topic #5 effective stress concept - reza s. ashtiani...vadose zone: by definition, vadose zone is...

Topic #5

Effective Stress Concept

CE3348 Spring 2019

Exam #1 Grade Distribution

100908070605040302010

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50

40

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Grades

Frequency

CE

33

48

-Geo

tech

nic

al

Eng

inee

rin

g

Ph

ysic

al P

rop

erti

es o

f So

ils

Weight-Volume Relations Phase Diagrams

Soil Texture Atterberg Limits

Aggregate Geometry (AIMS)

Soil Classification USCS Method

AASHTO Method

Soil Compaction

Mec

ha

nic

al A

na

lysi

s o

f So

ils

Effective Stress Pore Water Pressure Concept

Geostatic Stresses Effective Stress

Calculations

Flow of Water in Soils

One Dimensional Flow Theory Darcy’s Law

Two Dimensional Flow Theory Flow Nets

External Stresses Boussinesq Theory Newmark Method

Westergaard Theory

Shear Strength of Soils

Mohr-Coulomb Theory

Direct Shear Test

Triaxial Tests

Consolidated Drained (CD)

Consolidated Undrained (CU)

Unconsolidated Undrained (UU)

Mohr Circle

Settlement Analysis

Immediate Settlement

Primary Consolidation

Secondary Compression

Time Rate of Settlement

Decomposition of Total Stress to

Effective Stress and Pore Water Pressure

This slide shows a saturated soil element carrying a total vertical stress of σz. This

total stress is carried by both the pore water pressure, u, and the effective stress,

σ’z. The effective stress is the portion carried by the interparticle contact forces.

4


Case 1: Dry Sand under Load (Q)

The load applied at the surface of the soil is transferred

to the soil grains in the mold through their points of

contact.

If the load is quite considerable, it would result in the

compression of the soil mass in the mold.

If the cross sectional area of the cylinder is A, the

average stress at any level X-Y may be written as:

Since this stress is responsible for the deformation of the

soil mass, it is termed the intergranular or effective stress.

We may therefore write:

A

Qa

' a

5


Case 2: Fully Saturated Sand under Load (Q)

Closed Valve

If the same load Q is placed on the piston, this load will

not be transmitted to the soil grains in the same way as

in the case 1.

If we assume that water is incompressible, the external

load will be transmitted to the water in the pores.

The pressure that is developed in the water is called the

pore water pressure or neutral stress (u). This pore

water pressure prevents the compression of the soil

mass. For this specific case, pore pressure can be

calculated as:

A

Qu

6


Case 3: Fully Saturated Sand under Load (Q)

Open Valve

If the valve provided in the piston is opened, immediately there

will be expulsion of water through the hole in the piston. The

flow of water continues for some time and then stops.

The expulsion of water from the pores decreases the pore water

pressure and correspondingly increases the intergranular

pressure. At any stage the total pressure Q/A is divided between

water and the points of contact of grains. A new equation may

therefore be written as:

Total Pressure = Intergranular Pressure + Pore Water Pressure

Final equilibrium will be reached when there is no

expulsion of water. At this stage, the pore water pressure

u=0. All the pressure will be carried by the soil grains.

Therefore: '

u '

7

Effective Stress Concept

As in other materials, stresses may act in soils as a result of an external load and the volumetric weight of the material itself.

The pressure transmitted through grain to grain at the contact points through a soil mass is termed as intergranular or effective pressure (’). It is known as effective pressure since this pressure is responsible for the decrease in the void ratio or increase in the frictional resistance of a soil mass.

If the pores of a soil mass are filled with water (saturated state) and if the pressure induced into the pore water tries to separate the grains, this pressure is termed as positive pore water pressure (u). The effect of this pressure is to decrease the frictional resistance of the soil mass.

8

P

N

P Interparticle force

P

T

= + u

uA

ΣP'

A

P

c

N

9

Particles in Solid Contact

= + u

11

Summary of Important Points

Effective stress represents the actual intergranular pressure that

occurs between soil particles. This effective stress is the stress

that influences shear strength of the soil and volume changes or

settlements.

