version: 26. September, 2016,...

IAEA, TECDOC, Chapter 4Boris Jeremic

Draft Writeup (in progress, total up to 30 pages)

version: 26. September, 2016, 17:51

Contents

4 Free Field Ground Motions and Site Response Analysis 34.1 Free Field Ground Motion Analysis and Site Response Analysis . . . . . . . . . . . . . . 34.2 Seismic wave fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4.2.1 Recorded Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Ergodic Assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3D (6D) versus 1D Records/Motions. . . . . . . . . . . . . . . . . . . . . . . . 64.2.2 Analytical and Numerical (Synthetic) Earthquake Models . . . . . . . . . . . . . 7

Analytic Earthquake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Numerical Earthquake Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.2.3 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Uncertain Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Uncertain Path (Rock). . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Uncertain Site (Soil). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Dynamic Soil Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.1 Dynamic Soil Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Plastification – Stiffness Reduction. . . . . . . . . . . . . . . . . . . . . . 9Volume Change during Shearing. . . . . . . . . . . . . . . . . . . . . . . 10Effective Stress and Pore Fluid Pressures. . . . . . . . . . . . . . . . . . 11

4.3.2 Material Modeling of Dynamic Soil Behavior . . . . . . . . . . . . . . . . . . . . 124.4 Spatial Variability of Ground Motion and Seismic Wave Incoherence . . . . . . . . . . . 12

4.4.1 Incoherence Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Incoherence in 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Theoretical Assumptions behind SVGM Models. . . . . . . . . . . . . . . 15

4.5 Analysis Models and Modelling Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 151D Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163D Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173D versus 1D Seismic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Propagation of Higher Frequency Seismic Motions . . . . . . . . . . . . . . . . 20Material Modeling and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 233D vs 3×1D vs 1D Material Behavior and Wave Propagation Models . . . . . . 23

4.6 Limitations of Time and Frequency Domain Methods . . . . . . . . . . . . . . . . . . . 27

Chapter 4

Free Field Ground Motions and Site

Response Analysis

(seismic wave fields) (30 Pages) (Jeremic)

4.1 Free Field Ground Motion Analysis and Site Response Analysis

Free field motions can be developed using two main approaches

• Using Ground Motion Prediction (GMP) equations (Boore and Atkinson, 2008), and

• Using Large Scale Regional (LSR) simulations (Bao et al., 1998; Bielak et al., 1999; Taborda and

Bielak, 2011)

Both approaches assume elastic material behavior for the near field region for which free field motions

are developed.

Site response analysis uses free field motions and adds influences of nonlinear soil layers close to

the surface. Influence of close to surface soil layers is usually done in 1D. Three dimensional (3D) site

response analysis is desirable if free field motions feature full 3D wave fields (which they do most of

the time), and if local geology/site conditions are not horizontal soil layers, if topography is not flat

(presence of hills, valleys, and sloping ground) (Bielak et al., 2000; Assimaki and Kausel, 2007; Assimaki

et al., 2003).

Material models for site response can be equivalent linear, nonlinear or elastic-plastic.

• Equivalent linear models are in fact linear elastic models with adjusted elastic stiffness that repre-

sents (certain percentage of a ) secant stiffness of largest shear strain reached for given motions.

Determining such linear elastic stiffness requires an iterative process (trial and error). This mod-

eling approach is fairly simple, there is significant experience in professional practice and it works

well for 1D analysis and for 1D states of stress and strain. Potential issues with this modeling

approach are that it is not taking into account soil volume change (hence it favors total stress

analysis, see details about total stress analysis in section 6.4), and it is not useable for 3D analysis.

3

• Nonlinear material models are representing 1D stress strain (usually in shear, τ−−γ) response using

nonlinear functions. There are a number of nonlinear elastic models used, for example Ramberg-

Osgood (Ueng and Chen, 1992), Hyperbolic (Kramer, 1996), and others. Calibrating modulus

reduction and damping curves using nonlinear models is not that hard (Ueng and Chen, 1992),

however there is less experience in professional practice with these types of models. Potential

issues with this modeling approach are similar (same) as for equivalent linear modeling that it is

not taking into account soil volume change as it is essentially based on a nonlinear elastic model,

• Elastic-plastic material models are usually full 3D models that can be used for 1D or 3D analysis.

A number of models are available (Prevost and Popescu, 1996; Mroz et al., 1979; Elgamal et al.,

2002; Pisano and Jeremic, 2014; Dafalias and Manzari, 2004). Use of full 3D material models,

if properly calibrated, can work well in 1D as well as in 3D. Potential issues with these models is

that calibration usually requires a number of in situ and laboratory tests. In addition, there is far

less experience in professional practice with elastic-plastic modeling.

It is important to note that realistic seismic motions always have 3D features. That is seismic motions

feature three translations and three rotations at each point on the surface and shallow depth. Rotations

are present in shallow soil layers due to Rayleigh and Love surface waves, which diminish with depth as

a function of their wave length (Aki and Richards, 2002).

For probabilistic site response analysis, it is important not to count uncertainties twice (Abrahamson,

2010). This double counting of uncertainties, stems from accounting for uncertainties in both free field

analysis (using GMP equations for soil) and then also adding uncertainties during site response analysis

for top soil layers.

Chapter 5.3 presents site response analysis in more details.

4.2 Seismic wave fields

Earthquake motions at the location of interest are affected by a number of factors (Aki and Richards,

2002; Kramer, 1996; Semblat and Pecker, 2009). Three main influences are

• Earthquake Source: the slippage (shear failure) of a fault. Initial slip propagates along the fault

and radiates mechanical, earthquake waves. Depending on predominant movement along the failed

fault, we distinguish (a) dip slip and (b) strike slip sources. Size and amount of slip on fault will

release different amounts of energy, and will control the earthquake magnitude. Depth of source

will also control magnitude. Shallow sources will produce much larger motions at the surface,

however their effects will be localized (example of recent earthquakes in Po river valley in Italy,

• Earthquake Wave Path: mechanical waves propagate from the fault slip zone in all directions.

Some of those (body) waves travel upward toward surface, through stiff rock at depth and, close

to surface, soil layers. Spatial distribution (deep geology) and stiffness of rock will control wave

paths, which will affect magnitude of waves that radiate toward surface. Body waves are (P)

Primary waves (compressional waves, fastest) and (S) Secondary waves (shear waves, slower).

Secondary waves that feature particle movements in a vertical plane (polarized vertically) are

called SV waves, while secondary waves that feature particle movements in a horizontal plane

(polarized horizontally) are called S¿H waves.

