THIN LAYERED ELEMENT METHOD

B2. MOTION OF THE SURROUNDING SOIL

Many wave motion problems regarding soil-circular embedded foundation systems are best described in terms of cylindrical coordinates. The governing Eqs. are thus given as:

(A+2G) 08 _ 2G

ow,

^{+2G Owe}

=

p

o'u,

or r oe

^0,

ot'

(A+2G).!. 08 _2G

ow,

^+2G

ow, =

^p

o'u

^{e (B.1)}

r oe

^0,

or ot'

(A+2G) 08 _ 2G

~(rw )

^+2G

ow, _

^p

o'u,

OZ r Or e r iJ(J -

ot'

where,

2m =.!.

^{OU, _ oUe}

, ri5fJ 0,'

I 0 ( lou,

2w, = -;Or rUe) - -; iJ(J

Lame's constants A and G are complex numbers whose imaginary parts describe internal damping of soil. The soil-foundation system is assumed to be excited in x direction (Figure B.I), and the displacement components of soil are described in terms of potential functions as:

[U,]

_{Un =}^[cose _sinO ⁰

][V,

_v(}

1

_eM!

U, 0 cose v,

(B.2)

in which, Hj2l (ar) is the second kind Hankel function of orderj.

Substituting Eqs. (B.2) and (B.3) into Eq. (B.1), one obtains:

a'G(IjJ+rp)-G-2

d'

(ljJ+rp)_pw2(IjJ+rp)=0 (B.4) dz

d² dW

a2(A +2G)(IjJ-rp) - G-2 (1jJ-rp)+a(A +G)- - pw'(IjJ-rp)

=

O(B.5)

dz dz

Figure B.2: Soil slice

Layer boundary forces-displacements relationship is obtained by using Galarkin's method. First, surface tractions in p-direction on both boundaries (j

=

I,2) of a sub- layer with the thickness H are denoted by Sjp (Figure B.2). Unit nodal-point displacement causes the sub-layer to be deformed into a prescribed shape ofN^{j •} Thus, the equilibrium condition of virtual work is obtained by multiplying the equation of motion (Eq. (B.4), (B.5) or (B.6)) by Nj and integrating it over the entire extent of the element Has:

Where, F^p is the Eq. of motion, and

(B.7)

{1/2-Z/ H N -

j - 1/2+z/ H

(j

=

I) (j =2)

{K

^{jp} =}element stiffness matrix to be obtained,

{U}

⁼nodal point (boundary surface) displacements.

Just for preliminary arrangement, stress components are described III terms of potential functions:

au

= G(~; +~;) ⁼ G[(:~ - ~

^{W)H2 (2) (ar)}

+(~:+~

^{W)HO (Z)(a r)]}

a,e

= G(~~~ ^{+ ~:) =} G[(:~-

~W)H2(2)(ar)+(~:

+ ~

W) Ho(Z\a r)]

(B.8), (B.9), (B.l 0)

Eqs. (B.8), (B.9) and (B. I 0) are rewritten in the following matrix form as:

Hz⁽²⁾ (a y) +Ho(Z)(ay) 0 Hz ⁽²⁾(a y) - Ho (Z)(a y) 0

o

^2H/2)(ay)

(B.11), (B.12)

From Eq. (B.3), displacements are also described in similar marmer as:

Hz (Z)(a y) +Ho (Z)(a y) 0 Hz (Z)(a y) - Ho⁽²⁾(a y) 0

o

^2H1(2)(ay)

(B. 13)

It is noted that matrices in Eqs. (B. 11) and (B.13) are identical with each other, which fact eventually allows Galarkin's method (Eq. (B.7)) to be applied to the transformed Eqs. described in terms of the transformed traction

(0'1 O'z 0'3)

^{and the}

displacement

(t/J -

^If/

t/J +

^If/

W).

