visone-santucci

7/30/2019 Visone-Santucci

http://slidepdf.com/reader/full/visone-santucci 1/14

1 INTRODUCTION.

Embedded walls are relatively thin walls of steel,

reinforced concrete or timber, supported by anchor-ages, struts and/or passive earth pressure (EC7-1,2002). The bending capacity of such walls plays asignificant role in the support of the retained materialwhile the role of the self-weight of the wall is insig-nificant. Examples of such walls include: cantileversteel sheet pile walls, anchored or strutted steel orconcrete sheet pile walls, diaphragm walls, etc.

They are used throughout seismically active areasand frequently represent key elements of ports andharbors, transportation systems, lifelines, and otherconstruction facilities. Earthquakes have caused per-manent deformation of retaining structures in many

historical earthquakes. In some cases, these deforma-tions were negligibly small; in others they caused sig-nificant damage. In some cases, retaining structureshave collapsed during earthquakes, with disastrousphysical and economic consequences.

In this note, after a brief review of the main seis-mic earth pressures theories, the application of thepseudostatic approach to the analysis of embeddedretaining walls, taking into account the indications of the European seismic codes, is summarized. Finally,some critical considerations on certain assumptionssuggested by the EC8 Part 5 (2003) are done and the

improvements contained in the latest Italian BuildingCodes are recalled.

2 MAIN SEISMIC EARTH PRESSURESTHEORIES.

The seismic behaviour of retaining walls dependson the total lateral earth pressures that develop dur-ing earthquake shaking. These total pressures in-clude the static gravitational pressures that exists be-fore an earthquake occurs, and transient dynamicpressures induced by the earthquake. The responseof a wall is influenced by both. Here, assuming thatthe static theories of Rankine (1857), Coulomb(1776), Caquot & Kerisel (1948), Sokolowskii

(1965), Chen & Liu (1990) and Lancellotta (2002)are known for sake of brevity, a review of dynamicearth pressures theories is done.

In the literature, different notation was used forthe definition of the problem geometry and thestrength parameters of the backfill. In order to avoidconfusion on the symbols, in this chapter are signed:

γ – unit weight of the soilφ' - friction angle of the soilc – cohesion of the soilΨ – dilation angle of the soil

A review of design methods for retaining structures under seismicloadings

C. Visone & F. Santucci de MagistrisStructural and Geotechnical Dynamic Lab StreGa, University of Molise, Termoli (CB), Italy

ABSTRACT: The earth retaining structures frequently represent key elements of ports and harbors, transpor-tation systems, lifelines and other constructed facilities. Earthquakes might cause permanent deformations of retaining structures and even failures. In some cases, these deformations originated significant damages withdisastrous physical and economic consequences. For gravity walls, the dynamic earth pressures acting on thewall can be evaluated by using the Mononobe-Okabe method, while Newmark rigid sliding block scheme issuitable to predict the displacements after the shaking, as demonstrated by several experimental tests. Instead,

this simplified approach is not very useful for embedded retaining walls for various reasons. Many researchersare interested to this topic. Advanced numerical analyses, centrifuge modeling, in-situ monitoring of full-scalemodel are the main developing research activities on this subject. Here, after a brief review on the fundamen-tal seismic earth pressures theories, the application of the pseudostatic approach to the analysis of embeddedretaining walls, as prescribed by the European Codes, is highlighted. Finally, some considerations on the cer-tain limitations of this approach is done and the indications given by the latest Italian Building Codes (D.M.14/01/2008) are summarized.



ε – inclination angle of the backfill respect tohorizontal

β - inclination angle of the wall internal face re-spect to vertical

δ – soil-wall friction angleΨW – dilative component of the soil-wall friction

angleα – angle of the planar failure surface respect to

horizontal

θ – inclination angle of the seismic coefficient k with the vertical.Figure 1 illustrates the assumed symbology. The

subscript E indicates the seismic conditions, both foractive and passive earth pressure states. In the fol-lowing, the static loading system is denoted withoutthe subscript.

ε

α

β

θ

φW

k gh

E

k gv

Rδ

SE

Figure 1. Utilized symbols for the geometry of the problem.

2.1 Mononobe & Okabe method.

Okabe (1926) and Mononobe & Matsuo (1929)developed the basis of a pseudostatic analysis of

seismic earth pressures on retaining structures thathas become popularly known as the Mononobe-Okabe (M-O) method. The M-O method is a directextension of the static Coulomb theory to pseu-dostatic conditions. In a M-O analysis, pseudostaticaccelerations are applied to a Coulomb active (orpassive) wedge. The pseudostatic soil thrust is thenobtained from the force equilibrium of the wedge.

In addition to those under static conditions, theforces acting on an active wedge in a dry cohe-sionless backfill are constituted by horizontal andvertical pseudostatic forces, whose magnitudes arerelated to the mass of the wedge by the pseudostaticaccelerations ah = k h⋅ g and av = k v⋅ g. The total activethrust can be expressed in a form similar to that de-veloped for static conditions, that is:

( )v AE AE k H K S −= 12

1 2γ (1)

where the dynamic active earth pressure coeffi-cient, K AE , is given by:

( )

( )

( ) ( )( ) ( )

2

2

2

coscos

'sin'sin1

1

coscoscos

'cos

−++

−−++

×

×++

−−=

β ε θ β δ

θ ε φ φ δ

θ β δ β θ

θ β φ AE K

(2)

In Equation (2) φ'−ε ≥θ, and θ = tan-1

[k h /(1-k v)].

