lecture37-seismic retaining walls part2

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Seismic Response of Retaining Walls

Part-II

Lecture-37

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Dynamic Response of Retaining Walls

The dynamic response of even simplest type of retaining wall is quite complex. Wall movement and pressure depends on the response of the soil underlying the wall, the response of the backfill, the inertial and flexural response of the wall itself, and the nature of the input motions. Most of the current understanding of the dynamic response of retaining wall has come from the model test and numerical analyses. These tests and analyses, the majority of which involved gravity wall indicate that

1. Wall can move by translation and or by rotation. The relative amounts of translation and rotation depend on the design of the wall; one or the other may predominate for some wall, and both may occur for others.

2. Magnitude and distribution of dynamic wall pressure are influenced by the mode of wall movement (e.g. translation, rotation about the base, or rotation about the top).

3. Maximum soil thrust acting on the wall generally occurs when the wall has translated or rotated towards the backfill (when the inertial force on the wall is directed towards the backfill). The minimum soil thrust occurs when the wall has translated or rotated away from the backfill.

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4. The shape of the earth pressure distribution on the back of the wall changes as the wall moves. The point of application of the soil thrust therefore, moves up and down along the back of the wall. The position of the soil thrust in highest when the wall moves towards the soil and lowest when the wall moves outwards.

5. Dynamic wall pressures are influenced by the dynamic response of the wall and backfill and can increase significantly near the natural frequency of the wall-backfill system. Permanent wall displacement also increases at frequency of the wall-backfill system. Dynamic response effect can also cause deflections of different parts of the wall to be out of phase. This effect can be particularly significant for wall that penetrates into the foundation soil when the backfill soil moves out of phase with the foundation soils.

6. Increased residual pressures may remain on the wall after an episode of strong shaking has ended.

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According to Nazarian and Hadjian (1979) stated, the solution methods used for the dynamic behavior of retaining walls can be divided into four categories:

1.Fully Plastic Solutions: This kind of analytical approach assumes that there is a definite failure surface developed behind the retaining wall. The pseudo-static analysis of the wedge behind the wall gives analytical expressions about the amount and the point of application of the total dynamic thrust. The most famous method is Mononobe – Okabe method.

2. Solutions based on Elastic Wave Theory: This approach assumes that the whole walland backfill system moves in elastic range.

3. Inelastic Dynamic Solutions: These methods are based on the elasto-plastic models.The knowledge is still not enough. Solutions are complex and mostly based on FiniteElement Methods.

4. Experiments and Field Observations


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Mononobe-Okabe (M-O) Method

The most commonly adopted method for determining the dynamic lateral pressure on retaining structures was developed by Mononobe (1929) and Okabe (1926). The method was developed for dry cohesionless materials and was based on the assumption that:

1. The wall yields sufficiently to produce minimum active pressure

2. When the minimum active pressure in attained, a soil wedge behind the is at the point of incipient failure and the maximum shear strength is mobilized along the potential sliding surface.

3. The soil behind the wall behaves as a rigid body so that acceleration are uniform throughout the mass; thus the effect of the earthquake motion can be represented by the inertia forces kh W and kv W, where W is the weight of the sliding wedge. kh g and kv g are the horizontal and vertical components of the earthquake acceleration at the base of the wall

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Fig: (a) Forces acting on active edge (b) Force Polygon

f = internal angle of friction of the soil , = the inclination angle of the ground surface above the retaining structure, b = the slope angle from the horizontal minus 90°, δ = Angle of wall friction and kh and kv = horizontal and vertical pseudo-static coefficients.

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In effect, the active pressure during the earthquake PAE is computed by the Coulomb theory except that the additional forces kh W and kv W are included in the computation. Determining the critical sliding surface is the usual way and the active pressure corresponding to this surface lead to the following expression:

PAE = ½ KAE g H2 (1-kv)

22

2

coscossinsin

1coscoscos

cos

AEK

g = unit weight of soil, H = height of wall, f = internal angle of friction of the soil , = the inclination angle of the ground surface above the retaining structure, b = the slope angle from the horizontal minus 90°, δ = Angle of wall friction and kh and kv = horizontal and vertical pseudo-static coefficients.

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The inclination of the critical failure surface AE assuming active earth pressure conditions is



E

EAE C

C

2

11 )tan(tan

v

h

kk

1tan 1

)cot()tan(1cot(tantan1 EC

)cot(tantan12 EC

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Theobservations of the experimental studies by various investigators with respect to the M-O method are as below:

(1)Lateral earth pressure coefficients for the cohesionless backfill computed from the Mononobe-Okabe analysis are in reasonably good agreement with values developed from small scale model tests.

