. If G was measured at strains greater than 1x10-5, this value should be used instead of Gmax.

When Vs is used in calculations or as input, the corrections are as in equation 11b.

vsd = vsmax (c fc c) -(11b) Where

c = edwhenMeasur

servicein5.0

0

5.00

-(11c)

(Since G varies as the sq. root of mean principal effective principal stress, high levels of accuracy is not be necessary.)

And fc = Fe-withJackUp / Fe-prior (11d)

Change in Fe is not very sensitive to small changes in e. It can be neglected, unless significant compaction or consolidation occurs during pre-load.

Change in G due to earthquake strain is discussed in the next paragraph.

Correction for Strain c Shear modulus reduces with increasing strain as seen in fig Fig 2.

Reduction of shear modulus with strain [7]

Fig 2

Graphs by Vucetic and Dobry[8], containing reduction factors due to increased strain for various plasticity indices and mean effective confining pressures are given by many authors. Ishibashi & Zhang [9] have produced

Cor

rect

ion

fact

or

c ( G

/Gm

ax)

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expressions, which provide the same results. These can be easily incorporated into calculations or programs and are given below, in preference to the graphs by Vucetic and Dobry.

The correction factor c is given by: c = K(,PI) mm(,PI)-m0 (12)

Where K((,PI) is given by:

( )

++=492.0

000102.0lntanh15.0),( PInPIK (13)

And

( )

=3.14.0

0 0145.0exp000556.0lntanh1272.0, PImPIm

(14)

n(PI) is given in table 5 below.

Plasticity Index PI n(PI)

PI = 0 (sandy soils) 0.0

0 < PI 15 3.3710-6PI1.404 15 < PI 70 (medium plastic soils) 7.010-7PI1.976

PI > 70 (high plastic soils) 2.710-5PI1.115 Table 5

If shear modulus, say Gint was measured at strains higher than 1x10-5, obtain reduction factor for both strains, ie d (=Gd / Gmax) and int (=Gint / Gmax). The correction factor is d / int. Corrections for Inaccuracies It should be recognised that there is scope for large inaccuracies in dynamic shear modulus calculated or measured using methods mentioned in the above paragraphs. This can be recognised by using a range of values for G in calculating foundation stiffness.

Method Used Range for G Range for vs

In-situ -25% to +25% -10% to +10%

Laboratory -40% to +50% -20% to +20%

Empirical -50% to +80% -30% to +30%

Table 6

Internal Damping Internal damping is due to energy lost due to plastic behaviour of soil. It is small component of total damping which consists of both internal and radiation damping, higher inaccuracies may be permitted. Radiation damping and the method of combining it with internal damping is explained later.

Ishibashi & Zhang have produced an expression for internal damping as a function of c (or G/Gmax) and PI, obtained from eq. 12 or other sources, which can be easily incorporated into calculations or programs and is given below.

( )( )( )23.1 547.0547.110145.0exp11665.0 ccPI ++= (15) Alternatively:

Since Internal or hysteretic damping is small in relation to the radiation damping to which it is added and can be assumed to be 6% irrespective of the type of soil or magnitude of earthquake.

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Stiffness and Radiation Damping For the seismic analysis of the jack-ups, the effect of soil can be modelled as springs and dampers. It is preferable to calculate stiffness and damping using dedicated software taking into account the variability of soil conditions with depth and different layers. Output is normally provided for a range of frequencies. The choice of stiffness and damping from the range of values require approximate assessment of first natural frequencies involving sliding, rocking and vertical movement of the spud can. Table 7 contains these frequencies for a range of jack-ups and can be used as a guide to choosing appropriate frequencies.

Stiffness and damping can also be calculated using simplified expressions and graphs produced by many authors. Their accuracy is limited due to the lack of ability to take account of the variability of soil layers. However, even if shear wave velocities for different layers were measured in-situ, there are uncertainties about the values to be used for operating conditions due to variations caused by pre-load, operating load and anticipated earthquake strain. Hence using simplified expressions or graphs to calculate foundations stiffness would be adequate as long as the limitations are appreciated.

Since most of the energy is radiated away as surface waves, the calculated radiation damping should to be reduced substantially to account for the reflection from adjoining spud cans and other foundations. Hence the use of simplified expressions for radiation damping is acceptable, provided suitable reduction factors are applied.