The critical failure surface, which is flatter than thecritical failure surface for static conditions, is in-clined (Zarrabi-Kashani, 1979) at an angle:

( )

+−−−+−= −

E

E AE

C

C

2

11 'tantan'

ε θ φ θ φ α (3)

where:

( ) ( ) ( )[ ]

( ) ( )[ ]

( ) ( ) ( )[ ]{ } β θ φ ε θ φ β θ δ

β θ φ β θ δ

β θ φ ε θ φ ε θ φ

−−+−−+++=

−−+++×

×−−+−−−−=

'cot'tantan1

'cottan1

'cot'tan'tan

2

1

E

E

C

C

(4)

Although the M-O analysis implies that the totalactive thrust should act at a point H/3 above the baseof a wall of height H , experimental results suggestthat it actually acts at a higher points under dynamicloading conditions. The total active thrust, S AE , canbe divided into a static component, S A, and a dy-namic component, ∆S AE :

AE A AE S S S ∆+= (5)

The static component is known to act at H/3above the base of the wall. According to Seed &

Whitman (1970) the dynamic component acts at ap-proximately 0.6 ⋅ H . On this basis, the total activethrust will act at a height h:

( )

AE

AE A

S

H S H S h

6.03 ∆+⋅= (6)

above the base of the wall. The value of h de-pends on the relative magnitudes of S A and S AE : it of-ten ends up near to the mid-height of the wall. M-Oanalyses show that k v, if assumed to be as one-half totwo-thirds the value of k h, affects S AE by less than10%. Seed & Whitman (1970) concluded that verti-

cal accelerations can be ignored when the M-Omethod is used to estimate S AE for typical wall de-signs.

The total passive thrust on a wall retaining a drycohesionless backfill is given by:

( )vPE PE k H K S −= 12

1 2γ (7)

where the dynamic passive earth pressure coeffi-cient, K PE , is given by:

'



( )

( )

( ) ( )( ) ( )

2

2

2

coscos

'sin'sin1

1

coscoscos

'cos

−+−

−+++

×

×+−

−+=

β ε θ β δ

θ ε φ φ δ

θ β δ β θ

θ β φ PE K

(8)

The critical failure surface for M-O passive con-

ditions is inclined from horizontal by an angle:

( )

+++−+−= −

E

E PE

C

C

4

31 'tantan'

ε θ φ φ θ α (9)

where:

( ) ( ) ( )[ ]

( ) ( )[ ]

( ) ( ) ( )[ ]{ } β θ φ ε θ φ β θ δ

β θ φ β θ δ

β θ φ ε θ φ ε θ φ

+−++−−++=

+−+−+×

×+−++−+−=

'cot'tantan1

'cottan1

'cot'tan'tan

4

3

E

E

C

C

(10)The total passive thrust can also be divided

(Towhata & Islam, 1987) into static and dynamiccomponents:

PE PPE S S S ∆+= (11)

Note that the dynamic component acts in the op-posite direction of the static component, thus reduc-ing the available passive resistance.

Although conceptually simple, the M-O analysisprovides a useful means of estimating earthquake-induced loads on retaining walls. A positive horizon-

tal acceleration coefficient causes the total activethrust to exceed the static active thrust and the totalpassive thrust to be lesser than the static passivethrust. Since the stability of a particular wall is gen-erally reduced by an increase in active thrust and/or adecrease in passive thrust, the M-O method producesseismic loads that are more critical than the staticloads acting prior an earthquake. The effects of dis-tributed load and discrete surface loads and irregularbackfill surfaces are easily considered by modifyingthe free-body diagram of the active or passivewedge. In such cases, Equations (2) and (8) nolonger apply. The total thrusts must be obtained from

the analysis of a number of potential failure planes.Being an extension of the Coulomb analysis,

however, the M-O method is subject to all of thelimitations of the pseudostatic analyses and of theCoulomb theory. The determination of the appropri-ate pseudostatic coefficient is difficult and the analy-sis is not suitable for soils that experience significantloss of strength during earthquakes (e.g., liquefiablesoils). Just as the Coulomb theory does under staticconditions, the M-O analysis will overpredict the ac-tual total passive thrust, particularly for δ > φ '/2. For

these reasons, the M-O method should be used andinterpreted carefully.

2.2 Upper-bound limit analysis solution.

By equating the incremental external work to theincremental internal energy dissipation associated toa translational wall movement and a φ-spiral log-sandwich mechanism of failure proposed by Chen &

Rosenfarb (1973), Chang (1981) has deduced aseismic active and passive earth pressure formula-tions in which the soil thrust can be expressed interms of equivalent coefficients of seismic earthpressure, K AE and K PE , as:

2

2

1 H K S E E γ = (12)

The seismic active earth pressure coefficient K AE is:

Ac Aq A AE N H

c N

H

q N K

γ γ γ

22++= (13)

where γ is the unit weight of the backfill material, H the vertical height of the wall, q is the uniformsurcharge acting on the surface of the backfill, c isthe soil cohesion. N Aγ, N Aq and N Ac are three coeffi-cients for which closed form expressions can befound in Chen & Liu (1990). The most critical K AE -value can be obtained by a maximization with re-spect to ς and χ shown in Figure 2.

a)

ε

θ = π/2 − φ1

β

χζ

θ = π/2 + φ2

Stress Characteristics= Velocity Characteristics

I

II

III

b)

ε

θ = π/2 + φ1

β

χζ

θ = π/2 − φ2

Stress Characteristics= Velocity Characteristics

I

II

III

Figure 2. Log-sandwich failure mechanisms for lateral earthpressure limit analysis (Chen & Liu, 1990).

'

'

'

'



At the same manner, the seismic passive earth

pressure coefficient K PE is given by the following re-lationship:

PcPqPPE N H

c N

H

q N K

γ γ γ

22++= (14)

Again, the expressions of the three coefficients N Aγ, N Aq and N Ac can be found in Chen & Liu (1990).

The most critical K PE -value can be obtained by aminimization procedure with respect to the angles ς and χ shown in Figure 2

For practical purposes, the author has calculatedsome values of the seismic earth pressure coeffi-cients reported in tables (Chang, 1981, as quoted byChen & Liu, 1990).

In the next Tables 1 and 2 some of them aresummarized.

Table 1. Values of the seismic active earth pressure coefficientgiven by the upper-bound method for log-sandwich failuremechanisms (Chang, 1981 as quoted by Chen & Liu, 1990).

