seismic analysis of some geotechnical problems – pseudo-dynamic approach seismic analysis of some...

Seismic Analysis of Seismic Analysis of

Some Some

Geotechnical Problems Geotechnical Problems

– Pseudo-dynamic – Pseudo-dynamic

Approach Approach

Dr. Priyanka Ghosh

Assistant Professor

Dept. of Civil Engineering

Indian Institute of Technology, Kanpur

INDIA

Organisation

Introduction to Pseudo-dynamic Approach and Upper Bound

Limit Analysis

Seismic Bearing Capacity of Strip Footing using Upper Bound

Limit Analysis

Seismic Vertical Uplift Capacity of Horizontal Strip Anchors

using Upper Bound Limit Analysis

Seismic Active Earth Pressure Behind Non-vertical Retaining

Wall using Limit Equilibrium Method

Seismic Active Earth Pressure on Walls with Bilinear Backface

using Limit Equilibrium Method

Seismic Passive Earth Pressure Behind Non-vertical Retaining

Wall using Limit Equilibrium Method

Conclusions

Introduction to

Pseudo-dynamic Approach

and Upper Bound Limit

Analysis

Pseudo-dynamic ApproachPseudo-static Approach Pseudo-dynamic Approach

The dynamic loading induced by earthquake is considered as time independent, which ultimately assumes that the magnitude and phase of acceleration is uniform throughout the soil mass

The time and phase difference due to finite primary and shear wave velocity can be considered

Generally does not consider the amplification of vibration which takes place towards the ground surface

Considers the amplification of excitation

For a Sinusoidal Base Shaking, the Acceleration at any Depth z below the Ground Surface and Time t

Mass of the Shaded Element m(z) and Total Weight of the Failure Wedge W

Total Horizontal Seismic Inertia Force Qh(t)

Where, wavelength of the shear wave = TVs

Total Vertical Seismic Inertia Force Qv(t)

Where, wavelength of the primary wave = TVp

Upper Bound Limit Analysis

Theorem: If a compatible mechanism of plastic deformation

, , is assumed, which satisfies the condition = 0 on the

displacement boundary Su; then the loads Ti, Fi determined by

equating the rate at which the external forces do work to the

rate of internal dissipation of energy will be either higher or

equal to the actual limit load.

*p

ij *p

iv *p

iv

Equation

V S V

*p

ii

*p

ii

*p

ij

*p

ij dVvFdSvTdV

*p

ij

*p

iv

*p

ij

= displacement rate

= plastic strain rate compatible with

displacement rate

= stress tensor associated with plastic strain

rateTi = external force on the surface S

Fi = body forces in a body of volume V

Seismic Bearing Capacity of

Strip Footing using Upper

Bound Limit Analysis

Acta Geotechnica (Springer Pub.), 2008, Vol. 3, No. 2, pp 115-123.

Footing

Pu

Pu

B

C

D

Qh1

Qv1

W1

U1

U21

U2

Qh2

Qv2

W2z

z

dzdz

z

Vs, Vp

ah = hg

ah = fahg

(a)

A

b

U2

U21

U1

(b) Collapse mechanism and velocity

hodograph Acta Geotechnica (Springer Pub.), 2008, Vol. 3, No. 2, pp 115-123.

Variation of NE with h and v for different values of with H/ = 0.3,

H/ = 0.16 for (a) fa = 1.0, (b) fa = 1.2


Effect of soil amplification on NE for different values of h with

= 30o, v = 0.5h, H/ = 0.3, H/ = 0.16

0

10

20

30

40

0 0.1 0.2 0.3 0.4

fa = 1.0

fa = 1.2

fa = 1.4

fa = 1.6

fa = 1.8

fa = 2.0

NE

h

fa = 1.0

fa = 1.2

fa = 1.4

fa = 1.6

fa = 1.8

fa = 2.0


Comparison of NE with fa = 1.0, v = 0.0, H/ = 0.3 and

H/ = 0.16 for (a) = 30o, (b) = 40o


Seismic Vertical Uplift Capacity

of Horizontal Strip Anchors

using Upper Bound Limit

Analysis

Computers and Geotechnics (Elsevier Pub.), 2009, Vol. 36, No. 1-2, pp 342-351.

