CE 5101 Lecture 5 CE 5101 Lecture 5 CE 5101 Lecture 5 CE 5101 Lecture 5 –––– SETTLEMENTS SETTLEMENTS SETTLEMENTS SETTLEMENTS and Stress Distributionand Stress Distributionand Stress Distributionand Stress Distribution

Sep 2013

Prof Harry Tan

9/11/2013 1

Outline

9/11/2013 2

• Foundation Requirements

• Elastic Stress Distribution

• Concept of Effective Stress

• Settlements of Soils - Immediate, Delayed, and Creep Compression

• Hand Calculations

• SPREADSHEET Calculations (UNISETTLE)

• Finite Element Analysis (PLAXIS)

Requirements for Foundation Design

• Adequate Safety (degree of utilisation of soil strength)

• Acceptable Deformations(Movements and Settlements limits)

9/11/2013 3

What is Adequate Safety(Lambe and Whitman Pg 196)

AA

-2.000 -1.000 0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

9.000

Plastic Points

Plastic Mohr-Coulomb point Tension cut-off point

E

Stress

Strain

2c

Failure Load = 866 kPa

E = 34.48 MN/m2

c=167.6 kN/m2

νννν = 0.3

Yielded

Zone

9/11/2013 4

0 200 400 600 800 1.00E+03

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

Footing Load (kPa)

Footing CL Settlement [m]

Chart 1

Point A

Load vs Settlement Behaviour of Flexible Footing

First

Yield Local Shear

Failure

General Shear

Failure

Settlement (m)

Load (kPa)

9/11/2013 5

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Settlement (m)

Factor of Strength Reduction

Chart 2

Point A

Factor of Safety as Measure of Degree of Utilisation of Soil Strength

FS = 4.338; q = 200 kPa; qult = 868 kPa

FS = 2.164; q = 400 kPa; qult = 866 kPa

FS = 1.291; q = 670 kPa; qult = 865 kPa

9/11/2013 6

What is Allowable Settlements

Effects on:

• Appearance of Structure

• Utility of Structure

• Damage to Structure

9/11/2013 7

Types of Settlements

ρUniform settlement

Non-uniform settlement

Angular distortion = δδδδ/L

δ

L

L

δ

Angular distortion =

δδδδ/(L/2)

9/11/2013 8

Allowable Settlements

Controlled by type of settlements (Sowers 1962)

• Total settlements

• Tilting

• Differential settlements

9/11/2013 9

Limiting Angular Distortions (Bjerrum, 1963)

1/150 - Considerable cracking in panel and brick walls, safe limit

for flexible brick wall h/l<1/4, limit where structural damage of

general buildings is to be feared

1/250 - Limit where tilting of high rigid buildings become visible

1/300 - Limit where first cracking in wall panel expected,

difficulties with overhead cranes expected

9/11/2013 10

1/500 - Safe limit for buildings where cracking is not

permissible

1/600 - Limit of danger for frames with diagonals

1/750 - Limits where difficulties with machinery sensitive to

settlements are to be feared (high tech plants)

Limiting Angular Distortions (Bjerrum ,1963)

Max Distortion (δδδδ/L) vs Max Differential Settlement :

1/500 vs 25 mm

1/300 vs 45 mm

1/200 vs 70 mm

1/100 vs 150 mm

For Stiff Footing,

Max Diff Sett = 1/4 Max Sett

For Flexible Footing,

Max Diff Sett = 1/2 Max Sett

Typical Foundation Design,

Max Diff Sett < 25 mm

9/11/2013 11

9/11/2013 12

Limiting Angular Distortions (Bjerrum ,1963)

200 mm

350 mm

100 mm

• Stiff Footing

max diff sett =1/4

max sett

• Flexible Footing

max diff sett=1/2

max sett

9/11/2013 13

• ER2010 – Review Paper by

Ed Cording et al –

Assessment of excavation-

induced building damage

• Damage is due to combined

effects of:

• Angular distortion; β• Lateral strain

• Bending strain

Basic Soil Phase Relationships

Air

Water

Solids

MASSVOLUME

0

Mw Mt

Ms

Mw

Va

Vw

Vs

Vv

Vt

Densities (kg/m3)