φφφφ 20° 30° 40° 50°β

δ 0° 10° 20° 0° 15° 30° 0° 20° 40° 0° 25° 50°

-30° 0.77 0.74 0.76 0.62 0.61 0.67 0.49 0.50 0.62 0.38 0.42 0.65-15° 0.60 0.56 0.56 0.45 0.42 0.44 0.33 0.32 0.36 0.23 0.23 0.31

0° 0.49 0.45 0.43 0.33 0.30 0.30 0.22 0.20 0.21 0.13 0.13 0.1515° 0.41 0.37 0.34 0.24 0.21 0.21 0.13 0.12 0.12 0.06 0.06 0.0630°

k h = 0

0.34 0.29 0.27 0.17 0.14 0.13 0.07 0.05 0.05 0.01 0.01 0.01-30° 0.84 0.84 0.89 0.69 0.70 0.81 0.56 0.59 0.79 0.44 0.50 0.53-15° 0.68 0.65 0.66 0.51 0.50 0.53 0.39 0.33 0.45 0.28 0.29 0.410° 0.57 0.53 0.52 0.40 0.37 0.37 0.27 0.25 0.26 0.17 0.17 0.21

15° 0.49 0.45 0.43 0.31 0.27 0.27 0.18 0.16 0.17 0.09 0.09 0.1030°

k h = 0.1

0.44 0.38 0.36 0.23 0.20 0.18 0.10 0.09 0.09 0.04 0.03 0.03-30° 0.96 1.00 1.12 0.78 0.83 1.02 0.63 0.71 1.07 0.51 0.62 1.58-15° 0.78 0.78 0.82 0.59 0.60 0.66 0.45 0.47 0.58 0.34 0.37 0.550° 0.67 0.65 0.65 0.47 0.45 0.47 0.33 0.32 0.36 0.22 0.22 0.28

15° 0.61 0.56 0.55 0.38 0.35 0.35 0.23 0.21 0.23 0.13 0.13 0.1530°

k h = 0.2

0.56 0.51 0.48 0.31 0.27 0.26 0.15 0.13 0.14 0.06 0.06 0.06-30° 1.16 1.30 1.54 0.90 1.01 1.38 0.73 0.87 1.53 0.60 0.77 2.31-15° 0.95 1.00 1.10 0.70 0.73 0.86 0.53 0.57 0.77 0.40 0.46 0.780° 0.83 0.84 0.88 0.57 0.56 0.61 0.40 0.40 0.47 0.28 0.29 0.39

15° 0.77 0.75 0.75 0.48 0.45 0.46 0.30 0.28 0.31 0.13 0.17 0.2130°

k h = 0.3

0.75 0.70 0.68 0.40 0.36 0.36 0.21 0.19 0.20 0.10 0.09 0.10

Table 2. Values of the seismic passive earth pressure coeffi-cient given by the upper-bound method for log-sandwich failuremechanisms (Chang, 1981 as quoted by Chen & Liu, 1990).

φφφφ 20° 30° 40° 50°β

δ 0° 10° 20° 0° 15° 30° 0° 20° 40° 0° 25° 50°-30° 1.74 2.00 2.29 2.15 2.82 3.77 2.71 4.23 7.45 3.48 7.39 20.18

-15° 1.78 2.16 2.56 2.38 3.42 4.57 3.26 6.08 11.67 4.63 13.12 41.270° 2.04 2.58 3.17 3.00 4.71 7.10 4.60 10.09 20.91 7.55 28.68 9 8.06

15° 2.61 3.45 4.39 4.35 7.42 11.79 7.80 19.67 43.09 15.98 75.20 267.6930°

k h = 0

3.79 5.27 6.96 7.38 13.67 22.70 16.15 45.47 103.16 43.72 234.22 848.58-30° 1.66 1.86 2.10 2.09 2.67 3.52 2.66 4.10 7.04 3.45 7.12 19.25-15° 1.68 1.98 2.33 2.28 3.20 4.52 3.16 5.76 10.97 4.52 12.56 39.420° 1.89 2.35 2.86 2.82 4.37 6.55 4.38 9.49 19.66 7.27 27.37 93.61

15° 2.38 3.11 3.92 4.04 6.82 10.81 7.36 18.40 40.44 15.27 71.53 255.47

30°

k h = 0.1

3.39 4.68 6.16 6.77 12.51 20.74 15.11 42.60 96.72 41.63 223.34 809.77-30° 1.56 1.70 1.87 2.01 2.49 3.24 2.59 3.90 6.61 3.40 6.85 18.32

-15° 1.56 1.78 2.06 2.16 2.96 4.13 3.04 5.41 10.25 4.41 12.01 37.520° 1.71 2.08 2.50 2.63 4.00 5.95 4.15 8.86 18.33 7.00 25.95 89.09

15° 2.11 2.71 3.39 3.71 6.20 9.78 6.90 17.12 37.57 14.51 67.81 243.1330°

k h = 0.2

2.95 4.01 5.24 6.15 11.24 18.66 14.02 39.57 89.78 39.41 211.94 770.53-30° 1.39 1.46 1.56 1.91 2.30 2.94 2.51 3.68 6.16 3.35 6.56 17.53

-15° 1.37 1.51 1.71 2.02 2.69 3.71 2.91 5.06 9.50 4.29 11.42 3 5.540° 1.48 1.73 2.04 2.42 3.59 5.30 3.91 8.20 16.97 6.69 24.51 84.32

15° 1.77 2.21 2.71 3.34 5.50 8.64 6.42 15.73 34.61 13.75 64.09 230.0430°

k h = 0.3

2.40 3.19 4.10 5.45 9.89 16.41 12.94 36.27 82.68 37.13 200.35 729.04

2.3 Lower-bound limit analysis solution.

Consider a soil surface, sloping at an angle ε withrespect to the horizontal, subjected to the verticalbody force γ, due to gravity, and to the horizontalbody force k h⋅ γ, which represents the seismic coeffi-cient (positive assumed if the inertia force is towardsthe backfill). In order to compute the passive resis-

tance on a vertical wall of roughness δ, imaginetransforming the problem geometry trough a rigid ro-tation θ , given by:

hk 1tan−=θ (15)

θ represents the obliquity of the body force perunit volume in the presence of seismic action, and itis also noted that the presence of a vertical compo-nent of the inertia forces could be taken into account

by assuming:

v

h

k

k

±= −

1tan

1θ (16)

where k v is the coefficient of vertical acceleration.

