SD Lecture06 Infinite Media

7/28/2019 SD Lecture06 Infinite Media

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Soil Dynamics

Lecture 06

Stress Waves in Infinite Media

Luis A. Prieto-Portar, August 2006.


2/19

The equation of motion of a stress wave in an elastic medium.

Consider an element of an elastic medium, as shown below, with all the possiblestresses on each of its six faces.

x


3/19

The equation of motion can be found through a summation of the forces along all

three axes, and using Newtons second law (F = ma),

( )( )( )2

2

yxzxxx x zx zx yx yx

yx

Consider the displacement u in t he x direction,

dx dydz dz dxdy dy dxdz x z y

udx dy dz

t

Simplifying, and expanding to all three axes,

x

+ + + + +

=

+

2

2

2

2

2

2

x zx

xy y zy

yzxz z

uy z t

v

x y z tw

x y z t

+ =

+ + =

+ + =


4/19

Compression stress waves (P-waves, or Primary waves or Dilatational waves).

The stress wave of motion in the x-direction was developed on the previous slide,

( ) ( ) ( )

( )

2

2

2

2

2

2

2

2

2

yx zxx

yx yx zx zx x x

x yx zx

yx zx

x

u

t x y z

Re call that ,

G and G and G therefore ,

u

G G Gt x y z

and again recall that ,

v u u w

andx y z x

u v uG G

t x y x y

= + +

= = = +

= + + +

= + = +

= + + +

u wG

z z x

+ +


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( )

2 2 2 2 2 2 2

2 2 2 2 2

2 2 2

2

2 2 2 22 2

2 2 2 2

Simplifying,

u u v w u u uGt x x x y x z x y z

but

u v wx x y x z x

therefore,simplifying and extending to all three axes,

uG G u wheret x x y z

= + + + + + +

+ + =

= + + = + +

( )

( )

22

2

22

2

vG G v gradient squared or " del" squared

t y

wG G w

t z

= + +

= + +


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( )

( ) ( )

2 2 2 2

22 2 2 2

2

2 2 22

22 2 2

2

2

2P

Differentiating these three differential equations w / rt x, y and z and adding,

u v w u v wG Gt x y z x y z x y z

or

G G Gt

Gv wh

t

+ + = + + + + + +

= + + = +

+= =

2P

Gere v

+=

Notice that vp is the primary wave velocity (also known as the compression wave, orthe dilatational wave, or theP-wave).

Compare vp with vc (the longitudinal compression wave along a rod) equal to,

( ) ( )1 1 2

2c P

E

( rem em b er th a t )

E Gv w h ich is o b vio u sly sm a ller th an v

= +

+= =


7/19

Distortional waves (S-waves, or Shear waves).

If we differentiate with respect to y and z (instead of x, as before) we obtain,

( )

( )

2 22

2

2 22

2

22

2

5

w wG G

t y y z y

and

v vG G

t z z y z

Subtracting the sec ond from the first ,w v w v

Gt y z y z

wBut we showed in Lecture # thaty

= + +

= + +

=

2

2

2

2

xx

vz

therefore G t

=

=


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2 2 2 2

2

22 2

2

xx xs s

S

y

s

Therefore,

G Gv where vt

This is the equation that represents the equation for the

distorsional waves and their velocity for propagation is v whichis known as the shear wave or S wave.

Similarly,

vt

= = =

=

2

2 2

2

y

zzsv

t =


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Thus far, we have derived the equations of motion for primary (or P-waves) and

shear (or S-waves). We have also found that they travel at different velocities.

( ) ( ) ( )

( )( ) ( )

( )

( )( )

2

1 1 2 1

1

1 1 2

2 1

2 1

1 2

P

P

s

P s

P

s

P S

G E Ev however and G

therefore,

Ev

Similarly ,

G Ev

Combining both v and v ,

vThe next slide shows the result of plotting this ratio.

v

Note that for all values of , the ratio v / i

+= = =

+ +

=

+

= =+

=

s always greater than unity.


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12/19

Biot (1956) studied the effects of wave propagation through saturated soils (that is,

through the skeleton of solids with the pores filled with water).