The transformed tractions and the displacements are described on the layer boundaries as:

(B.14)

The transformed displacements within the sub-layer are described In terms of prescribed shape functions N^I and N, as:

1 z

N =---

I Z H

1 z

N, =-+-Z H (B. IS)

As an example, Galarkin's method is applied to Eq. (B.S) in which Eq. (B.1S) is substituted. Eq. (B.7) thus is written for j

=

1 as:

r::NJ;dZ= ^a'(A+2G)~ (~I-<p,)+a'(A+2G)~ (~, -<p,)+ ~(r/J,-<Pl)- ~(r/J, -<p,) + a A;

G

(W, -

ff't)- pm' ~ (~, -

<p,)-

pm' ~

(r/J,- <p,)

+G ~ (~-

<p)I,

(B.16)

A similar expression is obtained forj

=

Z. From Eq. (B.14), these expressions for j

=

1 and j

=

Z are arranged in the following matrix form as:

a'(A

+ZG)

H[Z l][rfi, - '1',]

^{+G [ 1}

-I][~I - '1'1] .

6 I Z ~,- '1',

H -

^{1 1}

rfi,-

^'1',

+a[-(A-G)/Z (A+G)/Z][~]_pm' H[Z l][rfil

^-'I',

]=[~II]

^(B.17)

- (A

+ G)/Z

(A -

G)/Z

W,

6 1 Z

rfi,-

'1', ^0"1

Similarly, one obtains the following equation from Eq. (B.4):

(a'[A,r

^+[G,r

-W'[Mr){~+q>}' = {O'2}

^(B.18)

where,

[

A

]'

=GH-

H

[2 1] []'

G =-

G [ 1 - I]

, 6 12 ' ., H-I 1

[Mr=PH~[2 I], {~+tpr=[~I+tpl]

6 1 2 ^~2 +tp,

The above matrices with superscript e imply that they are expressions for a particular sub-layer element. Using similar notations, transformed Eqs. (B.5) and (B.6) are written as:

(a'[Apr

^+[G,r

-W2[Mr){~-tpr -a([Brf {Wr = {a\}

-a[Bn~-tpr

+

(a2[A,J' +(A+2G)~G n-w2[Mr]{Wr

⁼

{iT,}

(B.19), (B.20)

where,

The global equation of motion is then assembled with all element matrices provided.

In this procedure, the layer-boundary displacements are arranged in the following order as:

{~-tp} =(~I -tpp~, -tp"""~N -tpN)'

{~+

tp} = (~1

+

tpp~,

+

tp,," "~N

+

tpN)' {W} = (rJ';.w, ,... ,W

^N)'

(B.21)

In which, the subscripts, 1,2, ... , N denote the numbers of layer boundaries starting from the ground surface. The displacement vectors are rewritten by using simpler notations as:

{qi-tp} = {X}, {qi+tp} = {y}, {w} = {z}

^(8.22)

The global equations of motion are thus obtained by putting element matrices with the superscript e at the proper positions in the global matrix (Fig. 8.3) as:

(/3' [A, j

+

[G,j-

m'

[MJ){y} = 0

(a'[ Ap]

+

[G,j- m'[ MJ){X} - a[BnZ} = 0 - a[B]{X}

+

(a'[ A, j

+

[Gp]- m'[MJ){Z} =

0

Figure B.3: Global matrix

(8.23), (8.24), (8.25)

The above equations are solved in the manner of an eigen-value problem, and it is noted that Eq. (8.23), of which eigen values are denoted by /3, is completely independent of the mutually coupled Eqs. (8.24) and (8.25) having the common eigen-values of

a .

Elimination of

{Z}

in Eqs. (8.24) and (8.25) leads to:

(a.8.

+

a'

8, +8^0){

X} =

0 8.

= [A,j([BtnAp]

8, =

[A,j([BtnG,j+[Gp]([BtnAp]

- m'[A,]([ Btr [M]- m'[ M]([ Bt nAp] - [Bj 8

⁰

= ([Gp] - m'[ M])([ Bt n[G,j- m'[MJ)

A similar expression is also obtained by eliminating

{X}:

(8.26), (8.27)

(a'S, +a

²

S, +So){Z}

⁼

0

s, = [Ap][BnA,]

S,

= [Ap][BnGp]

+

[G,][BnA,]

- (i)2[

Ap][ Bn ^{M] -}

^(i)2[

M] [Bn A,] - [Bj'

So = ([G,] -

^(i)2[

Mj)[ Br'([ ^G

^{p]- (i)'}

^[MJ)

Eq. (B.26) is written in the following 2Nx2N characteristic Eqs.:

([0 So] '[So 0 ][{X}]

So S, -a 0

-S,

{X}

^{=0 (B.29)}

Since Eq. (B.29) contains complex Lame's constants, 2N complex squared eigen- values, a', are obtained. This means that there exist total 4N solutions of a .The condition that the soil displacement converges on zero at r

--+

⁰⁰ requires that the appropriate eigen-values must have negative imaginary parts. Thus, imposing this requirement yields the number of appropriate eigen-values to be cut by half (2N).