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• Site Response: seismic waves propagating from rock and deep soil layers surface and create surface

seismic waves. Surface waves and shallow body waves are responsible for soil-structure interaction

effects. local site conditions (type and distribution of soil near surface) and local geology (rock

basins, inclined rock layers, etc.) and local topography can have significant influence on seismic

motions at the location of interest.

In general, seismic motions at surface and shallow depths consist of (shallow) body waves (P, SH, SV)

and surface waves (Rayleigh, Love, etc.). It is possible to analyze SSI effects using a 1D simplification,

where a full 3D wave field is replaced with a 1D wave field, however it is advisable that effects of this

simplification on SSI response be carefully assessed.

A usual assumption about of seismic waves (P and S) is based on Snell’s law about wave refraction.

The assumption is that seismic waves travel from great depth (many kilometers) and as they travel

through horizontally layered media (rock and soil layers), where each layer features different wave velocity,

waves will bend toward vertical (Aki and Richards, 2002). However, even if rock and soil layers are ideally

horizontal, and if the earthquake source is very deep, seismic waves will be few degrees (depending on

layering usually 5o − 10o off vertical when they reach surface. Most commonly, the stiffness of layers

increases with depth. This change in stiffness results in seismic wave refraction, as shown in Figure 4.1.

This usually small deviation from vertical might not be important for practical purposes.

soil/

rock

stiffness

rupturing faultseismic w

ave propagation

Figure 4.1: Propagation of seismic waves in nearly horizontal local geology, with stiffness of soil/rock

layers increasing with depth, and refraction of waves toward the vertical direction. Figure from Jeremic

(2016).

Other, much more important source of seismic wave deviations from vertical is the fact that rock

and soil layers are usually not horizontal. A number of different geologic history effects contribute to

non-horizontal distribution of layers. Figure 4.2 shows one such (imaginary but not unrealistic) case

where inclined soil/rock layers contribute to bending seismic wave propagation to horizontal direction.

It is important to note that in both horizontal and non-horizontal soil/rock layered case, surface

waves are created and propagate/transmit most of the earthquake energy at the surface.

4.2.1 Recorded Data

There exist a large number of recorded earthquake motions. Most records feature data in three per-

pendicular directions, East-West (E-W), North-South (N-S) and Up-Down (U-D). Number of recorded

strong motions, is (much) smaller. A number of strong motion databases (publicly available) exist,

mainly in the east and south of Asia, west cost of north and south America, and Europe. There are

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rupturing fault

seismic wave propagatio

n

soil/rock

stiffness

Figure 4.2: Propagation of seismic waves in inclined local geology, with stiffness of soil/rock layers

increasing through geologic layers, and refraction of waves away from the vertical direction. Figure from

Jeremic (2016).

regions of world that are not well covered with recording stations. These same regions are also seismically

fairly inactive. However, in some of those regions, return periods of (large) earthquakes are long, and

recording of even small events would greatly help gain knowledge about tectonic activity and geology.

Ergodic Assumption. Development of models for predicting seismic motions based on empirical evi-

dence (recorded motions) rely on Ergodic assumption. Ergodic assumption allows statistical data (earth-

quake recordings) obtained at one (or few) worldwide location(s), over a long period of time, to be used

at other specific locations at certain times. This assumption allows for exchange of average of process

parameters over statistical ensemble (many realizations, as in many recordings of earthquakes) is the

same as an average of process parameters over time.

While ergodic assumptions is frequently used, there are issues that need to be addressed when it is

applied to earthquake motion records. For example, earthquake records from different geological settings

are used to develop GMP equations for specific geologic settings (again, different from those where

recordings were made) at the location of interest.

3D (6D) versus 1D Records/Motions.

Recordings of earthquakes around the world show that earthquakes are almost always featuring all three

components (E-W, N-S, U-D). There are very few known recorded events where one of the components

was not present or is present in much smaller magnitude. Presence of two horizontal components (E-W,

N-S) of similar amplitude and appearing at about the same time is quite expectable. The four cardinal

directions (North, East, South and West) which humans use to orient recorded motions have little to

do with the earthquakes mechanics. The third direction, Up-Down is different. Presence of the vertical

motions before main horizontal motions appear signify arrival of Primary (P) waves (hence the name). In

addition, presence of vertical motions at about the same time when horizontal motions appear, signifies

Rayleigh surface waves. On the other hand, lack (or very reduced amplitude) of vertical motions at about

the same time when horizontal motions are present signifies that Rayleigh surface waves are not present.

This is a very rare event, that the combination of source, path and local site conditions produce a plane

shear (S) waves that surfaces (almost) vertically. One such example (again, very rare) is a recording

LSST07 from Lotung recording array in Taiwan (Tseng et al., 1991). Figure 4.3 shows three directional

recording of earthquake LSST07 that occurred on May 20th, 1986, at the SMART-1 Array at Lotung,

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Taiwan.

Figure 4.3: Acceleration time history LSST07 recorded at SMART-1 Array at Lotung, Taiwan, on

May 20th, 1986. This recording was at location FA25. Note the (almost complete) absence of vertical

motions. signifying absence of Rayleigh waves. Figure from Tseng et al. (1991).

Note almost complete lack of vertical motions at around the time of occurrence of two components

of horizontal motions, signifies absence of Rayleigh surface waves. In other words, a plane shear wave

front was propagating vertically and surfaced as a plane shear wave front. Other recordings, at locations

FA15 and FA35 for event LSST07 reveal almost identical earthquake shear wave front surfacing at the

same time (Tseng et al., 1991).

On the other hand, recording at the very same location, for a different earthquake (different source,

different path) (LSST12, occurring on July 30th 1986) revels quit different wave field at the surface, as

shown in Figure 4.4.

4.2.2 Analytical and Numerical (Synthetic) Earthquake Models

In addition to a significant number of recorded earthquakes (in regions with frequent earthquakes and

good (dense) instrumentation), analytic and numerical modeling offers another source of high quality

seismic motions.

Analytic Earthquake Models

There exist a number of analytic solutions for wave propagation in uniform and layered half space (Wolf,

1988; Kausel, 2006). Analytic solutions do exist for idealized geology, and linear elastic material. While

geology is never ideal (uniform or horizontal, elastic layers), these analytic solution provide very useful set

of ground motions that can be used for verification and validation. In addition, thus produced motions

can be used to gain better understanding of soil structure interaction response for various types incoming

ground motions/wave types (Liang et al., 2013) (P, Sh, SV, Rayleigh, Love, etc.).

