CUIMR-Q � 90 � 00l C3LOAN COPY QHLY

WATERFRONT SHEET PILE WALLS SI:XeJECTED TOE 4&THQUABK SK&HNG: ANALYSIS METHOD

Final Beycet on Sea Grant eject 8/OK-8Submitted to Sea Grant Pmgram

Priacipa1 Investigator:

Toycehi Nogmai

8 'W ~l I A~ybyUniversi,ty of Camfornia at SanDieg~ La JoHa, CA 92t;83

TABLE OF COM.'%20 S

1. INTRODUCTION

1.1 Pevious Studies Related to Seismic Earth Pressure onRetaining walls

1.2 Observation of Earthquake Damages ofWaterfront Sheet Pile Walls

1.3 Objective of Study

2. FINITE ELEMENT FORMULATION.

2.1 Soil Elements.

2.2 Joint Elements at Soil-Wall Interface....

3. TIME INTEGRATION FOR SOLVING EQUATION.....

4. VERIFICATIONS

5. EXAMPLE OF COMPUTED BEHAVIOR OF WATERFRONTSHEET PILE SUBJECTED TO EARTHQUAKE SM&XlVG........

REFERENCES

L. IKIRODUCTIGN

L.l Previous Studies Related to Seismic Earth Pres~~xm~ on Retaining%alls

Seismic effects on rigid retaining walls i.e. gravity walls! have been

studied both theoretically and experimentally. Mononobe �919! and Okabe

�926! proposed a simplified formulation to evaluate the seismic lateral

earth pressure on a retaining wall backfilled with dry cohesionless

naturals Mononobe-Okabe method!. Their method is based on a plastic

limit condition and assumes a triangular distribution of earth pressure

along the waU height. According to this pressure distribution, the total

force is acting at the one-third of the wall height from the bottom. Using

small scale rigid wall models with dry sand backfill, the earth pressure due

to dynamic ground. shaking was investigated experimentaHy by using a

shake table Mononobe and Matsuo, 1919; Jacobsen, 1939; Matsuo, 1941; and

Ishii, Arai and Tsuchida, 1960!. The experimental results showed that the

maximum lateral earth pressure was roughly equal to that predicted by the

Mononobe-Okabe, but the location of the total lateral force was higher than

one-third of the wall height. It was also shown that the increase of the

permanent lateral earth pressure aRer the shaking was substantially

higher than the maximum dynamic transient pressure during shaking.

Prakash and Basavanna �969! and Seed and Whitman �970! modified the

Mononobe-Okabe method and assumed a more realistic distribution of

earthquake lateral pressure. Sabzevari and. Ghahramani �974! used the

incremental rigid plasticity theory to obtain analytical expressions and

construct stress displacement and velocity Belds behind the wall. This

method enables us to compute the lateral earth pressure for various types ofrigid wall displacement.

Formulations based on elastic analysis were also developed for theevaluation of the dynamic lateral earth pressure. Matsuo and Ohara �965}and Tajimi �973! solved elastic wave equations to obtain the formulations.Scott �973! modeled the soil and wall system as an elastic vertical shearbeam and a rigid wall connected each other by distributed horizontalWinkler springs and developed the formulae to compute the dynamic earthpressure. Given earthquake excitation at the base of the beam, the beamsimulates the free-6eld earthquake ground motion, and Kinkier springsproduce the soil-wall interaction force i.e. lateral earth pressure!. Thoseformulations based on elastic analysis generally estimate the maximumlateral earth pressure significantly higher than those estimated by theformulations based on a plastic limit condition. Elastic analysis may bejusti6ed if the lateral response of the waH is not large enough to produce aplastic limit condition behind the wall.

Lt depends on the displacement of the wall whether the backfiH soil isin an elastic condition or plastic limit condition. The nonlinear Rnite

element method is a convenient analysis tool to evaluate the lateral earthpressure since the conditions in the soil are automatically adjustedaccording to the magnitude of strains developed in the soil. Potts andPourier �986! used a Gnite element method and. found that the

displacement pattern of the wall highly afFects the amount of the waHdisplacement required to fully mobilize the plastic limit condition behindthe waH and. the pressure distribution. Those indicate that the wall

Qexibility aFects the lateral earth pi'essure on the waH. Kuralowicz and

Tejchman ].984! investigated experimentally the static earth pressure on

rigid and. flexible sheet-pile walls in cohesionless soil. It was observed that

the distribution shape of the lateral earth pressure on a flexible wall was

nonlinear, although that on a, rigid wall was nearly triangular. Ayala,

Rorno and Goinez �985! investigated the maximum bending moment of

flexible walls induced by seismic earth pressure by using the Rnite element

method. Flexibility of the waH was found to afYect the maximum moment.

Kurata, Arai and Yokoi �965! conducted shaking table tests with a

small scale anchored sheet-pile wall model in dry cohesionless soil. They

observed that the ground shaking increased the lateral earth pressure

substantially in the vicinity of the anchor level but did. not increase much in

the middle part of the waH. This is due to redistribution of the lateral earth

pressure resulting from Qexural deformation of the wall. The total lateral

earth pressure observed during shaking was smaller than that computed

by the Mononobe-Okabe method. It wa.s also observed that the increase of

permanent 1ateral earth pressure due to the permanent soil deformation

was much larger than the transient dynamic earth pressure. Murphy

�960! also conducted model tests on an anchored sheet-pile wall subjected

to earthquake shaking. It was pointed out that the wall translated outward

without tilting until the anchor puH failed. As essentially planar slip

surface was observed in the backfiH soil. Simitar planer slip surface was

observed by Bolton and Steehnan �985!.

