ABSTRACT
BNLNURBZ6 16 6 4
CONVOLUTIONDECONVOLUTION IN DIGES
A.J. Philippacopoulos Department of Advanced Technology
Brookhaven National Laboratory Upton, NY 11973
Nikolaos Simos
Department of Advanced Technology Brookhaven National Laboratory
Upton, NY 11973
1.0 INTRODUCTION
Convolution and deconvolution operations is by all means a very important aspect of SSI analysis since it influences the input to the seismic analysis. This paper documents some of the convolution/deconvolution pro cedures which have been implemented into the DIGES code. The 1D propagation of shear and dilatational waves in typical layered configurations involving a stack of layers overlying a rock is treated by DIGES in a sim ilar fashion to that of available coda, e.g. CARES, SHAKE. For certain configurations, however, there is no need to perform such andyses since the correspond ing solutions can be obtained in analytic form. Typical cases involve deposits which can be modeled by a uni form halfspace or simple layered halfjpaces. For such cases DIGES uses closdform solutions. These so lutions are given for one as well as two dimensional deconvolution. The typ;iof waves considered include P, SV and SH waves. The nonvertical incidence is given special attention since deconvolution can be defined dif ferently depending on the problem of interest. For all wave cases considered, corresponding transfer functions ake presented in closedform. Transient solutions are obtained in the frequency domain. Finally, a variety of forms are considered for representing the free field motion both in terms of deterministic as well as prob abilistic representations. These include (a) acceleration time histories, (b) response spectra (c) Fourier spectra and (d) crossspectral densities.
1.1 ProbIem Definition The need to develop acceptable floor response spectra has been an ongoing process. Such spectra are affiliated with seismic loads that the structure is subjected to and they represent the prediction of the response of various elevations within the structure that in turn can be ntilised to predict the response of sensitive equipment ra t ing on a particular elevation. These seismic loads are conventionally ex pressed in the fotm of design response spectra for a number of reasons.
Consequently, the development of computational schemes which can incorporate the information or as sessment pertaining to the seismic load and, in conjnnc tion with the dynamic characteristics of the structure, predict the elevation spectral responses has been the fo cus of earthquake response prediction. The definition of the seismic load, which determines the theoretical basis of the link between excitation and response, has been deduced from both deterministic as well as stochastic models.
On one hand, the deterministic approaches seek to assess the elevation response due to a prescribed ground excitation or dynamic load on the structure itself. On the other hand, the stochastic approaches attempt to define the floor response to ai1 anticipated grvuiili exci tation that belongs to a family of earthquakes which in turn is described by turget response or power spectra.
Within the stochastic processes, however, the statis tics that accompany the definition of the ground excita tion are umally carried over to the floor response with an ensemble of realizations of the stochastic process that defines the ground excitation. This simiilation of earth quakes procedure that attempts to match the statistics of the target spectrum has been used extensively both by directly linking the target response spectruin to an artificial earthquake or by implementing the con straint of the power spectral density function of the ground motion. The latter earthquake simulation pro cess, more sophisticated in nature, matches some of the statistics of the target spectrum with realizations (sam ple earthquakes) deduced from the power spectrum of the stochastic process.
The direct link between a stochastic characteriza tion of the ground excitation and the stochastic floor response has received less attention. Through this pro cess, the statistical properties of an anticipated family of earthquakes, expressed in i ts power spectrum, are transferred to the structure of deterministically defined dynamic properties.
1.2 DIGES Proflle The present effort has been undertaken so that an
efficient theoretical/computational tool can be devised such that seismic problems of concern to the regulatory agencies can be effectively treated. In this study, the direct link between the input excitation and the output response in the stochastic sexise is explored. This dimension of the seismic analysis, along with the earthquake simulation procedures and the deterministic seismic and dynamic response of the structure, define the DIGES computational domain.