7

Figure 4.4: Acceleration time history LSST12 recorded at SMART-1 Array at Lotung, Taiwan, on July

30th, 1986. This recording was at location FA25. Figure from Tseng et al. (1991).

Numerical Earthquake Models

In recent years, with the rise of high performance computing, it became possible to develop large scale

models, that take into account regional geology (Bao et al., 1998; Bielak et al., 1999; Taborda and

Bielak, 2011; Cui et al., 2009; Bielak et al., 2010, 2000; Restrepo and Bielak, 2014; Bao et al., 1996;

Xu et al., 2002). Large scale regional models that encompass source (fault) and geology in great detail,

are able to model seismic motions of up to 2Hz. There are currently new projects (US-DOE) that that

will extend modeling frequency to over 10Hz for large scale regions. Improvement in modeling and in

ground motion predictions is predicated by fairly detailed knowledge of geology for large scale region,

and in particular for the vicinity of location of interest. Free field ground motions obtained using large

scale regional models have been validated (Taborda and Bielak, 2013; Dreger et al., 2015; Rodgers et al.,

2008; Aagaard et al., 2010; Pitarka et al., 2013, 2015, 2016) and show great promise for development

of free field motions.

It is important to note, again, that accurate modeling of ground motions in large scale regions is

predicated by knowledge of regional and local geology. Large scale regional models make assumption

of elastic material behavior, with (seismic quality) factor Q representing attenuation of waves due to

viscous (velocity proportional) and material (hysteretic, displacement proportional) effects. With that

in mind, effects of softer, surface soil layers are not well represented. In order to account for close to

surface soil layers, nonlinear site response analysis has to be performed in 1D or better yet in 3D.

4.2.3 Uncertainties

Earthquakes start at the rupture zone (seismic source), propagates through the rock to the surface soils

layers. All three components in this process, the source, the path through the rock and the site response

(soil) feature significant uncertainties, which contribute to the uncertainty of ground motions. These

8

uncertainties require use of Probabilistic Seismic Hazard Analysis approach to characterizing uncertainty

in earthquake motions (Hanks and Cornell; Bazzurro and Cornell, 2004; Budnitz et al., 1998; Stepp

et al., 2001)

Uncertain Sources. Seismic source(s) feature a number of uncertainties. Location(s) of the source,

magnitude the earthquake that can be produced, return period (probability of generating an earthquake

in within certain period), are all uncertain (Kramer, 1996), and need to be properly taken into account.

Source uncertainty is controlled mainly by the stress drop function (Silva, 1993; Toro et al., 1997).

Uncertain Path (Rock). Seismic waves propagate through uncertain rock (path) to surface layers.

Path uncertainty is controlled by the uncertainty in crustal (deep rock) shear wave velocity, near site

anelastic attenuation and crustal damping factor (Silva, 1993; Toro et al., 1997). These parameters are

usually assumed to be log normal distributed and are calibrated based on available information/data, site

specific measurements and regional seismic information. Both previous uncertainties can be combined

into a model that accounts for free field motions (Boore, 2003; Boore et al., 1978).

Uncertain Site (Soil). Once such uncertain seismic motions reach surface layers (soil), they propagate

through uncertain soil (Roblee et al., 1996; Silva et al., 1996). Uncertain soil adds additional uncertainty

to seismic motions response. Soil material properties can exhibit significant uncertainties and need to be

carefully evaluated (Phoon and Kulhawy, 1999a,b; DeGroot and Baecher, 1993; Baecher and Christian,

2003)

Chapter 5 (sections 5.2 and 5.6) provide details of taking into account uncertainties from three

sources into account.

4.3 Dynamic Soil Properties

4.3.1 Dynamic Soil Behavior

Soil is an elastic plastic material. It can shown, using micromechanics, that soil (or any particulate ma-

terial) does not have an elastic range (Mindlin and Deresiewicz, 1953). However, for practical purposes,

soils can sometimes be assumed to behave as an elastic material. This is true if strain imposed on soils

are small. Sometimes soil behavior can also be assumed to be (so called) equivalent linear. In most

cases soils do behave as truly elastic-plastic (nonlinear, inelastic material). Discussion about different

soil modeling regimes is available in section 3.2.1.

Plastification – Stiffness Reduction. In reality, soil is best modeled as an elastic-plastic material

(Muir Wood, 1990; Mroz and Norris, 1982; Prevost and Popescu, 1996; Mroz et al., 1979; Zienkiewicz

et al., 1999; Nayak and Zienkiewicz, 1972). There are two main models of soil deformation, compression

and shearing. For compression loading, soil can exhibit nonlinear elastic response, up to a point of

compressive yield. Soil is usually nonlinear (inelastic) from the very beginning of shearing.

Tangent stiffness of soil can rapidly change (reduce) from relatively stiff to perfectly plastic, and

even to negative (softening), if localization of deformation occurs within soil volume. Upon reloading,

soil tends regain part or all of its stiffness, and then yield, reduce stiffness on the unloading branch.

9

Figure 4.5 shows pure shear response of soil subjected to two different shear strain levels. It is important

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

−5

0

5

γ [%]

τ [k

Pa]

frictionalfrictional+viscous

−20 −15 −10 −5 0 5 10 15 20−100

−50

0

50

100

γ [%]

τ [k

Pa]

frictionalfrictional+viscous

Figure 4.5: Predicted pure shear response of soil at two different shear strain amplitudes, from Pisano

and Jeremic (2014).

to note that for very small shear strain cycles (left Figure 4.5) response is almost elastic, with a very

small hysteretic loop (almost no energy dissipation). On the other hand, large shear strain cycles (right

Figure 4.5) there is significant loss of (tangent) stiffness, as well as a significant energy dissipation (large

hysteretic loop).

Volume Change during Shearing. One of the most important features of soil behavior is the presence

of volumetric response during shearing (Muir Wood, 1990). Dilatancy, as it is called, is a result of

a rearrangement of particles on a micro-mechanical scale. The results is that during pure shearing

deformation, soil can increase volume (dilatancy or positive dilatancy) or reduce volume (compression or

negative dilatancy). Incorporating volume change information in soil modeling can be very important.

If models that are used to model soil do not allow for modeling of dilatancy (positive or negative),

potentially significant modeling uncertainty is introduced, and results of dynamic soil behavior might

not be accurate. An example below illustrates differences in soil response to 1D wave propagation when

dilatancy is taken and not taken into account (Jeremic, 2016).

Constrained deformation is any deformation where there is a constraint to free deformation of soil.