Total, ρρρρt = Mt/Vt

Dry, ρρρρd = Ms/Vt

Solids, ρρρρs = Ms/Vs

Water, ρρρρt = Mw/Vw

Saturated, ρρρρsat

= (Ms+Mw+ ρρρρw Va)/Vt

Ratios

Water content, w = Mw/Ms

Void ratio, e = Vv/Vs

Porosity, n = Vv/Vt

Degree of saturation = Vw/Vv9/11/2013 14

Basic Soil Phase RelationshipsDefine Vs=1, Vv=e

eS ρρρρw = w ρρρρs

Air

Water

Solids

MASSVOLUME

0ρρρρt

ρρρρs

Mw

(1-e)S

eS

1

e

1+e

Densities (kg/m3)

Total, ρρρρt = ρρρρs (1+w)/(1+e)

= ρρρρd (1+w)

Dry, ρρρρd = ρρρρs /(1+e)

Saturated, ρρρρsat

= (Ms+Mw+ ρρρρw Va)/VtRatios

Water content, w = ρρρρt/ ρρρρd - 1

Void ratio, e = ρρρρs/ρρρρd - 1

Porosity, n = e/(1+e)

Degree of saturation = S = wρρρρs / eρρρρw 9/11/2013 15

Common soil mineral densities

Mineral Type Solid Density, kg/m3

Calcite 2800

Quartz 2670

Mica 2800

Pyrite 5000

Kaolinite 2650

Montmorillonite 2750

Illite 27009/11/2013 16

Typical range of saturated densities

Soil Type Saturated Density, kg/m3

Sands;gravels 1900-2300

Silts 1500-1900

Soft Clays 1300-1800

Firm Clays 1600-2100

Peat 1000-1200

Organic Silt 1200-1900

Granular Fill 1800-22009/11/2013 17

The Textbooks on Foundations — They come no better

This is one of the few showing

more than one soil layer

9/11/2013 18

The Reality — With a bit of needed “W” add-on

9/11/2013 19

The Reality — Getting closer, at least

9/11/2013 20

So, with this food for thought,

on to the Fundamental

Principles

9/11/2013 21

Determining the effective stress is the key to geotechnical analysis

• The not-so-good method:

)1(' iwt −−= γγγ

9/11/2013 22

h∆=∆ '' γσ

)'(' hz ∆∑= γσ

γ’ = buoyant

unit weight

It is much better to determine, separately, the total stress and the pore pressure. The effective stress is then the total stress minus the pore pressure.

uz −= σσ '

)( hz ∆∑= γσ

9/11/2013 23

Determining pore pressure

u = γw h

The height of the column of water (h; the head representing the water pressure) is

usually not the distance to the ground surface nor, even, the distance to the

groundwater table. For this reason, the height is usually referred to as the “phreatic

height” or the “piezometric height” to separate it from the depth below the groundwater

table or depth below the ground surface.

The pore pressure distribution is determined by applying the facts that

(1) in stationary conditions, the pore pressure distribution can be assumed to be linear

in each individual soil layer

(2) in pervious soil layers that are “sandwiched” between less pervious layers, the pore

pressure is hydrostatic (that is, the vertical gradient is unity)

SAND

Hydrostatic distribution

CLAY

Non-hydrostatic distribution,

but linear

SAND

Hydrostatic distribution

Artesian phreatic head

GW

DEPTH

PRESSURE

9/11/2013 24

Distribution of stress below a a small load area

9/11/2013 25

)()(0

zLzB

LBqqz

+×+

××=

The 2:1 method

The 2:1-method can only be used for distributions directly under the center

of the footprint of the loaded area. It cannot be used to combine (add)

stresses from adjacent load areas unless they all have the same center. it is

then only applicable under the area with the smallest footprint.

Boussinesq Method for stress from a point load

9/11/2013 26

2/522

3

)(2

3

zr

zQqz

+=

π

Newmark’s method for stress from a loaded area

Newmark (1935) integrated the Boussinesq equation over a finite area and

obtained a relation for the stress under the corner of a uniformly loaded

rectangular area, for example, a footing

9/11/2013 27

π40

CBAIqqz

+×=×=

2222

22

1

12

nmnm

nmmnA

+++

++=

1

222

22

++

++=

nm

nmB

−++

++=

2222

22

1

12arctan

nmnm

nmmnC

m = x/z

n = y/z

x = length of the loaded area

y = width of the loaded area

z = depth to the point under the corner

where the stress is calculated

(1)