When the backGD soil is submerged, a portion of the water in the

pores of the soil mass moves with the lateral soil skeleton restrained water!

and the other portion moves freely during ground shaking free water!.

The hydrodynamic pressure in the bacldill against the wall is due to the

motion of the free water, and this depends on the volume of the free water.

Anzo �936! formulated the hydrodynamic pressure in the submerged

backfi11 against the wall. The formulation was based on a rigid soil

skeleton, Darcy's flow and Navier-Stokes equations. According to this

formulation, the hydrodynamic pressure in the soil never exceeds the value

computed by Westergard's formula and is smaller for a lower permeability.

iVIatsuo and Ohara j.965! conducted model tests on the wall with

submerged backfill sand. The observed hydrodynamic pressure was

somewhat larger than that computed by Westergard's formula.

Liquefaction was observed in some of those tests. Ishibashi, Matsuzawa

and Kawamura �985! proposed the method estimate the dynamic pressure

against the wall retaining submerged. backs soil. The method adopts the

Mononobe-Okabe method with the soil weight increased by the restrained

water weight! for earth pressure and simplified Anzo's formula for

hydrodynamic pressure.

Towhata and Islam �987! viewed that the magnitude of permanent

wall displacement was the most critical phenomena for the judgement of

the degree of seismic damage of anchored bulkheads. Based on this view,they developed a simple method to compute the lateral movement of

anchored bulkheads induced by earthquake shaking assuming a

triangular sliding wedge behind the wall. They considered that the e8ect of

liquefaction in backfill was particularly important and thus was taken into

account in the method.

12 Ok~ation of Earthquake Damages to Waterfront Sheet Pile Walls

The Niigata earthquake in 1964 caused substantial damages toretaining structures. Most of those damages were closely connected to the

development of excess pore water pressure associated with liquefactionprocess.

The ministry of Transportation �964! published a detailed report on

earthquake damages in Niigata harbor facilities. The report indicates that

substantial translation of walls occurred toward the mater at the mall next

to Yamanoshota Wharf, due to loose reclaimed backfiHs. The North and

South Wharfs were sufFered from excessive two-meter movement of their

walls. Similar movement was detected in the Bandai-Jima area as well.

The North bank of the Shinano River completely collapsed, accompanied byoverall displacement of backfill toward the river. Although the reportconcluded. that most sheet-pile wall damages were due to tilting, it alsomentioned that lateral translation took place in walls of shallow

embedment. At the areas of al1 those anchored bulkheads, evidences of

liquefaction mere reported.

13 Objective of the Study

The objective of the project is to develop an analysis tool for computingthe behavior and conditions of waterfront sheet pile walls during and rightafter the earthquake ground shaking. Key factors for the judgement of theseismic performance of waterfront anchor bulkheads are bending momentin the mall and permanent lateral movement of waH, induced by

earthquake shaking. According to the previous section, those are affected

mainly by elastic and inelastic soil behavior, excess porewater pressure,

flexibility of wall, motions of backfill and wall, and hdro-dynamic pressure

at both front and back sides of the wall. The finite element method may be

the most convenient analysis method to consider rationally all of those

important factors affecting the performance of walls, and thus was

developed in the project.

Fig. 1.1 shows finite element models of a waterfront sheet pile waH.

The effect of water at the front side of the wall is taken into account by added

masses attached at the nodes located at the water-wall and water-soil

interfaces. Those masses are active for motions only in the component

normal to the interface. Soil elements are formulated treating the soil

mass as a water-saturated two phase mixture. This enables soil elements

to simulate the coupling between porewater motions and soil frame

motions, generation and dissipation of excess porewater pressure in a soil

mass, and effects of excess porewater pressure on the soil frame behavior.

Discontinuity of displacements develops at the soil-wall interface due to

large difFerence in the stiffnesses between those of the waH and soil and due

to sliding and separation between the waQ and soil. This requires special

e1ements, termed joint elements, to accomodate such behaviors at the

interface.

I. FIMIIR RIRMENT FORt IULITION

2.1 Soil Elements

Submerged soil is treated. as a Quid-saturated porous medium. The

nonlinear behavior is implemented in the stress-strain relationship of thesoil frame.

�! De6nitions of Valuables

Porosity n is defined as

Vn�

V + V, �.1!

where Vf = volume or pores or fluid; and. Vs = volume of solid.

Relation given in Eq. 1 can be transformed to density relationship as

p = � � n!p, + iipf �.2!

7

where p, ps and pt= mass densities of solid-Quid mixture, solid and fiuid,

respectively. ln order to describe the deformation of a mixture,

deformations of the two phases are identifled separately. The componentsof deformation of the so1id can be denoted. by u; i = 1, 2, 3! and componentsof displacement of the Quid. can be denoted by U; i = 1, 2, 3!. i denotes

coordinate axes. The amount of Quid displaced from the solid skeleton, Q;,is obtained as

Q. = A,u U, - u,! �-3!

where A; = area of mixture n.ormal to the ith direction.

face of the mixture is obtained as

%, = � = u U. - u.! �-4!

The strains in the solid skeleton are dered as

�.5a!

or

<e> � D.j r�.5b!

where

={ ll 22 33 21 23 31}

0 oBx! 3Q8

ax,

0 0 0Bx

0

a a

B.] =

The displacement of fluid relative to solid skeleton, averaged over the

� 9!

Strains in the solid grains induced by pore pressure can be expressed byusing the bulk modulus for solid as

�,10!

Deformation of solid skeleton caused by the effective stress a;J' is related to cfJ!< through the constitutive terms or D1JQ1

�.1 j.!

