Soil

Soil-Structure Interaction

Large civil structures such as concrete dams, nuclear power plants, high-rise buildings and

bridges are massive enough that their vibration due to earthquake excitation affects the motion of

the soil or rock supporting them, which in turn further affects the motion of the structure itself.

This interaction between the structure and the soil needs to be modelled accurately in order to

design earthquake resistant structures and to correctly evaluate the earthquale safety of existing

structures.

Historically, engineering analysis of such soil-structure interaction has had several impediments:

(i) limited knowledge of the relevant earthquake faults and of the regional geological features

required to fully characterize the incoming earthquake ground motion, (ii) lack of accurate

earthquake input methods in existing analysis software, and (iii) inability to efficiently model the

unbounded soil domain.

Background of PML

LS-DYNA now has a novel method for soil-structure interaction analysis that applies the

earthquake forces in an efficient and rational manner and models the unbounded domain

accurately at low computational cost, given a free-field ground motion characterizing an

earthquake. It uses the effective seismic input method to incorporate the earthquake forces into

the soil-structure model, using only the free-field ground motion at the soil-structure interface,

and not requiring any deconvolution down to depth unlike older methods of earthquake input.

The unbounded domain is modeled using perfectly matched layers, which absorbs the outward-

propagating waves almost perfectly with only a slight increase in cost from the classical Lysmer

dashpot boundaries. These pages explain and demonstrate these techniques for seismic soil-

structure interaction analysis in LS-DYNA.

PML was originally developed for electromagnetic waves in seminal works by Bérenger and

Chew in 1994, and followed up by extensive investigation of electromagnetic PMLs by

numerous researchers, as well as extensions to other fields such as elastic waves for seismic

applications.

Most of these formulations and implementations used finite-difference split-field methods to

implement the PML, which had two disadvantages: (i) the finite-difference methods could not be

used easily with finite-element models for structures, and (ii) the split-field formulation often led

to long-time instability.

These shortcomings were rectified for elastic PMLs by Basu and Chopra [2003, 2004, 2009] by

developing a displacement-based finite-element implementation that allowed explicit analysis,

thus enabling realistic analysis of three-dimensional soil-structure systems.

Scattering Analysis Framework

A scattering analysis framework, developed by Bielak and co-workers as part of the effective

seismic input method, is adopted as the approach for soil-structure interaction analysis in LS-

DYNA. This approach considers soil-structure interaction to be caused by the scattering of the

free-field ground motion by the presence of the structure, and the appropriate earthquake forces

and the use of an absorbing boundary follow rationally from this viewpoint.

Consider the following two independent states as part of a thought experiment:

(a) soil is excited by an earthquake in the absence of the structure, and

(b) the structure disturbs and scatters the incoming earthquake wave.

Note that both these states cannot occur in reality: it can be either one or the other.

If we take the difference of the foundation motion in the two states, we are left with only the

scattered motion, having eliminated both the earthquake source and the incoming wave. The

scattered motion — because it is generated solely by the structure — propagates entirely outward

from the structure.

Now the unbounded domain can be replaced by a truncated bounded domain, but without only

simple boundary conditions, the outer boundary will reflect spurious waves back to the structure.

This may be avoided by using an absorbing boundary to reduce the wave reflection and

appropriately model the unbounded domain beyond.

We shall use perfectly matched layers as the absorbing boundary, and the scattered-wave

formulation will provide the equivalent earthquake forces, termed as the effective seismic input.

PML Absorbing Boundary

A perfectly matched layer (PML) is an absorbing layer model that — when placed next to an

elastic bounded domain — absorbs nearly perfectly all waves traveling outward from the

bounded domain, without any reflection from the interface between the bounded domain and the

PML.

The outgoing wave is absorbed and attenuated in the PML. There is some reflection of the wave

from the fixed outer boundary of the PML, but that reflected wave can be made as small as

desired. Therefore, an accurate model of an unbounded domain may be obtained even if the PML

is placed very close to the excitation.