Total water pressure at a point is termed the neutral stress (u),

for it acts equally in all directions (hydrostatic stress state).

It is important to recognize that the neutral stress in saturated

state acts to reduce the intergranular stress that develops

between soil particles. This condition frequently has an adverse

effect on the strength of a soil.

12

Capillary Rise and

the Effective Stresses in Soils

13

Definitions of Zones

Notice that the soil can

be saturated above the

phreatic surface in the

zone of capillary rise.

The zone of capillary

rise may be only a

fraction of an inch in

sands, but can be several

feet in clay or silty soils.

The phreatic zone is

that zone below the

ground water table.

14

Menisci Formation in Glass Tubes

The attraction between the water and the

capillary tube affects the shape of the air–

water interface at the top of the column of

water.

For water and glass, the shape is concave

downward; that is, the water surface is

lower at the center of the column than at

the walls of the tube. The resulting curved

liquid surface is termed the meniscus.

15

Capillary Rise

Notice that the pore water pressure above the

free water surface is negative.

w

c

cw

y

d

CosTh

CosTdhd

F

4

4

0

2

Notice that the height of

capillary rise is inversely

related to the tube diameter.

Surface Tension acting around the

circumference of the tube should be

in equilibrium with the weight of the

water rose in the glass tube.

16

Capillary Rise and Particle Size

Fines Silts and Clays

Sands Gravels

Fines Silts and Clays

Sands Gravels

Reminder: The inter-connected voids in particulate soils

resemble nano-tubes that acts as channels of water.

Menisci Radius and

Inclination of Capillary Tube

In the capillary tube, at

the level equal to the free

surface of the water in

the supply pan, the

hydrostatic pressure of

the water is zero.

Hydrostatic pressures

increase below that free

water surface (u= w . Z).

19

Expected Height of Capillary Rise

At normal room temperature (ambient

temperature), the magnitude of the surface

tension is close to 0.005 lb/ft or 0.064 N/m.

– Therefore:

dh

d

CosTh

c

w

c

31

4

In this equation both the diameter of the capillary

tube and the height of capillary rise are in mm.

20

Vadose Zone (Summary)

Ground Water Table (GWT):

Equilibrium elevation at which

the pore water pressure is equal to

atmospheric pressure.

Vadose Zone: By definition, vadose

zone is the zone above the ground

water table (GWT).

Vadose zone can be partially

saturated or fully saturated due to

capillary action.

Observation wells (bore holes) are

used to determine the depth of the

vadose zone.

Notice that the pore water

pressure is negative in

vadose zone. (+)

(-)

U=w.Z

21

Within the zone of capillary rise, a plot of

pore water pressure versus elevation is

simply an extension of the plot below the

groundwater table.

Above the zone of capillary rise, the pore

water accumulates near the particle contact

points and surface tension forces develop

between these pockets of pore water and the

adjacent solid particles. These forces produce

tensile stresses (negative pore water

pressures) in the water. The magnitude of the

negative pore water pressures in this zone

depends on the soil type and other factors,

and can be much greater (i.e., more negative)

than those in the capillary zone.

22

Apparent Cohesion

In partially saturated granular soils, the

surface tension phenomenon contributes to

the strength of the soil mass.

The available water collects between where

soil particles touch, forming a wedge of

moisture but leaving the center portion of

the void filled with air. Thus, an air–water

interface, or meniscus, is formed.

The surface tension in the meniscus imposes

a compressive force onto the soil particles,

increasing the friction between particles and

improves the shear strength of the soil.

The strength gain in partially saturated granular soils due to surface

tension phenomenon is termed apparent cohesion.

23

Pore Water Pressure and Strength

Negative pore water pressure tends to pull the particles

together therefore it will enhance the shear strength of soils.

24

Sand Castle

Sand castle is an example of

how the surface tensional

forces pulls the sand grains

together.

