Response of floating platforms to seismic excitation

Response of floating platforms to seismic excitation M. AROCKIASAMY, K. MUNASWAMY, A. S. J. SWAMIDAS Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St John's, Newfoundland

and D. V. R E D D Y Faculty of Engineering, Florida Atlantic University, Boca Raton, Florida, USA The nonlinear transient response of floating platforms to seaquake-induced excitation is studied taking into account cavitation effects. The motion transmitted to the LPG is assumed to consist only of vertical pulses causing the same transient water pressures at all points of the LPG bottom. The model is a hybrid lumped parameter (structure and water) - continuum with variable damping (soilbase). The water column below the platform is assumed to act as an elastic bar with its vertical sides restrained (by imaginary membranes) against horizontal displacement - 'trapped water model'. The nonlinearity associated with cavitation effects of the seawater medium is simulated by elastic bilinear springs, which permit gaps to occur when the spring tensions reach cutoff values. The gap opening and closing provides for the cavitation effect. Parametric studies of water depth, damping ratios, and soil depths are carried out for Taft earthquake excitation input. Key Words: floating platforms, seaquake, trapped water model, cavitation, liquefied petroleum gas storage platforms, seismic excitation.

INTRODUCTION The paper describes an analytical study of the nonlinear transient response of a floating platform subjected to strong boundary-induced excitation due to an earthquake. Both incident and reflecting waves can cause negative overpressures which exceed the hydrostatic value, thereby leading to cavitation. Collapse of the large cavities results in a 'jump or slapdown' of the platform. This phenomenon is of considerable significance for floating structures in moderately deep waters such as tension leg and liquefied petroleum gas storage (LPG) platforms, and semisubmersibles. The effect of the submarine earthquake is not dependent on the epicenter's location on the land onshore or below the seabed. In the current investigation travelling shock waves are referred to as a seaquake, without tsunami effects. Energy from earthquakes propagates through rock and soil in the form of either compressional or shear waves. A train of shock waves generated in the water by the seismic motion of the seafloor consists solely of compressional waves since water cannot transmit shear waves. The sources of compressional waves are the P and S waves beneath the seabed, and surface waves on the seafloor (Rayleigh and Love). The compressional waves are refracted at the seafloor and travel almost vertically. REVIEW OF LITERATURE

General observation Available literature on seaquake-induced damage is confreed to ships. Richter, 1 and Rudolph 2-4 reported that the Received May 1983. Written discussion closesMay 1984.

54

Engineering Analysis, 1984, Vol. 1, No. 1

seaquake effects on ships even in water depths of more than 40 fathoms, were similar to those encountered when passing over coral reefs. Von Huene s has described the effects of the 1964 Alaska earthquake as the sensation of running aground, shock-like depth charges and heavy vibrations. Water surface effects include a seaquake generated chop of 1-1.5 ft and sharp needle-like cones of water about 1 ft high in the surrounding area. Varied damage and sea conditions have been reported since 1900. In some instances the ship was lifted upwards and the shock effect was similar to striking and dragging over soft ground, even though the water depth was as much as 2400 fathoms. One of the most important incidents reported by William. son et al. 6 w a s the damage to a Norwegian tanker in 2000 fathoms of water off the coast of Portugal in 1969. Damage included toppling of masts, wrecking of radio equipment, overturning of refrigerators, and extensive hull damage. More documented evidences are well described in a recent paper by Knut Hove et al. 7 The literature contains relatively little information on the propagation of such acoustic waves or their effects on ships or floating structures.