Figure 1. The physical system
Figure 1 depicts the physical system whose response to the action of dynamic loads is sought. Participating in the generic physical system a r t the superstruckure, which is the focus of the resulting response, the founda tion and the soil tiiediuni the foundar.ion/siiperst.ritclnre is resting on. Further, the different dynamic loads that can excite the physical system are shown. The overall description of DIGES can be seeii in Figure 2 where the general capabilities are listed.
I
Figure 2. Overall description of DIGES According to Fig. 2, analyses of both stochastic and
deterministic nature can be undertaken. While in the deterministic analysis the consideration of dynamic su perstructure loads has been inipleniented (an important element of dynamic analysis) alongside with the classical treatment of defined ground motion, the stochastic anal ysis mode incorporates both the earthquake sirnulation and the direct transferring of stochastic properties.
The stochastic mode, which implenients both the simulation and the direct stochastic transferring) seeks to evaluate elevation response spectra induced by ground excitations that can be defined as both target response spectra or crossspectral densities of the stochastic pro cess describing the excitation.
The direct stochastic mode determines the cross spectral density matrix of the response [ @ y ( ~ ) ] for a stochastic process with cross spectral density @ ~ K ( W ) ] .
For a statistic process that defines the free field in terms of target response spectra, a corrsistent cross spectral matrix is formed arid eventually transferred to the elevation. The samukataon seeks the elevation response spectra through by utilizing statistical properties of the responses a t the same elevation due to an ensemble of ground accelerations whose response spectra match the target spectrum over some of its statistic properties.
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As shown, both simulation procedures are impiemeiited (one leads to ground motions from a response spectrum throrigh its power spectriim end t h e other to ground motions directly from the response spectrum).
2. FREEFIELD EARTHQUAKE A freefield earthquake may be in the form of
a response spectrum, vower spectrum or time varying acceleration.
Response Spectra to Power Spectra The response spectrum characterizing the freefield
motion R S t ( w , ( ) is known for the frequency range of interest. This spectrum could also be called target Response Spectrum. Assume that the power spec trum consistent with the target response spectrum is 'Pt(w, A) where {A) is a vector of parameters that are specific of the power spectrum. These parameters de fine the shape of the analytical expression of the pJd and they are unknown until the consistency between the power and the response spectra is achieved. Over the years, several closed form expressions that can describe the power spectrum of earthquake ground accelerations have been proposed. Some of these PSD forms are listed below:
1. Kanai Tajird
ii. RuioPenoien
Superposition 9.9
111.
(1 + 4 i 3 5
1  e k
+ 4c; ( :)2
Eort hquake Simulation If the simulation option is selected to transfer
the freefield earthquake to tlie structure it is irxiplied that an ensemble of generated earlhqriakes will be trans ferred bv utilizing the Transfer Function of the system H ( w ) according to the reiationship
where F,(w) and F,(w) are the Fourier expansions of the input and output respectively.
The artificial earthquakes can be generated from either a power or a response spectrum character izing the Gecfield stochastic process.
PSD Based Ground Accel. Simulation
be generated from the form A time history g ( t ) of an artificial acceleration can
1 9 ( t ) = c [. 2 JCO~ (wit + 4 8 )
i= 1 (2)
where, W ; = iAu Aw = 3 while w, is a cutoff frequency above which the power spectrum is assumed to vanish, N is the number of uniform frequency increments, {&) is a vector of random phase angles unifordy distributed between 0 and 2~ (different choices of the vector of random phase angles will lead to a different simulated earthquake that has both the mean and the autocorrelation of the stochastic process described by the P s D of the stochastic freefield), 'Pa(w) is the power spectral density of the process and ( ( t ) is R modiilatins function that i~ttrndrices the nonstationarity in the generated record. It should be noted that the simulated earthquake g ( t ) is periodic with a period To =  2 ~ AW
Simulation Based on Response Spectra Simulated earthquakes that belong to the family
represented by the target response spectrum can assume the form,
N
i= 1
where C i ( W ) is the amplitudes of the i t h contributing sinusoid while 4i and ( ( t ) are the same as above.