Keep in mind that you need

to wet the sand to be able to

build a sand castle. Dry sand

will just simply flow.

25

geometry

Surface Tension and Water Bug

27

Geostatic Stresses in Soils

28

Imaginary Column of Soil to Compute

the Geostatic Vertical Stress (σz)

In a soil mass, the vertical stress caused

by the soil at a point below the surface is

equal to the weight of the soil lying

directly above the point. Vertical stress

thus increases as the depth of the soil

overburden increases.

The vertical stress can be calculated as

the weight of a “column” of soil

extending above a unit area.

σv1 = γ1H1

σv2 = γ1H1 + γ2H2

σv3 = γ1H1 + γ2H2 + γ3H3

Point 1

Point 2

Point 3

29

Point 1

Point 2

Point 3

(a) Saturated soil due to capillary action (use sat) (b) Submerged soil (use ’)

Where the soil surface is below water (such as in oceans and lakes), the

effective stress should be computed by using the submerged or effective soil

unit weight multiplied by the depth measured from the soil surface.

30

Forces Acting on an Element of Soil When the Seepage Forces

are Acting Vertical Upward

Buoyancy Effect

When a soil exists below the groundwater table,

the submerged soil particles are subject to a

buoyant force resulting from the hydrostatic water

pressure, the same phenomenon that acts

on any submerged solid. Reminder

’ = sat - w Submerged unit weight of soil

31

Variation of Stresses with Depth

Plots of the variations of the total stress, pore water pressure, and effective stress, respectively, with depth for a submerged layer of soil placed in a tank with no seepage.

- u = 32

A piezometer is a device used to

measure static liquid pressure in

a system by measuring the height

to which a column of the liquid

rises against gravity.

Piezometers are widely used to

measure the pressure head (or

more precisely, the piezometric

head) of groundwater at a

specific point.

33

Pore Water Pressure Measurement Piezometers

Types of Piezometers Standpipe Piezometers

The standpipe piezometer is the most basic of piezometer types. It consists of a filter tip joined to a riser pipe. Water

flow from the surrounding soil into the standpipe. Readings are obtained with a water level indicator.

Advantages: Simple, reliable, not electrical, no calibrated components.

Limitations: Accuracy depends on skill of operator; reading requires a man on site; remote reading not possible;

slower to show changes in pore-water pressure.

Vibrating Wire Piezometers

The vibrating wire piezometer is the most commonly used piezometer and is suitable for almost all applications. It

consists of a vibrating wire pressure transducer and signal cable. It can be installed in a borehole, embedded in fill, or

suspended in a standpipe. Readings are obtained with a portable readout or a data logger.

Advantages: Easy to read, very accurate; good response time in all soils; easy to automate; reliable remote readings.

Limitations: Must be protected from electrical transients.

Pneumatic Piezometers

The pneumatic piezometer operates by gas pressure. It consists of a pneumatic pressure transducer and pneumatic

tubing. It can be installed in a borehole, embedded in fill, or suspended in a standpipe. Readings are obtained with a

pneumatic indicator.

Advantages: Reliable, remote reading possible, not electrical, indicator can be calibrated at any time.

Limitations: Accuracy depends on skill of operator; difficult and expensive to automate, so reading requires man on

site; reading time increases with length of tubing; pneumatic tubing can be blocked by condensation if not frequently

charged with dry nitrogen gas.

34

Effect of Ground Water Table (GWT)

A- Water Table Level at Ground Surface

The submerged weight of the soil (’), is

termed the effective soil weight, and the

subsurface stress that results is termed the

effective stress.

Effective stress represents the actual

intergranular pressure that occurs between

soil particles.