PML Theory

Consider a semi-infinite rod — a simple model of an unbounded half-space — where only

rightward waves are allowed:

The equations for the elastic medium of this rod can be converted into equations for a perfectly

matched medium (PMM), which is mathematically designed to damp out waves using a damping

function that increases in the unbounded direction:

This PMM may be placed next to a bounded elastic rod to absorb and damp out all waves

traveling outward from the bounded medium:

The medium is mathematically designed not to reflect any portion of the waves at its interface to

the elastic rod, this being the perfect matching property of the medium.

This PMM may be truncated where the wave is sufficiently damped, to give the perfectly

matched layer:

There will be some reflection from the truncated end of the PML, but the amplitude of the

reflected wave, given by

is controlled by and , and can be made as small as desired.

The attenuation function is typically chosen as

Typically, works best for finite-element analysis, and may be chosen from simplified

discrete analysis. LS-DYNA automatically chooses an optimal value of according to the depth

of the layer.

The depth of the layer may be chosen so that the layer is about 5–8 elements deep, with the

mesh density in the PML chosen to be similar to that in the elastic medium.

PML in LS-DYNA

PML has been implemented in LS-DYNA for elastic, fluid and acoustic media, and may be used

through one of the following cards:

o MAT_PML_ELASTIC (MAT_230), corresponding to MAT_ELASTIC (MAT_001)

o MAT_PML_ELASTIC_FLUID (MAT_230_FLUID), corresponding to

MAT_ELASTIC_FLUID (MAT_001_FLUID)

o MAT_PML_ACOUSTIC (MAT_231), corresponding to MAT_ACOUSTIC (MAT_090)

o MAT_PML_ORTHO/ANISOTROPIC (MAT_245), corresponding to

MAT_ORTHO/ANISOTROPIC_ELASTIC (MAT_002 and MAT_002_ANIS)

o MAT_PML_NULL (MAT_246), corresponding to MAT_NULL (MAT_009); to be used

only with EOS_LINEAR_POLYNOMIAL or EOS_GRUNEISEN.

Please see the latest draft manual for the input format of each card.

Each PML material is meant to be placed next to — and absorbs waves from — a bounded

domain composed of the material to which it corresponds. The material in the bounded domain

near the PML should be linear, and the material constants of the PML should match those of the

linear bounded material. To facilitate this, the input format of each PML material is largely

similar to the corresponding linear material.

Some further requirements for the PML materials are:

o The PML material should form a parallelepiped box around the bounded domain, and the

box should be aligned with the coordinate axes.

o The outer boundary of the PML should be fixed.

o The PML layer may typically have 5–8 elements through the depth.

o The PML material should not be subjected to any static load.

Accuracy of PML

Two examples are presented hereto demonstrate the accuracy of a PML model: the first gives a

visual demonstration of the absorption of waves by the PML, and the second shows the efficacy

of the PML model even with small bounded domains.

Consider a half-space, with a uniform vertical force applied over a square area on its surface:

We first choose the following PML model — with 5 elements through the PML — to

demonstrate the wave absorption:

The wave propagation may be seen in the following movie: (note the dark band in the PML in

the edges)

However, the PML is most effective when it is close to the excitation:

The following figure shows the above PML in cross-section, with 8 elements through the PML,

along with a dashpot model of the same size used for comparison.

An extended mesh model is used as a benchmark:

We apply a vertical force:

and calculate the vertical displacements at the center and at the corner of the area:

Clearly, the PML model produces accurate results, borne out by the computed error in the

results:

Model Center displacement Corner displacement

PML 5% 6%

Dashpots 46% 85%

But more striking is the cost of the PML model, which is found to be similar to the dashpot

model, but a tiny fraction of the cost of the extended mesh model:

Model Elements Time steps Wall-clock time

PML 4 thousand 600 30 secs

Dashpots 4 thousand 900 15 secs

Extd. mesh 10 million 900 35 proc-hrs

The PML and dashpot results were obtained from LS-DYNA running on a desktop workstation,

whereas the extd. mesh results required a specially parallelised and optimised code running on a

supercomputer.

Clearly, PML guarantees accurate results at low cost. A slightly shallower PML, e.g. one 5-

elements deep, would still have produced close to accurate results.

We may also mention here that:

Long-time stability of this PML has been verified numerically.