Combining Eqs. 7 through ll, the foUowing relations are obtained:

PijkI kj 3K ijkl kl P ij �.12!

which is rearranged as

a,. = 0,.�, e�, + ap5., �.13!

where

10

Similarly, strains are also decomposed into two parts. Part of the strain iscaused by deformation of grains due to pore pressure, and the part iscaused by deformation of grains and skeleton due to effective stress. Thiscan. be expressed as

�! Amount of fluid flowing out of 2, = change of volume of pores!;

�! Volume change in solid due to p, = p�-n!/K>!;

�! Volume change due to p, = n~!; and

�! volume change of solid. due to effective stress, < = a; /3

Thus, the continuity condition. is formulated as

e., = �-n! + � + 3KP AP ii1E �.15!

Rewriting 4�" in Eq. 2.15 with Eqs. 2.9, 2.10 and 2.11 leads to

1-n

F =Q .p+ � { "ld kl ill P! - Lj 3K,

or

11

Continuity of Quid Qow wi11 be formulated in order to obtain a relation

for pore pressure taking into the volumetric change of each phase. Thetotal volume change of the mixture, e;;, consists of the sum of the followingparts:

P =QC+ aug! �.16!

where

1 n an+-Q Kf K,

Equations for constitutive relation of porous material aresummarized as

�,13!

P =Q E+ aQ�! �.16!

i ' ijkA>9K,

1 n an+��.17!

d<.. = D..q! +q a' dp 5.. �.18!

�.18!

where

12

For nonlinear behavior, the constitutive relation needs to be

expressed in incremental form. The incremental forms of Eqs. 2.13 and2.16 are

ij ijkl kl

9Ks � 20!

ep [de,! + r> dp � 2~!

gp -gt r> dE!+ Jg]

where

1 n 1n� � � +�,23!

and

T da! -f«�«zz <zz «,z zz

T<«k = ~zz ~zz ~zz ~zz ~zz ~zz} �.24!

rk I I I 0 0 0!

�! +ammic Egmlibrimn Kyxation

Two equations are required to describe the equilibrium conditions of

two phase mixture. 'We wiH look at the equilibrium con.ditions in the

13

and D<;>g~ is incremental constitutive tensor for skeleton. The value a~ is

nearly equal to 1 since the second term at the right hand side of Zq. 20 isvery small compared with the erst term Zienkeiewicz at al., 1984!. Zqs.2.18 and 19 with a = 1 can be rewritten in a matrix form as

mixture as a whole and in the fluid phase first.. Masses of a solid phaseand fluid phase in a unit volume of mixture are respectively l-n!ps and

npg. Therefore, the equilibrium conditions in mixture as a whole are

described as

a, + I-n!p,b, + np,b. - � � n!p,ii, - np,U. =0 �.25!

w =n U- u! =k.. h.<J ~ 3 � 26!

where "~ = 3 w U u!/Bt; k;j = component of permeability tensor; h>j =gradient of fluid head in the jth direction. Under the steady state

conditions, the gradient of the head is composed of pressure gradient, bodyforce gradient and inertia of fluid. This is written as

h. =p, + p<b, - pfU, �,27!

Combining Eqs. 2.26 and 2.27, equilibrium conditions in fluid phase can bedescribed as

p . + p<b, = pfU, + k,. n U. - ia,! � 28!

Substitution of AU; in Eq. 2.28 into Eq. 2.25 yields

14

where b; = ith component of body force per unit mass; u = B~u/Bt2; and U =

32U/Bt2. The equilibrium conditions of fluid phase will be developed from thegeneralized Darcy's equation described by

o; + pb, - I-n!p,ii, + k,. n U. - u.! -np, - npfb. = 0 � 29!

Using a;> expressed by Eq. 2.7, Zq. 2.29 results in equilibrium condition in

solid skeleton expressed as

a; + I n!p;+ I n!pub; I n!p ii+k; n U; i.! =0 �.30!

Hereafter, Eqs. 2.30 and 2. 28 will used for two equilibrium equations

for two phase mixture. Those will be rewritten in a matrix form as

T T TfIU a'! + �-n! ~! p+ !-n!p, b! - !-n!p, !!+ n' k! 1 !! � ! = 0 �.3 !

~! p+ pf ] pf U j >I:kl � - u! = �! � 32!

@uk D-l fo'J + I-n! V! p+ Ii-n!p, b! - I-n!p, ii! +n Ik] U-uj dQ=0�.33!

where < = the domain of interest; �ul = components of virtual

displacement compatible with u!. Similarly, application of virtual workprinciple to Eq. 2.32 results in

15

�! Finite Element Formulation for Numerical Evaluation

The principle of virtual work requires that for an arbitrary virtual

displacement, the work done through Eqs. 2.31 and 2.32 over the domain of

interest must be equal to zero. This requirement on. Eq. 2.31 is given by

The domain is divided into finite elements. Within an element, the

variables are approximated in terms of the values of variables at nodal

points such that

u! = [K u! <! = ll:N! U! �.39!

where [+ = matrix of interpolation function. for {u! and U!; <! and U! =

vectors of nodal variables of u! and U!, respectively. Then, since ~ is

function of spatial coordinates only, it is also approximated that

� 40!

Similarly, �u! and. {5U! are approximated by

{5u! =PU{5u!. {5U! = PU{5U! �.41!

Using j de6ned by Eq. 2.6a with w! = n{U-u!, Eq. 2.22 can be

rewritten as

dp = Q r da + dg!= $ r LJ�u + n ~ dU - n V �u !=Q l-n!{V! du! +Qn{7! dU! �.42!