If the total weight of soil as it exists above

the water table is t (the wet weight of soil

before its submerged), and the unit weight

of soil at saturated state is sat , the effective

stress can be calculated as: 𝜎′𝑧 = 𝛾

𝑆𝑎𝑡𝑧 -𝛾

𝑤𝑧

𝜎′𝑧 = (𝛾𝑆𝑎𝑡

-𝛾𝑤) 𝑧 = 𝛾′ 𝑧

𝜎′𝑧 = 𝜎𝑧- 𝑢

35

Effect of ground Water Table (GWT)

Example #1

A- Water Table Level at Ground Surface

36


B- Water Table Level above Ground

Surface

In general, to calculate the effective stress,

simply calculate the total soil pressure

(disregarding buoyancy effects) and then

subtract the hydrostatic pressure (the neutral

pressure) at the point being analyzed.

The pore water pressure (u) is the unit

weight of water w multiplied by the depth

below the water table.

𝜎′𝑧 = [𝛾𝑆𝑎𝑡𝑧 + 𝛾

𝑤ℎ] − 𝛾𝑤(𝑧 + ℎ)

= 𝛾𝑆𝑎𝑡𝑧 -𝛾

𝑤𝑧 = (𝛾

𝑆𝑎𝑡-𝛾𝑤)𝑧

𝜎′𝑧 = 𝜎𝑧- 𝑢

𝜎′𝑧 = 𝛾′𝑧

37


Example #2

B- Water Table Level above Ground Surface

38


C- Water Table Level Below Ground Level

Above the water table, the effective soil

weight is the total soil weight, including the

pore water; below the water table, the

effective soil weight is the submerged or

buoyant weight.

The pore water pressure u is the unit weight

of water w multiplied by the depth below

the water table.

𝜎′𝑧 = 𝛾𝑡𝑧𝑎+ 𝛾𝑆𝑎𝑡𝑧𝑏- 𝛾𝑤𝑧𝑏

= 𝛾𝑡𝑧𝑎 + (𝛾𝑠𝑎𝑡-𝛾𝑤)𝑧𝑏 𝜎′𝑧 = 𝛾𝑡𝑧𝑎 + 𝛾′𝑧𝑏

A

za

zb 𝜎′𝑧 = 𝜎𝑧- 𝑢

39


Example #3

C- Water Table Level below Grand Level

40

Horizontal Stress (’h)

Horizontal stresses can be calculated from the vertical stresses.

The ratio of the lateral effective stress (horizontal stress) to

vertical effective stress is called coefficient of lateral earth

pressure (Ko).

'

'

v

ho

pressuresoilVertical

pressuresoilHorizontalK

’z

Z

’h

41

Coefficient of Lateral Earth Pressure (Ko)

In a horizontal, uniform soil mass, the lateral movement of the soil at any

depth is not possible, because the confining pressure is equal in all

horizontal directions, therefore, a state of static equilibrium exists and

the soil is in the at-rest condition.

The magnitude of K0 for a given soil mass is affected by the soil

deposit’s stress history.

Soils that have been subjected to heavy loading at some time in their

history, such as now-dense granular (sand or gravel) deposits would have

had to develop resistance to high lateral stress in order to remain stable.

Deposits that have not been exposed to heavy loading, such as loose

granular soils and soft clays, would not have developed high lateral

strength.

Typically, dense granular soils end up having lower values of K0

compared to loose granular soils and soft clays.

42

How to Calculate Horizontal Stresses

I. Calculate vertical effective stress (’z )

II. Calculate horizontal effective stress (’h )

’h = ’z x Ko

III. Remember that pore water pressure is the same in horizontal and vertical directions (hydrostatic stress state), therefore:

uh = uz = u

IV. Calculate horizontal total stress (h)

h = ’h + u

43

Example #4

The water table in an 8m thick silty sand deposit lies at a depth 3m below the ground level. The entire soil above the water table is saturated through capillary rise. The saturated unit weight is 18.8 KN/m3. Plot the variation of total and effective vertical stresses and pore water pressure with depth.