Constrained deformation is usually connected to the presence of pore fluid (quite incompressible) inside

the soil. However, there are other examples of constrained deformation that are present in SSI analysis:

• Constitutive testing under complete stress control is an example of NON-constrained deformation.

• 1D wave propagation is an example of a constrained deformation, soil cannot deform horizontally

as shear waves propagate vertically. Complementary shear will created dilatancy or compression in

horizontal direction, thus increasing horizontal stresses for dilatant soil and decreasing horizontal

stress for compressive soil

• Soil that is fully saturated with water will have constrained deformation, as water is fairly incom-

pressible and any shearing will either created excess pore fluid pressures (positive or negative), and,

with a low permeability of soil, that excess pore fluid pressure might take long time to dissipate

Use of G/Gmax and damping curves for describing and calibrating material behavior of soil is missing a

very important (crucial) information about soil/rock volume change during shearing deformation. Volume

10

change data is very important for soil behavior. It is important to emphasize that soil behavior is very

much a function of volumetric response during shear. During shearing of soil there are two essential

types of soil behavior:

• Dilative (dense) soils will increase volume due to shearing

• Compressive (loose) soils will decrease volume due to shearing

The soil volume response, that is not provided by G/Gmax and damping curves data can significantly

affect affect response due to volume constraints of soil. For example, for one dimensional site response

(1D wave propagation, vertically propagating (SV) shear waves) the soil will try to change its volume

(dilate if it is dense or compress if it is loose). However, such volume change can only happen vertically

(since there is no constraint (foundation for example) on top, while horizontally the soil will be constraint

by other soil. That means that any intended volume change in horizontal direction will be resisted by

change in (increase for dense and decrease for loose soil) horizontal stress. For example for dilative

(dense) soil, additional horizontal stress will contribute to the increase in mean pressure (confinement)

of the soil, thus increasing the stiffness of that soil. It is the opposite for compressive soil where shearing

will result in a reduction of confinement stress. Figure 4.6 shows three responses for no-volume change

(left), compressive (middle) and dilative (right) soil with full volume constraint, resulting in changes in

stiffness for compressive (reduction in stiffness), and dilative (increase in stiffness).

-20-15-10-5 0 5

10 15 20

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01

τ [k

Pa]

γ [-]

z = H/2

-10

-5

0

5

10

15

-0.02-0.015-0.01-0.005 0 0.005 0.01 0.015

τ [k

Pa]

γ [-]

z = H/2

-50-40-30-20-10

0 10 20 30 40 50

-0.035-0.03-0.025-0.02-0.015-0.01-0.005 0 0.005 0.01

τ [k

Pa]

γ [-]

z = H/2

Figure 4.6: Constitutive Cyclic response of soils with constraint volumetric deformation: (left) no volume

change (soil is at the critical state); (middle) compressive response with decrease in stiffness; (right)

dilative response with increase in stiffness.

Changes in stiffness of soil during shearing deformation will influence wave propagation and amplifi-

cation of different frequencies. Figure 4.7 shows response of no-volume change soil (as it is/should be

assumed if only G/Gmax and damping curves are available, with no volume change data) and a response

of a dilative soil which stiffens up during shaking due to restricted intent to dilate. It is clear that dilative

soil will show significant amplification of higher frequencies.

Effective Stress and Pore Fluid Pressures. Mechanical behavior of soil depends exclusively on the

effective stress (Muir Wood, 1990). Effective stress σ′ij depends on total stress σij (from applied loads,

self weight, etc.) and the pore fluid pressure p:

σ′ij = σij − δijp (4.1)

11

-8-6-4-2 0 2 4 6 8

10

0 5 10 15 20 25 30 35 40

Acc

eler

atio

n [m

/s2 ]

Time [s]

no dilatancy vs dilatancy

surf - no dilsurf - dil

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14

Am

plitu

de [m

/s]

Frequency [Hz]

no dilatancy vs dilatancy

basesurf - no dil

surf - dil

Figure 4.7: One dimensional seismic wave propagation through no-volume change and dilative soil.

Please note the (significant) increase in frequency of motions for dilative soil. Left plot is a time history

of motions, while the right plot shows amplitudes at different frequencies.

where δij is Kronecker delta used to distribute pore fluid pressure to diagonal members of stress tensor

(δij = 1, when i=j, and δij = 0, when i 6= j), Section 6.4.2 provides more information about the use of

effective stress principle in modeling.

As described in some detail in section 6.4.11, pore fluid pressure generation and dissipation can

(significantly) reduce (for increase in pore fluid pressure) and increase (for reduction in pore fluid pressure)

stiffness of soil material. Soil can even completely liquefy (become a heavy fluid) if pore fluid pressure

becomes equal to the mean confining stress (Jeremic et al., 2008). Loss of stiffness due to increase in

pore fluid pressure can significantly influence results of seismic wave propagation, as described by Taiebat

et al. (2010).

4.3.2 Material Modeling of Dynamic Soil Behavior

Material modeling of dynamic soil behavior is done in accordance with expected strain level within soil

during shaking. Section 3.2 describes in some detail, material models used for dynamic soil behavior.

4.4 Spatial Variability of Ground Motion and Seismic Wave Incoherence

Seismic motion incoherence is a phenomena that results in spatial variability of ground motions over

small distances. Significant work has been done in researching seismic motion incoherence over the last

few decades (Abrahamson et al., 1991; Roblee et al., 1996; Abrahamson, 1992a, 2005, 1992b; Zerva and

Zervas, 2002; Liao and Zerva, 2006; Zerva, 2009)

The main sources of lack of spatial correlation (Zerva, 2009) are

• Attenuation effects, are responsible for change in amplitude and phase of seismic motions due

to the distance between observation points and losses (damping, energy dissipation) that seismic

wave experiences along that distance. This is a significant source of lack of correlation for long

structures (bridges), however for NPPSSS it is not of much significance.

12

• Wave passage effects, contribute to lack of correlation due to difference in recorded wave field at

two location points as the (surface) wave travels, propagates from first to second point.

• Scattering effects, are responsible to lack of correlation by creating a scattered wave field. Scat-

tering is due to (unknown or not known enough) subsurface geologic features that contribute to

modification of wave field.

• Extended source effects contribute to lack of correlation by creating a complex wave source field,

as the fault ruptures, rupture propagates and generate seismic sources along the fault. Seismic

energy is thus emitted from different points (along the rupturing fault) and will have different

travel path and timing as it makes it observation points.

Figure 4.8 shows an illustration of main sources of lack of correlation.