This study showed that there were two compressive waves and one shear wave

through the saturated media. The two compressive waves were construed to befluidwaves (that is, transmitted through the fluid) and aframe wave (transmitted throughthe skeleton of solid particles). Obviously, the shear wave can not flow through the

water which has zero shear capacity. Therefore, the shear wave is solely dependent on

the properties of the soil skeleton.

The next slide shows Biots theory prediction of the compressive frame wave velocities

in dry and saturated sands, performed by Hardin and Richart in 1963 with quartz

sands. In addition, as a comparison, are the plots for the experimental longitudinal

wave velocities for dry and saturated Ottawa sands. At equal confining pressures, the

difference of the wave velocities between dry and saturated samples is negligible. This

small difference may due to the unit weight of the soil.

The velocity of the compression waves vw through water isand is about 4,800 ft/s, and whereBw is the bulk modulus of

water andw is the density of water.

ww

w

v

=


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vp

vc


14/19


15/19

Example #1.

Calculate the propagation velocities for theP,SandR waves in the figure shown below.The letters mark the time of arrival. What are the maximum strains if the maximum = 0.5 in/s for theR wave at 1,000 feet and the particle velocities are 0.17, 0.25 and 0.50in/s respectively?

AtR = 1,000 ft Parrives at 0.13 secSarrives at 0.29 secR arrives at 0.71 sec

R = 1,000 ft


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(a) The propagation velocity is the ratio of the distance from the blast (1,000 feet)

divided by the time of arrival of the first part of each particular wave, 0.13, 0.29 and

0.71 seconds for the compressive (P-wave), the shear (S-wave) and the Rayleigh (R-wave) respectively.

Therefore, thepropagation velocities are 7,692 ft/s, 3,448 ft/s and 1,408 ft/s respectively.

(b) Since longitudinal strains are the ratio of the particle and propagation velocities(see slide #15) for the compressive and Rayleigh waves, and the ratio of the particle to

twice the propagation velocity for the shear wave, the strains can be found by dividing

the appropriate particle velocities by the appropriate propagation velocities.

The particle velocities are 0.17 in/s, 0.25 in/s and 0.5 in/s for theP,SandR-waves,respectively.

The resulting strains are 1.8, 3.0 and 30 (in/in x 10-6) respectively.


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Example #2.

Calculate the transient compressive stress for the wave shown in the figure below, if the

elastic modulus of the soil is 8 ksi when its density is = 1.76 g/cm3; assume = 0.2.

Strain in in/in versus time in sec.

Particle velocity in in/s versus time.

Acceleration in in/s2 versus time.

Displacement u in inches versus time.


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( )( ) ( )

3 34 2 4

3 2

2

1 76 2 54 1 121 6 10

453 6 1 1 32 2

1 8 000

1 1 2

mf

m m m f

P

Convert from metric to English units,( . g / cm )( . cm ) ( ft / in )

. x lb s / in( . g / lb )( in ) ( . ft / s )( lb / lb )

The primary ( P wave ) and s hear ( S wave ) velocities are,

E , lb / inv

= =

= =

+

( )( ) ( )

( ) ( )

( )

4 2 4

1 0 2

1 6 10 1 0 2 1 2 0 2

7 453 621

0

0 32

3

0 3 8 000

7 453P

.

. x lb s / in . x .

, in / s ft / s

The max imum particle velocity u . in / s , therefore ,

. in / s , psiuE

v , in / s . psi

+

= =

=

= = =


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References.

Dowding C.H., Construction Vibrations, Prentice Hall, Upper Saddle River, New

Jersey, 1996;

Das, B., Principles of Soil Dynamics, PWS-Kent Publishing Co., Boston, 1993;

Richart F.E., Hall J.R., Woods R.D., Vibrations of Soils and Foundations, Prentice-

Hall Inc., New Jersey, 1970;

Humar J.L., Dynamics of Structures, Prentice-Hall, New Jersey, 1990;

Prakash S., Soil Dynamics, McGraw-Hill, New York, 1981;

SD Lecture06 Infinite Media

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