This paper provides design charts and expressions to be used for calculating stiffness and damping based on the work by Novak and Beredugo [10,11]. Different authors [12,13,14,15] have provided simplified expressions and graphs to be used to calculate foundation stiffness and damping. These results are not significantly different to those given by Navak and Beredugo. The work by Novak and Beredugo are chosen because the simplified polynomials can be incorporated into calculations or computer programs and they allow different soil properties to be used for soil surrounding the spud-can. In addition these authors have also validated their work by full scale tests.

Methods for calculating stiffness and damping for torsional mode or rotation of spudcan about the vertical axis is not presented. This mode is unlikely to be excited as long as the legs are not eccentric to the spud can. When the spacing between legs is large in relation to lateral dimensions of spud can, the contribution of torsional foundation stiffness of individual spudcan foundation in resisting the global torsionl rotation of the jack-up is small. Since the effect of coupling between horizontal and rocking modes are small, it is not included either. However simple expressions and guidance is provided for those who want to incorporate the effects of coupling. Variables

The stiffness and damping varies with the dimensionless frequency a0, which itself is a function of spud-can size, soil properties (Vs or G and ) and the frequency of excitation. The applicable range of these variables are considered in the following paragraphs in order to consider the simplest methods.

Dimensionless frequency a0 is defined by:

Gra 00 = -(16a)

or a0 = r0/vs -(16b) where r0 is the radius for circular or nearly circular spudcans. For other shapes of the spudcans, the equivalent radius can be defined by:

=Ar0 -(17a) for vertical and horizontal modes and

r0 = 4 4I -(17b) for rocking mode.

Shear wave velocity of soils

Shear wave velocity Vs of clay varies from around 60 m/s for very soft clay to 150m/s for firm clays. For sand, the values varies from around 120m/s for loose sands to 400m/s for dense sands. Gravels can have Vs ranging from 300m/s to 500m/s. Hence the range of shear wave velocities to be considered is between 60m/s to 400m/s.

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Variation of Spud can shapes and Sizes

For most jack-ups it is reasonable to consider the spud-cans to be circular. Their lateral dimension varies from 6m to 18m, giving a range of r0 from 3.0m to 9.0m.

Frequencies of Interest

Table 7 provides natural frequencies for jack-ups operating in various locations during the last 10 years. Using the range of frequencies, spudcan sizes and shear modulus, the dimensional frequencies a0 were assessed to be between 0.04 and 1.6 and the charts to assess stiffness and damping are presented for this range.

Water depth (m)

Mode 119 m

Maersk Gallant at

Goldeneye

119 m Maersk XL

at Goldeneye

98.5 m Maersk Giant

at Otway

92.7 m Magellan

at Elgin A

79.0 m Rowan Gorilla

at Ekofisk

Horizontal Hz 0.15 0.13 0.12 0.14 0.36

Torsion Hz 0.34 0.15 0.27 0.16

Vertical Hz 1.5 1.3 1.7 1.5 2.1

1 st Bending Hz 1.3-1.4 1.3-1.8 1.4-1.5 1.6-1.7 2.2-3.8

Table 7

Symbols

Fig 3a shows symbols used in the expressions used to calculate stiffness and damping.

Gs

GD

HarmonicForce

Reaction Fig 3a Fig 3b

Kz C z

K h

C h

C&

K

X

Z

Y

Fig 3c Fig 3d

The stiffness Kz given in equation 18 is the real component of the reaction to the unit harmonic force on the rigid mass-less disk, shown in Fig 3b. Cz is the imaginary component. Figs 3b shows excitation and reaction in the vertical direction only. Fig 3c shows one horizontal spring Kh with the corresponding dampers Ch in direction X. For most spudcan configurations, the horizontal stiffness and damping in the perpendicular Y direction will be the same. This also applies to the rotational spring and damper K & C about X and Y axis.

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Vertical Stiffness and Radiation Damping

The vertical stiffness Kz and damping Cz can be calculated using equations 18 and 19. The values of Cz1, Cz2, Sz1 and Sz2 given in Fig 4 as a function of dimensionless frequency a0, which can be calculated using the estimate of vertical natural frequency of the jack-up. D is the penetration as shown in fig 3a, G is the dynamic shear modulus in soil layers below and Gs - Dynamic shear modulus of embedded soil.

)( 10

10 zs

zz SrD

GG

CGrK += -(18)

and

)( 20

20

zs

zz SrD

GG

CGr

C += -(19)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.60

2

4

6

8

10

12.389

0

a 0

0

a 0

0

1.60 a 0

Val

ues

of C

& S

Dimensionless Frequency a0

C 2 = 0

.5

C 2 = 0

.2 5

C1 = 0 .5

C1 = 0 .2 5

S1

S2

Fig 4

Horizontal Stiffness and Radiation Damping

Horizontal stiffness and damping can be calculated from equations 20 and 21.