• Eq. 1 does not result in correct stress values near the ground surface. To represent the stress near the ground surface, Newmark’s integration applies an additional equation:

9/11/2013 28

π

π

40

CBAIqqz

−+×=×=

For where: m2 + n2 + 1 ≤≤≤≤ m2 n2

(2)

Stress distribution below the center of a square 3 m wide footing

0 20 40 60 80 100

-6

-4

-2

0

STRESS (KPa)

DE

PTH

(m

)

0.01 0.10 1.00 10.00

0.00

0.05

0.10

0.15

0.20

0.25

m and n (m = n)

INF

LU

EN

CE

FA

CTO

R,

I

9/11/2013 29

Eq. (1)

Eq. (2)

Eq. (1)

Eq. (2)

Comparison between Boussinesq, Westergaard, and 2:1 distributions

0

1

2

3

4

5

0 25 50 75 100

STRESS (%)D

EP

TH

(d

iam

ete

rs)

Boussinesq

Westergaard

2:1

0

1

2

3

4

5

0 25 50 75 100

SETTLEMENT (%)

DE

PT

H (d

iam

ete

rs)

Boussinesq

Westergaard

2:1

9/11/2013 30

0

1

2

3

4

5

0 25 50 75 100

STRESS (%)D

EP

TH

(d

iam

ete

rs)

2:1

Westergaard

Boussinesq

0

1

2

3

4

5

0 25 50 75 100

SETTLEMENT (%)

DE

PT

H (d

iam

ete

rs)

Boussinesq

Westergaard

2:1

9/11/2013 31

0

1

2

3

4

5

0 25 50 75 100

STRESS (%)D

EP

TH

(d

iam

ete

rs)

2:1

Westergaard

Boussinesq

Characteristic

Point; 0.37b

from center

0

1

2

3

4

5

0 25 50 75 100

SETTLEMENT (%)

DE

PT

H (d

iam

ete

rs)

Boussinesq

Westergaard

2:1Characteristic

Point; 0.37b

from center

Below the characteristic point, stresses for flexible and stiff footings are equal9/11/2013 32

Now, if the settlement distributions are so

similar, why do we persist in using

Boussinesq stress distribution instead of

the much simpler 2:1 distribution?

Because a footing is not alone in this world;

near by, there are other footings, and fills,

and excavation, etc., for example:

9/11/2013 33

The settlement imposed

outside the loaded

foundation is often critical

0

1

2

3

4

5

0 25 50 75 100

SETTLEMENT (%)

DE

PT

H (d

iam

ete

rs)

Boussinesq

Outside Point Boussinesq

Center Point

Loaded

area

9/11/2013 34

Calculations using Boussinesq distribution can be used to determine how stress

applied to the soil from one building may affect an adjacent existing building.

EXISTING

ADJACENT

BUILDING

NEW

BUILDING

WITH LARGE

LOAD OVER

FOOTPRINT

AREA

2 m2 m 4 m

0

5

10

15

20

25

30

0 20 40 60 80 100

STRESS (%)

DE

PT

H (m

)STRESSES ADDED

TO THOSE UNDER

THE FOOTPRINT OF

THE ADJACENT

BUILDING

STRESSES

UNDER THE

FOOTPRINT

OT THE

LOADED

BUILDING

STRESSES

UNDER AREA

BETWEEN THE

TWO BUILDINGS

9/11/2013 35

Calculation of Stress Distribution

Depth σσσσ0 u0 σσσσ’0 σσσσ1 u1 σσσσ’1

(m) (KPa) (KPa) (KPa (KPa) (KPa) (KPa)