Failure mechanism and associated forces


Variation of fE with h for different

values of fa, and v with = 20o,

H/ = 0.3 and H/ = 0.16


2b

Pf u

E

b

H

5

6

7

8

9

10

0.0 0.1 0.2 0.3 0.4

Fig. 5. Effect of soil amplification on fE for different values of h with = 30o, v = 0.5h, = 3.0, H/ = 0.3 and H/ = 0.16.

fa = 1.0 (upper most) 1.2 1.4 1.6 1.8 2.0 (lower most)

fE

h

Effect of soil amplification on fE for different values of h with

= 30o, v = 0.5h, = 3.0, H/ = 0.3 and H/ = 0.16


5

6

7

8

9

10

0 0.1 0.2 0.3 0.4

v/h = 0.00 (upper most) 0.25 0.50 0.75 1.00 (lower most)

fE

h

Fig. 6. Effect of v on fE for different values of h with = 30o, fa = 1.4, = 3.0, H/ = 0.3 and H/ = 0.16.Effect of v on fE for different values of h with = 30o, fa = 1.4,

= 3.0, H/ = 0.3 and H/ = 0.16


Geometry of the failure patterns for

different values of with fa = 1.4, = 3.0,

v = 0.5h, H/ = 0.3 and H/= 0.16


h

fE

Present analysis Kumar (2001) Choudhury & Subba Rao (2004)

30o

0.0 1.577 1.577 1.071

0.1 1.571 1.566 1.057

0.2 1.553 1.544 1.028

0.3 1.520 1.499 0.986

40o

0.0 1.839 1.839 1.543

0.1 1.835 1.832 1.457

0.2 1.821 1.815 1.386

0.3 1.798 1.786 1.286

50o

0.0 2.192 2.192 1.986

0.1 2.189 2.187 1.828

0.2 2.179 2.174 1.657

0.3 2.163 2.155 1.514

Comparison of fE for fa = 1.0, v = 0.0, = 3.0, H/= 0.3 and H/ = 0.16


Seismic Active Earth Pressure

Behind Non-vertical Retaining

Wall using Limit Equilibrium

Method

Canadian Geotechnical Journal, 2008, Vol. 45, No. 1, pp 117-123.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.1 0.2 0.3 0.4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.1 0.2 0.3 0.4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0 0.1 0.2 0.3 0.4

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.1 0.2 0.3 0.4

h h h

Kae Kae Kae

= 10o

5o

0o

-5o

-10o (lower most)

= 10o

5o

0o

-5o

-10o (lower most)

= 10o

5o

0o

-5o

-10o (lower most)

= 0.0 = 0.5 =

Fig. 2. Variation of active pressure coefficient Kae with h for = 30o, v = 0.5h, H/ = 0.3 and H/ = 0.16 (a) fa = 1.0, (b) fa = 1.4.

h hh

Kae KaeKae

= 10o

5o

0o

-5o

-10o (lower most)

= 10o

5o

0o

-5o

-10o (lower most)

= 10o

5o

0o

-5o

-10o (lower most)

= 0.0 = 0.5 =

(a)

(b)

Variation of Kae with h for= 30o, v = 0.5h, H/ = 0.3 and H/= 0.16 (a) fa = 1.0, (b) fa = 1.4 Canadian Geotechnical Journal, 2008, Vol. 45, No. 1, pp 117-123.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

fa=1.0

fa=1.2

fa=1.4

fa=1.6

fa = 1.8

pae/H

z/H

Fig. 3. Normalized seismic active earth pressure distribution for different values of fa ( = 30o, = 0.5, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16).

fa = 1.0

fa = 1.4

fa = 1.2

fa = 1.6

fa = 1.8

Normalized seismic active earth pressure distribution for different values of fa

( = 30o, = 0.5, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16) Canadian Geotechnical Journal, 2008, Vol. 45, No. 1, pp 117-123.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

phi=20

phi=30

phi=40

phi=50

= 20o

pae/H

z/H

Fig. 4. Normalized seismic active earth pressure distribution for different values of ( = 0.5, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16, fa = 1.4).