The problem of deriving the passive resistance

acting on a rough vertical wall in seismic conditions

can be dealt with the wall tilted from the vertical by

the angle θ and interacting with a backfill of slope

ε*= ε – θ . The resulting vertical body force is repre-

sented by the vector 21* hk += γ γ , which can be

thought of as a properly scaled gravity body force (in

the presence of vertical acceleration it would

be ( ) 221* hv k k +±= γ γ .

As in static conditions, it can be considered tworegions, one placed near to the wall in which thestress state is affected from the soil-wall friction andone with the half space stress conditions, divided bya fan of stress discontinuities. By determining theshift between the two extreme Mohr circles of thestress states in the two regions for this problem ge-ometry, Lancellotta (2007) has deduced a closed

form for the seismic passive earth pressure coeffi-cient K PE , that is:

( ) ( ) 'tan

22

22

sin'sincos

sin'sincos

cos

φ

δ φ δ

θ ε φ θ ε

δ

⋅

−+×

×−−−−

= aPE eK (17)

where:

( )( ) θ θ ε δ

φ

θ ε

φ

δ 2

'sin

sinsin

'sin

sinsin

11 +−++

−+

= −−

a

(18)

It is useful to remember that the values given bythe Equation (17) represent the normal componentsto the vertical wall of the seismic passive coeffi-cients. The total earth pressure coefficients can beobtained by dividing for cos(δ) the results of the re-lationship.



2.4 Comparisons between the different methods.

In Figures 3 and 4 the values calculated with thedifferent methods previously recalled of the normalcomponents of the seismic active and passive earthpressure coefficients exerted on vertical walls byhorizontal backfills are compared.

For the active case, the K AEn values obtained bylimit equilibrium and the limit analysis methods arepractically identical. This is due to the fact that,

when the wall is approximately vertical and theslope angle of the backfill is larger than zero, themost critical failure is practically planar.

a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Seismic horizontal coefficient, k h

S e i s m i c n o r m a l a c t i v e e a r t h p r e s s u r e

c o e f f i c i e n t , K A E n

φ=20°

φ=30°

φ=40°

M-O

Upper Bound

δ = 0

b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3




φ=20°

φ=30°

φ=40°

M-O

Upper Bound

δ = φ/2

c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3




φ=20°

φ=30°

φ=40°

M-O

Upper Bound

δ = φ

Figure 3. Comparisons between the normal components of theactive earth pressure coefficients given by the various methodsfor horizontal backfills sustained by vertical walls: a) δ=0; b)δ=φ/2; c) δ=φ.

For the passive case, the most critical sliding sur-face is much different from a planar surface as is as-sumed in the M-O analysis. The K PEn values are se-riously overestimated by the M-O method. They are,in most cases, higher than those obtained by the limit

analysis. This is especially the case when the wall isrough and the angle of wall repose is large. The con-dition φ ‘ = δ = 40° carries out very high K PEn valueslarger than 20, unreported in Figure 4c. For smoothwalls, the potential sliding surface is practically pla-nar and the different methods give almost identicalresults.

2.5 Effects of water on the wall pressures.

The procedures for estimation of seismic loads onretaining walls described in the preceding sectionshave been limited to cases of dry backfills.

a)

0

2

4

6

8

10

12

14

16

18

20

0 0.05 0.1 0.15 0.2 0.25 0.3


S e i s m i c n o r m a l p a s s i v e e a r t h p r e s s u r e

c o e f f i c i e n t , K P E n

φ=20°

φ=30°

φ=40°

δ = 0M-O

Upper Bound

Lower Bound

b)

0

2

4

6

8

10

12

14

16

18

20

0 0.05 0.1 0.15 0.2 0.25 0.3




φ=20°

φ=30°

φ=40°

δ = φ/2M-O

Upper Bound

Lower Bound

c)

0

2

4

6

8

10

12

14

16

18

20

0 0.05 0.1 0.15 0.2 0.25 0.3




φ=20°

φ=30°

φ=40°

δ = φM-O

Upper Bound

Lower Bound

Figure 4. Comparisons between the normal components of thepassive earth pressure coefficients given by the various meth-ods for horizontal backfills sustained by vertical walls: a) δ=0;b) δ=φ/2; c) δ=φ.

Most retaining walls are designed with drains toprevent water pressure building up within the back-fill. This is not possible, however, for retaining wallsin waterfront areas, where most earthquake-inducedwall failures have been observed.



The presence of water plays a strong role in de-termining the loads on waterfront retaining wallsboth during and after earthquakes. Water outboard of a retaining wall can exert dynamic pressures on theface of the wall. Water within a backfill can also af-fect the dynamic pressures that act on the back of thewall.

Therefore, properly accounting for the effects of water is essential for the seismic design of retaining

structures, especially in waterfront areas. Since fewwaterfront retaining structures are completely im-permeable, the water level in the backfill is usuallyat approximately the same level as the free wateroutboard of the wall. Backfill water levels generallylag behind changes in outboard water level: the dif-ference in water level depends on the permeability of the wall and the backfill and on the rate at which theoutboard water level changes. The total water pres-sures that act on retaining walls in the absence of seepage within the backfill can be divided in twocomponents: hydrostatic pressure, which increaseslinearly with the depth and acts on the wall before,

during and after the earthquake shaking, and hydro-dynamic pressure, which results from the dynamicresponse of the water itself.

