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Soil-structure interaction — an engineering evaluation
Nuclear Engineering and Design 38 (1976) 267-272 © North-Holland Publishing Company
S O I L - S T R U C T U R E INTERACTION - AN ENGINEERING EVALUATION * A.H. HADJIAN Bechtel Power Corporation, Los Angeles PowerDivision, Norwalk, California 90650, USA Received 14 June 1976
The two methods of analysis for soil-structure interaction, the impedance and the finite element methods, are reviewed with regard to their present capabilities to address the significant factors of the problem. The objective of the paper is to evaluate if an adequate engineering solution to the problem is provided by either approach. Questions related to the reduction of seismic motions with depth, scattering of incident waves, the three-dimensionality of the real problem, soil damping, strain dependency of soil properties and the uncertainties associated with all of the above are discussed in sufficient detail. All conclusions made are based on referenced material. It appears that, although both methods as presently practised have not yet completely solved the problem, the impedance approach has come closer to addressing the more significant issues. Because of this finding, in addition to its simplicity and low cost, the impedance approach is the preferred engineering method for soil-structure interaction. 1. Introduction
pedances are not calculated via a finite-element soil model. In the field of engineering mechanics, the finiteelement method and analytical approaches have played complementary roles. The finite-element method is usually checked against available analytical methods, numerical stability and convergence studies are conducted, and then the method is extended and used for problems that would be extremely difficult or impossible to solve b y analytical methods. Sometimes this extension is also corroborated b y laboratory controlled tests. In short, b o t h numerical and laboratory experimentation have been the basis of the acceptance of the finite-element approach which is destined to play a significant role in engineering mechanics. Thus, the engineer usually has available to him both acceptable solution capabilities, and he makes the decision to use one or the other depending on the requirements of his particular problem. In the field o f soil-structure interaction analysis this collaboration has been lacking. There is today an unnecessary confrontation between the two methods o f analysis. This can be traced in part to the assumption that the finite-element approach is appropriate for solving soil-structure interaction problems without first showing that the adopted models and methods can reproduce the results of a simple class o f
Analytical work conducted during the last decade shows that for the massive structures of nuclear power plants, s o i l - s t r u c t u r e interaction during earthquakes is an important consideration. Fig. 1 shows how the change o f the foundation soil properties under a typical containment structure affects the system fundamental frequency, the maximum floor acceleration, and the maximum amplitude o f the in-structure response spectrum at the top of the containment. Two methods of analysis have evolved for the evaluation o f the effects of s o i l - s t r u c t u r e interaction: the impedance and the finite-element methods [1 ]. The impedance approach is also referred to as the continuum or the lumped parameter method. It is essentially a substructure interaction analysis technique [2], in contrast to the commonly used direct finiteelement analysis. With comparable models and calculational techniques the direct and the substructure analyses would give identical results [3]. In this paper the impedance approach refers to the substructure interaction analysis technique where the foundation im* Paper U2/1 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguards and Containment Structures CELCALAP, Berlin, 8-11 September 1975. 267
268
A.H. Had]ian / S o i l - s t r u c t u r e interaction - an engineering evaluation 8
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Fig. 1. Typical containment responseparameters, normalized to lg ground motion.
problems that have been solved analytically. Thus, and perhaps inevitably, differences between the two solutions have been reported. Instead of attempting to find the causes of these differences, the analytical approaches have been assumed inadequate. Unfortunately experimental or i n - s i t u measurements have not been sufficient to substantiate either method of analysis. This lack of factual data has been one of the most important factors contributing to the continued controversy that has beset this aspect of earthquake engineering. It must be emphasized that there is nothing inherent in the finite-element method that makes it unsuitable for soil-structure interaction analysis. On the contrary, as in other fields of engineering mechanics, it has the potential of becoming an important and versatile tool in resolving some of the more difficult problems in soil-structure interaction.