When the complete ensemble of generated earth quakes has been transferred to the structure, the re sponse of the system at any d.0.f. can be then seen as a single Response Spectrum which is deduced from the average of the ensemble of response spectra each deterministic process will provide,
(4)
along with the statistical properties of the ensemble of amplifications ab every specified frequency w i t [ m + a] * RS(w;) where CT is the number of standard deviations from the mean.
3. FOUNDATION INPUT MOTIONS The foundation input motion represents the re
sponse of the rigid and m a d e m foundation to the freefield motion. Generally, the response of the foun dation depends on its geometry, the properties of the interacting soil and the nature of the impinging seismic waves. A scattering transfer function a ( w ) links the freefield motion u$ with the foundation input motion VG
V G = H (Cd) @ ( 5 ) Described below are three different approaches that are extensively utilized in determining the foundation input motion:
a. Freefield applied directly
This case represents early stages of seismic analyses of building foundation systems according to which the criteria motion was directly applied at the bottom of the soil springs and i t reflects primarily cases involving surface foundations.
where [IJ and [O) are [3 x 31 unit and null matrices respectively.
b. Convolution/ Deconvolut ion
In this case the foundation input motion is the freefield motion a t some depth, depending on the char acteristics of the embedded foundation. The freefieid motion at a given depth is obtained through convolu tion or deconvolution depending on whether or not the
criteria motion is treated as an oritcrop motion or a surface (or near surface for very soft top layers) motion.
Convolution in Uniform Soil Deposits
Inclined SHwave
Figure 3 depicts the incidence of an inclined SH wave propagating in a uniform soil deposit. The dis placement vector associated with such wave is of the form
( 8 )  ~ ~ i k ( r * P  c i ' ) d +,4  where p = sindeil + cosei2 = propagation vector d = i 3 = direction of particle motion T = zlil + x2i2 = position vector A, k, C. velocity respectively
are the amplitude, wavenumber and phase
The requirement that the surface is free of traction yields that the reflected wave is in phase with the incident wave. The total displacement is then
( 9 ) ik (z l r in@c , t )
213 ( X I , 2 2 ; w , t ) = 2 A c o s ~ e
Thus, the transfer function between the displace ment (or acceleration) a t a depth zz = h and its counterpart at the surface is expressed in the form
w h ah =  c o d
cs Inclined Pwaves
From the incidence of an inclined Pwave (seen in Fig. 4) and the condition that the surface is traction free, the displacement vectors associated with the vari ous waves have as follows:
Incident Pwave:
I
Reflected Pwave:
cp, c.9 : P and SV wave velocity respectively Ai, Az, A3 : Wave amplitudes p i , p 2 , p3 : propagation vectors of incident P, reflected P and reflected SVwave
while, sindo = ko = sk and s = 9 .9 C*
The amplitudes A i , & and A3 satisfy the relation
 [ ^ + p i n 2 9 2pc0s20
where q1 = 2 qz = 2 and A, p are the Lame constants of the halfspace.
The total displacement due to the incident and reflected waves is simply the superposition of the three displacement vectors. Based on the above relations various transfer functions of interest can be deduced. Specifically, the transfer function between the horizontal and the vertical displacement a t the frecsurface will be of the form
( 1 2 ) fLrllzz=o  (1 + q1) sin9 + q2cos@0  U2(122=o (1  Q1) COS@ f q Z S 2 n @ O
The transfer function between the vertical displace ment a t a depth h f i o e h e free surface and the vertical displacement at the surface is of the form
uZJJza=h   "21122=0
( e  j a L  q1 eiOL) cos9 + q2eia# sin00 (1  q1) cos9 + q2sin00 (13)
Similarly, the transfer function between the hori zontal displacement at a depth h and the horizontal displacement a t the surface takes the form
where aL = Whco8@ and a, = $cos80 repre sent dimensionless frequencies for P and sv waves respectively.