Fluid-structure coupling The structure-fluid coupling was considered with different degrees of simplification- incompressible fluid with a flexible body, and compressible fluid with a rigid body, Reddy et al.,a Arockiasamy et al. 9'1° Dynamic structure-fluid-surface wave interaction during an earthquake has been studied by many researchers. The fluid and the free surface wave are described by Eulerian variables (pressure unknowns) and the deformable solid by a Lagrangian formulation (displacement unknowns). Dubois and de Rouvray 11 used the Lagrangian approach for the fluid and 0264-682X/841010054-07 $02.00 © 1984 CML Publications

Response o f floating platforms to seismic excitation: M. Arockiasamy et aL

free surface waves, and eliminated the fluid Eulerian unknowns leading to a symmetric matrix equation for the dynamic structure-fluid-surface wave condition. The generalized fluid pressure was used as a Lagrangian multiplier for the constraint equation on the fluid loading. Mass and stiffness matrices were generated for the deformable solid. Ham d-i et al. 1~ presented a variational principle in terms of displacements for the vibration of coupled fluidstructure systems. The kinematic interface and boundary conditions were taken into account using the displacement variables, and a penalty functional used for imposing irrotational motion in the fluid. A computer program, MOSAS, for determining the motion and stress behavior of floating structures and semisubmersible units in irregular sea states has been described by Van Opstal et al. 13 Zienkiewicz and Bettess 14 and Newton Is reported analyses of coupled fluid-structure problems incorporating surface waves and radiation boundary conditions. A comprehensive review of fluid-structure response analysis of floating plants, including the effects of mooring, has been presented by Thangam Babu and ReddyJ 6 Thangam Babu et alJ 7 studied the transient response of a LPG platform, similar to that used in the Java Sea at the Ardjuna field. The work described by Thangam Babu et al. 17 was a follow-up of plane strain f'mite element analysis of the response of an offshore floating nuclear plant to seismic forces by Reddy et alJ s More recently Thangam Babu and Reddy ~ described a numerical integration scheme for solving coupled equations of fluid-structure interaction systems. The occurrence of finite.amplitude effects before cavitation in fluids under high static pressure has been discussed by Blackstock 2° in his extensive theoretical treatment of the problem of plane progressive sound waves. Tokuo Yamamoto, 21 has studied the harmonic propagation of gravity waves and acoustic waves in the ocean by vertical oscillation of a block of ocean floor. The non-linear nature of the exact wave equation for an adiabatic condition was presented by McDaniel) 2 Interaction between structures and bilinear fluids was investigated by Bleich and Sandler) 3 The problem of one-dimensional wave propagation in a bilinear fluid was considered by a characteristic approach. The physics of acoustic cavitation in liquids has been described comprehensively by Flynn) 4

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MODELING

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The floating platform The floating platform is represented by two masses, one representing the platform base, and the other the rest of the structure, which are connected by a spring and dashpot to simulate the elasticity of the base. The water column The connecting elastic spring stiffnesses for the water column are based on the compressibility of the water, and the dashpots are assumed to provide an appropriate percentage of critical damping. The water column below the structure acts as an elastic bar with its vertical sides restrained (by imaginary membranes) against horizontal displacement- 'trapped water model', refs. 6 and 25, hereafter termed the 'Williamson-Kennedy' model for ease of reference. The non-linearity associated with cavitation effects of the seawater medium will be simulated by elastic bilinear springs, which permit gaps to occur when the spring tensions reach cutoff values, Fig. 1. The gap opening and closing provides for the cavitation effect.

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Engineering Analysis, 1984, Vol. 1, No. 1

55

Response o f floating platforms to seismic excitation: M. Arockiasamy et at The soil medium The soil medium is treated as a number of parallel layers as shown in Fig. 2. The wave propagation through the soil medium is considered unidirectional and the soil represented by uniaxial elements. The vertical displacement u over a layer is assumed to vary linearly and given by: u = u/ (l _ Z_L~+ Zj h// u/+l h5

(1)

axx = (X + 2/a) exx = CA + 2u)(uj +1- uj)/hi

(2)

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where Oxx, exx = stress and strain in the vertical direction, ;%/a = Lame's constants, and hi = thickness of the segment. The strain energy can be written as:

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1 {u}r[k]{u} 1 (X+ 2/a) (uj+l_ui)2 (3) 2 2 hi The element stiffness matrix derived by minimization of equation (3) is: ~+2p[ 1 [k]= hi --1