Layer 1 Sandy silt ρρρρ = 2,000 kg/m3

0.00 0.0 0.0 0.0 30.0 0.0 30.0

GWT 1.00 20.0 0.0 20.0 48.4 0.0 48.4

2.00 40.0 10.0 30.0 66.9 10.0 56.9

3.00 60.0 20.0 40.0 85.6 20.0 65.6

4.00 80.0 30.0 50.0 104.3 30.0 74.3

Layer 2 Soft Clay ρρρρ = 1,700 kg/m3

4.00 80.0 30.0 50.0 104.3 30.0 74.3

5.00 97.0 40.0 57.0 120.1 43.5 76.6

6.00 114.0 50.0 64.0 136.0 57.1 79.0

7.00 131.0 60.0 71.0 152.0 70.6 81.4

8.00 148.0 70.0 78.0 168.1 84.1 84.0

9.00 165.0 80.0 85.0 184.2 97.6 86.6

10.00 182.0 90.0 92.0 200.4 111.2 89.2

11.00 199.0 100.0 99.0 216.6 124.7 91.9

12.00 216.0 110.0 106.0 232.9 138.2 94.6

13.00 233.0 120.0 113.0 249.2 151.8 97.4

14.00 250.0 130.0 120.0 265.6 165.3 100.3

15.00 267.0 140.0 127.0 281.9 178.8 103.1

16.00 284.0 150.0 134.0 298.4 192.4 106.0

17.00 301.0 160.0 141.0 314.8 205.9 109.0

18.00 318.0 170.0 148.0 331.3 219.4 111.9

19.00 335.0 180.0 155.0 347.9 232.9 114.9

20.00 352.0 190.0 162.0 364.4 246.5 117.9

21.00 369.0 200.0 169.0 381.0 260.0 121.0

Layer 3 Silty Sand ρρρρ = 2,100 kg/m3

21.00 369.0 200.0 169.0 381.0 260.0 121.0

22.00 390.0 210.0 180.0 401.6 270.0 131.6

23.00 411.0 220.0 191.0 422.2 280.0 142.2

24.00 432.0 230.0 202.0 442.8 290.0 152.8

25.00 453.0 240.0 213.0 463.4 300.0 163.4

0

5

10

15

20

25

0 100 200 300 400 500

STRESS (KPa)

DE

PT

H (

m)

SAND

CLAY

SAND

9/11/2013 36

HYDROSTATIC PORE PRESSURE DISTRIBUTION

Calculation of Stress DistributionSTRESS DISTRIBUTION (2:1 METHOD) BELOW CENTER OF EARTH FILL (Calculations by means of UniSettle)

ORIGINAL CONDITION (no earth fill) FINAL CONDITION (with earth fill and artesian pore pressure in sand)

Depth σσσσ0 u0 σσσσ’0 σσσσ1 u1 σσσσ’1

(m) (KPa) (KPa) (KPa (KPa) (KPa) (KPa)

Layer 1 Sandy silt ρρρρ = 2,000 kg/m3

0.00 0.0 0.0 0.0 30.0 0.0 30.0

GWT 1.00 20.0 0.0 20.0 48.4 0.0 48.4

2.00 40.0 10.0 30.0 66.9 10.0 56.9

3.00 60.0 20.0 40.0 85.6 20.0 65.6

4.00 80.0 30.0 50.0 104.3 30.0 74.3

Layer 2 Soft Clay ρρρρ = 1,700 kg/m3

4.00 80.0 30.0 50.0 104.3 30.0 74.3

5.00 97.0 40.0 57.0 120.1 43.5 76.6

6.00 114.0 50.0 64.0 136.0 57.1 79.0

7.00 131.0 60.0 71.0 152.0 70.6 81.4

8.00 148.0 70.0 78.0 168.1 84.1 84.0

9.00 165.0 80.0 85.0 184.2 97.6 86.6

10.00 182.0 90.0 92.0 200.4 111.2 89.2

11.00 199.0 100.0 99.0 216.6 124.7 91.9

12.00 216.0 110.0 106.0 232.9 138.2 94.6

13.00 233.0 120.0 113.0 249.2 151.8 97.4

14.00 250.0 130.0 120.0 265.6 165.3 100.3

15.00 267.0 140.0 127.0 281.9 178.8 103.1

16.00 284.0 150.0 134.0 298.4 192.4 106.0

17.00 301.0 160.0 141.0 314.8 205.9 109.0

18.00 318.0 170.0 148.0 331.3 219.4 111.9

19.00 335.0 180.0 155.0 347.9 232.9 114.9

20.00 352.0 190.0 162.0 364.4 246.5 117.9

21.00 369.0 200.0 169.0 381.0 260.0 121.0

Layer 3 Silty Sand ρρρρ = 2,100 kg/m3

21.00 369.0 200.0 169.0 381.0 260.0 121.0

22.00 390.0 210.0 180.0 401.6 270.0 131.6

23.00 411.0 220.0 191.0 422.2 280.0 142.2

24.00 432.0 230.0 202.0 442.8 290.0 152.8

25.00 453.0 240.0 213.0 463.4 300.0 163.4

9/11/2013 37

Aquifer with

artesian head

Stress Distribution

0

5

10

15

20

25

0 100 200 300 400 500

STRESS (KPa)

