Disaster Mitigation Geotechnology

3, 4

Analytical methods for evaluating

earthquake-induced damages

How to guarantee the performance? (Slide from last week)

- Prediction of movements

• Simple pseudo-static (準静的) analysis

Uses seismic coefficient, kh, to express seismic motions

• Uncoupled dynamic analysis (what does ‘coupling’ mean?)

Newmark’s method

• Full dynamic analysis

Finite element method, finite difference method, etc.

- Evaluation of liquefaction potential

Pseudo-Static Analysis Method (準静的解析法)

Replace dynamic loads with static loads

Calculation is then same as static analysis

Adopted mainly for simple structures under L1 actions.

The conversion method differs between design codes:

Some consider structure geometry, frequency characteristics,

duration etc. (e.g. Port and Harbour Design Code)

Input acceleration Seismic coefficient kh (震度係数)

0 50 100 150 200 250 300-600

-400

-200

0

200

400

600

0 50 100 150 200 250 300-150

-100

-50

0

50

100

150

0 50 100 150 200 250 300-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

amax

=394 Gal

Accele

ration [

cm

/se

c2 ]

Time [Sec]

Velo

city

[cm

/se

c]

Honmoku L2 NS

Dis

plac

em

ent

[cm

]

Time [Sec]

khW

W

P(kh) u(kh)

Example of kh

Expression in terms of Peak Ground Acceleration (PGA)

g

akh

max

3

1

max

3

1

g

akh

ga 2.0max

ga 2.0max

Noda (1975)

maxa : PGA

Case studies of 129 gravity-type

quay walls in 12 earthquakes

Proved valid for anchored

sheet-pile walls (110 case

histories: Kitajima & Uwabe,

1979)

Analysis: Example of retaining structures against sliding

Pseudo-static limit equilibrium

Pa

d Rh

Rv

d

W

khW

psw+pdw

- Body force on wall

- Dynamic earth pressure

Mononobe-Okabe Method

(for cohesionless soils)

Treatment of pore water

-Dynamic water pressure, pdw

Westergaard equation

Note!

Earth pressure is considered not to be affected by the wall’s

behaviour. In other words, soil-structure interaction is not

considered. Its influence shall be discussed later.

Mononobe-Okabe Method (物部・岡部の地震時土圧)

Pa

d

q

i

a f

R

W

a = tan-1kh

H H’

2

2

22

)sin()sin(

)sin()sin(1)sin(sin

)(sin

2

1

i

iHP ta

qdq

ffddqq

fq

2

2

22

)sin()sin(

)sin()sin(1)sin(sincos

)(sin

2

1

i

iHP ta

qadq

affdadqqa

afq

Coulomb active earth

pressure (i.e. a=0)

M-O active earth

pressure (i.e. a≠0)

Mononobe-Okabe Method

Simple if soil is dry.

What is it is saturated with water? How to treat pore pressure?

Use ‘Apparent’ seismic coefficient, kh’

Basic idea:

• Calculate vertical stress in terms of effective stress

(i.e. subtract water pressure from overburden pressure)

• Increase kh by multiplying sat / ’ (i.e. kh’ = kh sat / ’ )

It is assumed that pore water moves with soil

vv

vhahh kK )(uvv

ukK vhah )(

Dry soil Saturated soil

Westergaard Equation (Westergaard, 1933)

Closed-form equation for dynamic water pressure

Hygkp whdw 8

7 2

12

7gHkp whdw

pdw

H

(Static)

psw psw+pdw psw+pdw

Total water pressure

Depth

Acc. + -

Equation for performance evaluation against sliding

For L1 earthquake, the pseudo-static analysis may be used.

dwddddd FdwHaBVdd PPPPPPWf

df

dd BVd PPW

dwddd FdwH PPPP

a

L.H.S.: Resisting forces

: Friction coefficient

: Wall weight + vertical E.P. – Buoyant force

: Structure analysis factor = 1

: Horizontal E.P. + Static W.P

+ Dynamic W.P. + Inertia force on wall

R.H.S.: Driving forces

Similar analysis required for bearing capacity, overall stability, etc.

What about factor of safety (安全率)?

Recent design codes: based on reliability analysis (信頼性解析)

‘Level 1’ Reliability design

Factor each design parameter by to take account of

variability

e.g.

Friction coefficient, fd: to be multiplied by = 0.55

Earth forces: to be multiplied by = 1.15

The factors are determined by existing data and statistical

analysis to satisfy preset degree of reliability.

This is in contrast to conventional ‘overall’ factor of safety.