2.5.1 Water outboard of the wall.Hydrodynamic water pressure results from the

dynamic response of a body of water. For retainingwalls, hydrodynamic pressures are usually estimatedfrom Westergaard’s solution (Westergaard, 1931)for the case of a vertical, rigid dam retaining a semi-infinite reservoir of water that is excited by har-monic, horizontal motion of its rigid base. Wester-

gaard showed that the hydrodynamic pressure ampli-tude increased with the square root of water depthwhen the motion is applied at a frequency lower thanthe fundamental frequency of the reservoir, f 0 =V P / 4H , where V P is the P-wave velocity of water(about 1400 m/s) and H is the depth of water in thereservoir (the natural frequency of a 10m-deep reser-voir, for example, would be over 35 Hz, well abovethe frequencies of interest for earthquakes). Wester-gaard computed the amplitude of the hydrodynamicpressure as:

H z

g

a p ww

hw γ

8

7= (19)

The resultant hydrodynamic thrust is given by

2

12

7 H

g

a p w

hw γ = (20)

The total water pressure on the face of the wall isthe sum of the hydrostatic and hydrodynamic waterpressures. Similarly, the total lateral thrust due to thewater is equal to the sum of hydrostatic and hydro-dynamic thrusts.

Another important consideration in the design of a waterfront retaining wall is the potential for rapiddrawdown of the water outboard of the wall. Earth-quakes occurring near large bodies of water often in-duce long-period motion of the water, such as tsu-namis or seiches, that cause the water surface tomove up and down. While the upward movements of water outboard of a retaining wall will generally tendto stabilize the wall (assuming that it does not rise

above the level of the top of the wall), downwardmovements can create a destabilizing rapid draw-down conditions. When liquefiable soils exist underrelatively high levels of initial shear stress, failurescan be triggered by very small changes in waterlevel. Such failures, can originate in the soils adja-cent to or beneath the retaining structure rather thanin the backfill.

2.5.2 Water in backfill.The presence of water in the backfill behind a re-

taining wall can influence the seismic loads that acton the wall in three ways:

1. by altering the inertial forces within the backfill:2. by developing hydrodynamic pressures within the

backfill: and,3. by allowing excess pore water pressure generation

due to cyclic straining of the backfill soils.The inertial forces in saturated soils depend on

the relative movement between the backfill soil par-ticles and the pore water that surrounds them. If, as itis usually true, the permeability of the soil is smallenough (typically around k ≤ 10

-5m/s or so) so that

the pore water moves with the soil during the earth-quake shaking (no relative movement of soil and wa-

ter, or restrained pore water conditions), the inertialforces will be proportional to the total unit weight of the soil. If the permeability of the backfill soil is veryhigh, however, the pore water may remain essen-tially stationary while the soil skeleton moves back and forth (the soil particles move through the porewater in free pore water conditions). In such cases,inertial forces will be proportional to the buoyant (orsubmerged) unit weight of the soil. Hydrodynamicwater pressures can also develop under free pore wa-ter conditions and must be added to the computedsoil and hydrostatic pressures to obtain the totalloading on the wall.

For restrained pore water conditions, the M-Omethod can be modified to account for the presenceof pore water within the backfill (Matsuzawa et al.,1985). Representing the excess of pore water pres-sure in the backfill by the pore pressure ratio, r u =∆u / p’0, the active soil thrust acting on a yielding wallcan be computed from Equation (1) using:

( )ub r −= 1γ γ (21)

and



h

d

H

AK γ

PK γ

PKAK γ

AK γ d K γ (h+d)P

d '

z '

O

Figure 6. Earth pressures distributions assumed in limit equilib-rium method.

To eliminate stresses discontinuities in corre-spondence of the rotation point and to obtain a sim-plified shape of the pressures distributions, differentsimplifications and assumptions were proposed inliterature. The main of which are plotted in Figure 7aand 7b.

In the first, the net pressure distribution is simpli-

fied by a rectilinear shape. It is assumed that the pas-sive resistance below the dredge level is fully mobi-lized. The rotation point coincides with the zero netpressure point. At the bottom of the wall the soilstrengths, active and passive, are mobilized and thenet pressure assumes the values reported in Figure7a.

The limit depth d can be evaluated imposingtranslation and moment equilibrium. In this manner,a system of two equations of second and third degreeis obtained and the two unknowns, the depth of thepoint of the inversion of pressures z’ and the limitdepth of embedment d , may be calculated using:

+

−

+−

=

h

d

K

K

h

d

h

d

K

K

h

z

A

P

A

P

211

1'

22

(24)

+

−

+−

=

h

d

K

K

h

d

h

d

K

K

h

z

A

P

A

P

211

1'

33

(25)

The second method, commonly used in U.K. anddescribed in Padfield & Mair (1984), assumes thatthe net pressure distribution below the point of rota-tion can substituted with the net force R applied at adistance z’ = 0.2⋅d’ from the bottom of the wall.Writing the moment equilibrium around the point O,one has an equation of the third degree with the sin-gle unknown d:

a)

AK γ

P(K - K ) γ A

[K (h+d) - K d] γ P A

h

d d '

z '

b)

h

d

AK γ

PK γ

d '

R

0 . 2

d '

Figure 7. Simplified earth pressures distributions: a) FullMethod; b) Blum Method

1

2.1

3 −=

AP K K

hd (26)

The main problem for the design of embeddedwalls is then the right choice of the earth pressurecoefficients K A and K P when the soil-wall friction δ would be considered. It is well-recognized that theCoulomb theory provides unrealistic values of the

passive earth pressure coefficient when δ > φ '/2. Dif-ferent suggestions can be found in the literature(Padfield & Mair, 1984; Terzaghi, 1954; Teng,1962). Since knowledge on this field is limited, inthe current practice is commonly adopted δ A = 2/3 φ ' for the active case and δP = 0, for the passive case. Inthis manner, passive resistance of soil on the dredgeside of reinforced concrete walls, realized with pilesor diaphragm, is largely underestimated. Padfield &Mair (1984) assert that reasonable values of the soil-wall friction for the calculation of the earth pressurecoefficients are δ A = 2/3 φ ' and δP = 1/2 φ ' .