(2)In case of unanchored retaining structures, the increase in the lateral pressure due to the base excitation are greater at the top of the wall and the resultant increment acts at the height varying from 0.5H to 0.67H above the base of the wall.

(3) The increase in the lateral pressure due to dynamic effect may be accompanied by an outward movement of the wall, the amount of movement increasing with the magnitude of the base acceleration.

(4) After a retaining structures with a granular backfill materials has been subjected to a base excitation, there is a residual pressure which is substantially greater than the initial pressure before the base excitation; this residual is also a substantial portion of the maximum pressure developed during the excitation.

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Steps involved in Mononobe-Okabe analysis

1. Compute KA and PA

2. Compute y (for known kh and kv)

3. Compute KAE and PAE

4. Compute PAE = PAE - PA

5. Find out the location of action of resultant thrust

height of resultant from base

h = [(PAH/3)+ (PAE 0.6H)]/PAE

6. Find out the overturning moment Mo = PAE cos δ h

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Mononobe-Okabe (M-O) Method for passive earth pressure conditions

Fig: (a) Forces acting on passive edge (b) Force Polygon

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In effect, the passive pressure during the earthquake PPE is computed by the Coulomb theory except that the additional forces kh W and kv W are included in the computation. Determining the critical sliding surface is the usual way and the active pressure corresponding to this surface lead to the following expression:

PPE = ½ KPE g H2 (1-kv)

22

2

coscossinsin

1coscoscos

cos

PEK

g = unit weight of soil, H = height of wall, f = internal angle of friction of the soil , = the inclination angle of the ground surface above the retaining structure, b = the slope angle from the horizontal minus 90°, δ = Angle of wall friction and kh and kv = horizontal and vertical pseudo-static coefficients.


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The inclination of the critical failure surface AE assuming passive earth pressure conditions is


E

EPE C

C

4

31 )tan(tan

v

h

kk

1tan 1

)cot()tan(1cot(tantan3 EC

)cot(tantan14 EC


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Steedman-Zeng Method

The M-O method accounts for the dynamic nature of the earthquake loading in a very approximate way.

Steedman and Zeng (1990) proposed a simple pseudo-dynamic analysis of seismic earth pressures.

This method accounts for the phase difference and amplification effects within the backfill behind a retaining wall.

Fig: Wall geometry and notation for Steedman-Zeng Method

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Steedman-Zeng MethodIf the active wedge shown in figure (previous slide) is considered, acceleration at a depth z can be expressed as follows:

whereah = amplitude of harmonic horizontal input accelerationw = cyclic frequency of harmonic input motiont = timevs= velocity of vertically propagating harmonic shear waveH = height of the wallZ = depth

The mass of a thin element in the active wedge is:

whereg = unit weight of the backfill materialg = gravitational acceleration

sh v

zHtatza sin),(

dzzH

gzm

tan

)(

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Steedman-Zeng Method

The total inertial force acting on the wall can therefore be given as:

Whereand

By resolving the forces on the wedge we can obtain the total soil thrust as follows:

If we differentiate the total soil thrust we can obtain the total earth pressure distribution as follows:

The height of application of total thrust varies with time according to the following formula:

)sin(sincos2tan4

),()()( 20

tHg

adztzazmtQ h

H

h

/2 svsvH

t

)cos()sin()cos()(

)(

WtQtP h

AE

s

hAEAE v

zt

zkzztP

tP

sin)cos(

)cos(tan)cos(

)sin(tan

)()(

)sin(sincos2)cos(cossin2cos2 222

tHtHH

Hhd

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Effects of Wall Stiffness

Velestos and Younan (1997) stated that “the existing elastic solutions are limited tonondeflecting rigid walls and do not provide for the important effect of wall flexibility.”

According to their research on rigid walls which are elastically constrained against rotation at their base, both the magnitudes and distributions of the dynamic wall pressures and forces are quite sensitive to the flexibility of the base constraint.

For realistic wall flexibilities the total wall force or base shear is one-half or less that obtained for a fixed-based, rigid wall, and the corresponding reduction in the overturning base moment is even larger. With the information that has been presented, the precise dependence of these critical forces on the flexibilities of the wall and its base may be evaluated readily.

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Effects of Water

The methods explained for estimating the seismic loads on retaining walls are applicable only for retaining walls with dry backfills. Most of the retaining walls are provided with drains to prevent the building of water pressure.

However, it is not possible to stop the building of water pressure in retaining walls built in waterfront areas, where most earthquake induced failures are observed.

Water outboard of retaining wall can exert dynamic pressures on the face of the wall.

Water in the backfill can also exert pressure on the back of the wall.

Proper consideration of effect of water is essential while estimating the seismic loads on waterfront retaining walls.