)( 10

10 hs

hh SrD

GGCGrK += -(20)

and

)( 20

20

hs

hh SrD

GGCGrC +=

-(21)

Ch2, Sh1 and Sh2 are given in Fig 5. Cx1 is 4.571 for = 0.0 and 5.333 for = 0.5

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0 0.05 0.1 0.15 0.20

1

2

3

4

543

0

0

0

0

0

0.20 a 0

Val

ues

of C

& S

Dimensionless Frequency a0

S1 = 0 .2 5

S1 = 0 .4

S2 = 0 .4S2 = 0

.2 5

C2 = 0 .5C2 = 0

Fig 5

Rocking Stiffness and Radiation Damping

Rocking stiffness and damping can also be expressed by the following equations by Beredugo and Novak, as simplified by Prakash and Puri to avoid coupling and higher order terms.

)]3

([ 120

2

10

13

0 hs S

rDS

rD

GGCGrK ++= -(22)

and

)]31([ 22

0

2

20

2

30

hs S

rDS

rD

GGCrGC ++=

-(23)

C1, C2, S1 and S2 are given in Fig 6.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

553

0

a 0.

a 0.

0.

0.

1.50 a 0.

Val

ues

of C

& S

Dimensionless Frequency a0

S1

S2

C1

C2

Fig 6

Horizontal & Rocking Coupling

Coupling between horizontal and rocking modes is small and no detailed guidance is given. Those who want to calculate it can incorporate it within the stiffness and damping matrices, or if the foundation is modelled as discrete springs and dampers as shown in Fig 3c, these can be located at distances Lk and Lc, below the spud can, effectively extending the legs. Lk and Lc are defined in equations 24 and 25. When backflow is expected, the effect of coupling could be reduced substantially.

Lk = Kh/Kh -(24) Lc = Ch/Ch -(25)

Where Kh and Kh are cross and sliding stiffness respectively and Ch and Ch are cross and sliding dampers. The extended members Lk and LC are normally modelled as rigid elements and they overlap with each other. In the absence of a calculated values, the following, by Roesset [12] can be used to calculate cross stiffness and damping:

Kh = Kh = Kh (0.4D -0.03r0) -(26)

Ch = Ch = Ch (0.4D -0.03r0) -(27) Where D is the embedment and r0 is its equivalent radius. In discrete modelling this extends the length of leg by 0.4D-0.03r0, which is insignificant for a jack-up. Variable Coil Conditions or Soil Stratum Resting on Rock

Stiffness

For layer of soil with thickness H exceeding 2r0, Kausel and Roesset [13] have given the following values for static stiffness for horizontal and rocking modes.

+= HrGrKh 2

128 00

-(28)

( )

+= HrGrK

61

138 0

30

-(29)

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Kausel and Ushijima [17] have provided the following expression for vertical stiffness.

+= HrGrK z 00

28.11

14

-(30)

Equations 28 to 30 are valid for a0 = 0. The first part of these expressions 28 0Gr , ( )13

8 30Gr and 1

4 0Gr are the static

stiffness of foundation on semi infinite media and will give the same stiffness as equations 18, 20 and 22 and using C1 only for a0 =0 and S1 =0. Scrutiny of the work by Kausel and Ushijima shows that equations 18, 20 and 22 can still be used, provided the curve for C1 in figs 4, 5 and 6 are raised by the factors

+

Hr028.11

,

+Hr

21 0 and

+

Hr

61 0

respectively. Since this procedure does not take into account of variations due to reflection or resonance, the error can be up to 20%. For variable soil conditions Roesset [12] considers that it is adequate to use the adjusted shear modulus at depth r0/2 in stiffness calculations.

Reduction in Radiation Damping

Unless the soil properties are uniform for depths considerable greater than 8r0, the theoretical values calculated using equations 19, 21 and 23 are not valid. Even for uniform soils, the dynamic shear modulus will vary due to increase in mean effective stress with depth. Load from the spudcan will also increase it. Reflections from the adjoining spud cans and other foundations will also reduce the radiation damping. Hence it is recommended that the calculated radiation damping to the values given in table 8.

