Quasistatic Poroelasticity - Continuum Mechanics of Anisotropic Materials - page 222

Example 8.10.2
Determine the vertical surface settlement of a layer of poroelastic material resting
on a stiff impermeable base subjected to a constant surface loading T 33 ¼ P

( t ) p o h ( t ). The layer, illustrated in Fig. 8.4 , is in the x 1 , x 2 plane and the x 3

positive coordinate direction is downward. The conditions for the drainage of the
layer are described in Example 8.10.1.
Solution : Since the applied stress at the surface is constant for times greater than
zero, it follows that equation ( 8.65 ) becomes

¼

2 p

@ p

@

c @

t ¼

x 3 ; @

where the initial pore pressure p I is obtained from the formula for the pore
pressure in an undrained isotropic and compressible medium, p

ÞU T

¼ð S = 3 ¼

ð S = 3 Þ tr T where

a
C d K d

S ¼ (8.49)

thus

¼ a

P o
C d K d

p I

p ð x 3 ; Þ¼

0

in one of the initial conditions on the pressure. The other is that @ p =@ x 3 ¼ 0at

x 3 ¼ L .

Assuming separation of variables, p

2

¼ X ð x 3 Þ T ð t Þ , it follows from @ p =@ t ¼ c ð@

x 3 Þ that p =@

2 X

cT @

1 T

@ X @

1

2

t ¼ x 3 ¼b @

or

2 X

T @

1 T

@ t ¼ @

log T
@

@

2 2 X

¼ c b ;

x 3 þ b

¼ 0

t @

thus performing the integration,

2

b

C e c

t

T ð t Þ¼ ; X ð x 3 Þ¼ A cos b x 3 þ B sin b x 3

and

2

b

e c

t

p ð x 3 ; t Þ¼ð A cos b x 3 þ B sin b x 3 Þ :