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Displacement based design methods

Richard and Elms Method

This method considers the wall inertia effect and therefore suitable for gravity type retaining walls. For no wall movement the required weight of the wall is expressed as follows:

whereg = unit weight of the wall materialH = height of the wallb = the slope angle from the horizontal minus 90°δ = Angle of wall friction

Richard and Elms (1979) state that for the use of KAE considering the maximum ground acceleration leads to uneconomical design of wall. Therefore using smaller wall dimensions is more feasible where the wall is allowed to move for a small amount.

tantan

tan)sin()cos(21 2

b

bAEw KHW

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Displacement based design methodsRichard and Elms Method

For the calculation of the total relative displacement of a wall following relation is given:

whereA=maximum acceleration coefficient; V=maximum velocity; g = Acceleration due to gravity; N=maximum critical acceleration coefficient and defined as follows:

Following steps are followed in the design of gravity retaining walls by using Richards and Elms method:

1. An allowable design displacement D is chosen.2. N is calculated for known values of A and V3. KAE is calculated from Mononobe-Okabe formulation. Instead of kh, Nis used for the calculation of q .4. The required wall weight (Ww) is computed from the expression given in previous slide.

42

087.0

AN

AgV

D

4/12087.0

DAgV

AN

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Example Problem

5 mDry silty sandg = 17 kN/m3

f= 34d = 17

Compute the overturning moment about the base for the wall shown in figure for pseudo-static seismic coefficients kh = 0.15 and kv = 0.075

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Solution

The static active pressure coefficient on the wall using Coulomb’s theory is:

22

2

coscossinsin

1coscos

cos

AK

256.0

00cos017cos034sin3417sin

1017cos)0(cos

034cos2

2

2

AK

The static active thrust on the wall is PA = ½ KA g H2

kN/m4.54517256.021 2 AP

2.9075.0115.0

tan1

tan 11

v

h

kk

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Solution

The total active thrust is computed as:

362.0

00cos2.9017cos2.9034sin3417sin

12.9017cos)0(cos)2.9cos(

2.9034cos2

2

2

AEK

PAE = ½ KAE g H2K(1-kv)

kN/m15.71)075.01(517362.021 2 AEP

22

2

coscossinsin

1coscoscos

cos

AEK

The dynamic component of the total thrust is:

PAE = PAE – PA = 71.15-54.4 = 16.75 kN/m

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Solution

The static active thrust acts at a height of H/3 above the base of the wall, where as the dynamic thrust acts at 0.6H above the base of the wall. Hence the total thrust acts at a height h above the wall, where

m98.115.71

)56.0(75.1635

4.54)6.0(3

AE

AEA

P

HPH

Ph

Because only the horizontal component of the total active thrust contributes to the overturning moment about the base, the overturning moment is given by

m/mkN72.13498.117cos15.71cos)()( hPhPM AEhAEo

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Exercise Problems

1. A yielding wall sloped at 10° has to be constructed to retain a 12 m column of sand having unit weight of 18 kN/m3 and friction angle of 42 and wall friction of 18. Compute dynamic thrust on the wall and overturning moment about the base of the wall under seismic loading, for pseudo-static seismic coefficients kh = 0.25 and kv = 0.125 .

2. The wall shown in figure is subjected to a peak horizontal acceleration of 0.28 g. Assuming a pseudo-static coefficient kh = 0.38 amax/g, compute the total active thrust on the wall, height of the resultant active thrust and total overturning moment about the base of the wall.

4 m

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Dry Sandg = 18 kN/m3

f= 38d = 20

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Kramer, S.L. (1996) Geotechnical Earthquake Engineering, Prentice Hall.

Day, R.W. (2001) Geotechnical Earthquake Engineering Handbook, McGraw-Hill.

Richards, R. and Elms, D.G. (1979) Seismic behavior of gravity retaining walls,

Journal of the Geotechnical Engineering Division, 105 (GT4) 449–464.

Zeng, X. and Steedman, R.S. (2000) Rotating block method for seismic

displacement of gravity walls, Journal of Geotechnical and Geoenvironmental

Engineering, 126 (8), 709–717.

Newmark, N.M. (1965) Effects of earthquake on dams and embankments,

Geotechnique, 15, 139–160.

NCHRP report on Seismic Analysis and Design of Retaining Walls, Buried

structures, Slopes, and Embankments: http://

onlinepubs.trb.org/onlinepubs/nchrp/nchrp_rpt_611.pdf

(accessed on 14 April 2012)

References

http://onlinepubs.trb.org/onlinepubs/nchrp/nchrp_rpt_611.pdf

http://onlinepubs.trb.org/onlinepubs/nchrp/nchrp_rpt_611.pdf

lecture37-seismic retaining walls part2

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