The critical time-step of the PML for explicit analysis is the same as that for the

corresponding elastic element.

PML References

U. Basu and A. K. Chopra. Perfectly matched layers for time-harmonic elastodynamics of

unbounded domains: theory and finite-element implementation. Computer Methods in

Applied Mechanics and Engineering, 192(11–12):1337–1375, March 2003.

U. Basu and A. K. Chopra. Perfectly matched layers for transient elastodynamics of

unbounded domains. International Journal for Numerical Methods in Engineering,

59(8):1039–1074, February 2004. Erratum: Ibid. 61(1):156–157, September 2004.

U. Basu. Explicit finite element perfectly matched layer for transient three-dimensional

elastic waves. International Journal for Numerical Methods in Engineering, 77(2):151–

176, January 2009.

Effective Seismic Input

When incorporating earthquake excitation into the soil-structure model, we not only have to

apply effective forces that are equivalent to the incoming earthquake, but we also have to

account for the non-linearity of the structure, or in other words, start the transient earthquake

analysis from a static state of the structure.

The effective forces to be applied to the model are derived from the scattering analysis

framework described earlier, wherein we consider the difference between the ground motion

with the structure and without, and this scattered motion - entirely outgoing from the structure -

is absorbed by a PML boundary.

The non-linear analysis of the structure can also be incorporated in the same context, as

described late

Seismic Input Theory

Effective seismic input

We consider two possible states of the soil domain during an earthquake: one with the structure,

and one without.

Consider first the structure in an earthquake, for which the equations of motion of the free body

are:

where is the total motion of the system, and is the reaction force on the structure from the

soil.

For the associated soil domain, the free-body equations of motion are:

where and is the earthquake force.

The same soil domain in the absence of the structure will be governed by the following equation:

where is the free-field ground motion.

The scattered motion in the soil domain is obtained by taking the difference between the two, on

all nodes other than those on the interface with the structure:

When put together with the equation of motion for the structure, the equations for the whole

system become:

wherein the right-hand side give the effective earthquake forces that are equivalent to —-

these depend only on the free-field ground motion at the interface, and because of the sparsity of

the mass and stiffness matrices, are confined to one layer of elements around the soil-structure

interface.

In other words, using the scattered motion in the soil domain creates a discontinuity at the

interface with the structure, where the total motion is used, and this discontinuity creates

effective forces at the interface. The discontinuity is exactly the free-field ground motion at the

interface, and thus effective forces depend solely on that free-field ground motion.

This is the effective seismic input method developed by Bielak and co-workers - it directly uses

the free-field earthquake ground motions at the soil-structure and does not require their

deconvolution down to depth.

Non-linear analysis of the structure

Since the goal of transient soil-structure interaction analysis is to predict the non-linear

behaviour of the structure, the transient analysis needs to start from a static state of the structure.

Furthermore, the soil itself may behave non-linearly, and this needs to be accounted for in the

analysis. However, the soil domain itself is

a. linear by assumption, in order to allow calculating the scattered motion by subtraction,

and

b. incapable of carrying any static load, because (i) the static state is eliminated in

calculating the scattered motion, and (ii) the PML is meant to absorb only wave motion

and cannot support static loads.

This conflict may be resolved as follows:

1. Assume that all the non-linearity in the soil is limited to a region near the structure, and define

the generalized structure to be the physical structure itself along with this non-linear part of the

soil. The rest of the soil domain is then linear and can be taken to be the soil domain for the

purpose of the interaction analysis.

2. For the analysis, first calculate the static reactions at the base of the generalized structure by a

static analysis, and apply those reactions at the base during the transient analysis to support the

weight of the structure and non-linear soil.

Seismic Input in LS-DYNA

Effective seismic input has been implemented in LS-DYNA, with INTERFACE_SSI cards used

to identify the soil-structure interface, and LOAD_SEISMIC_SSI used to specify the ground

motion on such an interface. Typically, only ground acceleration histories are required to specify

the ground motion, but if ground velocity and displacement curves are also available from signal

processing of the accelerograms, then the ground motion may be specified using

DEFINE_GROUND_MOTION.