17

Substitution of Eq. 2.42 and Eqs. 2. 39 through Eq. 2.4L into Zqs.2. 35 and 2.36leads to

Avl U! - C,] !+ C,! U! :K2] !+t:K3] ~! = <v! � 44!

where

t'MJ = �- !p, N] N] dQ

Mvl=~pf W N] d~T

C,] =n 'N] I'k] [N] dQ

T

[K,] =�-rl! Q <! [N] �! IN] dQ

[K ] = !-n!nQ|t[V! [N]! V! [N] dQ

T

[K,]=n Q ~! [N] V'! >]aO

f ! = l-n!P, [NJ b! dQ+ [N] t! dI + I-a! {iV] p! dlA r r

fv! = npf [N] �! dQ+ n [N] f!! dI

18

[Mg u! + [C] u! - [C~] !! +f[Bg u! dQ+ [K] [u! + [K ] U! = [ ] Zu3!

Combining Eqs. 2.43 and 2.44, we have

[M] dvj + [Cj dvj + [K] dvj - O] c9.j = dfj

where 0 M�

19

dh

dv! =dU

df� >! =

dfU

Incremental equation of equilibrium can be written as

[M] hdv! + [C] duiv! + [K] kdv! - Pg &! = hdf! �.45!

�! ShessWtxain Constitutive Equation

Elasto-plastic behavior is considered in the stress-strain relationship

It is assumed that the following requirements are met:

of the porous skeleton. In this report four di8erent yieM criteria, the

Trescas, Von Mises, Mohr-Coulomb and Dru.cker-Prager criteria, are

employed. The latter two are applicable to concrete, rocks and soils.

l. An explicit relationship between stress and strain must be

formulated to describe material behavior under elastic conditions,

i.e. before the onset of plastic deformation.

2. A yield criterion indicating the stress level at which plastic flow

comments must be postulated.

3. A relationship between stress and strain must be developed for post

yield behavior, i.e. when the deformation is made up of both elastic

and plastic components.

Before the onset of plastic yielding, the relationship between stress

and strain is given by the standard linear elastic expression.

�.80!ij ijkl kl

where D;>g1 = tensor of elastic constant. For an isotropic material, D;>g1 isexpressed in the form of

D, ki = X5.�.5�! + u.5.�5.! + ph, 5.� �,81!

where A. and p. =Lame's constants; and 51> = Kronecker delta deemed by

1 if i=j5

0 if iwj �.82!

a. Yield Cxiteria

The yield criterion determines the stress level at which plastic

deformation begins. 76th the stress invariants de6ned by

21

�.84}

the yield criterion can be written in the form of

f{J ', J '! = S s! �.85!

where f and S = some functions to be determined experimentally; s =hardening parameter; J2' and J3' = the second and third invariants of the

deviatoric stresses expressed by

G;. = G.. � � 5.. Zkkij ij 3 lj kk {2.86!

Tr a Id ' ri: This states that the yielding begins when themaximum shear stress reaches to a certain value or

a - cr = S s!1 3 �.87!

for ai > a2 > cd ~here cubi, o2, and a3 = principle stresses.

J '! = S s! �.88}

The second deviatoric stress invariant, J2', can. be explicitly written as

22

J =a..l ii

J = � a..o..12 2 1J 1J

J = � cr, a. cr.13 3 ij jk kl

VnM' ' d

reaches to a value

n: This states that yielding occurs when Jg'

�.89!

Yield criteria, Eq. 2.88 may be further written as

5=&3$ s! � 90!

where < is termed effective stress, generalized stress or equivalent stress,

expressed as

<z= J3 J.'! = � {a,.' cr..'! �.91!

This is a generalization of the Coulomb

friction failure law de6ned by

�.93!

where z and cr�= shear and normal stresses, respectively; c = cohesion; and

p = internal friction. angle. For a'y > a'2 > cry, Eq. 2.93 can be rewritten as

�.94!

or rearranging

cr, � o'>! = 2c cos Q- cz,+a>! sin P �.95!

: This is a modi5cation of the Von loses

23

yield criterion for making it dependent on hydrostatic pressure like the

Mohr-Coulomb criteria. The influence of a hydrostatic stress component on.

�.96!

The yield surface defined by Zq. 2.96 in the principal stress space is a

circular cone whereas that defined by the Mohr-Coulomb is a conical yield

surface whose normal section is an irregular hexagon. The followingexpressions for a and S makes the Drucker-Prager circle coincide with the

outer apices of the Mohr-Coulomb hexagon at any section.:

2 siii PC=

l 3 {3 - sin !! �.97!

6c cos 4

l3 �- sin 4! � 98!

Coincidence with the inner apices of the Mohr-Coulomb hexagon isprovided by

2 siii P

P3 � + sin !! �.99!

S 6c coshR3�+ sin 4! �.10G!

W h St

If f = s represents a yield surface, the elastic and plastic

conditions exist respectively when f < s and f = s. At plastic state, the

24

yielding was introduced by inclusion of an additional term in the Von Misesexpression given

incremental change in the yield function due to an incremental stress

change is

�. 105!

Then if:

elastic unloading occurs elastic behavior! and the stress

point returns inside the yield surface.

neutral loading plastic behavior for a perfect plastic

material! and stress point remains on the yield surface.

df>0 plastic loading plastic behavior for a strain hardening

material! and the stress point remains on the expanding

yield surface.

c. Zlasto-P13stic StxmsHtrain Relationship

da,. = da,.!, + de,,!> �.106!

The elastic strain increment is related to the stress increment by Zq. 2.80.

Decomposing the stress terms into their deviatoric and hydrostatic

components, the elastic component of the incremental strain can be

expressed as

25

After initial yielding the material behavior will be partly elastic and

partly plastic. During any increment of stress, the changes of strain are

assumed. to be divisible into elastic and plastic components such that

dq a = H' q! �. 114!

For the u.niaxial case ay = a, az = a3 = 0!, Eq. 2.91 leads to incremental

effective stress and effective strain expressed respectively as

da = � {da..' da..' ! = da3 13 1J �.115!