Bica & Clayton (1992) have collected a series of experimental data of collapse of embedded walls andhave proposed an expression for the preliminary de-sign in simple soil conditions:

°−−

⋅= 18

30'

3

2φ

eFS h

d (27)

In Figure 8, the relationship (27) for the case of limit conditions is compared with a series of numeri-cal and experimental results of failure taken from theliterature. It can be seen the good agreement between



the relationship and the numerical and experimentaldata given by the different authors.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

25 30 35 40 45 50

Friction Angle, φφφφ (°)

L i m i t D e p t h R

a t i o o f E m b e d m e n t , d / h

Empirical Formula

Day (1999)

Pane & Tamagnini (2004)

Fourie & Potts (1989)

Rowe (1951)

Bica & Clayton (1998)

Bransby & Milligan (1975)

Lyndon & Pearson (1984)

King & McLoughlin (1992)

▲ Numerical Analyses

■ 1-g Model

● Centrifuge Model

Figure 8. Experimental and numerical limit depth ratios of em-bedment at collapse for free embedded walls.

Adopting the Coulomb and Lancellotta theoriesfor the evaluation of the active and passive earthpressure coefficients K A and K P, respectively, and

assuming the soil-wall friction values suggested byPadfiled & Mair (1984), the full and Blum methodsgive limit depth ratios of embedment in relation tothe soil friction angle φ ' plotted in Figure 9. In thesame Figure the d/h ratios at failure evaluated withthe two limit equilibrium methods and adopting thesoil-wall friction angles currently utilized in the de-sign are reported.

It can be noted the large overestimation of theneeded depth of embedment d when the soil-wallfriction is not considered for the calculation of thepassive resistances. The Blum method gives moreconservative values than the full method for which,

if it is applied by adopting Padfield & Mair (1984)indications, the results are close to those experimen-tally and numerically estimated.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

25 30 35 40 45 50

Friction Angle, φφφφ(°)

L i m

i t d e p t h r a t i o o f e m b e d m e n t , d / h

Equation (27)

Blum method

Full method

δA = 2/3 φ; δP = 1/2 φ

δA = 2/3 φ; δP = 0

Figure 9. Limit depth ratios of embedment at collapse for freeembedded walls computed with limit equilibrium methods.

Five methods are used in design to incorporate afactor of safety against collapse. These involve in-creasing embedment depth, reducing the strength pa-rameters, reducing the passive pressure coefficient,

reducing the net passive pressure or reducing netavailable passive pressure (Padfield & Mair, 1984).

The maximum bending moment M max acting onthe wall depends to the soil and wall properties andto the depth of embedment d , for a given retainingheight h.

Bica & Clayton (1992), on the basis of a collec-tion of experimental results, have proposed the ap-proximated relationship for the computation of M max

represented in Figure 10 with some numerical andexperimental data published in the literature:

2

3

2

16

30'

3max 095.0

−

°−−

⋅⋅=

h

d

eeh

M φ

γ (28)

Near to each point is reported the depth ratio d/h.The values given by Equation (28) are often conser-vative when compared with those corresponding tolimit conditions for the wall. It can be seen the in-crease of bending moment with depth of embedmentfor a fixed value of friction angle. This fact contraststhe design recommendations that M max should beevaluated for a safety factor equal to 1. This factorshould be greater than 1 when M max is computed.

0

0.03

0.06

0.09

0.12

0.15

25 30 35 40 45 50

Friction Angle, φφφφ (°)

N o

r m a l i z e d M a x i m u m B e n d i n g M o m e n t ,

M m a x

/ γ γγ γ h 3

Equation (28)

Day (1999)

Fourie & Potts (1989)

Rowe (1951)

Bica & Clayton (1998)

Lyndon & Pearson (1984)

King & McLoughlin (1992)

▲ Numerical Analyses

■ 1-g Model

● Centrifuge Model

1.26

0.89

1.0

1.5

0.670.920.7

1.06

0.42

0.52

1.31

0.54

0.39

0.38

0.7

1.06

0.27

1.0

1.5

0.67

1.0

1.5

0.67 0.23

0.29

Figure 10. Experimental and numerical normalized maximumbending moment for free embedded walls.

Figure 11 shows the comparisons between thenormalized maximum bending moment M max / γh

3

computed with Equation (28) and those obtained bythe limit equilibrium method with the following ex-pression:

( )[ ]33

max 6 xK xhK M

P A

−+= (29)

where x is the depth from the dredge level inwhich the shear force is zero:

1

1

−=

AP K K h

x(30)



0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

25 30 35 40 45 50


N

o r m a l i z e d m a x i m u m b e n d i n g m o m e n t ,

M m a x

/ γ γγ γ h 3

Equation (28)

Limit equilibrium

δA = 2/3 φ; δP = 1/2 φ

δA = 2/3 φ; δP = 0

Figure 11. Normalized maximum bending moment for free em-bedded walls at collapse computed with limit equilibriummethod.

The values obtained by assuming the soil-wallfrictions suggested by Padfield & Mair (1984) arelightly underestimated respect to those predictedwith the empirical relationship (28), while, adoptingδ A = 2/3 φ ' and δP = 0, limit equilibrium provides re-alistic maximum bending moment at collapse. It

should be remembered that, if a safety factor isadopted on the design of the depth of embedment,the actual depth ratio d/h should be utilized for theestimation of M max.

The Equations previously recalled are valid fordry homogeneous soils with constant values of K A and K P. Limit equilibrium of embedded retainingwalls in layered saturated soils is commonly studiedby using a hybrid approach in which active and pas-sive horizontal effective stresses are computed mul-tiplying vertical effective stresses by active and pas-sive earth pressure coefficients given by the theories.

3.3 Seismic design of free embedded walls.

In the EC8 Part 5 (2003) is described a simplifiedpseudostatic approach to analyze the safety condi-tions of retaining walls. The seismic increments of earth pressures may be computed with the M-Omethod. Its application for rigid structures is moreprompt than for embedded walls for which the sta-bility is mainly due to the passive resistance of thesoil in the embedded portion. As for the Coulombtheory in static conditions, the M-O theory givesvery high values for passive earth pressure coeffi-

cient when the soil-wall friction is considered. Forthis reason, the evaluation of passive pressure shouldbe conducted assuming zero soil-wall friction.