Rationally take account of variability of individual parameters.

Toward the Reliability Based Design

From conventional design method

to a new design method

Slope stability

Circle arc slip analysis based on Modified

Fellenius Method

W

x

a

b

Fs for clay slope

xW

cRF

u

s

Here, R is a radius of circle slip, cu is undrained shear strength, is

base length of a slice, W is weight of a slice, x is arm length

between a slice and the circle center.

FS > 1.30 is required R

W

l

x

a

b

R

W

l

x

a

b

In reliability based design, partial factor is

introduced for each parameter as follows:

Here, subscript k means characteristic value

xWcR F Wkcuuk s

Subscript k and d mean characteristic value

and design value respectively.

cuukud cc

Wkd WW

xWcR F dud s

Using the design value, it becomes xWcR WkFcuuk s

xW

cRF

u

s

xWcR F dud s

Reliability based design with partial safety factors

Conventional design (based on overall safety

factor)

Using design value with subscript d

uukud ccc WWW kd

• Replace dynamic loads with equivalent static loads.

• Consider individual phenomena (earth pressure, water

pressure, wall vibration, etc.) separately. This ignores soil-

structure interactions.

• Mostly based on classical theories. Applicable only to

relatively simple problems.

• Less subjective and simple to perform: different engineers

would come up with a same answer.

• No scope for deformation prediction. Mostly used for L1

earthquakes to check structural integrity.

• Evolving to reliability-based design from deterministic design.

Pseudo-Static Analysis Method: Summary

Newmark’s Method (ニューマーク法)

• Extension of limit equilibrium to calculate displacement

• Considers acceleration time-series, not kh.

PIANC (2001)

Acceleration

Violation of threshold

means failure in static

limit equilibrium.

If violation lasts for

finite duration, it

results in finite

displacement

This method can be applied to

slope stability problems as well as

retaining structures.

Acceleration

PIANC (2001)

The ‘threshold (閾値)’

acceleration is derived from

static equilibrium

Relative movement of mass

stops when its velocity becomes

same as that for base again.

Application to retaining structures

• More advanced modelling has been attempted recently

- Consideration of strain-softening in backfill

- Deformation of foundation layers

• Difficulty

Threshold setting

affects the outcomes

significantly.

• This method, though

more realistic than

pseudo-static method,

still rules out the soil-

structure interactions.

Richards & Elms, 1979

Dynamic Analysis (動的解析)

Full dynamic analysis usually requires computer.

Finite element method (有限要素法) is most frequently used.

Sea

Sand

Cement-treated

Quay wall Boulder fill

Dynamic Finite Element Analysis

• Formulation

(Soil-water) Coupled analysis (土水連成解析):

- Describe soil’s stress-strain relationships by effective stress

- Pore water pressure changes are computed

Non-coupled analysis

- Total stress approach

Small-strain formulation and large-strain formulation

• Constitutive model (構成モデル)

Model to describe soil’s stress-strain relationships

A variety of models exist;

The simplest: Isotropic elasticity (2 parameters: E & n)

Complex: Material parameters as many as 15

To be determined based on field/laboratory tests

Standard flow of dynamic FEA

Model the domain - Determine domain size

- 2-D or 3-D?

- Boundary conditions?

- Generate appropriate mesh

Assign constitutive models

and parameters to materials

Input earthquake motion Normally give at domain base

Time-marching computation Finite different scheme

Field/Lab tests

Geological /

geotechnical information - Ground conditions

- Construction sequence

Input

Input

Stress, strain,

pore water pressure,

displacement, etc.

at any given point

Output

Importance of constitutive model

• Many factors affect outcomes of FEA.

• Constitutive modelling is one of the most influential.

• There as many constitutive models as number of researchers;

• You need to be clear of what feature you exactly need.

For dynamic analysis, cyclic behaviour of soil needs to be simulated.

Consider cyclic loading:

Time

Cyclic stress-strain relationship of soil

Non-linearity + hysteresis

(Iwasaki et al., 1978)

Definition of

damping ratio

Hysteresis

Cyclic stress-strain relationship in simplest models

Real behaviour

Irrecoverable

deformation for

relatively small

strains

Linear elasticity

No damage.

Deformation

recoverable

Morh-Coulomb

with linear elasticity

No damage unless

ultimate strength is

reached

f tanc

Some simple models to describe cyclic stress-strain behaviour

1. Skeleton curve (骨格曲線)

• Hyperbolic model (双曲線モデル)

(Kondner, 1963; Hardine & Drnevich, 1973)

• Ramberg-Osgood model (Jennings, 1964)

r

y y y

a

max max

1

G

maxG

max

Some simple models to describe cyclic stress-strain behaviour

2. Masing Rule (メーシング則)

A

O

B

Skeleton

curve

For loading reversal at A,

expand OA by factor of 2,

and rotate 180o to create AB.