2. Engineeringmethod In an attempt to evaluate this whole question it is deemed necessary to review in a few paragraphs the basic concepts of the engineering method. The method of reasoning used by engineers to obtain the answer to questions or the solution to problems has been successful simply because it is organized common sense which is applied in a systematic fashion. The first step in the engineering method is to clarify the statement of the problem. The next step is to identify the factors or the variables of the problem, which can
be defined as anything that influences the answer. This step also includes the determination of appropriate relationships between the variables, and between the variables and the problem. And finally, the classification of the variables into a hierarchy so that proper emphasis can be placed on those variables that are more significant. Evaluation of the relationships between the factors and the question can be accomplished either by computation or judgement. Before methematics became an engineering tool all evaluations must have been done by judgement. With the advant of computers, the potential exists for shifting the emphasis to the other extreme, with the result that the more complex the calculations the less judgement would be brought to bear on the solution of the problem. The sheer volume of the calculations would be substituted for the judgemental evaluation phase. A computer printout could become today's sacred cow. The final step in the engineering method is the consideration of uncertainties. To minimize the uncertainties inherent in the use of formulae it is incumbent on the engineer to have a clear knowledge of the assumptions on which the formulae are based as well as the actual conditions to which they are being applied. 3. Review of
the more significant factors
One of the interesting features of the development of the finite-element method for soil-structure inter-
A.H. Had]ian /Soil-structure interaction - an engineering evaluation
action has been the selective emphasis on those factors of the problem that could be inherently accounted for by the finite-element method at the expense of identifying the significant factors of the problem and developing the proper relationships. As our understanding of the real problem progressed and the analytical methods matured, the emphasis was shifted to secondary factors with little or no impact on the overall response of the soil-structure system. In the following some of the more significant issues are explored. 3.1. Reduction o f seismic motions with depth
The finite-element method implicitly or explicitly deconvolves the ground surface motion to the base of the soil model by the laws of one-dimensional wave propagation. Thus, for embedded structures, the motion at the base level of structures becomes a uniquely defined motion. This modification is deterministic and is a simple function of the assumed soil properties above the base level. Thus, this deconvolution process is philosophically in complete contradiction to the definition of ground motions for the design of nuclear power plant as broad band response spectra. The use of the broad band response spectra justifiably results from our imperfect understanding of the seismic site response phenomenon; yet the method of analysis adopted violates this intention by creating a unique motion throughout the soil profile under consideration. The seriousness of this violation of the probabilistic nature of the design input motion as a simple characteristic of the top 20-50 ft of overburden should be eveluated in light of the results of many studies. It has been concluded many times that, due to the importance of the source mechanism, the transmission path, geologic discontinuities and the general topography 'the site response problem is still a long way from complete resolution. In particular the one-dimensional body wave solutions, even when fully refined, will nbt solve the total problem' [4]. 3.2. Scattering o f incident waves
The usual assumption made today in the analysis of soil-structure interaction is that the incident motion corresponds to a vertically propagating body
269
wave. Thus, for the surface structure no scattering of waves is possible. For the embedded structure scattering of vertically propagating waves is possible and the finite-element method automatically takes this into account [5]. However, results of recent studies [6,7] show that the more significant effects are those due to obliquely incident waves. Scattering of oblique waves is the significant factor whether or not the foundation is embedded. For the vertically propagating wave model, the scattering effects due to embedment are secondary factors. Thus, performing a finiteelement analysis to account for scattering effects due to vertically propagating waves cannot be justified, since the significant factor that needs attention is the scattering due to inclined waves. As yet, inclined waves, with Or without embedment of structures, are not routinely incorporated in any type of analysis. This is an area where much work remains to be done. 3.3. Reduction o f the three-dimensional problem to a two-dimensional model
Fig. 2 shows a typical plan of a nuclear power plant. There are circular, rectangular and arbitrary shaped foundations. Today finite-element analysis is done either for axisymmetric structures or plane strain models. The issues will be explored for both a single structure and the plant complex. For the single structure resort to finite-element methods does not seem to be justified since today the general three-dimensional problem of arbitrary foundation shape and the layered site with material damping (frequency dependent and frequency independent) can be solved by the impedance approach [8,9]. The question of embedment effects has also been adequately resolved [10-13]. Thus, the severe approximation to plane strain models of the general three-dimensional problem is not necessary nor justified. For the entire facility questions of structurestructure interaction seem to indicate the use of the finite-element approach. Fig. 3(a) shows an idealized plan of three structures. However, to do a two-dimensional finite-element analysis of these structures torsional effects must be ignored in the first place. Additionally, the structures have to be moved around and foundation dimensions changed so that a plane strain model can be synthesized [figs. 3(b) and 3(c)]. If structure-structure interaction were a significant fac-
2 70
A.H. Hadrian / S o i l - structure in teraction - an engineering evaluation 124.0'
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117,0'
~LCONTAINMENT FUELBLDG TURBINEBLDG
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._
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[..o.. I Fig. 2. Plan of a typical nuclear power plant. tor, the 'massaging' of the problem to fit the analysis capability has modified the objectives of the analysis beyond acceptable limits, thus negating the purpose of the analysis. The logic of such an analysis is in most cases self-defeating [14]. Any realistic structurestructure interaction of nuclear power plant structures could only be done in three dimensions. Another important point to evaluate is whether structure-structure interaction is a significant factor. Despite reported evaluations of the problem [15], it can be argued that the model significantly exaggerates the structure-structure interaction effects. First, the energy radiated from the infinitely long structures (two-dimensional plane strain analysis) must always
be reflected between neighboring structures, whereas in a three-dimensional environment this reflection is only minimal (fig. 4). Secondly, it can be shown that shear, compression and Rayleigh waves attenuate differently in two-dimensional and three-dimensional space. Thus in two-dimensional space Rayleigh waves do not decay whereas in the three-dimensional environment these waves decay according to r-1/2, where r is the distance from the source. Additionally the decay of both shear and compressional waves is greater in the three-dimensional problem r -2 than in the twodimensional problem x - 3 / 2 . These arguments and the study of neighboring structures during earthquakes leads to the conclusion that structure-structure interaction is in all probability one of the less significant factors of soil-structure interaction and should not be the reason for disrupting a significant first order factor, namely, the three-dimensionality of the problem. 3.4. H y s t e r e t i c d a m p i n g in soils
(a)
(b)
(c)
Fig. 3. Idealized plant rearrangements (b, c and d are possible analysis widths).