In the pseudostatic analyses, the seismic actionscan be represented by a set of horizontal and verticalstatic forces equal to the product of the gravityforces and a seismic coefficient. For non-gravitywalls, the effects of vertical acceleration can be ne-glected. In the absence of specific studies, the hori-zontal seismic coefficient k h can be taken as:

g

a

r

S k

gh = (31)

where S is the soil factor that depends to the seismiczone and considering the local amplification due tothe stratified subsoil and to the topographic effects,ag is the reference peak ground acceleration on typeA ground, g is the gravity acceleration and the factorr is a function of the displacement that the wall can

accept. For non gravity walls, the prescribed value isr = 1 (EC8 Part 5, Table 7.1).Furthermore, for walls not higher than 10m, the

seismic coefficient can be assumed constant alongthe height.

The point of application of the force due to thedynamic earth pressures should be taken at mid-height of the wall, in the absence of a more detailedstudy taking into account of the relative stiffness, thetype of movements and the relative mass of the re-taining structure.

Assuming that the position of the point of rotationO near to the bottom of the wall is the same of the

static condition, the application of the Blum methodto search the seismic limit equilibrium of a free em-bedded wall can be conducted adopting the loadingsystem represented in Figure 12.

The earth pressure thrusts have the following ex-pressions:

( )

( ) ( )2

2

'2

1

'2

1

d hK K S

d hK S

A AE AE

A A

+−=∆

+=

γ

γ

(32)

( ) 2

2

'2

1

'21

d K K S

d K S

PPE PE

PP

γ

γ

−=∆

=

(33)

in which the earth pressure coefficients with the sub-script E are referred to the seismic conditions whilethose without the subscript E are the static coeffi-cients.

h

d PS d '

R

0 . 2

d '

AS

AE∆S

PE∆S

Figure 12. Earth pressures on a free embedded wall subjectedto seismic loadings according to EC8-5 pseudostatic analysis.



The moment equilibrium of the forces around thepoint O provides a simple relationship for the limitdepth of embedment:

13

3

2.1

3 −−

−=

A AE

PPE

K K

K K

hd (34)

If the seismic horizontal coefficient k h = 0 (static

conditions), the seismic earth pressure coefficientsare equal to the corresponding static values and,then, Equation (34) becomes equal to the (26).

Graphical representations of the (34) in a semilogarithmic scale for different values of k h is shownin Figure 13. The soil-wall friction angles used forthe calculation of the earth pressure coefficients arethose suggested to EC8-5 and currently adopted (δ A = 2/3 φ ', δP = 0) and those suggested to Padfield &Mair (1984) (δ A = 2/3 φ ' , δP = 1/2 φ).

As noted above for the static conditions, the EC8-5 indications on the soil-wall friction conduct to avery conservative design of the depth of embedment,

underestimating the soil passive resistance. The useof the Blum method with the seismic passive earthpressure coefficient given by the lower-bound limitmethod proposed by Lancellotta (2007) allows to es-tablish more reasonable depths of embedment forcantilever walls.

The maximum bending moment can be computedas:

( ) ( ) ( )( )

( ) 23

23max

'

4

1

6

1

'4

1

6

1

xd K K xK

xhd hK K xhK M

PPE P

A AE A

γ γ

γ γ

−−−

+++−++=

(35)

where x is the depth from the dredge level at whichthe shear force is zero and can be evaluated byequaling the two members of the force equilibriumequation:

( ) ( )( )[ ]( )

( )[ ] xd K K xK

xhd hK K xhK

PPE P

A AE A

'

'

−+=

=++−++(36)

In Figure 14 are plotted in a semi logarithmicscale the values calculated for different seismic hori-

zontal coefficients k h and for the two soil wall-friction conditions. While the assumption of theEC8-5 on the soil-wall friction is conservative forthe computation of the depth of embedment, theevaluation of maximum bending moment with thelimit equilibrium method is more safe if δP is takenequal to zero.

a)

0.1

1

10

25 30 35 40 45 50


L i m i t d e p t h r a t i o o f e m b e d m e n t , d / h Equation (27)

k h = 0

k h = 0.1

k h = 0.3

k h = 0.2

b)

0.1

1

10

25 30 35 40 45 50Friction Angle, φφφφ(°)

L i m i t d e p t h r a t i o o f e m b e d m e n t , d / h Equation (27)

k h = 0

k h = 0.1

k h = 0.3

k h = 0.2

Figure 13. Limit depth ratios of embedment given by the Blummethod for the EC8-5 seismic loadings: a) δA = 2/3 φ, δP = 0; b)δA = 2/3 φ, δP = 1/2 φ.

3.4 The recent Italian Building Code.

The new Italian Building Code (NTC, 2008) in-troduced some innovations on the seismic design of embedded walls to eliminate some discrepancies ex-

isting on the application of the pseudostatic analysesfor embedded walls (see for instance Callisto, 2006).

The pseudostatic analysis of an embedded retain-ing wall should be carried out assuming that the soilinteracting with the wall is subjected to a value of the horizontal acceleration which is:

a) constant in space and time (this is implicit in apseudostatic analysis);b) equal to the peak acceleration expected at thesoil surface.Deformability of the soil can produce amplifica-

tion of acceleration, that is incorporated into the soilfactor S , but that can be better evaluated through asite response analysis.



0.1

0.2

0.3

0.6

0.5

0.4

0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.00.3

H/λ

E q u i v a l e n t s e i s m i c a c t i v e e a r t h p r e s s u r e

c o e f f i c i e n t , K A E

kh = 0.15

kh = 0.25

kh = 0.35

φ = 33°

δ = φ/3

Figure 16. Influence of the ratio between the height of the wallH and the wavelength λ of a harmonic wave on the seismic ac-tive earth pressure coefficient (Steedman & Zeng, 1990).

It should be clear that coefficient r in equation(31) depends on the displacements that the structurecan accept with no loss of strength. That is, it may beacceptable that over a small temporal period duringan earthquake the acceleration could be higher than acritical value producing limit conditions, provided

that this will lead to acceptable displacements andthat these displacements do not produce any strengthdegradation. This is equivalent to state that the be-haviour of the structure should be ductile, i.e. thatstrength should not drop as the displacements in-crease.