��

��

Attenuation

Wave Front(s)

1 2

Fault

Figure 4.8: Four main sources contributing to the lack of correlation of seismic waves as measured at

two observation points. Figure from Jeremic (2016).

It is important to note that SVGM developed in this report due take into account wave passage

effects (effectively removing them from SVGM). This is very important if ground motions (seed ground

motions used as a basis for development of SVGM) are developed as full 3D, inclined seismic motions,

as they will be for this effort.

4.4.1 Incoherence Modeling

Early studies concluded that the correlation of motions increases as the separation distance between

observation points decreases. In addition to that, correlation increased for decrease in frequency of

observed motions. Most theoretical and empirical studies of spatially variable ground motions (SVGM)

have focused on the stochastic and deterministic Fourier phase variability expressed in the form of ”lagged

coherency” and apparent shear wave propagation velocity, respectively. The mathematical definition of

coherency (denoted γ) is given below in equation:

γjk(f) =Sjk(f)

(Sjj(f)Skk(f))1/2(4.2)

Coherency is a dimensionless complex-valued number that represents variations in Fourier phase

between two signals. Perfectly coherent signals have identical phase angles and a coherency of unity.

13

Lagged coherency is the amplitude of coherency, and represents the contributions of stochastic processes

only (no wave passage). Wave passage effects are typically expressed in the form of an apparent wave

propagation velocity.

Lagged coherency does not remove a common wave velocity over all frequencies. Alternatively,

plane wave coherency is defined as the real part of complex coherency after removing single plane-wave

velocity for all frequencies. Recent simulation methods of SVGM prefer the use of plane-wave coherency

as it can be paired with a consistent wave velocity. An additional benefit is that plane-wave coherency

captures random variations in the plane-wave while lagged coherency does not. Zerva (2009) has called

the variations ’arrival time perturbations’.

Most often, γ is related to the dimensionless ratio of station separation distance ξ to wavelength

λ. The functional form most often utilized is exponential (Loh and Yeh, 1988; Oliveira et al., 1991;

Harichandran and Vanmarcke, 1986). The second type of functional form relates γ to frequency and ξ

independently, without assuming they are related through wavelength. This formulation was motivated

by the study of ground motion array data from Lotung, Taiwan (SMART-1 and Large Scale Seismic Test,

LSST, arrays), from which Abrahamson (1985, 1992a) found γ at short distances (ξ < 200m) to not be

dependent on wavelength. Wavelength-dependence was found at larger distances (ξ = 400 to 1000m).

For SVGM effects over the lateral dimensions of typical structures (e.g., < 200m), non-wavelength

dependent models (Abrahamson, 1992a, 2007; Ancheta, 2010) are used.

Moreover, there is a strong probabilistic nature of this phenomena, as significant uncertainty is

present in relation to all four sources of lack of correlation, mentioned above. A number of excellent

references are available on the subject of incoherent seismic motions (Abrahamson et al., 1991; Roblee

et al., 1996; Abrahamson, 1992a, 2005, 1992b; Zerva and Zervas, 2002; Liao and Zerva, 2006; Zerva,

2009).

It is very important to note that all current models for modeling incoherent seismic motions make

an ergodic assumption. This is very important as all the models assume that a variability of seismic

motions at a single site – source combination will be the same as variability in the ground motions from

a data set that was collected over different site and source locations. Unfortunately, there does not exist

a large enough data set for any particular location of interest (location of a nuclear facility), that can be

used to develop site specific incoherence models.

Incoherence in 3D. Empirical SVGM models are developed for surface motions only. This is based

on a fact that a vast majority of measured motions are surface motions, and that those motions were

used for SVGM model developments. Development of incoherent motions for three dimensional soil/rock

volumes creates difficulties.

A 2 dimensional wave-field can be developed, as proposed by Abrahamson (1993), by realizing

that all three spatial axes (radial horizontal direction, transverse horizontal direction, and the vertical

direction) do exhibit incoherence. Existence of three spatial directions of incoherence requires existence

of data in order to develop models for all three directions. Abrahamson (1992a) investigated incoherence

of a large set of 3-component motions recorded by the Large Scale Seismic Tests (LSST) array in

Taiwan. Abrahamson (1992a) concluded that there was little difference in the radial and transverse lagged

coherency computed from the LSST array selected events. Therefore, the horizontal coherency models

by Abrahamson (1992a) and subsequent models (Abrahamson, 2005, 2007) assumed the horizontal

coherency model may apply to any azimuth. Coherency models using the vertical component of array

14

data are independently developed from the horizontal.

Currently, there are no studies of coherency effects with depth (i.e. shallow site response domain).

One possible solution is to utilize the simulation method developed by Ancheta et al. (2013) and the

incoherence functions for the horizontal and vertical directions developed for a hard rock site by Abra-

hamson (2005) to create a full 3-D set of incoherent strong motions. In this approach, motions at each

depth are assumed independent. This assumes that incoherence functions may apply at any depth within

the near surface domain (< 100m). Therefore, by randomizing the energy at each depth, a set of full

3-D incoherent ground motion are created.

Theoretical Assumptions behind SVGM Models. It is very important to noted that the use of

SVGM models is based on the ergodic assumption. Ergodic assumptions allows statistical data obtained

at one (or few) location(s) over a period of time to be used at other locations at certain times. For

example, data on SVGM obtained from a Lotung site in Taiwan, over long period of time (dozens of

years) is developed into a statistical model of SVGM and then used for other locations around the world.

Ergodic assumption cannot be proven to be accurate (or to hold) at all, unless more data becomes

available. However, ergodic assumption is regularly used for SVGM models.

Very recently, a number of smaller and larger earthquakes in the areas with good instrumentation

were used to test the ergodic assumption. As an example, Parkfield, California recordings were used

to test ergodic assumption for models developed using data from Lotung and Pinyon Flat measuring

stations. Konakli et al. (2013) shows good matching of incoherent data for Parkfield, using models

developed at Lotung and Pinyon Flat for nodal separation distances up to 100m. This was one of

the first independent validations of family of models developed by Abrahamson and co-workers. This

validation gives us confidence that assumed ergodicity of SVGM models does hold for practical purposes

of developed SVGM models.

Recent reports by Jeremic et al. (2011); Jeremic (2016) present detailed account of incoherence

modeling.

4.5 Analysis Models and Modelling Assumptions

Development of analysis models for free field ground motions and site response analysis require an analyst

to make modeling assumptions. Assumptions must be made about the treatment of 3D or 2D or 1D

seismic wave field (see sections 4.2.1, 4.2.2), treatment of uncertainties (see section 4.2.3), material

modeling for soil (see section 4.3), possible spatial variability of seismic motions (see section 4.4).