Fur vertical incidence ( 8 = 0) the transfer function matrix reduces to a single relationship between the vertical displacement at a depth h and the vertical displacement a t the free surface, specifically,
C L
Inclined SHwave in SoiI Deposit Overly ing a Rock Formation
In such formation, Figure 5, the halfspace repre sents the rock underlying the soil. Thus, the transfer functions will represent relations between the base rock (outcropping) motion and the motion in the soil de posit. The displacement vectors induced by the various waves involved have as follows:
Incident S H wave:
12) = A2eik('.acs,Rt) u3 Refiacted SH acroa interface:
213 (4) = A3eiko(r'*Plcg,st) where, C S , R , C S , S : SHwave velocities in the rock and soil respectively
T , T : position vectors WRT coordinate systems p i , .. , p4 : propagation vectors 41, .., A4 : wave amplitudes k, Lo : wave numbers ps, p~ : shear moduli of soil and rock respectively
I
Continuity of total stresses ~ 2 3 and total displace ments ~3 across the interface yields,
where, W H aH =  sindo = %sine J S , R C ~ & P S
= c s , s c o s @ ~ R cs,s CS,R Based on the above relations the following transfer
functions can be deduced:
motion at depth zi =  h motion at ~ u r f a c e HI (w) =
motion ut depth zh =  h motion at interface H2 ( w ) =
eiah + eiah  eiaH + e  i a ~
motion at depth zi =  h at interface (no top soil) H3 ( w ) =
Considering that the rock is sufficiently stiff, the latter transfer function represents the relation between the motion a t depth h and the outcropping motion. One should further note that the following relations hold between the transfer functions above:
H: ( w ) HF ( w ) = H,O ( w ) HP ( w ) = 1
H: ( W ~ H ? ( w ) # 1 The three amplitudes&l , A2 and A3 are related with two equations. The third equation required to completely define them depends on the selected input to the system. For example, if the outcropping motion
and the system is completely defined.
is known then A1 =
c: Kinematic Interact ion
In the case of foundation input motion which incor porates kinematic interaction effects due to the scatter ing of the seismic waves by the rigid foundation, H ( w )
is a [6 x 31 frequency dependent niattix containing the scattering coefficienLs which depend on the types of seis mic waves considered, the properties of the underlying medium and the geometry of the foundation itself.
When kinematic interaction is considered, the 3 ~ 3 HTotnl submatrix in H ( w ) is no longer a null matrix. Specifically, i t contains scattering coefficients relating the rocking and torsional motion of the foun dation due to the horizontal and vertical components of the freefield motion u;.
The dependency of the foundation input motion on the geometry of the massless foundation and the interface condition with the soil, on the properties and the stratigraphy of the underlying soil and on the nature of waves l e d to a complex problem. While in the generic complex configuration techniques in boundary integral or finite element methods need to be employed, for surface foundations with simple geometries (circular or rectangular) that rest on uniform hdfipaces and are subjected to the action of plane waves analytical closed form expressions of the scattering coefficients have been deduced by various researchers (Ref. 7, 8, 11).
Specifically, for a rectangular foundation ( 2 ~ z 2 b ) resting on a half space and subjected to an incident nonvertical wave with propagation in the 2 1 direction and vertical angle of incidence dv (an gle between the normal to the propagation and the horizontal) which leads to apparent surface velocity
(velocity of propagation = P, S or RayIeigh depending on the nature cif waves), the scattering functions are in the form,
velocity of propagation coreV ca =
where, a = dimension normal to propagation and p = E. Similarly, for a circular foundation of radius r:
It should be noted that when the interface condition is not relazed there exists a n additional rocking scatter ing function &,,, = 0 which for surface foundations is small and for practical purposes can be ignored.