= 30o

= 40o

= 50o

Normalized seismic active earth pressure distribution for different values of

( = 0.5, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16, fa = 1.4) Canadian Geotechnical Journal, 2008, Vol. 45, No. 1, pp 117-123.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

theta=-15

theta=-10

theta=-5

theta=0

theta=5

theta=10

theta=15

= 0o

pae/H

z/H

Fig. 5. Normalized seismic active earth pressure distribution for different values of ( = 30o, = 0.5, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16, fa = 1.4)

= 5o

= 10o

= 15o

= -15o

= -10o

= -5o

Normalized seismic active earth pressure distribution for different values of

( = 30o, = 0.5, h = 0.2, v = 0.5h, H/= 0.3, H/= 0.16, fa = 1.4) Canadian Geotechnical Journal, 2008, Vol. 45, No. 1, pp 117-123.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

phi=20, del=0

phi=20, del=0.5phi

phi=20, del=phi

phi=30, del=0

phi=30, del=0.5phi

phi=30, del=phi

= 0

pae/H

z/H

Fig. 6. Normalized seismic active earth pressure distribution for different values of and ( = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16, fa = 1.4)

= 20o

= 30o

= 0.5

=

Normalized seismic active earth pressure distribution for different values of and

( = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16, fa = 1.4)


Geometry of the failure patterns

for different values of h with fa

= 1.4, = 10o, = 0.5, v = 0.5h,

H/ = 0.3 and H/ = 0.16


Comparison of Kae for

v = 0.5h, H/ =

0.3, H/ = 0.16 and fa

= 1.0


Seismic Active Earth Pressure

on Walls with Bilinear

Backface using Limit

Equilibrium Method

Computers and Geotechnics (Elsevier Pub.), (In press).

22 2

2 aeae

P tK

H

11 2

1

2 aeae

P tK

H



Variation of active pressure coefficients Kae1 and Kae2 with h for = 30˚, H1/H = 1/3, v = 0.5h, fa

= 1.4, H/TVs = 0.3 and H/TVp = 0.16: (a) θ2 =100˚ (b) θ1 = 75˚

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

h

Kae

1

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

h

Kae

1

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2

h

Kae

1

θ1 = 90 (upper most) 75 60 45

δ1 = δ2 = 0.5

θ1 = 90 (upper most) 75 60 45

δ1 = δ2 =

θ1 = 90 (upper most) 75 60 45

δ1 = δ2 = 0

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0

h

Kae

2

0.0 0.1 0.2 0.3 0.40.2

0.4

0.6

0.8

1.0

1.2

1.4

h

Kae

2

0.0 0.1 0.2 0.3 0.40.0

0.4

0.8

1.2

1.6

2.0

2.4

h

Kae

2

θ2 = 120 (upper most) 110 100 90

δ1 = δ2 = 0 δ1 = δ2 = 0.5

θ2 = 120 (upper most) 110 100 90

δ1 = δ2 =

θ2 = 120 (upper most) 110 100 90


Variation of Kae1 and Kae2 for different

combinations of 1 and 2 with = 30˚,

1 = 2 = 0.5, H1/H = 1/3, v = 0.5h, fa = 1.4,

H/TVs = 0.3 and H/TVp = 0.16

(a) Kae1 (b) Kae2



Normalized pae distribution for different fa (= 30˚, 1 = 2 = 0.5, θ1 = 75˚, θ2

= 100˚, H1/H =1/3, h = 0.2, v = 0.5h, H/TVs = 0.3 and H/TVp = 0.16)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pae

/H

z/H

fa = 1.0

fa = 1.2

fa = 1.4

fa = 1.6

fa = 1.8


Normalized pae distribution for different (1 = 2 = 0.5, θ1 = 75˚, θ2 = 100˚, H1/H