This section is used to address aspects of modeling assumptions. The idea is not to cover all possible

(more or less problematic) modeling assumptions (simplification), rather to point to and analyze some

commonly made modeling assumptions. It is assumed that the analyst will have proper expertise to

address all modeling assumptions that are made and that introduce modeling uncertainty (inaccuracies)

in final results.

Addressed in this section are issues related to free field modeling assumptions. Firstly, a brief

description is given of modeling in 1D and in 3D. Then, addressed is the use of 1D seismic motions

assumptions, in light of full 3D seismic motions (Abell et al., 2016). Next, an assumption of adequate

propagation of high frequencies through models (finite element mesh size/resolution) is addressed as

15

well (Watanabe et al., 2016). There are number of other issues that can influence results (for example,

nonlinear SSI of NPPs Orbovic et al. (2015)) however they will not be elaborated upon here.

1D Models.

One dimensional (1D) wave propagation of seismic motions is a commonly made assumption in free

ground motion and site response modeling. This assumptions stems from an assumption that horizontal

soil layers are located on top of (also) horizontally layered rock layers. Snell’s law (Aki and Richards,

2002) can then be used to show that refraction of any incident ways will produce (almost) vertical

seismic waves at the surface. Using Snell’s law will not produce vertical waves, however, for a large

number of horizontal layers, with stiffness of layers increasing with depth, waves will be slightly off

vertical (less than 10o in ideal, horizontal layered systems with deep sources). It is noted that with this

ideal setup, with many horizontal layers of soil and rock and with the increase of stiffness with depth.

waves will indeed become more and more vertical as they propagate toward surface (but never fully

vertical!). However, seismic waves will rarely (never really) become fully 1D. Rather, seismic wave will

feature particle motions in a horizontal plane, and there will not be a single plane to which these particle

motions will be restricted. In addition, even if it is assumed that seismic waves propagating from great

depth, are almost vertical at the surface, incoherence of waves at the surface at close distances, really

questions the vertical wave propagation assumption. Nonetheless, much of the free and site response

analysis is done in 1D.

It is worth reviewing 1D site response procedures as they are currently done in professional practice

and in research. Currently most frequently used procedures for analyzing 1D wave propagation are based

on the so called Equivalent Linear Analysis (ELA). Such procedures require input data in the form of

shear wave velocity profile (1D), density of soil, stiffness reduction (G/Gmax) and damping (D) curves

for each layer. The ELA procedures are in reality linear elastic computations where linear (so called strain

compatible) elastic constants are set to secant values of stiffness for (a ratio of) highest achieved strain

value for given seismic wave input. This means that for different seismic input, different material (elastic

stiffness) parameters need to be calibrated from G/Gmax and D curves. In addition, damping is really

modeled as viscous damping, whereas most damping in soil is frictional (material) damping (Ostadan

et al., 2004). Due to secant choice of (linear elastic) stiffness for soil layers, the ELA procedures are

also known to bias estimates of site amplification (Rathje and Kottke, 2008). This bias becomes more

significant with stronger seismic motions. In addition, with the ELA procedures, there is no volume

change modeling, which is quite important to response of constrained soil systems (di Prisco et al.,

2012).

Instead of Equivalent Linear Analysis approach, a 1D nonlinear approach is also possible. Here, 1D

nonlinear material models are used (described in section 3.2). Nonlinear 1D models of soil (nonlinear

elastic, with a switch to account for unloading-reloading cycles) are used for this modeling. Nonlinear

1D models overcome biased estimates of site amplification, as they model stiffness in a more accurate

wave. While modeling with nonlinear models is better than with ELA procedures, it is noted, that volume

change information is still missing. It is also noted that 1D models used here only work for 1D, that is,

extension to 2D or 3D is rather impossible to be done in a consistent way.

16

3D Models.

In reality, seismic motions are always three dimensional, featuring body and surface waves (see more in

section 4.2). However, development of input, free field motions for a 3D analysis is not an easy task.

Recent large scale, regional models (Bao et al., 1998; Bielak et al., 1999; Taborda and Bielak, 2011;

Cui et al., 2009; Bielak et al., 2010, 2000; Restrepo and Bielak, 2014; Bao et al., 1996; Xu et al., 2002;

Taborda and Bielak, 2013; Dreger et al., 2015; Rodgers et al., 2008; Aagaard et al., 2010; Pitarka et al.,

2013, 2015, 2016) have shown great promise in developing (very) realistic free field ground motions.

What is necessary for these models to be successfully used is the detailed knowledge about the deep

and shallow geology as well as a local site conditions (nonlinear soil properties in 3D). This data is

unfortunately usually not available.

When the data is available, a better understanding of dynamic response of an NPP can be developed.

Developed nonlinear, 3D response will not suffer from numerous modeling uncertainties (1D vs 3D

motions, elastic vs nonlinear/inelastic soil in 3D, soil volumetric response during shearing, influence of

pore fluid, etc.). Recent US-NRC, CNSC and US-DOE projects have developed and are developing a

number of nonlinear, 3D earthquake soil structure interaction procedures, that rely on full 3D seismic

wave fields (free field and site response) and it is anticipated that this trend will only accelerate as

benefits of (a more) accurate modeling become understood.

3D versus 1D Seismic Models

We start by pointing out one of the biggest simplifying assumptions made, that of a presence and use

of 1D seismic waves. As pointed out in section 4.2.1 above, world wide records do not show evidence

of 1D seismic waves. It must be noted, that an assumption of neglecting full 3D seismic wave field and

replacing it with a 1D wave field can sometimes be appropriate. However such assumption should be

carefully made, taking into account possible intended and unintended consequences.

Present below is a simple study that emphasizes differences between 3D and 1D wave fields, and

points out how this assumptions (3D to 1D) affects response of a generic model NPP. It is noted that

a more detailed analysis of 3D vs 3×1D vs 1D seismic wave effects on SSI response of an NPP is is

provided by (Abell et al., 2016).

Assume that a small scale regional models is developed in two dimensions (2D). Model consists of

three layers with stiffness increasing with depth. Model extends for 2000m in the horizontal direction

and 750m in depth. A point source is at the depth of 400m, slightly off center to the left. Location of

interest (location of an NPP) is at the surface, slightly to the right of center.