1

d =g � { de,,'! de,,'! f =defz�.116!

Substitution of E ps. 2.115 and 2.116 into Eq. 2.114 leads to

da da I E T

�.117!de - dec dada da

Adopting the strain hardening hypothesis, Zq. 2. 85 can be rewrittenas

F a, s! = f a! - S s! = 0 �.121!

DifFerentiating Eq. 2.121, we have

dF = da+ � ds =0BF= aa a. �.122!

or

27

where ET = tangent modulus. Thus, the hardening function H' = can. bedetermined experimentally from a simple uniaxial yield test.

�.123!

where

~F BF 3F BF BF BF a! = Ba,' 3 z ' 3a,' 8 'W '&t�y �.124!

A= � � ds1 8F= ~as �-125!

The vector a! is termed the flow vector.

Eq. 2.112 can be rewritten as

T

de! = D] «!+~Ia~7 �.126!

Premultiplying both sides of Zq. 2.126 by a!T D] and. eliminating a!T ds by

use of Eq. 2.123, the plastic multiplier d~ can be expressed as

a! D] de! �.127!A + a! D] a!

epde =D de �.128!

where

28

Substitution of Eq. 2.127 into Eq. 2.126, with dDT = aTD and dD = Da, results

D'=DA+dD a

7 �.129!

work hardening hypothesis, ds is

ds=o de �.130!

Zq. 2.126 can be rewritten for uniaxial condition as

A= 1 dods~ ds �.131!

Employing the normality condition in Eq. 2.130 results in

ds =&a a' � 132!

Eq. 2.124 for uniaxial condition is

da�.133!

Using Euler's theorezn applicable to all homogeneous function of order one,

we have

�.13$

or

a~~= a �.135!

Bf3�

The uniaxial conditions, cry = a and a~ = ag = 0, are considered. With

Substituting Eqs. 2.133 and 2.135 into Eqs. 2.132 and 2.131, ~ and A are

found to be

�.136!

A=H'� 137!

Thus, A is the second slope of the uniaxial stress-strain curve and can be

deternnned experimentally from Eq. 2. 117.

It is written as

sin 38 =-�2

J~'! ~�.»8!

Then, the total principal stresses can be expressed by

�.139!sin 8+~" !

3

The focus yield criteria can now be rewritten in terms of Jy, J g' and 8 asfollows:

Tresca yield criterion1

2{J>'! = a s! �.140!

Von Mises yield criterion

30

I

2 J2'!sin 8+ 2z!

sin 8

1

1J 3

1

bohr-Coulomb yield criterionI

� J sin P+ J '! {cos 8 - � sin 8 sin g! = c cos $1 13 j.

r3�. 142!

Drucker-Prager yield criterion1

aJ, + J '! =S�.143!

where e and S are given in Zqs. 2.97 through 2.100.

In order to calculate the Dep matrix given in Eq. 2.129, it is requiredto express the flow vector a and the elasticity matrix D in a form suitable for

numerical computation. The explicit form of the elasticity matrix D forplane strain condition can be written as

1 N 0 N

N 1 0 N

�.144!0 0 � -NO2v

N N 0 1

where N = v/ l-v!. The flow vector a for plane strain condition can bewritten as

BF BF BF a! = Ba ' Bcz' >

x y xy {2.146!

The specific form of the vector, a!, is given by

31

a! = C, a<! + C2 q! + C3 Q t',2 147!

where

a ! =�,1,0,1!

a ! = ~~ o�', a�', 2x��, a,' !'7 J '

�.148!

~! = +y' +z' + ~ +x' +z' + +z ~xy ~+x' +y' ~xy +

and the deviatoric stress invariants are

1 2r = ' ~ '! + ~ '! + a '! + ~ '2 z xy {2.149!

�.150!

Tresca yield criterion

C =0I

C> = 2 cos 8 � + tan e tan 36! g.151'

lY sineo 3e

Von Mises yield criterion

32

The constants Cl, C2 and C3 are to de6ne the yield surface and are given asfollows:.

22 Joint Element at Soil-WaH Interface

The nodal displacement vector in the s-n local coordinate system is

defined as

�.157!

and also the strain vector as

�.158!

where y, e and m are respectively average difFerential displacements in s

and n directions and angular opening between the two sides, expressed by

U.+U ~ ti.+usj sl si sk

7 2 2u .+U

nj nl6

U -+u ~ni ak �.159!2

Q -U U,. -Unl nk ~niCO�

34

A discontinuity of the displacements develops at the soil-wall

interface due to large difference between the stiffnesses of the wall and soil

and due to sliding and separation between the waO and soil. The joint

element originally developed by Goodman �976! can simulates such soil-

wall interface behaviors. Referring to Fig. 2.1, the joint element is

formulated in the following:

�.160!

where

02

0I

2

0IL

0I2

02

0I2 �.161!

0 - � 0IL

The stress vector in the joint element is dered as

{ a! = c�,, a�, Mo!T

�.162!

where s�s, a�and Mp are respectively shear and normal stresses and

monMnt, expressed by

g= � f +f!Isj sl

a�= � f, + f,! �.163!

The nodal force vector is de6ned as

�.164!

The relationship defined in Zq. 2.159 can be rewritten in the matrix formsuch that

Eqs. 2.I63 and 2.164 with fs; = -f» result in

� 165! fJ =%] o!

where

0 0L2

0 LL

0 02

L I02 L

0 02

0 L I2 L

0 02

0 L I2 L

�.166!

The stress-strain relations at the soil-wall interface are assumed to

be as shown in Fig. 2.2. Yield shear stress w> is assumed. to be

c + G� tan Q s< 0!