There was a time when the use of the finite-element solution was justified because of its capability to include material damping in the analysis, and until
A.H. Hadrian /Soil-structure interaction - an engineering evaluation
271
But to mix both effects, as is presently done, leaves much to be desired. 3.5. Strain dependent soil properties a. Tylil~ E l m r ~ /
b. l ' h m e - D i ~ r i ~ ~ 4
4 ©. Tw-0immmml Plan Fig. 4. Schematic representation of Wavereflection. very recently [ 15] radiation damping was claimed to be of minor importance in studies of earthquake response. This unusual conclusion results from the fact that material damping in the finite-element model acts in an undefined dual role: as absorber of radiated energy and hysteretic soil damping. Early solutions of the impedance functions did not account for the hysteretic damping in the soil; this effect was first explicitly introduced into the impedance calculations by Veletsos and Nair [16]. Thus, the analytical results do provide for a direct comparison between the two sources of energy dissipation. Depending on the circumstances, one or the other can become the predominant source of energy absorption. In general material damping considerations become important only for sites with significant layering. A similar separation of the role of material and radiation damping muct be achieved by the finite-element method if the proper numerical values are to be ascribed to the soil properties. One way to achieve this is to develop the capability of transmitting boundaries where the proper radiation effects would be modeled with zero material damping in the soil elements. Only then should the soil elements be given whatever is deemed the appropriate material damping.
It has been reported in the soils engineering literature that stiffness and damping properties of soils are highly strain dependent. Starting with this premise, two distinct strains, one due to the free field earthquake motions and the other due to the interaction between the structure and soil, should be recognized. The evidence to date from numerical studies shows that the former effects are significant and the latter effects relatively insignificant [17]. Finite-element models are inherently capable of incorporating both effects, whereas the imdepance calculations incorporate the strains due to the free field motions by the use of an auxiliary model. This model could be either what is today the basic approach in the finite-element method [18] or a more appropriate model as described in ref. [19]. Some engineers consider the insignificant interaction effects to be so important as to dictate the use of the finite-element method. In some of the presently used finite-element codes nonlinear effects are introduced via the equivalent stepwise linear solutions [18]. The important question that must be raised is whether the stepwise linear solution does in fact solve the hoped-for nonlinear problem. As it turns out 'the equivalent linear method significantly underestimates the intensity of motion for periods between 0.1 and 0.6 sec' [19]. 3. 6. Uncertainties
Uncertainties are an inherent part of engineering. One common way to take care of the problem is to conduct sensitivity studies and assess the possible impact of variabilities and not-so-well-known relations. This approach can be justified only if the tools available to the engineer can be used with reasonable cost and within schedule. The finite-element approach does not lend itself to this type of probing analysis, since about 80-90% of the effort goes into defining and solving the equations relating to the soil. This has a compounding effect in that details of the structure are sacrified and the high frequency response often compromised in an effort to develop a reasonable model.
272
A.H. Hadrian / S o i l - s t r u c t u r e interaction - an engineering evaluation
In the presence of uncertainties it is convenient to be able to judge if the different parts of the analysis are correct. This cannot be achieved conveniently with the one pass direct finite-element solutions since all that is available are the final results. In the impedance approach one can evaluate the impedances, the input motion as modified b y kinematic effects and the structural response. At each step the meaning and accuracy of the final results can be better evaluated.