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ANALYSIS

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where [M], [C'] and [K] = mass (lumped), damping and stiffness matrices respectively; {/i}, {fi} and {u} = column vectors of relative acceleration, velocity and displacements respectively with respect to the bedrock; {R(t)} = earthquake load vector, --{mz}az; and {Fc} = load vector of correction forces. The damping matrix for each element is obtained based on Rayleigh damping equations, Idriss et al. 26 and Reddy et al., 27 as: (7)

where [C]q, [m]q and [k]q = the damping, mass and stiffness matrices for element q and ~q and/$q = functions of the damping value and stiffness characteristics of the element q given by ~q = ~6oa and 13q= ~l/wl, in which Xq=damping ratio and w = u n d a m p e d fundamental frequency. The damping ratio ~ , is based on the strain developed within the soil element in contradistinction to constant values for the water column and the floating structure. The damping matrix for the entire structure-water-soil system is obtained by appropriate addition of the individual elemental damping matrices. The resulting dynamic equilibrium equations are solved by the Wilson-0 method to obtain the nodal displacements, accelerations, and the elemental stresses and strains. The forces in the springs representing the water column are modified at each time step to include the cavitation effects. The non-linear soil behavior is treated by2 an equivalent linear method presented by Seed and Idriss. a The approxi-

Engineering Analysis, 1984, Vol. 1, No. 1

i~5.00

(5) ~

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where Ps = mass density of the soil and As = unit area.

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Taft earthquake acceleration record

Figure 3.

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Acceleration spectra o f Taft earthquake record

mate relationship between the shear modulus and undrained shear strength for clays and the variation of shear strain with shear modulus and damping ratio, established by Seed and Idris, 29 are used. The effective shear strain amplitudes for each element are estimated and checked for strain compatibility with those reported by Lysmer et al. a° The properties of the soil elements, which do not exhibit compatible values, are modified and the procedure repeated until the shear moduli are compatible with strain amplitudes. The response, as obtained from the final iteration, is assumed to be the approximate non-linear value. NUMERICAL ILLUSTRATION AND DISCUSSION A LPG storage plaform similar to the ARCO-LPG (Atlantic Richfield Company Liquid Petroleum Gas) facility installed at the Ardjuna field off the Java Sea is chosen as the example problem. The mass, stiffness and damping values of the 'trapped water model' are calculated assuming the platform to be divided into major and minor parts with weights Ws and Wp connected by a spring kp and a dashpot

Cp:

Response or floating platforms to seismic excitation: M. Arockiasamy et aL Table 1. Responsevaluesfor ease (a)(i) for excitation inputs at seabed Water depth Fundamental frequency (Hz) Second modal frequency (Hz) Third modal frequency (Hz) Acceleration amplification of the platform deck Maximum pressure at the platform bottom (psi) Maximum displacement at the platform deck (in)

96 ft

116 ft

156 ft

240 ft

286 ft

3.75 19.43 56.99 4.29 (4.23) 28.55 (46.28)* (46.19)t" 0.518 (1.37)

3.71 14.81 43.11 4.56 (4.54) 28.91 (79.57)* (48.83)1" 0.586 (1.49)

3.63 10.14 29.02 4.30 (4.39) 28.69 (44.85)* (44.80)t 0.590 (1.46)

3.40 6.35 17.25 8.42 (7.44) 38.66 (439.3)* (61.19)I" 1.20 (2.30)

3.24 5.44 14.i3 9.71 (7.29) 39.76 (645.00)* (66.50)~" 1.33 (3.03)

Note: The values in parentheses refer to the magnified acceleration * The maximum pressures at the base during cavitation "~The maximum pressures at the base when cavitation occurs

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Figure 6. Responses to Taft earthquake acceleration input at 100 f t beh>w the seabed

Engineering Analysis, 1984, Vol. 1, No. l

57

Response o f floating platforms to seismic excitation: M. Arockiasamy et al. Table 2.

