geotechnical earthquake engineering...geotechnical earthquake engineering by dr. deepankar choudhury...

Geotechnical Earthquake

Engineering

by

Dr. Deepankar Choudhury Humboldt Fellow, JSPS Fellow, BOYSCAST Fellow

Professor

Department of Civil Engineering

IIT Bombay, Powai, Mumbai 400 076, India.

Email: [email protected]

URL: http://www.civil.iitb.ac.in/~dc/

Lecture – 32

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Module – 8

Site Response Analysis

3

Site Response - The Problem

Predicts the response of a soil deposit due to earthquake

excitation

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Site Response

Ideally, a complete ground response analysis should

include:

Rupture mechanism at source of an earthquake

(source)

Propagation of stress waves through the crust to the

top of bedrock beneath the site of interest (path)

How ground surface motion is influenced by the soils

that lie above the bedrock (site)

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In Reality,

Mechanism of fault rupture is very complicated and

difficult to predict in advance

Crustal velocity and damping characteristics are generally

poorly known

Nature of energy transmission between the source and site

is uncertain

Site Response

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6

Ground Response Analysis

In practice,

Seismic hazard analyses (probabilistic or

deterministic) are used to predict bedrock motions at the

location of the site.

Seismic hazard analyses rely on empirical attenuation

relationships to predict bedrock motion parameters.

Ground response problem becomes one of determining

response of soil deposit to the motion of the underlying

bedrock. IIT Bombay, DC

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Used to:

Predict ground surface motions, Time histories, Response

spectra, Scalar parameters

Evaluate dynamic stresses and strains, Liquefaction

hazards, Foundation loading

Evaluate ground failure potential, Instability of earth

structures, Response of various geotechnical structures like

retaining wall, earth dam, pile, various foundations etc.

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Definitions:

Rock outcropping motion the motion that would occur

where rock outcrops at a surface.

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Definitions:

Bedrock motion – the motion that occurs at bedrock overlain by a

soil deposit. Differs from rock outcrop motion due to lack of

free surface effect.

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Definitions:

Free surface motion – the motion that occurs at the surface of a

soil deposit

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Common situations # 1

Rock outcrop motion is known – usually obtained from attenuation

relationship (based on database of rock outcrop motions)

Free surface motion is to be determined

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Common situations # 2

Free surface motion is known – usually obtained from attenuation

relationship (based on database of soil outcrop motions)

Free surface motion is to be determined for site with different soil

conditions

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13 IIT Bombay, DC

Linear Analysis

A known time history of bedrock (input) motion is

represented as a Fourier series, usually using the FFT

Each term in the Fourier series of the bedrock (input)

motion is then multiplied by the transfer function to

produce the Fourier series of the ground surface (output)

motion

The ground surface (output) motion can then be expressed

in the time domain using the inverse FFT

14 IIT Bombay, DC

Transfer Function

The transfer function determines how each frequency in the

bedrock (input) motion is amplified, or de-amplified by the soil

deposit

A transfer function may be viewed as a filter that acts upon

some input signal to produce an output signal

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Input

Transfer

function

(filter) Output

Transfer Function Evaluation Uniform Undamped Soil on Rigid Rock

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Assume harmonic base motion,

Then, response should also be harmonic

Wave traveling in

– z direction

(upward)

Wave traveling in

+ z direction

(downward)

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Transfer Function Evaluation Uniform Undamped Soil on Rigid Rock

Displacement :

Stress:

At z = 0 (ground surface)

A = B

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Transfer Function Evaluation

Uniform Undamped Soil on Rigid Rock

, 22

ikz ikz

i te e

u z t A e

, 2 cos i tu z t A kz e

0, 2 1

2 cos cos,

i t

i t

u t AeF

A kH e kHu H t

Defining a transfer function as the ratio of the displacement at

the ground surface to the bedrock displacement

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Uniform Undamped Soil on Rigid Rock

As kH = wH/Vs goes to zero, denominator goes to zero Transfer

function goes to infinity

Natural frequencies

/

2n SV n H

Fundamental period

02 4

SS

HTV

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Uniform Damped Soil on Rigid Rock

How do we handle damping?