4. Conclusions
Some of the significant factors of the soil-structure interaction problem were explored in light of presently available calculational capabilities. It has been shown that, in general, the finite-element approach has emphasized the secondary factors of the problem at the expense of investigating the significant factors of the problem. Thus, emphasis is placed on modifying the design motion at a site by a simple onedimensional wave propagation model; the significant scattering effects due to inclined waves are completely ignored and instead the secondary scattering of vertically propagating waves due to embedment of structures is emphasized; severe approximations in modeling an inherently three-dimensional problem as twodimensional are adopted in the name of incorporating structure-structure interaction effects; adequate concern is not shown for the fact that a real plant layout has to be distorted out of reality to suit the analysis capability; the distinction between radiation and hysteretic soil damping effects has not been fully appreciated; and finally, an empirical solution to solve a nonlinear problem is introduced that is shown to have important limitations. The need for a simplified yet adequate engineering solution to the problem of soil-structure interaction is important in the design process. The interface problem between the participants of the total effort is so complicated that the analysis should start with a minimum data base. Under these circumstances and assuming that a correct and general three-dimensional finite element method for soil-structure interaction would eventually become available, its use in the design environment should be limited to special cases only.
Acknowledgements The author is indebted to J.E. Luco, C.B. Smith, R.P. Kennedy and A.K. Chopra for reading the manuscript and for making many valuable comments. References [1] A.H. Hadjian, J.E. Luco and N.C. Tsai, Nucl. Eng. Des. 31 (2) (1974) 151-167. [2] A.K. Vaish and A.K. Chopra, J. Eng. Mech. ASCE 100 (EM6) Dec. (1974) 1101-1116. [3] J.A. Gutierrez, EERC 76-9, Erathquake Engineering Research Center, University of California, Berkeley, Calif., Apr. (1976). [4] P.E. Salt, Bull. N. Z. Nat. Soc. Earthquake Eng. 7 (2) (1974) 62-77. [5] H.B. Seed, R.V. Whitman and J. Lysmer, Soil-structure interaction effects in the design of nuclear power plants, Symposium on Structural and Geotechnical Mechanics, University of Illinois, Urbana, Illinois, Oct. 1975. [6] J.E. Luco, Bull. Seismal. Soc. Amer. 66 (1) (1976) 109124. [7] J.E. Luco, Earthquake I~ng. Struc. Dyn. 4 (1976) 207219. [8] H.C. Wong, Report EERL 75-01, California Institute of Technology, Pasadena, May (1975). [9] J.E. Luco, Nucl. Eng. Des. 36 (1976) 325-340. [10] M. Novak, Earthquake Eng. Struc. Dyn. 3 (1974) 7996. [ 11 ] J.R. Hall Jr. and J.I. Kissenpfenning, Special topics on soil-structure interaction, Proc. ELCALAP Seminar, Berlin, Sept. (1975), Paper U2/2. [12] E. Kausel and J.M. Roesset, Soil-structure interaction for nuclear containment structures, Proc. ASCE Power Division Specialty Conferrence, Boulder, Colorado, Aug. (1974). [13] A.H. Hadjian, G.E. Howard and C.B. Smith, A comparison of experimental and theoretical investigations of embedment effects on seismic response, 3rd SMiRT Conf., London, Sept. (1975), Paper K2/5. [14] A.H. Hadjian, J.E. Luco and H.L. Wong, On the reduction of three-dimensional soil-structure interaction problems to two-dimensional models, Presented at the ASCE Structural Engineering Specialty Conference, Madison, Wisconsin, Aug. (1976). [15] H.B. Seed, J. Lysmer and R. Hwang, J. Geotech. Div., ASCE 101 (GT5) May (1975) 439-457. [16] A.S. Veletsos and V.V.D. Nair, J. Struct. Div., ASCE 101 (ST1), Proc. Paper 11460, Jan. (1975) 109-129. [17] J.T. Christian, Uncertainties in soil-structure interaction, Proc. 2nd, ASCE Specialty Conference on Structural Design of Nuclear Plant Facilities, Vol. III, New Orleans, Dec. (1975). [18] I.M. Idriss and B.H. Seed, J. Soil Mech. Found. Div., ASCE 94 (1968) 1003-1031. [19] W.B. Joyner and A.T.F. Chen, Bull. Seismol. Soc. Amer. 65 (5) (1975) 1315-1336.