0 e >0!�.167!

where c and ~ = cohesion and friction angle at the soil-wall interface,respectively. Furthermore, the following relationship is assumed:

4L k�m a<0!I 3

0 e >0!�.168!

36

at the soil-wall interface is obtained as

T11

d~! = ~f! p�! du'! = ~][C! L ]{du'! �.169!

where

�,170!

with

�.171!

Thus the stiFness matrix of the joint element in local coordinates is

�.172!

with

k, 0 -ks 0

0 2k 0 -2k�

-k, 0 k, 0 �.173!

37

With Zqs. 2.160 and 165, the incremental force-displacement relation

k, 0 -k, 0

0 0 0 0

k, 0 ks 0

0 0 0 0

�.174!

The displacement and force vectors in the local coordinate system are

expressed by those in the global coordinate system as

f~'l = f2 <!

t I = fT]f~] �.175!

where

ft] fo] f0] f0]

fo] «] fO] f0]

f0] f0] f~] fO] �.176!

fo] fO] fo] ft]

with

cos e sin a

�, 177!-sin a cos m

�.178!

Then, the tangential stiffness matrix in the global coordinate system can be

obtained as

3. TIME I%I'EGRATION FOR SOLVING EQUATION

Incremental equation of equilibrium equation for the problemconsidered is written in the form of

[M] {dii! + [C] du J + [Kj {du J = {dff

For the assumed linear variation of the acceleration. and the correspondingquadratic and cubic variations of the velocity and displacement, the Taylorexpansions at the end of the interval ht leads to the following equations for

increments of velocity and displacement:

huf = tijht hQ!2

�.3!

Hence, Eqs. 3.2 and 3.3 result in the incremental displacement and

incremental velocity expressed in terms of incremental displacement asfollows:

� 4!

�.5!

Introducing Wilson 9 method, Eqs. 3.4 and 3.5 result in

39

where

equation of motion:

�.9!

where

CI = KI+ � MI+NCI

ipj = Af! M] ! + 3 uj +[CI

ASolving Eq.3.9 for ~u, the incremental acceleration ~u is

�. 11!

Auj = � Du! - � u! -3 iij j

and. thus

Substituting Eqs. 3.7 and 3.8 into Eq. 3.1 leads to the following

The initial acceleration for next time step is given as

�. 14!

and the entire process may be repeated for any desired number of fime

increments to compute the response time history.

41

4. VRRIFICATIGNS

4.1 Static Pmblems

Young's modulus E! = 6000 psf

Poisson's ratio v! = 0.4

Coef. of permeability k! = 2.5 x 10< ft/day

Density of water ~ g! = 62.5 pcf

The time factor T is de6ned as

C t

H� I!

where c< = coeffxcient of consolidation; t = tizne; and H = height of the soil

column. The coefBcient of consolidation cv for the conditions considered in

this problem is 5.1428 x 10-2 ft2/day. Fig. 4.2 shows the variation. of the

displacements at surface node A! with time and Fig. 4.3 the variation of

excess porewater pressure along the axis of the column at various times.

Uniformly distributed vertical load is assumed to be applied on the

surface of a saturated homogeneous elastic soil deposits uderlain by

impervious base. This corresponds to a classical one dimensional

consolidation problem. Theoretical solution for this problem was presented

by Terzaghi �921!. For the analysis by the developed program, a soil

column, 7ft. high, and 1 ft. wide, is descretized as shown in Fig. 4.1. Thematerial properties are as follows:

Good agreements between the finite element results and analytical resultscan be seen.

A uniform strip load applied on a Quid-saturated homogeneous

elasto-porous half space is considered next. This problem has been solved

analytically by SchifFman et al. �969!. Finite element descretization of the

problem is shown in Fig. 4.4 and the material properties used are

Young's modulus K! = 1000 KN/m2

Poison's ratio v! = 0.

Unit weight of water p~! = 1000 kg/m3

The time factor T in this problem is defined as

�-~!

where d = half of the width of loaded area; and

2GI<

Pf �.3!

42 ~m:mac Problems

43

with 6 = shear modulus. Fig. 4.5 shows the computed variation of excess

pore pressure, beneath the center of loaded area, at T = 0.1. Good

agreements between the finite element results and analytical results can beseen.

A uniformly distributed vertical load is applied over the surface of a

homogeneous fluid-saturated elasto-porous half space. The following two

different loading time histories, step function and triangular impulse time

histories, are considered as shown in Fig. 4.6. Analytical solutions for

those cases were presented by Simon et al. �984!. In the 6nite element

computation, the maximum intensity of the load is 10 units and material

properties are as follow:

Young's modul~s K! = 3000 units

Poisson's ratio {v! = 0.2

Density of fluid {pf! = 0.2977

Density of solid p~! = 0.3101

Porosity n! = 0.333

Bulk density {p! = 0.306

CoeKcient of permeability k! = 0.004883

Bulk stiBness of solid grain K,! = 0.5005 x 104

Bulk stiffness of fluid +! = 0.6106 x 105

The responses are computed for the step loading time history by

using the finite element descretization shown in Fig. 4.7 with soil depth = 60units. The time step ht! is 0.002 time units. Figs. 4.8 and 4.9 show

respectively the comparisons of the solid and. fluid displacements at the

surface node A!. Figs. 4.10 and 4.11 show respectively the variations of

solid and fluid displacements along the depth at t = 0.06 time units.

The responses are also computed for the triangular impulse loading

time history by using the finite element descretization as shown in Fig. 4.7

with soil depth = 120 units. Material properties are same as those

aforementioned except K, = ~ for the present problem. The loadingduration = 2 d to! is 0.021 time units. The time step ht! is 0.002 time units.

