DISPLACEMENT FORCING OF WALL

y,v

H

0

Homogenous elastic soil (Plane strain)

~-Rotating

wall

Rigid boundary

L

In this form the solution clearly satisfies the displacement boundary conditions at x

=

^{0 and x}

=

L, Substitution of expressions (6.2) into the stress-strain relation for shear stress gives

(6. 3)

From expression (6, 3) it is evident that the stress boundary condi- tion ^(r

=

⁰⁾ at x

=

⁰ and x

=

^L is also satisfied by the assumed

xy

solution. Substitution of equations (6. 2) into equations (6.1) gives

00 2-

I ^{~ aa u~}

n=O y

( 6. 4)

00 2-

I l

^8a

:n

n=O Y

2 < 2

au: ^f

_ .!_

v ⁺

^{k -1)r ~}

k2 n k2 8y cos rx

=

^{u o(k2 2) - p sin px}

Hk2

Fourier series expansiora can be written for sin px and cos px as follows

in which

sin px = a₀

+ !

an cos n;:' n=l

00

cos px =

\ L

n=1

4 1

b . n!rX

n sin~

a =

n -;r(4n2 -1) n=i,2,3, ...

( 6. 5)

b = ⁴ n 2

, n = , , ,

1 2 3

...

n ,,. (4n -1)

Hence for a particular value of n (except the special case of n

=

⁰

which is treated later) equations (6. 4) give

a2u

_n

ay2 -

2 2- r k u -

n

,,- ob k2 2 2 uvn u n p y (k -1) r -

0y- = H

0 2

u a (k -2)p n

(6. 6)

It can be shown by substitution that a general form of the solution of equations (6. 6) is

u a 0

= Hp n [ B sinh ry + C ryery _{n n} + Dnrye -ry +

np~y J

r

u a 0

= __ n_

[-B

cosh ry -

Hp n

in which

k' = 3 - 4v

C (ry-k')ery + D (ry+k')e-ry

n n

(6. 7)

B , C , D n n n = constants determined by the boundary conditions at y = 0 and y = H

The boundary conditions at y = 0 are u = 0 and v = O, Hence from expressions (6. 2)

u

_n ⁽⁰⁾

=

⁰

u a 0

(6. 8)

v

^(O)

=

H n

n p

At y = H, er = 0 and T = O. From the stress-strain relations

y xy

and expressions (6. 2) it is readily shown that these boundary condi- tions require

( aun )

ay--rvn sinrx=O

(6. 9) 2 avn 2 - 0 2

k --..-- + (k -2)ru = u (k -2)pa

uy n n

The first equation of expressions (6. 8) is satisfied directly by expressions (6. 7). From the second equations of expressions (6. 7) and (6.8)

2

B _n = k'(C _n +D) -_n

[(P)

_r j(k2-2) - np(k_r 2

-1)\

^{+1] (6. 1 O)}

Substitution of expressions (6. 7) and (6. 10) into (6. 9) gives

C [ (2rH-k'+i)erH+2k'coshrH] + D [ (-2rH-k'+1)e-rH+ 2k'cosh rH]

n n

= 2d cosh rH - f

n n

(6.11) C [ (2rH-k'-1 )erH+2k'sinh rH] +D [ (2rH+k' +1)e -rH+ 2k' sinh rH]

n n

= 2d sinh rH - rHf

n n

in which

Solution of the linear algebraic equations (6. 11) gives

- (2rH+k'-1)(2dnsinhrH-rHfn)t - 2k'fn(sinhrH-rHcoshrHl]

(6. 12)

+ (2rH-k'-1)(2d cash rH-f ) I + 2k'f (sinh rH-rH cash rHl]

n n f n

in which

l

^{2 2 . 2 2 1}

.6 = 2 1 + (2rH) +k' +2k'(smh rH +cash rH)f

For the special case of n = 0 the partial differential equations reduce to a single equation given by

(6.13)

A solution of equation (6.13) that satisfies the'appropriate boundary conditions at y

=

^{0 and y}

=

^{H is}

( 6. i 4)

The complete displacement solution can be expressed in the

following dimensionless form,

CD

u(x,y) =

y_

cos px + 1 \

o H

Hpi,

a jB sinhry +c ryerY+n rye-ry

n n n n

u n=1

v(x,y)=

u 0

3

+

7

^{\sin rx}

r

a 2

1 .