DE

PT

H (

m)

0

5

10

15

20

25

0 100 200 300 400 500

STRESS (KPa)

DE

PT

H (

m)

SAND

CLAY

SAND

Stress from Fill

Artesian Pore Pressure Head

9/11/2013 38

The distribution for the hydrostatic case

Effective Stress Concept

Effective stress = Total stress - Pore pressure

σ σ σ σ’ = σ σ σ σ - u

Unit weight γγγγ = ρρρρg (kN/m3)

g assumed to be 10 m2/s

Total vertical stress or overburden stress,

σσσσz = γγγγ t.z

9/11/2013 39

Pore Water Pressure (PWP)

Pore water pressure u = uw = uss + uexc

uss = Steady state condition, hydrostatic or steady seepage

uexc = Excess pwp due to soil loading

Buoyant unit weight

γγγγ ‘ = γγγγ t - γγγγ w for hydrostatic condition

γγγγ ‘ = γγγγ t - γγγγ w + i γγγγ w

i = hydraulic gradient (head diff/ distance)

i is negative for upward artesian flow9/11/2013 40

Elastic Stress Distribution with Depth

LB

1(H):2(V)zL+z

B+z

Q

qo

qz

BL

Qqo =

))(( zLzB

BLqq oz

++=

• Only can be used for stress at centre of

loaded area

• Cannot use for combined effects of two or

more loaded areas, unless they have same

centres

9/11/2013 41

Boussinesq Distribution (1885)

Q

r

z

qz

Assumes: isotropic linear elastic halfspace,

Poisson ratio = 0.5

For Point load Q kN

2/522

3

)(2

)3(

zr

zQqz

+=

π

Integrate for Line Load, P kN/m

222

3

)(

2

rz

zPqz

+=

π

Integrate for Rectangular Area, get Fadum’s Influence Chart

Fig.1; for Circular Area, get Foster and Alvin Chart Fig.2

9/11/2013 42

Weastergaard Distribution (1938)

Q

r

z

qz

Assumes: isotropic linear elastic halfspace,

Poisson ratio = 0, rigid horizontal layers

For Point load Q kN

2/322 ))/(21( zrz

Qqz

+=

π

Differences with Boussinesq is small

For wide flexible loaded areas Westergaard method is preferred

9/11/2013 43

UniSettle 2.4 21 Jan 2000

EX02.STL page 1

Effective Stress Comparison, ( 4.11 , 8.11 )

-----------------------------------------------------------------

BOUSSINESQ WESTERGAARD 2:1

Depth Ini. Fin. Ini. Fin. Ini. Fin.

Stress Stress Stress Stress Stress Stress

(m) (kPa) (kPa) (kPa) (kPa) (kPa) (kPa)