Response values for case (a)(ii) for excitation inputs at bedrock 100 ft below seabed

Water depth Fundamental frequency (Hz) Second frequency (Hz) Third frequency (Hz) Acceleration amplification of the platform deck Maximum pressure at the platform bottom (psi) Maximum displacement at the platform deck (in)

96 ft

116 ft

156 ft

240 ft

286 ft

1.30 (1.02) 4.25 (3.77) 8.26 (4.78) 3.01 (2.04) 24.21 (28.06) 1.91 (3.88)

1.17 (0.94) 4.17 (3.76) 4.97 (4.59) 2.66 (1.60) 23.07 (36.49) 2.32 (4.93)

1.11 (0.91) 4.15 (3.76) 4.99 (4.53) 2.47 (1.59) 22.97 (26.36) 2.39 (5.09)

0.87 (0.65) 3.96 (3.54) 4.64 (4.08) 2.09 (1.56) 20.76 (23.30) 2.23 (6.15)

0.80 (0.58) 3.89 (3.48) 4.60 (3.98) 1.69 (1.56) 19.96 (22.20) 2.17 (6.79)

Note: The values in the brackets refer to the magnified acceleration input

Table 3. Response values for case (b) for excitation input (Taft earthquake) at various bedrock levels (water depth = 116 ft)

Bedrock depth below the seabed

0 ft

5 ft

10 ft

20 ft

30 ft

60 ft

100 ft

Fundamental frequency (Hz) Second frequency (Hz) Third frequency (Hz) Acceleration amplification at the platform deck Maximum pressures at the platform bottom (psi)

3.71 14.81 43.11 4.56 28.91

2.64 5.20 27.80 3.62 26.50

2.39 4.98 22.24 3.25 25.25

1.85 4.68 12.64 2.69 23.40

1.37 4.49 7.82 2.49 23.07

1.28 4.40 6.23 2.68 22.64

1.17 4.17 4.97 2.66 23.27

Total platform weight

152 320 kips

Ws

144 519 kips, i.e. 15.05 psi 7801 kips, i.e. 0.8613 psi 24.25 psi/unit width (based on deflection kp due to design sagging moment) 0.5% (damping coefficient) Cp Taft earthquake Excitation Cases considered (a) water depth 96-286 ft (i) bedrock at seabed, (ii) bedrock at 100 ft below the seafloor (b) water depth 116 ft and bedrock location below seabed at (i) 5 It, (ii) 10ft, (ill) 20ft, (iv) 30ft, (v) 60 ft and (vi) 100 ft The Taft earthquake (Fig. 3) forms the excitation input to the model. The power spectral density of the Taft earthquake is shown in Fig. 4. It could be seen that the Taft earthquake has significant energy input over a range of frequencies from 0.7 to 10.0 Hz and the peak of spectral energy is found to occur around 1.5 Hz. Table 1 gives the results of parametric studies carried out for different water depths for the case (a)(i). The acceleration amplification of the platform deck with respect to the input acceleration is of the order of 4.5 for smaller depths and increases to 9.7 with increasing water depths (Table 1). The peak pressures induced at the bottom ranges from 28.3 to 43.76 psi. For the example problem chosen, cavitation does not occur for the assumed input accelerations at the seabed - c a s e (a)(i). In order to study the effect of cavitation, the input acceleration records are magnified to a peak value of 0.45 g. Although the increase in the amplification of the acceleration output is small, peak pressures induced at the platform bottom seem to increase significantly for smaller depths and considerably for higher depths (Table 1). This may be due to the occurrence of cavitation. Cavitation occurs at locations