+

o [HZ 2 (k -2) - - sin px - p

Hp Hp k2

CD

~+ii

2 2H

J

(6. 15)

+-1 \

Hp

L,

a [-B cash ry - C (ry-k')ery+ D (ry+k')e-ry

n n n n

n=1

(in which r

=

^mr

^/L,

^p

=

^{11' /2L).}

The normal stress on the wall er (y) b is readily derived from x

the displacement solution and the stress-strain relations, and can be expressed as

Gu⁰

H (k2_-z) a p

2

0 k

.y

1T

+

I

a:r j zBnsinh ry + C (2ry-k'+3)ery n

n=1

2 . 3 t D (2ry+k'-3)e-ry +k np

YI

n

2 I

r

(6. 16)

The above solution has not been numerically evaluated; how- ever, it is expected that this can be performed with the aid of a simple digital computer program similar to that used in Section 2. 1

for the rigid-wall problem,

Inspection of expression (6.16) shows that the wall pressures for this problem are singular for Poisson's ratio equal to 0.5. It is of interest to note that the formal solution for the case L -

oo

is readily derived by taking Fourier transforms of equations(6 .1) with respect to x. The sine transform is used for the first equation, the cosine for the second, and in principle the solution is obtained in a manner similar to the bounded problem. It does not appear possible to express the Fourier integral of the formal solution in analytical form; however, approximate expressions and numerical methods can be applied to give solutions.

Solutions for forcing by a higher-order displacement function on the wall boundary can be evaluated by assuming a solution of the form

u(x,y) cos px

00

+

'>

_{_'.'-1}

u

_n ^(y)

n=O

sin rx

o

^{m-1)

Ioo

v(x,y) - - _u'--"y=-- sin px

+

vn{y) cos rx mHp

n=O inwhich m=2,3,4, ...

6. 2. STATIC FINITE ELEMENT SOLUTIONS

( 6. 1 7)

Pressures on the statically rotated wall of the problem shown in Fig. 6.1 and studied in the previous section were evaluated using the finite element method. The second-order quadrilateral element and the mesh described previously for the rigid-wall solutions

(Fig. 2. 7) were used. (The previously used antisymmetry is not applicable for this problem and so a mesh extending the full length of the problem was used.)

The wall pressures computed for various values of

~

^and

Poisson's ratio are shown in Figs. 6.2 and 6.:f. The wall forces and moments (about the base of the wall) were computed by numeri- cal integration of the finite element pressure distributions and are plotted in Fig. 6.4. The wall pressures, forces and moments are dependent on the soil elastic moduli (E or G) and the magnitude of the rotation 9. Consequently these variables appear in the dimen- sionless parameters used in the pressure and force plots. (In contrast the rigid-wall pressures and forces are independent of the soil moduli.)

The effect of assuming a bonded contact on the wall boundary was investigated by computing solutions with the wall boundary con-

o

ditions taken as u

=

^{~y and v}

=

0, The bonded contact pressures for

~

= 5.0 and Poisson's ratio values of 0.3 and 0.4 are compared with the smooth wall counterparts in Fig, 6. S. Except in the region at the top of the wall

(g

^> O. 8), where the bonded wall stresses be- come singular, agreement between the smooth and bonded pressure distributions is moderately good. In view of the fact that the develop- ment of the singularity will be limited by the nonuniform and non- linear properties of a typical soil, the difference between the results for the two contact assumptions is unlikely to be of practical sig- nificance.

*The pressure distributions shown in this section do not include stresses resulting from gravity forces in the soil.

SMOOTH CONTACT LENGTH/HEIGHT= I.O

q

•

-.so -.2s o.o o.2s a.so o.1s DIMENSIONLESS NORMAL STRESS

-.50

SMOOTH CONTACT LENGTH/HEIGHT 5.0

q

"l

0

:z: w

,.

,_o

o.o o.2s a.so o.1s DIMENSIONLESS NORMAL STRESS

v. 0·3 0-4 0-45

1.00 1.25

u~/E8

v• 0·3 0-4 0-45

1.00 1.25

u~/E8

Figure 6. 2 Pressure distributions on statically forced smooth rotating wall.

Figure 6. 3

0

0.0