=1/3, fa = 1.4, h = 0.2, v = 0.5h, H/TVs = 0.3 and H/TVp = 0.16)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pae

/H

z/H

= 20 = 30 = 40 = 50


Normalized pae distribution for different θ1 and θ2 ( = 30o, 1 = 2 = 0.5, H1/H =1/3,

fa = 1.4, h = 0.2, v = 0.5ah, H/TVs = 0.3 and H/TVp = 0.16)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pae

/H

z/H

1 = 60 ,

2 = 90

1 = 70 ,

2 = 100

1 = 80 ,

2 = 110

1 = 90 ,

2 = 120


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

pae

/H

z/H

1 =

2 = 0

1 =

2 = 0.5

1 =

2 =

= 20° = 30°

Normalized pae distribution for different wall friction and (θ1 = 75˚, θ2 = 100˚,

H1/H =1/3, fa = 1.4, h = 0.2, v = 0.5h, H/TVs = 0.3 and H/TVp = 0.16)


h

Present analysis

Greco [8] H/TVs = 0.3 H/TVs = 0.4 H/TVs = 0.5

H/TVp = 0.16 H/TVp = 0.21 H/TVp = 0.27

Kae1 Kae2 Kae1 Kae2 Kae1 Kae2 Kae1 Kae2

0.0 0.147 0.260 0.147 0.260 0.147 0.260 0.147 0.260

0.1 0.201 0.318 0.199 0.312 0.196 0.305 0.204 0.307

0.2 0.266 0.385 0.262 0.371 0.256 0.354 0.273 0.355

0.3 0.344 0.462 0.337 0.437 0.328 0.409 0.353 0.403

0.4 0.435 0.551 0.426 0.514 0.416 0.471 0.447 0.453

0.5 0.544 0.655 0.535 0.601 0.524 0.538 0.556 0.504

Comparison of Kae1 and Kae2 for H1/H = 1/2, = 36˚, 1 = 2 = 18˚,

θ1 = 75˚, θ2 = 105˚, v = 0.5h and fa =1.0

Seismic Passive Earth Pressure

Behind Non-vertical Retaining

Wall using Limit Equilibrium

Method

Geotechnical and Geological Engg. Journal (Springer Pub.), 2007, Vol. 25, No. 6, pp 693-703.

Variation of passive pressure coefficient Kpe with h for = 30o,

v = 0.5h, H/ = 0.3 and H/ = 0.16

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.1 0.2 0.3 0.4

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0.0 0.1 0.2 0.3 0.4

0.0

5.0

10.0

15.0

20.0

25.0

0.0 0.1 0.2 0.3 0.4h h h

Kpe Kpe Kpe

= -10o

-5o

0o

5o

10o (lower most)

= -10o

-5o

0o

5o

10o (lower most)

= -10o

-5o

0o

5o

10o (lower most)

= 0.0 = 0.5

=


Normalized ppe distribution for different values of

( = 0.5, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

phi=20

phi=30

phi=40

phi=50

= 20o

ppe/H

z/H

= 30o

= 40o

= 50o


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

theta=-15

theta=-10

theta=-5

theta=0

theta=5

theta=10

theta=15

= 0o

ppe/H

z/H

= 5o

= 10o

= 15o

= -5o

= -10o

= -15o


( = 30o, = 0.5, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16)


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0

del=0

del=0.25phi

del=0.5phi

del=0.75phi

del=phi

= 0

ppe/H

z/H

= 0.25

=

= 0.5

= 0.75


( = 30o, = 10o, h = 0.2, v = 0.5h, H/ = 0.3, H/ = 0.16)


Comparison of Kpe for = 0.5 = 0o, v = 0.0, H/ = 0.3 and H/ = 0.16

h Kpe

Present analysis

Chang (1981)

Soubra (2000) Lancellotta (2007)