Complex shear modulus 2 2

* * */SG V k

1/2* 2 */k G Complex Wave

Number

* * / 1S SV G V i

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*

*1

S

wk k i

V

21



Repeat analysis as before

Transfer function becomes

**

0, 2 1

cos, 2 cos

i t

i t

u t AeF

k Hu H t A k H e

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*

*

1 1,

coscos

S

F wk H wH

V

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2 22 2

1 1,

cos cos / /S S

F wkH kH wH V wH V

23



Note:

Natural frequencies still exist

Low natural frequencies strongly amplified

High natural frequencies weakly amplified

Very high frequencies de – amplified

Amplification strongly frequency - dependent

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Uniform Undamped Soil on Elastic Rock

su* *

,s si t k t i t k t

s s s su z t C e D e* *

,r ri t k t i t k t

r r r ru z t C e D e

, 0,s s r ru z H t u z t

, 0,s s r rz H t z t

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Maintaining equilibrium and compatibility of displacements

at the boundary, the amplitude of the transfer function can be

written as

2 2 2

1, 0

cos sins z s

F wk H k H

*

*s ss

zr sr

v

v

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Note:

Even with no soil damping, resonance cannot occur

Why???

Energy removed from soil layer by transmission into rock

Form of radiation damping

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Maintaining equilibrium and compatibility of displacements

at the boundary, the amplitude of the transfer function can be

written as

2 2 2

1, 0

cos sins z s

F wk H k H

*

*s ss

zr sr

v

v

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Note:

Even with no soil damping, resonance cannot occur

Why???

Energy removed from soil layer by transmission into rock

Form of radiation damping

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Layered, Damped Soil on Elastic Rock

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For layer j

From equilibrium

From compatibility

* *

, j j j jik z ik z i t

j j j ju z t A e B e e

* *

1 1j j j jik h ik h

j j j jA B A e B e

* ** *

1 1 * *

1 1

s j s jik h ik hj j

j j j j

j j

G kA B A e B e

G k

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If we know response at layer j (Aj and Bj are known), then

we have two equations with two unknowns (Aj+1 and Bj+1)

We can relate Aj+1 and Bj+1 to Aj and Bj by means of

recursive relationships

For layer j

From equilibrium

From compatibility

* *

, j j j jik z ik z i t

j j j ju z t A e B e e

* *

1 1j j j jik h ik h

j j j jA B A e B e

* ** *

1 1 * *

1 1

s j s jik h ik hj j

j j j j

j j

G kA B A e B e

G k

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Solving for the unknowns

* ** *

1

1 11 1

2 2

j j j jik h ik h

j j j j jA A e B e

* ** *

1

1 11 1

2 2

j j j jik h ik h

j j j j jB A e B e

1 1 1j jA a A1 1 1j jB b B

Or, relating the coefficients to those at the ground surface

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Then, a transfer function relating the motion in layer i to

the motion in layer j can be written as

If we know the motion at any layer, we can use this

transfer function to compute the corresponding motion at

any other layer

i i

ij

j j

a bF

a b

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Equivalent Linear Approach

The actual nonlinear hysteretic stress – strain behavior of

cyclically loaded soils can be approximated by equivalent linear

properties

Assume some initial strain and use to

estimate G and ξ

Determine peak strain and effective strain maxeff R

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Select properties based on updated strain level

Compute response with new properties and determine

resulting effective shear strain

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Repeat until computed effective strains are consistent with

assumed effective strains

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Non – linear Approach

Solve Wave equation incrementally

, ,i t t i tu uu

t t

2

2

u u

z t t

1, ,i t i t

z z

1, , 1, ,i t i t i t i tu u

z t

Approximate partial derivatives

Finite difference form

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then

, , 1, ,i t t i t i t i t

tu u

z

Velocity at time t+Δt can be calculated from velocity

and shear stress at time t

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Non-Linear Approach

Solve wave equation incrementally

Start with initial stiffness, Gmax

Compute response for small time step, Δt

Compute shear strain amplitude at end of time step

Use stress-strain model to find Gtan for next time step

Compute shear strain amplitude at end of next time step

Continue stepping through time for entire input motion

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Nonlinear response is simulated in

incrementally linear fashion

Material damping is taken care by

hysteretic response

Approach requires good model for

description of soil stress – strain

behaviour

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Non-Linear Stress-Strain Models

Two main types

Cyclic nonlinear models

Advanced constitutive models

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Non – Linear Stress – Strain Models

Cyclic nonlinear models

Requires :

• Backbone curve

• Unloading – reloading rules

• Pore pressure model

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Non – Linear Stress – Strain Models

Advanced constitutive models

Requires:

•Yield surfaces

•Hardening rule

•Failure surface

•Flow rule

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Non-Linear Stress-Strain Models

Cyclic nonlinear models Advantages:

Relatively simple

Small number of parameters

Disadvantages:

Simplistic representation of soil behavior

Cannot capture dilatancy effects

Advanced constitutive models Advantages:

Can better represent mechanics of yield, failure

Disadvantages:

Many parameters

Difficult to calibrate

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