S O I L - S T R U C T U R E INTERACTION - AN ENGINEERING EVALUATION * A.H. HADJIAN Bechtel Power Corporation, Los Angeles PowerDivision, Norwalk, California 90650, USA Received 14 June 1976
The two methods of analysis for soil-structure interaction, the impedance and the finite element methods, are reviewed with regard to their present capabilities to address the significant factors of the problem. The objective of the paper is to evaluate if an adequate engineering solution to the problem is provided by either approach. Questions related to the reduction of seismic motions with depth, scattering of incident waves, the three-dimensionality of the real problem, soil damping, strain dependency of soil properties and the uncertainties associated with all of the above are discussed in sufficient detail. All conclusions made are based on referenced material. It appears that, although both methods as presently practised have not yet completely solved the problem, the impedance approach has come closer to addressing the more significant issues. Because of this finding, in addition to its simplicity and low cost, the impedance approach is the preferred engineering method for soil-structure interaction. 1. Introduction
pedances are not calculated via a finite-element soil model. In the field of engineering mechanics, the finiteelement method and analytical approaches have played complementary roles. The finite-element method is usually checked against available analytical methods, numerical stability and convergence studies are conducted, and then the method is extended and used for problems that would be extremely difficult or impossible to solve b y analytical methods. Sometimes this extension is also corroborated b y laboratory controlled tests. In short, b o t h numerical and laboratory experimentation have been the basis of the acceptance of the finite-element approach which is destined to play a significant role in engineering mechanics. Thus, the engineer usually has available to him both acceptable solution capabilities, and he makes the decision to use one or the other depending on the requirements of his particular problem. In the field o f soil-structure interaction analysis this collaboration has been lacking. There is today an unnecessary confrontation between the two methods o f analysis. This can be traced in part to the assumption that the finite-element approach is appropriate for solving soil-structure interaction problems without first showing that the adopted models and methods can reproduce the results of a simple class o f
Analytical work conducted during the last decade shows that for the massive structures of nuclear power plants, s o i l - s t r u c t u r e interaction during earthquakes is an important consideration. Fig. 1 shows how the change o f the foundation soil properties under a typical containment structure affects the system fundamental frequency, the maximum floor acceleration, and the maximum amplitude o f the in-structure response spectrum at the top of the containment. Two methods of analysis have evolved for the evaluation o f the effects of s o i l - s t r u c t u r e interaction: the impedance and the finite-element methods [1 ]. The impedance approach is also referred to as the continuum or the lumped parameter method. It is essentially a substructure interaction analysis technique [2], in contrast to the commonly used direct finiteelement analysis. With comparable models and calculational techniques the direct and the substructure analyses would give identical results [3]. In this paper the impedance approach refers to the substructure interaction analysis technique where the foundation im* Paper U2/1 presented at the International Seminar on Extreme Load Conditions and Limit Analysis Procedures for Structural Reactor Safeguards and Containment Structures CELCALAP, Berlin, 8-11 September 1975. 267
268
A.H. Had]ian / S o i l - s t r u c t u r e interaction - an engineering evaluation 8
--160
4 ~
?6
-;120
I
p~z.,cELERATION
31
z o k-
<
~4 U < u. 2
-',.) 80~'~ Z 2
i|
-~
4oI
u.
1 0
1~
1~
2000
2500
~
SHEAR WAVE VELOCITY, FT/SEC
Fig. 1. Typical containment responseparameters, normalized to lg ground motion.
problems that have been solved analytically. Thus, and perhaps inevitably, differences between the two solutions have been reported. Instead of attempting to find the causes of these differences, the analytical approaches have been assumed inadequate. Unfortunately experimental or i n - s i t u measurements have not been sufficient to substantiate either method of analysis. This lack of factual data has been one of the most important factors contributing to the continued controversy that has beset this aspect of earthquake engineering. It must be emphasized that there is nothing inherent in the finite-element method that makes it unsuitable for soil-structure interaction analysis. On the contrary, as in other fields of engineering mechanics, it has the potential of becoming an important and versatile tool in resolving some of the more difficult problems in soil-structure interaction.