Fig 4. 12 sho~s the variation of displacement of solid and Fig. 4. 13 showsvariation of f1uid displacement at surface node A! with time. Variations of

the responses along the depth at t = 0.12 units are also shown in Fig. 4.14through 4.17 for soiM and Quid phases.

5. ZXAiVIPLE OF COMPUTED BEHAVIOR OF WATERFRONT SHZETPILE WALL clL'B JKCTED TO ErMTHQUAKE SKARBNG

In 1983, Nihonkai Chubu Earthquake of magnitude 7.7 hit northern

part of Japan. The earthquake caused damages to quay walls at the Akita

Port located about 100 km from the epicenter. Damages were associated

with liquefaction in backfill sand. Fig. 5.1 shows a cross section view of a

sheet pile of Oohama No. 2 wharf, at which the water depth is 10 m. This

sheet pile was displaced about 1.1 m to 1.8m towards the sea and was

cracked due to excessive moment at about 4m above the sea bed. Sand boils

and settlexnents wexe observed at the apron, indicating liquefaction in

bacldill induced during the earthquake shaking.

Based on information given by Tsuchida, et al. �985!, the soil at the

sheet pile is divided. into four homogeneous layers as shown in Fig. 5.1. Theproperties of each layer are G~~ = 65030 kpa, max. bulk modulus of soil

skeleton = 173400 kpa, P = 370, Kp = 2x106 kpa. The sheet pile is FSP-VEg type

of which second moment of area = 8.6xl0-4, cross section area = 0.306 m2

and density = 7.5 tfm3. The 6nite element mesh used is shown in Fig. 5.2.

The tie bar is idealized as massless beam elexnents and hinge connections

are assumed at the connections to sheet pile and anchor piles. Two anchor

piles and sheet pile are idealized by beam elements. The top ends of two

anchor piles are capped with a rigid mass.

Earthquake ground motions, recorded at nonlique6ed ground. at the

Akita Port, were used to determine input xnotions. The maximum

accelerations recorded are 190 gal and 205 gal in the NS and EW

seconds corresponds to the end of shaking. The stress due to the

maximum bending computed is 320,000 kpa whereas the failure strength of

steel of the sheet pile is 300,000 kpa. Seismic eFects significantly increasethe earth pressure and bending moment.

48

Anzo, Z., 1936, "Dynamic Water Pressure against Retaining Walls duringEarthquake," in Japanese!. Proc. 3rd Aaaual Meeting of Japanese Societyof Civil Engineers.

Ayala, G., Rorno, M.P., and Gomez, R., 1985,"Earthquake Analysis ofFlexible Retaining Walls," Proc. 5th Int. con. Numerical Methods inGeomechanics, Nagoya.

Bolton, M.D. aad Steedmaa, R.S., 1985,"Modelling the Seismic Resistanceof Retaining Structures, " Proc. 11th Int. Conf. Soil Mechanics andFoundation Engrg., Vol. 4, pp. 1845-1848.

Clough, G.W. and Duncan, J.M., 1971, "Finite Element Analysis ofRetaining Wall Behavior," J. Soil Mechaaics and Foundation Engrg.,ASCE, Vol.97, SM 12.

Goodmaa, R.E., Taylor, R.L., and Brekke, T.L., 1968, "A Model for theMechanics of Jointed Rock," J. Soil Mechanics and. Foundation Engrg.,Vol. 94, SM3.

Ishibashi, I., Maisuzawa, H., and Kawamura, M., 1985,"GenerahzedApparent Seismic Coef5cient for Dynamic Lateral Earth PressureDetermination, Proc. 2nd Int. Conf. Soil Dynamics and EarthquakeEngineering, Southampton, Vol. 6, pp. 33-42.

Ishii, Y., Arai, H., and. Tsuchida, H., 1960, "Material Earth Pressure in anEarthquake," Proc. 2nd World Conf. Earthquake Engrg, Tokyo, Japan.

Jacobsen, L.S., 1939, Described in The Kentucky Project, Technical ReportNo. 13, Tennessee Valley Authority, 1951.

Ku.ralowicz, Z. aad Tejchman, A., 1984, "Interaction between Flexible SheetPiling in Cohesioaless Soil," Proc. 6th Conf. Soil Mechanics andFoundation Engrg., New Zealand.

Matsuo, H., 1941, "Experimental Study on the Distribution of EarthPressure Actiag on a Vertical Wall duriag Earthquake Pressure Acting ona Vertical Wall duriag Earthquake, J. Japaaese Society of Civil Engineers,Vol. 27, No. 2.

Matsuo, H. and Ohara, S., 1960, "Lateral Earth Pressure and. Stability ofQuay Walls during Earthquakes," Proc. 3rd World Conf. EarthquakeEngrg., New Zealand.

Matsuo, H. and Ohara, S., 1965, "Dynamic Porewater Pressure Acting onQuay Walls during Earthquakes," Proc. 3rd World Conf. EarthquakeEngineering, New Zealand.

The Minister of Transportation, 1964, "Damages of Ports and HarborsCaused by the Niigata Earthquake," Japanese!.

Mononobe, N., 1929, "Earthquake-Proof Construction of Masnary Dams,"Proc. World Engineering Conf., Vol. 9, p. 275.

Mononobe, N. and Matsuo, H., 1929, "On the Determination of EarthPressure during Earthquakes," Proc. Engineering Conf., Vol. 9, p. 176.

Murphy, V.A., 1960, "The Effect of Ground Characteristics on the AseismicDesign of Structures," Proc. 2nd World Conf. Earthquake Engrg., Vol. 1,pp. 231-247.

Okabe, S., 1926, "General Theory of Earth Pressure," J. Japanese Society ofCivil Engineers, Vol. 12, No. 1.