25o

0.0 3.55 3.45 3.43 3.10

0.1 3.26 2.89 3.15 2.86

0.2 2.96 2.74 2.85 2.62

0.3 2.63 2.38 2.50 2.26

30o

0.0 4.98 4.64 4.69 4.29

0.1 4.60 4.29 4.35 3.93

0.2 4.21 3.93 3.99 3.57

0.3 3.80 3.45 3.59 3.21

35o

0.0 7.36 6.67 6.67 5.71

0.1 6.84 6.19 6.24 5.48

0.2 6.31 5.71 5.78 5.00

0.3 5.76 5.24 5.29 4.52

40o

0.0 11.77 10.00 9.99 8.33

0.1 11.00 9.29 9.40 7.86

0.2 10.21 8.57 8.79 7.26

0.3 9.41 8.10 8.15 6.67


h Kpe

Present analysis

Mononobe-Okabe method

Caquot and Kerisel (1948)

Zhu and Qian (2000)

20o

0

0.0 1.84 1.84 1.74 1.83

0.1 1.66 1.64 - -

0.2 1.45 1.40 - -

0.5

0.0 2.27 2.27 - 2.26

0.1 2.00 1.96 - -

0.2 1.70 1.62 - -

0.0 2.86 2.86 2.57 2.66

0.1 2.47 2.42 - -

0.2 2.04 1.93 - -

30o

0

0.0 2.54 2.54 2.33 2.51

0.2 2.07 2.02 - -

0.4 1.53 1.35 - -

0.5

0.0 3.80 3.80 - 3.73

0.2 2.96 2.85 - -

0.4 2.00 1.70 - -

0.0 6.45 6.45 4.98 5.20

0.2 4.79 4.58 - -

0.4 2.96 2.43 - -

Comparison of Kpe for = 0.5 = 10o, v = 0.5h, H/ = 0.3 and H/ = 0.16


Conclusions

The magnitude of NE decreases with increase in soil

amplification, shear and primary wave velocities,

which can not be predicted by the existing pseudo-

static approach

In the upper-bound solution, for higher values of , a

significant increase in NE was observed at lower

value of h

Strip Footing


The values of fE were found to decrease extensively

with increase in both h and v, and soil amplification

In presence of horizontal and vertical earthquake

acceleration, the present values were found to be the

highest

In presence of amplification of vibration, no significant

difference between present values and the existing

pseudo-static values was found except for higher

values of embedment ratio and h

Horizontal Strip Anchor


In presence of , the active earth pressure first

decreases with increase in up to z/H = 0.3 and then

increases significantly at higher depth with increase

in for a particular value of

The seismic active earth pressure distribution was

found to be non-linear behind the wall in pseudo-

dynamic analysis

The non-linearity of active earth pressure

distribution increases with the increase in

seismicity, which causes the point of application of

total active thrust to be shifted

Active Pressure on Cantilever Wall


It was found that the magnitude of seismic active earth

pressures for upper and lower parts of the wall increases with

an increase in the horizontal earthquake acceleration coefficient

h and the wall inclinations θ1 and θ2, respectively

Unlike the pseudo-static analysis, the seismic active earth

pressure distribution was found to be nonlinear behind the wall

in pseudo-dynamic analysis and the nonlinearity of seismic

active earth pressure distribution increases with an increase in

seismicity, which causes the point of application of the total

active thrust to be shifted

Wall with Bilinear Backface


It was found that the magnitude of seismic passive earth pressure decreases with the increase in the values of wall inclination , horizontal and vertical earthquake acceleration coefficients

In presence of , the passive earth pressure increases with the increase in for a particular value of

The present analysis adopted the Coulomb failure mechanism, which generally overestimates the passive pressure coefficient Kpe in case of a rough retaining wall and

the error generated by the Coulomb theory increases as the wall inclination increases in the inward direction

Passive Pressure on Cantilever Wall


"The concern for man and his destiny must be the chief interest of all "The concern for man and his destiny must be the chief interest of all technical efforts. Never forget this among your equations and diagrams“technical efforts. Never forget this among your equations and diagrams“

-Albert Einstein.-Albert Einstein.

seismic analysis of some geotechnical problems – pseudo-dynamic approach seismic analysis of some...

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