2. Engineeringmethod In an attempt to evaluate this whole question it is deemed necessary to review in a few paragraphs the basic concepts of the engineering method. The method of reasoning used by engineers to obtain the answer to questions or the solution to problems has been successful simply because it is organized common sense which is applied in a systematic fashion. The first step in the engineering method is to clarify the statement of the problem. The next step is to identify the factors or the variables of the problem, which can
be defined as anything that influences the answer. This step also includes the determination of appropriate relationships between the variables, and between the variables and the problem. And finally, the classification of the variables into a hierarchy so that proper emphasis can be placed on those variables that are more significant. Evaluation of the relationships between the factors and the question can be accomplished either by computation or judgement. Before methematics became an engineering tool all evaluations must have been done by judgement. With the advant of computers, the potential exists for shifting the emphasis to the other extreme, with the result that the more complex the calculations the less judgement would be brought to bear on the solution of the problem. The sheer volume of the calculations would be substituted for the judgemental evaluation phase. A computer printout could become today's sacred cow. The final step in the engineering method is the consideration of uncertainties. To minimize the uncertainties inherent in the use of formulae it is incumbent on the engineer to have a clear knowledge of the assumptions on which the formulae are based as well as the actual conditions to which they are being applied. 3. Review of
the more significant factors
One of the interesting features of the development of the finite-element method for soil-structure inter-
A.H. Had]ian /Soil-structure interaction - an engineering evaluation
action has been the selective emphasis on those factors of the problem that could be inherently accounted for by the finite-element method at the expense of identifying the significant factors of the problem and developing the proper relationships. As our understanding of the real problem progressed and the analytical methods matured, the emphasis was shifted to secondary factors with little or no impact on the overall response of the soil-structure system. In the following some of the more significant issues are explored. 3.1. Reduction o f seismic motions with depth
The finite-element method implicitly or explicitly deconvolves the ground surface motion to the base of the soil model by the laws of one-dimensional wave propagation. Thus, for embedded structures, the motion at the base level of structures becomes a uniquely defined motion. This modification is deterministic and is a simple function of the assumed soil properties above the base level. Thus, this deconvolution process is philosophically in complete contradiction to the definition of ground motions for the design of nuclear power plant as broad band response spectra. The use of the broad band response spectra justifiably results from our imperfect understanding of the seismic site response phenomenon; yet the method of analysis adopted violates this intention by creating a unique motion throughout the soil profile under consideration. The seriousness of this violation of the probabilistic nature of the design input motion as a simple characteristic of the top 20-50 ft of overburden should be eveluated in light of the results of many studies. It has been concluded many times that, due to the importance of the source mechanism, the transmission path, geologic discontinuities and the general topography 'the site response problem is still a long way from complete resolution. In particular the one-dimensional body wave solutions, even when fully refined, will nbt solve the total problem' [4]. 3.2. Scattering o f incident waves
The usual assumption made today in the analysis of soil-structure interaction is that the incident motion corresponds to a vertically propagating body
269
wave. Thus, for the surface structure no scattering of waves is possible. For the embedded structure scattering of vertically propagating waves is possible and the finite-element method automatically takes this into account [5]. However, results of recent studies [6,7] show that the more significant effects are those due to obliquely incident waves. Scattering of oblique waves is the significant factor whether or not the foundation is embedded. For the vertically propagating wave model, the scattering effects due to embedment are secondary factors. Thus, performing a finiteelement analysis to account for scattering effects due to vertically propagating waves cannot be justified, since the significant factor that needs attention is the scattering due to inclined waves. As yet, inclined waves, with Or without embedment of structures, are not routinely incorporated in any type of analysis. This is an area where much work remains to be done. 3.3. Reduction o f the three-dimensional problem to a two-dimensional model
Fig. 2 shows a typical plan of a nuclear power plant. There are circular, rectangular and arbitrary shaped foundations. Today finite-element analysis is done either for axisymmetric structures or plane strain models. The issues will be explored for both a single structure and the plant complex. For the single structure resort to finite-element methods does not seem to be justified since today the general three-dimensional problem of arbitrary foundation shape and the layered site with material damping (frequency dependent and frequency independent) can be solved by the impedance approach [8,9]. The question of embedment effects has also been adequately resolved [10-13]. Thus, the severe approximation to plane strain models of the general three-dimensional problem is not necessary nor justified. For the entire facility questions of structurestructure interaction seem to indicate the use of the finite-element approach. Fig. 3(a) shows an idealized plan of three structures. However, to do a two-dimensional finite-element analysis of these structures torsional effects must be ignored in the first place. Additionally, the structures have to be moved around and foundation dimensions changed so that a plane strain model can be synthesized [figs. 3(b) and 3(c)]. If structure-structure interaction were a significant fac-
2 70
A.H. Hadrian / S o i l - structure in teraction - an engineering evaluation 124.0'
t
85.5'
J !