where, p = 2 and f l = 2 and r = J1  p2 . For the circular case Lnco (9) has deduced the first approximation in the form
t *i
Rz3,, = 2Rzzz , z c m p ) z ip Fig. 5. Soil Overlying Rock: SHwaves
I
Fig. 3. Uniform Soil: SHWave Incidence
w
Fig. 6. Description of the SSI problem
Fig. 4. Uniform Soil: PWave Incidence
4. Numerical Examples and Summary Incident SHwave In the previous section a scattering matrix H(w)
has been formed lhat allows the transferring of the freefield motion to the foundation. In addressing the applicability of tthe three different approaches the fol lowing points are raised:
o The direct application of the freefield motion on the foundation results in only translational com ponents of the foundation motion regardless the soil stratigraphy, wavetype and the position of the foundation depth wise.
o The convolution/deconvolution approach does incorporate the stratigraphy by including the out crop motion, the wavetype and the position of the foundation mat but i t fails to introduce any torsional components on the foundation.
o The kinematic interaction approach incorpo rates the wavetype, the foundation geometry, the soil deposit stratigraphy (analytic closedform solu tions are possible only for uniform haifspace), and allows the torsional and rocking effects to appear in the lower half of the scattering matrix. For the simple model of a surface foundation resting on a halfspace and subjected to a nonvertically incident SHwave, the differences between the convolution approach and the kinematic interaction extend even to the upper part of the scattering matrix. Appar ently, the translational components of the freefield motion are altered for the kinematic interaction by the matrix coefficient 9. In order to address the basic SSI problem which
is defined as the transferring of a freefield motion (expressed in the form of a response or power spectrum) to a location on the structure, a simple structural model has been adopted (stick model on the foundation with a 6dof mass) and subjected to various nonvertically incident waves, Fig. 6. Assuming that the Geefield motion is a crossspectral density matrix *,(w), the rela tion 
The response spectra a t the mass point or Fig. 6 are evaluated for incident SBwaves a t diffcrent anglcs I $ , (8, = 90' represent vertical incidence and 0, = 0'' indicates surface SH waves). The input is a Reg. 1.60 response spectrum (single direction of particle motion) while the output response spectra include a translational component along 2 2 (Fig. 7) and a torsional component
due to the kinematic inkeraction (Fig. 8).
spectral Damprng = 2 p e m t
0.1 1 10 100 Aaq. (H3
Fig. 7. Translational RS at Mass Point
spstral DamPin((92 pL
+y (w) = H (&F4!= ( w ) H* (w)= (17) 0.1 1 10 100
Rcqw leads to the crossspectral density of the response @y(w). The diagonal elements of @=(w) are the power spectra of the particle motion in the freefield and the offdiagonal represent the crosscorrelation of the motion in the three directions. For a freefield ino tion represented by a response spectrum, say Reg. 1.60, a compatible power spectrum is generated and trans ferred to the structure through Eqn (17). The response power spectrum is finally converted to a compatible response spectrum.
~ i ~ . 8. ~ ~ ~ ~ i ~ ~ d RS at M~~~ Point
Incident Pwave References
The same model is subjected to a Pwave impinging a t different angles. Along with the vertical (Fig. 9) and horizontal component (Fig. lo ) , a rocking response spectrum about ax is z2 (Fig. 11) is evaluated.
Spectral Damping = 2 percent
01 1 10 100 Regw
Fig. 9. Vertical RS (inclined Pwave) SpCcLral Dampine = 2 percent
16
14
12 10
8 3 6 Y $ 4
2 0

0.1 1 10 Reqm
Fig. 10. Horiz. RS (inclined Pwave) sprrtrai Damplng = 2 p a m n t
2
1.6 5 = 1.4 e 2 12
2 08 8 0.6
02 0
1.8
4 1
Le 0.4
4
0.1 t 10 100 Req. 0
Fig. 11. Rocking RS (inclined Pwave)
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Acknowledgments This work was performed under the auspices of the NRC.