Figure 4.10 shows a snapshot of a full wave field, resulting from a small scale regional simulation,

from a point source (simplified), propagating P and S waves through layers. Wave field is 2D in this

case, however all the conclusion will apply to the 3D case as well,

It should be noted that regional simulation model shown in Figure 4.10 is rather simple, consisting of

a point source at shallow depth in a 3 layer elastic media. Waves propagate, refract at layer boundaries

(turn more ”vertical”) and, upon hitting the surface, create surface waves (in this case, Rayleigh waves).

In our case (as shown), out of plane translations and out of plane rotations are not developed, however this

simplification will not affect conclusions that will be drawn. A seismic wave field with full 3 translations

and 3 rotations (6D) will only emphasize differences that will be shown later.

Assume now that a developed wave field, which in this case is a 2D wave field, with horizontal and

17

Figure 4.9: Snapshot of a full 3D wave field

Figure 4.10: (Left) Snapshot of a wave field, with body and surface waves, resulting from a point source

at 45o off the point of interest, marked with a vertical line, down-left. This is a regional scale model of

a (simplified) point source (fault) with three soil layers. (Right) is a zoom in to a location of interest.

Figures are linked to animation of full wave propagation.

vertical translations and in plane rotations, are only recorded in one horizontal direction. From recorded

1D motions, one can develop a vertically propagating shear wave in 1D, that exactly models 1D recorded

motion. This is usually done using de-convolution Kramer (1996). Figure 4.11 illustrates the idea of

using a full 3D seismic wave field to develop a reduced, 1D wave field

Figure 4.11: Illustration of the idea of using a full wave field (in this case 2D) to develop a 1D seismic

wave field.

Two seismic wave fields, the original wave field and a subset 1D wave field now exist. The original

wave field includes body and surface waves, and features translational and rotational motions. On the

other hand, subset 1D wave field only has one component of motions, in this case an SV component

(vertically polarized component of S (Secondary) body waves).

Figure 4.12 show local free field model with 3D and 1D wave fields respectively.

Please note that seismic motions are input in an exact way, using the Domain Reduction Method

(Bielak et al., 2003; Yoshimura et al., 2003) (described in chapter 6.4.10) and how there are no waves

leaving the model out of DRM element layer (4th layer from side and lower boundaries). It is also

important to note that horizontal motions in one direction at the location of interest (in the middle of

the model) are exactly the same for both 2D case and for the 1D case.

Figure 4.13 shows a snapshot of an animation (available through a link within a figure) of difference

in response of an NPP excited with full seismic wave field (in this case 2D), and a response of the same

18

http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4

http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4

Figure 4.12: Left: Snapshot of a full 3D wave field, and Right: Snapshot of a reduced 1D wave field.

Wave fields are at the location of interest. Motions from a large scale model were input using DRM,

described in chapter 6 (6.4.10). Note body and surface waves in the full wave field (left) and just

body waves (vertically propagating) on the right. Also note that horizontal component in both wave

fields (left, 2D and right 1D), are exactly the same. Each figure is linked to an animation of the wave

propagation.

NPP to 1D seismic wave field.

Figure 4.13: Snapshot of a 3D vs 1D response of an NPP, upper left side is the response of the NPP

to full 3D wave field, lower right side is a response of an NPP to 1D wave field.

Figures 4.14 and 4.15 show displacement and acceleration response on top of containment building

for both 3D and 1D seismic wave fields.

A number of remarks can be made:

• Accelerations and displacements (motions, NPP response) of 6D and 1D cases are quite different.

In some cases 1D case gives bigger influences, while in other, 6D case gives bigger influences.

• Differences are particularly obvious in vertical direction, which are much bigger in 6D case.

• Some accelerations of 6D case are larger that those of a 1D case. On the other hand, some

19

http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4

http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4

http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4

Figure 4.14: Displacements response on top of a containment building for 3D and 1D seismic input.

Figure 4.15: Acceleration response on top of a containment building for 3D and 1D seismic input.

displacements of 1D case are larger than those of a 6D case. This just happens to be the case for

given source motions (a Ricker wavelet), for given geologic layering and for a given wave speed

(and length). There might (will) be cases (combinations of model parameters) where 1D motions

model will produce larger influences than 6D motions model, however motions will certainly again

be quite different. There will also be cases where 6D motions will produce larger influences than

1D motions. These differences will have to be analyzed on a case by case basis.

In conclusion, response of an NPP will be quite different when realistic 3D (6D) seismic motions are

used, as opposed to a case when 1D, simplified seismic motions are used. Recent paper, by Abell et al.

(2016) shows differences in dynamic behavior of NPPs same wave fields is used in full 3D, 1D and 3×1D

configurations.

Propagation of Higher Frequency Seismic Motions

Seismic waves of different frequencies need to be accurately propagated through the model/mesh. It is

known that mesh size can have significant effect on propagating seismic waves (Argyris and Mlejnek,

1991; Lysmer and Kuhlemeyer, 1969; Watanabe et al., 2016; Jeremic, 2016). Finite element model mesh

20

(nodes and element interpolation functions) needs to be able to approximate displacement/wave field

with certain required accuracy without discarding (filtering out) higher frequencies. For a given wave

length λ that is modeled, it is suggested/required to have at least 10 linear interpolation finite elements

(8 node bricks in 3D, where representative element size is ∆hLE ≤ λ/10) or at least 2 quadratic

interpolation finite elements (27 node bricks in 3D, where representative element size is ∆hQE ≤ λ/2)

for modeling wave propagation.

Since wave length λ is directly proportional to the wave velocity v and inversely proportional to the

frequency f , λ = v/f . we can devise a simple rule for appropriate size of finite elements for wave

propagation problems:

• Linear interpolation finite elements (1D 2-node truss, 2D 4-node quad, 3D 8-node brick) the

representative finite element size needs to satisfy the following condition

hLE ≤ v

10 fmax

• Quadratic interpolation finite elements (1D 3-node truss, 2D 9-node quad, 3D 27-node brick) the

representative finite element size needs to satisfy the following condition

hQE ≤ v

2 fmax

It is noted that while the rule for number of elements (or element size ∆h) can be used to delineate

models with proper and improper meshing, in reality having bigger finite element sizes than required by

the above rule will not filter out higher frequencies at once, rather they will slowly degrade with increase

in frequency content.

Simple analysis can be used to illustrate above rules (Jeremic, 2016). One dimensional column model

is used for this purpose, as shown in figure 4.16.