-----------------------------------------------------------------

Layer 1 Any Soil 0. kg/m^3

0.00 100.0 0.0 100.0 0.0 100.0 0.0

0.10 97.9 0.0 89.3 0.0 93.7 0.0

0.20 94.3 0.0 79.4 0.0 87.9 0.0

0.30 89.4 0.0 71.4 0.0 82.6 0.0

0.40 83.2 0.0 64.4 0.0 77.9 0.0

0.50 76.8 0.0 58.4 0.0 73.5 0.0

0.60 70.9 0.0 53.2 0.0 69.4 0.0

0.70 65.6 0.0 48.9 0.0 65.7 0.0

0.80 61.1 0.0 45.1 0.0 62.3 0.0

0.90 57.2 0.0 41.9 0.0 59.2 0.0

1.00 53.7 0.0 39.1 0.0 56.3 0.0

1.10 50.8 0.0 36.7 0.0 53.5 0.0

1.20 48.1 0.0 34.5 0.0 51.0 0.0

1.30 45.8 0.0 32.5 0.0 48.7 0.0

1.40 43.7 0.0 30.8 0.0 46.5 0.0

1.50 41.7 0.0 29.2 0.0 44.4 0.0

1.60 40.0 0.0 27.8 0.0 42.5 0.0

1.70 38.3 0.0 26.4 0.0 40.7 0.0

1.80 36.8 0.0 25.2 0.0 39.1 0.0

1.90 35.4 0.0 24.1 0.0 37.5 0.0

2.00 34.0 0.0 23.0 0.0 36.0 0.0

Comparison of stresses under a

3m square footing below its

characteristic point

Characteristic point is point

where vertical stress is equal for

both rigid and flexible footing,

this point is located at 0.37B

and 0.37L from centre of

rectangular footing, or 0.37R of

circular footing

Results show that under

characteristic point 2:1 method

is similar to Boussinesq result

9/11/2013 44

AA

BCDEFGHIJKLMNOPQRS

-1.000 0.000 1.000 2.000 3.000 4.000 5.000 6. 000

-0.000

1.000

2.000

3.000

4.000

5.000

E ffective mean stresses

Extrem e eff ective mean s tress -882.98*10-3

kN/m2

[ kN/m2]

A : -0.900

B : -0.850

C : -0.800

D : -0.750

E : -0.700

F : -0.650

G : -0.600

H : -0.550

I : -0.500

J : -0.450

K : -0.400

L : -0.350

M : -0.300

N : -0.250

O : -0.200

P : -0.150

Q : -0.100

R : -0.050

S : 0.000

T : 0.050

Elastic Stress Bulb for Circular Footing

9/11/2013 45

Elastic Settlement for Circular Flexible Load

q

R

D = ∞∞∞∞

Surface settlements is given by Terzaghi,1943 as:

3.),( FigseerfisI

IE

qR

µ

ρ

ρ

ρ=

Edge settlement = 0.7 centre settlement

Centre settlement is :

footingFlexibleforE

Rqz )1(2 2µρ −=

footingRigidforE

Rqz )1(

2

2µπ

ρ −=

9/11/2013 46

Flexible and Rigid Footing

AA*

0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96 1.04

9.76

9.84

9.92

10.00

10.08

10.16

10.24

10.32

10.40

10.48

A

A

-0.300 -0.200 -0.100 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100

9.300

9.400

9.500

9.600

9.700

9.800

9.900

10.000

10.100

10.200

Flexible Footing

E=1000 kPa, µµµµ=0, q=10 kPa, sett= 20 mm

-0.300 -0.200 -0.100 0.000 0. 100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100

9.200

9.300

9.400

9.500

9.600

9.700

9.800

9.900

10.000

10.100

10.200

AA*

0.08 0.16 0.24 0.32 0.40 0. 48 0.56 0.64 0.72 0.80 0.88 0.96 1.04

9.76

9.84

9.92

10.00

10.08

10.16

10.24

10.32

10.40

10.48

Rigid Footing

E=1000 kPa, µµµµ=0, q=10 kPa, sett= 16 mm

9/11/2013 47

Approximate Ratio at corner, centre and edge to average settlement

Flexible Load Area Rigid

Footing

Foundation

Depth

Corner/

Ave

Edge/

Ave

Centre/

Ave

Rigid/

Ave

H/L=∞∞∞∞ 0.6 0.9 1.2 0.9

H/L=1 0.5 0.7 1.3 0.8

H/L=1/4 0.4 0.7 1.3 0.8

9/11/2013 48

Elastic Settlement for Other Flexible Load

B

D = ∞∞∞∞

Corner settlements is given by Terzaghi,1943 as:

4.)/(

)1(2

FigseeLBfisI

IE

qB

ρ

ρ

µρ

−=

Points other than corner for any combination of

rectangles can be obtained by superposition

For centre of square loaded area:

)1(12.1 2µρ −=E

Bqz

qL

9/11/2013 49

Superpostion Principle for Rectangles

= + + +zσ

- - +=

zσ

9/11/2013 50

Elastic Settlement for Other Flexible Load

For Flexible Rectangular Loaded area, Corner settlement ρρρρ,

with finite depth of elastic layer, use Steinbrenner chart

5.