117,0'
~LCONTAINMENT FUELBLDG TURBINEBLDG
AUXILIARYBLDG ~[ I CONTROL8LOG
]I
',,.0' :1
._
I O'ESELI GENERATOR I 8Lo°
RAO ASTE,LO0
151.0'
,I ,~/
[..o.. I Fig. 2. Plan of a typical nuclear power plant. tor, the 'massaging' of the problem to fit the analysis capability has modified the objectives of the analysis beyond acceptable limits, thus negating the purpose of the analysis. The logic of such an analysis is in most cases self-defeating [14]. Any realistic structurestructure interaction of nuclear power plant structures could only be done in three dimensions. Another important point to evaluate is whether structure-structure interaction is a significant factor. Despite reported evaluations of the problem [15], it can be argued that the model significantly exaggerates the structure-structure interaction effects. First, the energy radiated from the infinitely long structures (two-dimensional plane strain analysis) must always
be reflected between neighboring structures, whereas in a three-dimensional environment this reflection is only minimal (fig. 4). Secondly, it can be shown that shear, compression and Rayleigh waves attenuate differently in two-dimensional and three-dimensional space. Thus in two-dimensional space Rayleigh waves do not decay whereas in the three-dimensional environment these waves decay according to r-1/2, where r is the distance from the source. Additionally the decay of both shear and compressional waves is greater in the three-dimensional problem r -2 than in the twodimensional problem x - 3 / 2 . These arguments and the study of neighboring structures during earthquakes leads to the conclusion that structure-structure interaction is in all probability one of the less significant factors of soil-structure interaction and should not be the reason for disrupting a significant first order factor, namely, the three-dimensionality of the problem. 3.4. H y s t e r e t i c d a m p i n g in soils
(a)
(b)
(c)
Fig. 3. Idealized plant rearrangements (b, c and d are possible analysis widths).
There was a time when the use of the finite-element solution was justified because of its capability to include material damping in the analysis, and until
A.H. Hadrian /Soil-structure interaction - an engineering evaluation
271
But to mix both effects, as is presently done, leaves much to be desired. 3.5. Strain dependent soil properties a. Tylil~ E l m r ~ /
b. l ' h m e - D i ~ r i ~ ~ 4
4 ©. Tw-0immmml Plan Fig. 4. Schematic representation of Wavereflection. very recently [ 15] radiation damping was claimed to be of minor importance in studies of earthquake response. This unusual conclusion results from the fact that material damping in the finite-element model acts in an undefined dual role: as absorber of radiated energy and hysteretic soil damping. Early solutions of the impedance functions did not account for the hysteretic damping in the soil; this effect was first explicitly introduced into the impedance calculations by Veletsos and Nair [16]. Thus, the analytical results do provide for a direct comparison between the two sources of energy dissipation. Depending on the circumstances, one or the other can become the predominant source of energy absorption. In general material damping considerations become important only for sites with significant layering. A similar separation of the role of material and radiation damping muct be achieved by the finite-element method if the proper numerical values are to be ascribed to the soil properties. One way to achieve this is to develop the capability of transmitting boundaries where the proper radiation effects would be modeled with zero material damping in the soil elements. Only then should the soil elements be given whatever is deemed the appropriate material damping.
It has been reported in the soils engineering literature that stiffness and damping properties of soils are highly strain dependent. Starting with this premise, two distinct strains, one due to the free field earthquake motions and the other due to the interaction between the structure and soil, should be recognized. The evidence to date from numerical studies shows that the former effects are significant and the latter effects relatively insignificant [17]. Finite-element models are inherently capable of incorporating both effects, whereas the imdepance calculations incorporate the strains due to the free field motions by the use of an auxiliary model. This model could be either what is today the basic approach in the finite-element method [18] or a more appropriate model as described in ref. [19]. Some engineers consider the insignificant interaction effects to be so important as to dictate the use of the finite-element method. In some of the presently used finite-element codes nonlinear effects are introduced via the equivalent stepwise linear solutions [18]. The important question that must be raised is whether the stepwise linear solution does in fact solve the hoped-for nonlinear problem. As it turns out 'the equivalent linear method significantly underestimates the intensity of motion for periods between 0.1 and 0.6 sec' [19]. 3. 6. Uncertainties
Uncertainties are an inherent part of engineering. One common way to take care of the problem is to conduct sensitivity studies and assess the possible impact of variabilities and not-so-well-known relations. This approach can be justified only if the tools available to the engineer can be used with reasonable cost and within schedule. The finite-element approach does not lend itself to this type of probing analysis, since about 80-90% of the effort goes into defining and solving the equations relating to the soil. This has a compounding effect in that details of the structure are sacrified and the high frequency response often compromised in an effort to develop a reasonable model.
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A.H. Hadrian / S o i l - s t r u c t u r e interaction - an engineering evaluation
In the presence of uncertainties it is convenient to be able to judge if the different parts of the analysis are correct. This cannot be achieved conveniently with the one pass direct finite-element solutions since all that is available are the final results. In the impedance approach one can evaluate the impedances, the input motion as modified b y kinematic effects and the structural response. At each step the meaning and accuracy of the final results can be better evaluated.