To account these aspects, in the latest ItalianBuilding Code NTC two coefficients were intro-duced. In the absence of specific studies, the seismichorizontal coefficient k h can be estimated with therelationship:

g

Sa

k g

h ⋅⋅= β α (37)

where α ≤ 1 and β ≤ 1 are factors for the deform-ability of the soil that interacts with the wall and forthe capability of the structure to accept displace-ments without losses of strength, respectively. Theirvalues are reported in the next Figures 17 and 18.

The points of application of the forces due to thedynamic earth pressures can be assumed to be thesame of the static earth thrusts, if the wall can acceptdisplacements. Instead they should be taken to lie atmid-height of the wall, in the absence of more de-

tailed studies, accounting for the relative stiffness,the type of movements and the relative mass of theretaining structure.

0.2

0.4

0.6

1.2

1.0

0.8

5

H (m)

α αα α

10 15 20 25 30 35 40 45 50

D

C

B

Ground type A

Figure 17. Diagram for the evaluation of the deformability fac-tor α (NTC, 2008)

0.2

0.4

0.6

1.0

0.8

us (m)

β

0.1 0.2 0.3

Figure 18. Diagram for the evaluation of displacements factor β (NTC, 2008)

4 REFERENCES.

Bica, A.V.D., Clayton, C.R.I. (1992). “The preliminary designof free embedded cantilever walls in granular soil. Proc. Int.Conf. on Retaining Structures, Cambridge

Callisto L. (2006). “Pseudo-static seismic design of embeddedretaining structures”, Workshop of ETC12 EvaluationCommittee for the Application of EC8, Athens, January 20-21.

Caquot A., Kerisel F. (1948). “Tables for the calculation of passive pressure, active pressure and bearing capacity of foundations”, Gauthier-Villars, Paris.

Chang M.F. (1981). “Static and seismic lateral earth pressureson rigid retaining structures”, PhD Thesis, School of CivilEngineering, Purdue University, West Lafayette, IN, 465pp.

Chen W.F., Rosenfarb J.L. (1973). “Limit analysis solutions of earth pressure problems”, Soils and Foundations, Vol. 13,No. 4, pp. 45-60

Chen W.F., X.L. Liu (1990). “Limit analysis in soil mechan-ics”, Elsevier, Amsterdam, 477 pp.Coulomb C.A. (1776). “ Essai sur une application des regles

des maximis et minimis a quelques problemes de statiquerelatifs a l’architecture”, Memoires de l’Academie Royalepres Divers Savants, Vol. 7

EN 1997-1 (2002). Eurocode 7 Geotechnical Design – Part 1:General Rules. CEN European Committee for Standardiza-tion, Bruxelles, Belgium.

EN 1998-5 (December 2003). Eurocode 8: Design of structuresfor earthquake resistance – Part 5: Foundations, retainingstructures and geotechnical aspects. CEN European Com-mittee for Standardization, Bruxelles, Belgium.



Kramer, S.L. (1996). “Geotechnical Earthquake Engineering”,Prentice Hall, Inc., Upper Saddle River, New Jersey, 653pp.

Lancellotta R. (2002). “Analytical solution of passive earthpressure”, Geotechnique, Vol. 52, No. 8, pp. 617-619

Lancellotta R. (2007). “Lower-Bound approach for seismicpassive earth resistance”. Geotechnique, Vol. 57, No. 3, pp.319-321

Matsuzawa, H., Ishibashi, I., Kawamura, M. (1985). “Dynamicsoil and water pressures of submerged soils”, Journal of Geotechnical Engineering, ASCE, Vol. 111, No. 10, pp.

1161-1176Mononobe N., Matsuo H. (1929).”On the determination of

earth pressures during earthquakes”, Proc. World Engineer-ing Conference, Vol. 9, pp. 176

NTC (2008) “Approvazione delle nuove norme tecniche per lecostruzioni”, Gazzetta Ufficiale della Repubblica Italiana,n. 29 del 4 febbraio 2008 - Suppl. Ordinario n. 30,http://www.cslp.it/cslp/index.php?option=com_docman&task=doc_download&gid=3269&Itemid=10 (in Italian).

Okabe S. (1926). “General theory of earth pressure”, Journal of Japanese Society of Civil Engineering, Vol. 12, No. 1

Padfield, C.J., Mair, R.J. (1984). “Design of retaining wallsembedded in stiff clays”, Report No. 104, London: Con-struction Industry Research and Information Association.

Rankine W. (1857). “On the stability of loose earth”, Philoso-

phical Transactions of the Royal Society of London, Vol.147

Seed H.B., Whitman R.V. (1970). “Design of earth retainingstructures for dynamic loads”, Proc. ASCE Specialty Con-ference on lateral stresses in the ground and design of earthretaining structures, pp. 103-147

Sokolovskii V.V. (1965). “Static of granular media”, PergamonPress, New York, NY, 232 pp.

Steedman R.S., Zeng X. (1990). “The seismic response of wa-terfront retaining walls”, Proc. ASCE Specialty Conferenceon Design and Performance of Earth Retaining Structures,Special Technical Publication 25, Cornell University,Ithaca, New York, pp.872-886

Teng, W. C. (1962). “Foundation Design”, Prentice Hall,Englewood Cliffs, NJ.

Terzaghi, K. (1954). “Anchored bulkheads”. Transactions,American Society of Civil Engineers, No 119, pp. 1243-1280

Towhata I., Islam S. (1987). “Prediction of lateral movement of anchored bulkheads induced by seismic liquefaction”, Soilsand Foundations, Vol. 27, No. 4, pp. 137-147

Westergaard, H. (1931). “Water pressure on dams duringearthquakes”, Transactions of ASCE, paper No. 1835,pp.418-433

Zarrabi-Kashani K. (1979). “Sliding of gravity retaining wallduring earthquakes considering vertical accelerations andchanging inclinations of failure surface” S.M. Thesis, De-partment of Civil Engineering, Massachussetts Institute of Technology, Cambridge, Massachussetts

visone-santucci

Documents