58

Engineering Analysis, 1984, Vol. 1, No. 1

close to the platform bottom with water depths around 100 ft. However, for large r depths, cavitation locations are 100ft and more from t h e p l a t f o r m bottom. Whenever cavitation occurs, there is a sudden rise in pressure to a magnitude six to eight times that without cavitation. It is found that frequencies decrease with increase in water depth. It could be seen from this table that the fundamental frequency for all the cases is within the maximum spectral energy range, whereas the second mode frequency for the last two cases only (240 and 286 ft water depths) lie in this range, and thus contribute to the acceleration amplification of the platform deck. Table 2 shows the acceleration amplification values at platform deck and peak pressures at platform bottom when the bedrock is located at 100 ft below the s e a b e d - case (a)(ii). For water depth change from 96 to 286 ft, the maximum acceleration of the platform deck reduces from 0.7 to 0.476 g and the induced peak pressures at the platform bottom also correspondingly decrease from 24.21 to 19.96 psi. This is probably due to alteration of frequency contents of the input earthquake motion, Fig. 3, caused by the non-linear behavior of the soil medium. Also no evidence of cavitation is found to occur even for the magnified acceleration input. The effect of the variation in the soil layer thickness is studied - case (b), with a water depth of 116 ft. The soil depth is varied from 0 to 100 ft and the results are shown in Table 3. The presence of even a relatively thin layer of soil at the seabed-water interface affects the response values considerably. Figures 5 and 6 show the response values of the platform for the actual Taft earthquake acceleration input at the seabed and at the bedrock 100 ft below it. The non-linear soil behavior causes an increase in the displacements, filters the medium and high frequency components of the displacement response, and decreases the accelerations and dynamic pressures experienced by the platform. Similar observations can be made from Figs. 7 and 8, which

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describe the response to magnified excitation. It is seen from Fig. 7 that cavitation causes a sudden pressure surge which drops to the normal steady-state value, on return to the no cavitation state.

CONCLUSION (1) Presence of a bedrock at the seabed level amplifies input seismic acceleration considerably at the platform deck. (2) The non-linearity of the soil reduces the accelerations and pressures at the platform base. (3) Due to cavitation, the platform base experiences very high transient dynamic pressures.

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ACKNOWLEDGEMENTS The authors would like to thank Dr G. R. Peters, Dean of the Faculty of Engineering and Applied Science, and Dr I. E. Rusted, Vice-President, Memorial University of Newfoundland, for the continued interest and encouragement. Appreciation is expressed to Dr S. Dunn, Department of Ocean Engineering, Florida Atlantic University, for encouraging the fourth author to complete the work on this project. The financial support provided by President, Dr L. Harris, Memorial University of Newfoundland, in the form of a President's Grant to the first author, is gratefully acknowledged. The authors are indebted to Ms Marilyn Warren, for the considerable care exercised in typing this manuscript.

EngineeringAnalysis, 1984, Vol. 1, No. 1

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Response o f floating platforms to seismic excitation: M. Arockiasamy et al.

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7

8

9

10

11

12 13

14 15

60

Richter, C. F. Elementary Seismology, W. H. Freeman Co., San Francisco, 1958 Rudolph, E. 0ber Submarine Erdeben und Eruptioner [Gerlands], Bei.trage zur Geophysik 1887, 1,133 Rudolph, E. Uber Submarine Erdeben und Vulkane (Zweiter Beitrage) [Gerlands], Beitrage zur Geophysik 1895, 2, 537 Rudolph, E. Uber Submarine Erbeden und Eruptioner [Gerlands], Beitrage zur Geophysik 1898, 3, 273 Von Huene, R. Seaquakes, The great Alaskan earthquake of 1964, Vol. 6, Oceanography and Coastal Engineering, National Academy of Science, 1972, pp. 13-18 Williamson, R. A., Kennedy, R. P., Bachman, R. E. and Chow, A. W. Response of a proposed nuclear-powered ship to watertransmitted earthquake vibrations, Technical Report, Nuclear and Systems Sciences Group, Holmes and Narver, Inc., 1975 Hove, K., Seines, P. B. and Bungum, H. Seaquakes: a potential threat to offshore structures, Proe. 3rd Int. Conf. on the Behaviour of Offshore Structures, MIT, Cambridge, Massachusetts, 1982 Reddy, D. V., Arocklasamy, M., Haldar, A. K. and Thangam Babu, P. V. Response of an offshore floating nuclear plant to seismic forces, Proc. Conf. Vibration in Nuclear Plants, Keswick, UK, 1978 Aroeklasamy, M., Reddy, D. V., Thangam Babu, P. V. and Haldar, A. K. Probabilistic response of floating nuclear plants to seismic forces, Proe. 5th Nat. Meeting of the Univ. Council for Earthquake Eng. Res., MIT, pp. 83--85, 1978 Arocklasamy, M., Thangam Babu, P. V. and Reddy, D. V. Probabilistic seismic fluid-structure interaction of floating nuclear plant platforms, Proc. 5th Conf. on Struct. Mech. Reactor Tech., Berlin, K4/7, pp. 1-8, 1979 Dubois, J. J. and de Rouvray, A. L. An improved fluid superelement for the coupled solid-fluid wave surface wave dynamic interaction problem, Earthquake Eng. and Struct. Dyn. 1978, 6, 235 Hamdi, M. L., Ousset, Y. and Verchery, G. A displacement method for the analysis of vibrations of coupled fluid-structure systems, Int. J. Num. Meth. in Eng. 1978, 13, 139 Van Opstal, G. H. C., Hans, D., Salmons, J. W. and Van Der Vlies, J. A. MOSAS: a motion and strength analysis for semisubmersible units and floating structures,/'roe. Offshore Tech. Conf., Houston, Texas, Paper No. OTC 2105, pp. 721-728, 1974 Zienkiewicz, O. C. and Bettess, P. Fluid-structure dynamic interaction and wave forces, an introduction to numerical treatment, Int. J. Num. Meth. Eng. 1978, 13, 1 Newton, B. E. Finite element analysis of two
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18