4. Conclusions
Some of the significant factors of the soil-structure interaction problem were explored in light of presently available calculational capabilities. It has been shown that, in general, the finite-element approach has emphasized the secondary factors of the problem at the expense of investigating the significant factors of the problem. Thus, emphasis is placed on modifying the design motion at a site by a simple onedimensional wave propagation model; the significant scattering effects due to inclined waves are completely ignored and instead the secondary scattering of vertically propagating waves due to embedment of structures is emphasized; severe approximations in modeling an inherently three-dimensional problem as twodimensional are adopted in the name of incorporating structure-structure interaction effects; adequate concern is not shown for the fact that a real plant layout has to be distorted out of reality to suit the analysis capability; the distinction between radiation and hysteretic soil damping effects has not been fully appreciated; and finally, an empirical solution to solve a nonlinear problem is introduced that is shown to have important limitations. The need for a simplified yet adequate engineering solution to the problem of soil-structure interaction is important in the design process. The interface problem between the participants of the total effort is so complicated that the analysis should start with a minimum data base. Under these circumstances and assuming that a correct and general three-dimensional finite element method for soil-structure interaction would eventually become available, its use in the design environment should be limited to special cases only.
Acknowledgements The author is indebted to J.E. Luco, C.B. Smith, R.P. Kennedy and A.K. Chopra for reading the manuscript and for making many valuable comments. References [1] A.H. Hadjian, J.E. Luco and N.C. Tsai, Nucl. Eng. Des. 31 (2) (1974) 151-167. [2] A.K. Vaish and A.K. Chopra, J. Eng. Mech. ASCE 100 (EM6) Dec. (1974) 1101-1116. [3] J.A. Gutierrez, EERC 76-9, Erathquake Engineering Research Center, University of California, Berkeley, Calif., Apr. (1976). [4] P.E. Salt, Bull. N. Z. Nat. Soc. Earthquake Eng. 7 (2) (1974) 62-77. [5] H.B. Seed, R.V. Whitman and J. Lysmer, Soil-structure interaction effects in the design of nuclear power plants, Symposium on Structural and Geotechnical Mechanics, University of Illinois, Urbana, Illinois, Oct. 1975. [6] J.E. Luco, Bull. Seismal. Soc. Amer. 66 (1) (1976) 109124. [7] J.E. Luco, Earthquake I~ng. Struc. Dyn. 4 (1976) 207219. [8] H.C. Wong, Report EERL 75-01, California Institute of Technology, Pasadena, May (1975). [9] J.E. Luco, Nucl. Eng. Des. 36 (1976) 325-340. [10] M. Novak, Earthquake Eng. Struc. Dyn. 3 (1974) 7996. [ 11 ] J.R. Hall Jr. and J.I. Kissenpfenning, Special topics on soil-structure interaction, Proc. ELCALAP Seminar, Berlin, Sept. (1975), Paper U2/2. [12] E. Kausel and J.M. Roesset, Soil-structure interaction for nuclear containment structures, Proc. ASCE Power Division Specialty Conferrence, Boulder, Colorado, Aug. (1974). [13] A.H. Hadjian, G.E. Howard and C.B. Smith, A comparison of experimental and theoretical investigations of embedment effects on seismic response, 3rd SMiRT Conf., London, Sept. (1975), Paper K2/5. [14] A.H. Hadjian, J.E. Luco and H.L. Wong, On the reduction of three-dimensional soil-structure interaction problems to two-dimensional models, Presented at the ASCE Structural Engineering Specialty Conference, Madison, Wisconsin, Aug. (1976). [15] H.B. Seed, J. Lysmer and R. Hwang, J. Geotech. Div., ASCE 101 (GT5) May (1975) 439-457. [16] A.S. Veletsos and V.V.D. Nair, J. Struct. Div., ASCE 101 (ST1), Proc. Paper 11460, Jan. (1975) 109-129. [17] J.T. Christian, Uncertainties in soil-structure interaction, Proc. 2nd, ASCE Specialty Conference on Structural Design of Nuclear Plant Facilities, Vol. III, New Orleans, Dec. (1975). [18] I.M. Idriss and B.H. Seed, J. Soil Mech. Found. Div., ASCE 94 (1968) 1003-1031. [19] W.B. Joyner and A.T.F. Chen, Bull. Seismol. Soc. Amer. 65 (5) (1975) 1315-1336.