Then it joins the skeleton

curve smoothly.

If reloading occurs at C, it

heads back to the most

recent reversal point (i.e. A)

Practical problems:

• What about irregular cycles with reducing amplitude?

• This scheme tends to predict too large a damping ratio

(i.e. the loop becomes too fat): Critical in assessing

vibration (hence strain) amplitude

Some countermeasures proposed for both problems.

C

Volumetric behaviour

Shear stress () – shear strain () relationship is not the only cyclic

feature!

Under drained conditions, volumetric strains accumulates

against cyclic shear.

Shahnazari & Towhata (2002)

Volumetric behaviour

Under undrained conditions, this may well lead to strength

reduction. The extreme is liquefaction.

(Ishihara, 1985; reproduced after Iai et al., 1991)

Relationship between volumetric strain and effective stress changes

Volumetric changes under drained conditions

Pore water pressure changes under undrained conditions

e

p

Drained conditions:

De due to plastic straining

Undrained conditions:

Because of plastic straining due to the cyclic loading, e

wants to decrease, but it cannot (De must be zero). So

p’ is forced to decrease instead.

Normal

Compression

Line

Volumetric behaviour: Can Cam Clay Model be useful?

Cam Clay Model

Imagine what response you get from the

Cam Clay Model against cyclic loading.

And what about the simple models

mentioned so far?

q

p

Stress

path

Plastic strain

increment

q

q

Critical State

p

Elastic

Hardening

0p

Volumetric behaviour

To simulate volumetric changes due to shear deformation,

flow rule (流れ則) is required.

0.00 0.01 0.02 0.03-0.02

-0.01

0.00

0.01

0.02

Volumetric strain, v

0.00 0.01 0.02 0.03-0.02

-0.01

0.00

0.01

0.02Simple shear

p0' = 98 [kPa]

Dr = 57 [%]

Shea

r str

ain

,

(Nishimura and Towhata, 2004)

pAssociated flow rule

p

Non-associated flow rule

Example:

Constitutive modelling - Summary

• Most of the simple models you learn at school are for monotonic

loading; not applicable to cyclic loading

• Cyclic features include:

- Hysteresis and damping

- Accumulation of volumetric changes (drained conditions) or pore

water pressure changes (undrained conditions).

• Mode advanced modelling is necessary. But this is a job for real

experts!

Example: Simulation of Quay Walls at Kobe

Code FLIP (Iai et al., 1998)

Finite Element Analysis

Towhata-Iai Model

Code FLAC (Commercial software: Finite Difference Analysis)

Pastor et al.’s model. Analysed by Dakoulas & Gazetas (2008)

Example: Simulation of Quay Walls at Kobe

Two different codes and models Different results

Horizontal

Disp. [m] Tilting [o]

FLIP 3.5 4

Dakoulas &

Gazetas, 2008 4.5 1

Parametric study

FEM computes all the

problems (sliding, tilting,

settlement, etc.) at once.

No need for evaluation

of individual mechanisms.

Also useful in exploring

influence of material

properties, ground

conditions, etc.

(Iai, et al., 1998)

Hor. Disp. = 1.6m

Hor. Disp. = 2.1m

Hor. Disp. = 2.5m

Another example: Railway embankment

A similar idea applies:

For L1 Earthquake:

Pseudo-static analysis with seismic coefficients

Check stability

For L2 Earthquake:

Newmark’s method

Compute displacement

(More exactly, rotation;

formulation in terms of

moment equilibrium)

W

khW

Dynamic finite element analysis of slopes

Example:

Model behaviour Computed FE mesh

deformation

(Wakai & Ugai, 2004)

Comparison of analytical methods

Pseudo-static

limit equilibrium

Newmark-type

method Dynamic FEM

Damage

mechanism

Assumed

individually

Assumed

individually Output as result

Soil behaviour Perfect

plasticity Perfect plasticity

Elasticity to elasto-

plasticity, refined

as much as

needed

Output Safe or not Rigid-body

movements

Stresses, strains,

displacements,

forces, PWP, etc.

Computation

ease and

performer-

dependency

Done by hand.

Most people

would output

same results

Simple program

or spreadsheet

computation.

Result depends

much on

condition setting

Complex program

needed (normally

blackbox). Outputs

depend on

modelling /

parameter setting