Potts, D.M. and Fourie, A.B., 1986, A Numerical Study of the ZQ'ects of WallDeformation on Earth Pressures. Int. J. Numerical and AnalyticalMethods in Geomechanics, Vol. 10, pp. 383-405.

Prakash, S. and Basavanna, B.M., 1969, "Earth Pressure DistributionBehind Retaining Walls during Earthquake," Proc. 4th WorM ConfEarthquake Engrg., Santiago, Chile.

Saabzevari, A. and Ghaahramani, A., 1974, "Dynamic Passive EarthPressure Problem., J. Geotechnical Kngrg., ASCE, Vol. 100, GTI, pp. 15-30.

Schifman, R.L., Chen, A.T. and Jordan, J.C., 1969, "An Analysis ofConsolidation Theories," Int. J. Soil Mechanics and FoundationEngineering, ASCE, Vol. 95, pp. 285-312.

Seed, H.B. and Whitman, R.V., 1970, "Design of Earth RetainingStructures for Dynamic Loads," Lateral Stresses in the Ground and Designof Earth-Retaining Structures, ASCE.

Simon, B.R., Zienkiewicz, O.C. and Paul, D.K, 1984, "An AnalyticalSolution for Transient Response of Saturated. Porous Elastic Solid, Int. J.Numerical and Analytical Methods in Geomechanics, Vol. 8, pp. 381-398.

Tajimi, H. 1973, Dynamic Earth Pressure on Basement Wali," Proc. 5thWorld Conf. Earthquake Engrg., Rome, Italy.

Terzaghi, K, 1925, "Principle of Soil Mechanics," Engrg. News Record.

50

Towhata, I. and Islam, S., 1987, "Prediction of Lateral Displacement ofAnchored Bulkheads Induced by Seismic Liquefaction," J. Soils andFoundation, Vol. 27, No.4, pp.137-147.

Tsuchida, H. et al., 1985, "Damage to Port Structures by the 1983 Nipponkai-Chubu Earthquake," Technical Note of the Port and Harbor ResearchInstitute, No. 511, p. 447

Zienkeiewicz, O.C. and Shiomi, T., 1984, "Dynamic Behavior of SaturatedPorous Media; The Generalized Biot Formulation and Its NumericalSolution," Int. J. Numerical and Analytical Methods in Geomechanics,Vol. 8, pp. 71-96.

+W0

4 m ~CO g!

oHW

q

O

' Fig. 2.1 Jaint Element

IFig. 2.2 Stress-Strain Relationship at Soil-WaH Interface

contact ~ ~ seper ation

<ns

OW go~

p =>m =>nCQ

O

'C

C!CO

j X~ XQ~8

+A

Q '4

B~k

O

O bnFG

M

T = 0.01

10050

Excess Pore Pressure psf!

Fig. 4.3 Variations of Excess Pore Pressure with Depthat Various Times Static 1-D problem!

0 0

alyticalu tion

I CQO 8

8

~

Q '49-8~

.8-

4

V

r j

Fig. 4.6 Loading Time Histories

qo

qp

q t!

0

q t!

0 0.012 units

3U3

4

4 O

p -X~ -Xg

~ 8'5 2O O

aA

~8g84

g~

88OPI

OV

0

'0 QCf}

O

-0.02

O cf

a

-0.04 .

0 0.05 0.15

Time

Fig. 4.8 Variation of Displacement of Solid at Surface with Time Step Function Loading!

xl 0-2

0.6

0

0.050 O,l 0

T1m e

Fig. 4.9 Variation of Displacement of Fluid at Surface with Time Step Function Loading!

x10-2

0.20 0.1 0.3

20

40

60

Fig. 4.11 Variation of Displacement of Fluid with Depth at t = 0.06 Step Function Loading!

-0.1

0

Disp1acement of Fluid

x10-2

0

-0.4

0.05 0.10 0.15

Time

Fig. 4.12 Variation of Displacement of Solid at Surface with Time Triangular Impulse Loading!

0.3

0

0.1 50.05 0.10

Time

Fig. 4.13 Variation of Displacement of Fluid at Surface with Time Triangular Impulse Loading!

Total Stress

100

40

120

Fig. 4.16 Variation of Total Stress with Depth at t = 0.12 Triangular Impulse Loading!

-. Pore Pressure '

-5 10 150

40

80

1ZO

Fig. 4.17 Variation of Excess Pore Pressure with Depth at4 = 0.12 Triangu1ar Impulse Loading!

.9

0

Q :8K

C!

C>

C!

LQ

P.

A 8

8 8 OO

cm!node A

0

-43

172

0

-172

. 207

0

-207

10

Tim e sec.!

, Fig. 5.4 Horizontal Displacement Time Histories at Various Locations

em!

194

0

-194

147

0

-147

1510

Tim e sec.!

15.8

0

Fig. 5.5 Vertical Displacement Time Histories at Various Locations

gal!

Fig. 5.6 Horizontal Acceleration Time Histories at Various Locations

189

0

-189,

218

0

-218

147

-147

0

Tim e Sec.!

10

node A

'I

node H

~ nodeEI

clem ent B27.9

0

-27.9

81.8

0

-81.8

91.7

-91.7

48.8

0

-48.8 !10

Time sec.!

Fig. 5.7 Excess Pore Pressure Time Histories at Various Locations

ha

K A

S OCQ

t

kpa m!1180

clem e

0

-1180

100

Time sec.!

Fig. 5.9 Bending Moment Time History at 6 m below the Sea Level

kpa!2000-200

P ressure on the % all

Fig. 5.11 Pressure Acting on Sheet Pile before andat 15 Seconds of Earthquake Shaking