Mode % of Radiation damping to be used in analysis

Vertical 40%

Sliding 25%

Rocking 25%

Table 8

For Soil Stratum Resting on Rock, or other hard strata, Dobry and Gazetas [15] recommend that the radiation damping is taken as zero for frequencies less than fs.

fs = Vs/4H -(31)

Inclusion of Internal Damping Internal or hysteresis damping is normally expressed as a fraction of critical damping, whereas the radiation damping Cz and Ch are expressed in dimension FL-1T and C is expressed in dimension FLT. Internal damping can be converted and added to radiation damping using the definition for critical damping as follows:

Czt = zCz + 2z/z -(32) Where Czt is the total damping to be used in the analysis, z is the reduction factor for radiation damping, is the damping ratio due to internal damping and the frequency z obtained from table 7 for vertical mode. Similarly the total damping for horizontal and rocking degrees of freedom can be expressed as follows :

Cht = hCh + 2h/h -(33) Ct = C + 2/ (34)

Conclusion Information from several sources were brought together in this paper with the hope of providing adequate guidance to those without specialist knowledge of soil and foundation dynamics. The author welcomes any comments in order that a more detailed guidance to be published by HSE, the authors employer can incorporate any improvements.

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References [1] Rix, G.J. AND Stoke, K.H. (1991). Correlation of initial tangent modulus and cone penetration resistance, Calibration Chamber Testing. International Symposium on Calibration Chamber Testing, A.B. Huang, ed., Elsevier Publishing, New York, pp. 351-362. [2] Mayne, P.W. and Rix, G.J. (1993). Gmax-qc relationship for clays, Geotechnical Testing Journal, ASTM, VOL. 16, NO. l, pp. 54-60. [3] Hardin, B.O. (1978) The nature of stress-strain behaviour for soils, ASCE Geotech. Eng. Div. Specialty Conf. Earthq. Eng. Soil Dyn, Vol. I, pp. 3-90. [4] Weiler, W.A. (1988). Small strain shear modulus of clay, Proceedings, ASCE Conference on Earthquake Engineering and Soil Dynamics II: Recent Advances in Ground Motion Evaluation, Geotechnical Special Publication 20, ASCE, New York, pp. 331-335. [5] Towhata I, Geotechnical Earthquake Engineering, Springer, 2008. [6] Seed, H.B., Wong, R.T., Idriss, I.M., and Tokimatsu, K. (1984). Moduli and damping factors for dynamic analyses of cohesionless soils, Journal of Geotechnical Engineering, ASCE, Vol. 112, No. 11, pp.1016-1032. [7] Imazu M and Fukutake K, 1986 Dynamic Shear Modulus and Damping Ratio of Gravel Materials, Proc. 21 Annual Convention of JSSMFE, pp 509-512. [8] VUCETIC, M. AND DOBRY, R. (1991). Effect of soil plasticity on cyclic response, Journal of Geotechnical Engineering, ASCE, Vol. 117, No. 1, pp. 89-107. [9] ISHIBASHI, I. AND ZHANG, X. (1993). Unified dynamic shear moduli and damping ratios of sand and clay, Soils and Foundations, Vol. 33, No. 1, pp. 182-191. [10] Novak M and Beredugo Y O, Vertical Vibration of Embedded Footings, Journal of Geotechnical Engineering, ASCE Dec 1972. [11] Beredugo Y O and Novak M, Coupled Horizontal and Rocking Vibration of Embedded Footings, Canadian Geotechnical Journal, 9,477 1972, pp 477-497. [12] Roesset J. M. (1980). The use of simple models in soil-structure interaction. In Civil Engineering and Nuclear Power, ASCE, No. 10/3, pp. 1-25. [13] Kausel E. and Roesset, J. M. (1975). Dynamic stiffness of circular foundations, J. Eng. Mech. Div. ASCE, 101(EMb), 771-85. [14] Gazetas, G. (1983). Analysis of machine foundation vibrations: state of the art, Int. J. Soil Dynamics. Earthquake Eng., 2(1), 2-42. [15] Dobby, R and Gazetas,G. (1985). Dynamic Stiffness and damping of foundations by simpler methods. In Vibration Problems in Geotechnical Engineering, ed. G.Gazetas and E.T. Selig, ASCE, pp.77-107 [16] Prakash S and Puri V K, Foundations for Machines: Analysis and Design, Wiley, 1988. [17] Kausel E. and Ushijima R (1979) Vertical and Torsional Stiffness of Cylindrical Footings, Research Report R7-6, Civil Engineering Dept., Massachusetts Institute of Technology, Cambridge.