19

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26 27

28 29 30

Thangam Babu, P. V. and Reddy, D. V. Fluid-structure interaction response analysis of floating plants including the effects of mooring, Ocean Engineering Journal 1980, 7 Thangam Babu, P. V., Arockiasamy, M. and Reddy, D. V. Coupled fluid-structure interaction of floating platforms, Proc. 8th Can. Cong. of Applied Mech., Moncton, pp. 633634, 1981 Reddy, D. V. Arockiasamy, M., Haldar, A. K. and Thangam Babu, P. V. Response of an offshore floating nuclear plant to seismic forces, Proc. Conf. Vibration in Nuclear Plants, Keswick, UK, 1978 Thangam Babu, P. V. and Reddy, D. V. A numerical integration scheme for solving coupled equations of fluid-structure interaction systems, Proe. Int. Conf. on Numerical Methods for Coupled Problems, Swansea, Great Britain, 1981 Blackstock, D. T. Propagation of plane sound waves of finite amplitude in nondissipative fluids, £ Acous. Soc. of America 1962, 34, 9 Tokuo Yamamoto, Gravity wave generated by submarine earthquakes, Int. J. Soil Dynamics and Earthquake Engineering 1982, 1 (2), 75 McDaniel, O. H. Harmonic distortion of spherical sound waves in water,£ Acous. Soc. of America 1965, 39, 644 Bleich, H. H. and Sandier, I. S. Interaction between structures and bilinear fluids, Int. J. Solids Struct. 1970, 00, 617 Flynn, H. G. Physics of acoustic cavitation in liquids, Physical Acoustics, VoL I, Part B, Mason, W. P., ed., Academic Press, pp. 57-172, 1964 Thangam Babu, P. V., Reddy, D. V. and Arockiasamy, M. Fluid-structure interaction of floating platforms, Proc. 6th National Meeting of the Universities Council for Earthquake Engineering Research, University of IlLinois, Urbana, IL, 1980 Idriss, I. M., Seed, H. B. and Serif, N. Seismic response by variable damping finite elements, J. Geotech. Eng. Div. ASCE 1974, 100 (GT1). 1 Reddy, D. V., Arockiasamy, M. and Haldar, A. K. Nonlinear seismic response analysis of a gravity monopod using MODSAP-IV, Proc. 3rd Canadian Conf. on Earthquake Engineering, Montreal, Canada, pp. 1343-1364, 1979 Seed, H. B. and Idriss, I. M. Influence of soil condition on ground motion during earthquakes, Proe. ASCE, SM1 1969, 95 (1). 99 Seed, H. B. and Idriss, I. M. Soil moduli and damping factors for dynamic response analysis, EERC Report No. 70-10, Univ. Calif., Berkeley, 1970 Lysmer, J., Udaka, T., Seed, H. B. and Hwang, R. LUSH: a computer program for complex response analysis of soilstructure systems, 1974, Report No. EERC 74.4, Univ. CaliL, Berkeley, 1974