B3. BOUNDARY CONDITIONS ON THE UPRIGHT CYLINDRICAL HOLLOW

A foundation with a circular cross-section is assumed to be embedded upright in the stratified soil. Thus, the force-displacement relationship is to be obtained on the wall of the cylindrical hollow. Since the displacement components on the wall are proportional to either cose or sine, displacements are described in terms of (v,

VB

^v,). Thus displacement vectors,

{v,} , {VB}' {v,,},

used in this formulation contain layer boundary displacement components. These displacement vectors are expressed in terms of the eigen-vectors ofthe stratified soil as:

{v,}

=~

I{H,(2)(aRa)-Ho1')(aRa)j{X}aqa

2 ^a

+ ~

I {H,(

²

)(/3 Ra)

⁺

H

^O(2)

(/3 Ra)){Y}

^pqp

p

{VB}

^{= ~}

I {H,I2)(a Ra)

+

H

^O

(2)(a Ra)j{ x} aqa

2 a

+ ~

I {H

²

1')(/3 Ra) - H ^o

⁽²⁾

(/3.Ro)){Y}

^pqp

p

{V,}

⁼

I H,i2) (a Ra){Z}aqa

a

(B.30)

where,

{X}

^a and

{z}

^a are the mutually coupled eigen-vectors corresponding to an eigen-value

a,

and the eigen-vector

{f}

p corresponds to an eigen-value fJ. The effective contributions of these eigen-vectors are denoted by qa and qp, respectively.

A modal matrix is defined as:

[X]=[{XL, {Xl, ,{XL]

[f] = [{fL, {fl" ,{fL ] [Z]=[{Z},. {Zl,. ",,{Zl,N]

(B.31)

Where, the dimensions of matrices

[X]

and

[Z]

are

N

x

2N,

whereas that of

[Z]

is Nx N. Eq. (B.30) is further simplified by virtue of the mathematical conveniences of Hankel functions, which are describedas:

H,(2)(a Ro)+Ho(2)(a Ro)

=

2 H,(2)(a Ro)

a To

H,c2)(a Ro)-Ho(2)(a Ro) _ 2

H1(2)(a Ro) - fa

aRo

(B.32)

Introducing the newly defined parameters shown above, Eq. (B.30) is rewritten in the following matrix form as:

[{v,}] [[X][fa] ^[Y] ][{i1a}]

{Ve} = [X] [Y][ I

p] {_}

{v,} [Z][aRo]

0 qp

(B.33)

It is assumed that the cross-section of the cylindrical hollow is kept completely circle.

This assumption requires the following Eq. to be satisfied.

(B.34)

Substituting Eq. (B.34) in Eq. (B.33) yields:

[X][l

+

fa ]{i1a}

^{+[Y][l +}

fp]{i1p}

^{=0 (B.35)}

From Eq. (35),

{i1a}

is described in terms of

{i1p}

^as:

Where, matrix [EJ has the dimension of N x2N. Eq. (B.33) is degenerated into 2N x2N matrix form as:

[{V,.}]=[[X][Ia]+[Y][E]].{_} =[J ]{-}

{v,} [Z][aRo]

qa ^H qa (B.37)

Tractions (p, PeP

J

on the wall of the cylindrical cavity are expressed as:

[:::] =

_[cose sine 0

1[~]

u" 0 cose P,

(B.38)

From Eq. (B.l3), one can describe the tractions in terms of the transformed displacements ~-If , ~+If and Was:

[,(1 0 ( ) ^{VB Ov,) GOV,]}

P,=- /I, -- rv, +-+- +2-

r or r OZ or

r==Ro

= (~ H2(2)(a

Ro) - (A

+2G)aH¹(2)(a Ro)}~ - tp)

+~

H,'2V3Ro)(~+tp)-,1, ~: H/2)(aRo)