Total height of the model is 1000m. Two models are built with element height/length of 20 m and

50 m, and each model has two different shear wave velocities (100 m/s and 1000 m/s). Density is set

to ρ = 2000 kg/m3, and Poisson’s ratio is ν = 0.3. Various cases are set and tested as shown in table

4.1. Both 8 node and 27 node brick elements are used for all models. Thus, total 24 parametric study

cases are inspected. Linear elastic elements are used for all analyses. All analyses are performed in time

domain with Newmark dynamic integrator without any numerical damping (γ = 0.5, and β = 0.25, no

numerical damping, unconditionally stable).

Ormsby wavelet (Ryan, 1994) is used as an input motion and imposed at the bottom of the model.

Ormsby wavelet, given by the equation 4.3 below

f(t) = A((πf4

2

f4 − f3sinc(πf4(t− ts))2 −

πf32

f4 − f3sinc(πf3(t− ts))2)

− (πf2

2

f2 − f1sinc(πf2(t− ts))2 −

πf12

f2 − f1sinc(πf1(t− ts))2) (4.3)

features a controllable flat frequency content. Here, f1 and f2 define the lower range frequency band,

f3 and f4 define the higher range frequency band, A is the amplitude of the function, and ts is the time

at which the maximum amplitude is occurring, and sinc(x) = sin(x)/x. Figure 4.17 shows an example

of Ormsby wavelet with flat frequency content from 5 Hz to 20 Hz.

21

Figure 4.16: One dimensional column test model to address the mesh size effect.

Figure 4.17: Ormsby wavelet in time and frequency domain with flat frequency content from 5 Hz to 20

Hz.

As an example of such an analysis, consider three cases, as shown in Figures 4.18, 4.19, and 4.20m

with cutoff frequencies of Ormsby wavelets set as 3Hz, 8Hz, and 15Hz. As shown in figure 4.18, case

1 and 7 (analysis using Ormsby wavelet with 3 Hz cutoff frequency) predict exactly identical results

to the analytic solution in both time and frequency domain. In this case, the number of nodes per

wavelength for both cases is over 10 and presented very accurate results (in both time and frequency

domain) are expected. Increasing cutoff frequency to 8Hz induces numerical errors, as shown in figure

4.19. In frequency domain, both 10 m and 20 m element height model with 27 node brick element

predict very accurate results. However, in time domain, asymmetric shape of time history displacements

are observed. Observations from top of 8 node brick element models show more numerical error in

both time and frequency domain due to the decreasing number of nodes per wavelength. Figure 4.20

shows analysis results with 15Hz cutoff frequency. Results from 27 node brick elements are accurate

22

Table 4.1: Analysis cases to determine a mesh size

Case Vs Cutoff Element Max. propagation

number (m/s) freq. (Hz) height (m) freq. (Hz)

1 1000 3 10 10

2 1000 8 10 10

3 1000 15 10 10

4 100 3 10 1

5 100 8 10 1

6 100 15 10 1

7 1000 3 20 5

8 1000 8 20 5

9 1000 15 20 5

10 100 3 20 0.5

11 100 8 20 0.5

12 100 15 20 0.5

in frequency domain but asymmetric shapes are observed in time domain. Decreasing amplitudes in

frequency domain along increasing frequencies are observed from 8 node brick element cases.

Material Modeling and Assumptions

Material models that are used for site response need to be chosen to have appropriate level of sophisti-

cation in order to model important aspects of response. For example, for site where it is certain that 1D

waves will model all important aspects of response, and that motions will not be large enough to excite

fully nonlinear response of soil, and where volumetric response of soil is not important (soil does not

feature volume change during shearing), 1D equivalent linear models can be used. On the other hand,

for sites where full 3D wave fields are expected to provide important aspects of response (3D wave fields

develop due to irregular geology, topography, seismic source characteristics/size, etc.), and where it is

expected that seismic motions will trigger full nonlinear/inelastic response of soil, full 3D elastic-plastic

material models need to be used. More details about material models that are used for site response

analysis are described in some detail in section 3.2, and in section 4.3, above.

3D vs 3×1D vs 1D Material Behavior and Wave Propagation Models

In general behavior of soil is three dimensional (3D) and very nonlinear/inelastic. In some cases, sim-

plifying assumptions can be made and soil response can be modeled in 2D or even in 1D. Modeling soil

response in 1D makes one important assumption, that volume of soil during shearing will not change

(there will be no dilation or compression). Usually this is only possible if soil (sand) is at the so called

critical state (Muir Wood, 1990) or if soil is a fully saturated clay, with low permeability, hence there is

no volume change (see more about modeling such soil in section 6.4.2).

If soils will be excited to feature a full nonlinear/inelastic response, full 3D analysis and full 3D

material models need to be used. This is true since for a full nonlinear/inelastic response it is not

appropriate to perform superposition, so superimposing 3×1D analysis, is not right.

23

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time (s)

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Figure 4.18: Comparison between (a) case 1 (top, Vs = 1000 m/s, 3 Hz, element size = 10m) and (b)

case 7 (bottom, Vs = 1000 m/s, 3 Hz, element size = 20m)

24

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26

Recent increase in use of 3×1D models (1D material models for 3D wave propagation) requires

further comments. Such models might be appropriate for seismic motions and behavior of soil that is

predominantly linear elastic, with small amount of nonlinearities. In addition, it should be noted that

vertical motions recorded on soil surface are usually a result of surface waves (Rayleigh). Only very early

vertical motions/wave arrivals are due to compressional, primary (P) waves. Modeling of P waves as

1D vertically propagating waves is appropriate. However, modeling of vertical components of surface

(Rayleigh) waves as vertically propagating 1D waves is not appropriate. Recent work by (Elgamal and

He, 2004) provides nice description of vertical wave/motions modeling problems.

4.6 Limitations of Time and Frequency Domain Methods

Limitations of time and frequency domain methods are twofold. First source of limitations is based on

(usual) lack of proper data for (a) deep and shallow geology, surface soil material, and (b) earthquake

wave fields. This source of limitations can be overcome by more detailed site and geologic investigation

(as recently done for the Diablo Canyon NPP in California NO REFERENCE). The second source of

limitations is based on the underlying formulations for both approaches. Time domain methods, with

nonlinear modeling, require sophistication from the analyst, including knowledge of nonlinear solutions

methods, elasto-plasticity, etc. Technical limitations on what (sophisticated) modeling can be done are

usually with the analyst and with the program that is used (different programs feature different level

of modeling and simulation sophistication). On the other hand, frequency domain modeling requires

significant sophistication on the analyst side in mathematics. In addition, frequency domain based

methods are limited to linear elastic material behavior (as they rely on the principle of superposition)

which limits their usability for seismic events where large nonlinearities are expected.

27

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