The variations on the INTERFACE_SSI cards (_AUX, _AUX_EMBEDDED and _STATIC) are

meant for different stages in the analysis. All the INTERFACE_SSI cards (except for _AUX)

create a tied-contact interface between two specified segment sets, the master surface being on

the soil side and the slave on the structure side.

Soil-structure interaction analysis under earthquake excitation may then be carried out in LS-

DYNA using these cards as follows:

0. Carry out a static analysis of the soil-structure system (e.g. using dynamic relaxation;

see *CONTROL_DYNAMIC_RELAXATION), with the soil-structure interface identified

using *INTERFACE_SSI_STATIC_ID.

Optionally, carry out a free-field analysis to record free-field motions on the future soil-

structure interface, using either *INTERFACE_SSI_AUX or

*INTERFACE_SSI_AUX_EMBEDDED, for surface-supported or embedded structures

respectively.

1. Carry out the transient analysis as a full-deck restart job (see *RESTART), with only

the structure initialized to its static stress state (see *STRESS_INITIALIZATION), and the

same soil-structure interface identified using *INTERFACE_SSI_ID with the same ID as in

static analysis:

a. The structure mesh must be identical to the one used for static analysis.

b. The soil mesh is expected to be different from the one used for static analysis,

especially because non-reflecting boundary models may be used for transient analysis.

c. The meshes for the structure and the soil need not match at the interface.

d. Only the structure must be subjected to static loads, via *LOAD_BODY_PARTS.

e. The earthquake ground motion is specified using *LOAD_SEISMIC_SSI, and/or read

from motions recorded from a previous analysis using *INTERFACE_SSI_AUX or

*INTERFACE_SSI_AUX_EMBEDDED.

Examples of seismic input in LS-DYNA

The figure above shows a model representing a building (red) upon soil — the blue part is the

elastic soil domain and the green part is the PML. The figure on the right shows the soil-structure

interface.

An example of a purely dynamic analysis of this system — without any gravity load — is given

in the input deck ssi-dynamic.k. Transient analysis of the system following an initial static

analysis is demonstrated by a pair of input decks: ssi-static.k and ssi-transient.k, to be run one

after the other. The ground motions used in the analysis are given in elcentro-x.ath, elcentro-

y.ath, and elcentro-z.ath.

Results from effective seismic input

The implementation of effective seismic input in LS-DYNA was validated by analysing a model

of the Morrow Point Dam and comparing the results against the response computed from EACD,

which using the substructure method for ground motion input and a boundary-element model for

the foundation rock, serves as a benchmark.

The upstream-downstream displacement amplitude at the center of the crest of the dam, as

computed from LS-DYNA and from EACD, are shown below. It is seen that the LS-DYNA

results closely match the semi-analytical results from EACD.

References for Seismic Input

1. I. Herrera and J. Bielak. Soil-structure interaction as a diffraction problem. In

Proceedings of the 6thWorld Conference on Earthquake Engineering, vol. 2., 1467– –

1472, New Delhi, India, 1977.

2. J. Bielak and P. Christiano. On the effective seismic input for non-linear soil-structure

interaction systems. Earthquake Engineering and Structural Dynamics, 12(1):107–119,

January 1984.

3. M. G. Cremonini, P. Christiano and J. Bielak. Implementation of effective seismic input

for soil structure interaction systems. Earthquake Engineering and Structural Dynamics,

16(4):615–625, May 1988.

4. J. Bielak, K. Loukakis, Y. Hisada and C. Yoshimura. Domain reduction method for three-

dimensional earthquake modeling in localized regions, Part I: Theory. Bulletin of the

Seismological Society of America, 93(2):817–824, April 2003.

5. C. Yoshimura, J. Bielak, Y. Hisada and A. Fernandez. Domain reduction method for

three-dimensional earthquake modeling in localized regions, Part II: Verification and

applications. Bulletin of the Seismological Society of America, 93(2):825–840, April

2003.

Dam analysis

Earthquake analysis of dams provide an added challenge beyond analysis of soil-structure

systems because here there are two unbounded domains — the water and the foundation rock,

interacting with each other as well as the dam — as opposed to only one unbounded domain in

soil-structure interaction.