5 m

3 m

44

5 m

3 m

0

1

2

3

4

5

6

7

8

-50 0 50 100 150 200

De

pth

(m

)

Stress (kPa)

Total Stress (kPa)

Pore Water Pressure (kPa)

Effective Stress (kPa)

Note the negative pore water

pressure above the ground level

due to capillary action.

sat 18.8 KN/m3

w 9.81 KN/m3

z (m) v u 'v

0 0 -29.43 29.43

1 18.8 -19.62 38.42

2 37.6 -9.81 47.41

3 56.4 0 56.4

4 75.2 9.81 65.39

5 94 19.62 74.38

6 112.8 29.43 83.37

7 131.6 39.24 92.36

8 150.4 49.05 101.35

45

𝜎𝑇𝑜𝑡𝑎𝑙 = 𝛾𝑖𝑍𝑖

𝑛

𝑖=1

Example #5

Calculate and Plot the variation of total, effective

vertical stresses, and pore water pressure with

depth for the following soil profile.

4 m

4 m

2 m

5 m

sat=19.8 KN/m3

sat=21.3 KN/m3

t=17.3 KN/m3

sat=17.8 KN/m3 Soil Layer #1

Soil Layer #2

Soil Layer #3

46

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 50 100 150 200 250 300 350

De

pth

(m

)

Stress (kPa)

Total Stress (kPa)

Pore Water Pressure (kPa)

Effective Stress (kPa)

The slope of the pore water pressure plot

stays constant with depth.

z (m) v u 'v

0 0 0 0

1 17.3 0 17.3

2 34.6 0 34.6

3 51.9 0 51.9

4 69.2 0 69.2

5 87 9.81 77.19

6 104.8 19.62 85.18

7 124.6 29.43 95.17

8 144.4 39.24 105.16

9 164.2 49.05 115.15

10 184 58.86 125.14

11 205.3 68.67 136.63

12 226.6 78.48 148.12

13 247.9 88.29 159.61

14 269.2 98.1 171.1

15 290.5 107.91 182.59

17.3

17.8

19.8

21.3

Notice the change in the slope

of the total stresses and

effective stresses due to

different layer properties.

47

Example #5-Solution

Example #6

Sensitivity analysis for the variation of effective

stress for different values of saturated unit

weight in the second layer.

4 m

4 m

2 m

5 m

sat= Variable

sat=21.7 KN/m3

t=17.3 KN/m3

sat=17.8 KN/m3 Soil Layer #1

Soil Layer #2

Soil Layer #3

48

Keep in mind that some of the assumed values of the saturated unit weights in this

example are not realistic and are exaggerated. The purpose of this example is to

demonstrate the impact of varying saturated unit weight on the distribution of the

effective stresses in the soil layer.

10 15 20 25 30 35

A B C D E F

z (m)

0 0 0 0 0 0 0

1 17.3 17.3 17.3 17.3 17.3 17.3

2 34.6 34.6 34.6 34.6 34.6 34.6

3 51.9 51.9 51.9 51.9 51.9 51.9

4 69.2 69.2 69.2 69.2 69.2 69.2

5 79.19 79.19 79.19 79.19 79.19 79.19

6 87.18 87.18 87.18 87.18 87.18 87.18

7 87.37 92.37 97.37 102.37 107.37 112.37

8 97.56 112.56 127.56 142.56 157.56 172.56

9 117.75 147.75 177.75 207.75 237.75 267.75

10 147.94 197.94 247.94 297.94 347.94 397.94

11 159.83 209.83 259.83 309.83 359.83 409.83

12 193.42 243.42 293.42 343.42 393.42 443.42

13 248.71 298.71 348.71 398.71 448.71 498.71

14 325.7 375.7 425.7 475.7 525.7 575.7

15 424.39 474.39 524.39 574.39 624.39 674.39

3 (kN/m3)

'v (kPa)

49

Example #6-Solution

Notice the change in the magnitude and the slope of the effective stress distribution lines

with increasing values of saturated unit weight in the second layer.

Soil Layer #2

7

7.5

8

8.5

9

9.5

10

0 50 100 150 200 250 300 350 400 450 500

De

pth

(m

)

Stress (kPa)

A

B

C

D

E

F

50

Example #6-Solution

topic #5 effective stress concept - reza s. ashtiani...vadose zone: by definition, vadose zone is...

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