We show next how the theory of effective seismic input for soil-structure interaction is extended

to the earthquake analysis of dams, and how the existing implementation in LS-DYNA is

repurposed for such analysis.

Dam analysis theory

As mentioned earlier, the effective seismic input method is derived by viewing soil-structure

interaction as a scattering problem, wherein the presence of the dam causes the scattering of the

free-field ground motion in the linear foundation domain. This is entirely analogous to acoustic

scattering, wherein acoustic waves in a linear acoustic fluid are made to scatter by the presence

of a solid body. In both cases, the waves in a linear background medium — the soil foundation or

the acoustic fluid — are scattered by a solid structure placed in the free-field, and the solution is

formulated by considering the scattered motion in the background medium, which replaces a

distant excitation source with equivalent effective forces at the interface with the structure.

This shows that this scattered-motion approach to solving a scattering problem depends only on

the linearity of the background medium, and not on the particular physical properties of the

medium. Therefore, for dam analysis, we can consider the water domain and the foundation rock

together as one free-field background domain, and the dam as the structure. However, we know

the free-field ground motion in the foundation rock only, not in the water, so to determine that,

we need an initial auxiliary analysis where the water domain plays the part of the structure. This

results in a two-step analysis procedure, where we first analyze the auxiliary water-foundation

rock to compute the free-field motion in the water, and then use this in the analysis of the whole

dam-water-rock system.

Note that there is a physical significance to the auxiliary system, even though it is not a physical

entity by itself. The earthquake ground motion affects the dam not only directly through its base,

but also through pressure waves in the water, excited by the earthquake at the reservoir bottom.

The auxiliary system brings in the effect of the far-field pressure waves in the final analysis of

the whole system without needing to model a large length of the reservoir.

Dam analysis example

Dam analysis can be formulated in LS-DYNA according to two-step approach outlined earlier

using the INTERFACE_SSI and LOAD_SEISMIC_SSI cards implemented for soil-structure

interaction.

Two preliminary analyses are required — a static analysis, and the auxiliary analysis to capture

the effect of upstream ground motion.

Static analysis

In static analysis, the foundation rock and water are constrained on the outer boundary, and they

are chosen to be sufficiently large to model the static stress distribution. This is not as onerous a

constraint as in dynamic analysis, where the outward propagating waves would have reflected

back from a fixed outer boundary. Only the dam and the water are subjected to gravity loading,

the reasoning being that the current state of the foundation is already its deformed shape due to

aeons of gravity acting upon it. The following figures show the model and the "soil-structure"

interface, specified using INTERFACE_SSI_STATIC to record the static reactions. In a manner

similar to soil-structure interaction analysis — where the nearby non-linear part of the soil was

taken as part of the generalized structure — here the near-field water, which may behave non-

linearly, is taken to be part of the structure, and the far-field water and the foundation rock are

together taken to be the "soil" somain. Of course, any nearby non-linear portion of the foundation

rock may also be taken to be part of the structure as necessary.

Auxiliary analysis

Auxiliary analysis involves only the far-field water and the foundation rock, with the water

domain acting as the structure, and the unbounded water and foundation rock truncated using

PML. The earthquake excitation is applied using LOAD_SEISMIC_SSI on a segment set at the

water-foundation interface away from the PML boundary, and the motions at the future "soil-

structure" interface are recorded using INTERFACE_SSI_AUX for use in the subsequent

transient analysis.

Transient analysis

In the transient analysis of the entire dam-water-rock system, the direct earthquake excitation at

the base of the dam is applied using LOAD_SEISMIC_SSI on a segment set, and the effect of

the upstream ground motion, computed in the auxiliary analysis at the "soil-structure" interface,

is incorporated using INTERFACE_SSI, which also reads in the static reactions computed in the

static analysis.

The corresponding example input files for these analyses are dam-static.k, dam-auxiliary.k and

dam-transient.k, using the ground motion files elcentro-x.ath, elcentro-y.ath, elcentro-z.ath.

Please also see the README file.

NOTE: Please ignore error messages from LS-Prepost flagging the INTERFACE_SSI cards as

invalid.