)21()1(

)1(

21

2

2

1

2

2

FigingivenFandF

FFI

IE

qB

µµµ

µρ

ρ

ρ

−−+−=

−=

B

Finite

D

qL

9/11/2013 51

Equivalent Footings for Pile Groups Settlements

L

2/3L

2

1Homogeneous

Clay

Equivalent Ftg

Ground level

L

2/3L

2

1

Soft Clay

Firm Layer

Equivalent Ftg

Ground level

9/11/2013 52

design analysis design analysis design analysis design analysis

isisisis

SettlementSettlementSettlementSettlement

9/11/2013 53

Movement, Settlement, and Creep

Movement occurs as a result of an increase of stress, but the term should be reservedto deformation due to increase of total stress. Movement is the result of a transfer ofstress to the soil (the movement occurs as necessary to build up the resistance to theload), and when the involved, or influenced, soil volume successively increases as thestress increases. For example, when adding load increments to a pile or to a plate in astatic loading test (where, erroneously, the term "settlement" is often used). As a term,movement is used when the involved, or influenced, soil volume increases as the loadincreases.

Settlement is volume reduction of the subsoil as a consequence of an increase in

effective stress. It consists of the sum of "elastic" compression and deformation due toconsolidation. The elastic compression is the compression of the soil grains (soilskeleton) and of any free gas present in the voids, The elastic compression is oftencalled "immediate settlement”. It occurs quickly and is normally small (theelastic compression is not associated with expulsion of water). The deformation due toconsolidation, on the other hand, is volume change due to the compression of the soilstructure associated with an expulsion of water—consolidation. In the process, theimposed stress, initially carried by the pore water, is transferred to the soil structure.Consolidation occurs quickly in coarse-grained soils, but slowly in fine-grained soils. Asa term, settlement is used when the involved, or influenced, soil volume stays constantas the effective stress increases.

9/11/2013 54

Movement, Settlement, and Creep

The term "settlement" is normally used for the deformation resulting from the

combined effect of load transfer, increase of effective stress, and creep during

long-term conditions. It is incorrect to use the term “settlement” to mean

movement due to increase of load such as in a loading test.

Creep is compression occurring without an increase of effective stress.

Creep is usually small, but may in some soils add significantly to the

compression of the soil skeleton and, thus, to the total deformation of the

soil. It is then acceptable to talk in terms of creep settlement.

9/11/2013 55

Strain

Linear Elastic Deformation (Hooke’s Law)

ε = induced strain in a soil layer

= imposed change of effective stress in the soil layer

E = elastic modulus of the soil layer (Young’s Modulus)

Young’s modulus is the modulus for when lateral expansion is allowed, which may be the case for soil loaded by a small footing, but not when the load is applied over a large area. In the latter case, the lateral expansion is constrained (or confined). The constrained modulus, D, is larger than the E-modulus. The constrained modulus is also called the “oedometer modulus”. For ideally elastic soils, the ratio between D and E is:

9/11/2013 56

)21()1(

)1(

νν

ν

−+

−=

E

D

'σ∆

E

'σε

∆=

ν = Poisson’s ratio

Stress-Strain

9/11/2013 57

Stress-strain behavior is non-linear for most soils. The

non-linearity cannot be disregarded when analyzing

compressible soils, such as silts and clays, that is, the

elastic modulus approach is not appropriate for these soils.

Non-linear stress-strain behavior of compressible soils, is

conventionally modeled as follows.

where ε = strain induced by increase of effective stress from σ‘0 to σ‘1Cc = compression index

e0 = void ratio

σ‘0 = original (or initial) effective stress

σ‘1 = final effective stress

CR = Compression Ratio = (MIT)

0

1

0

1

0 '

'lg

'

'lg

1 σ

σ

σ

σε CR

e

Cc =+

=

01 e

CCR c

+=

9/11/2013 58

In overconsolidated soils (most soils are)

9/11/2013 59

)'

'lg

'

'lg(

1

1 1

00 p

c

p

cr CCe σ

σ

σ

σε +

+=

where σ‘p = preconsolidation stress

Ccr = re-compression index

The Janbu Method

])'

'()

'

'[(

1 01 j

r

j

rmj σ

σ

σ

σε −=

9/11/2013 60

The Janbu tangent modulus approach, proposed by Janbu (1963; 1965; 1967; 1998),

and referenced by the Canadian Foundation Engineering Manual, CFEM (1985; 1992),

applies the same basic principles of linear and non-linear stress-strain behavior. The

method applies to all soils, clays as well as sand. By this method, the relation between

stress and strain is a function of two non-dimensional parameters which are unique for a

soil: a stress exponent, j, and a modulus number, m.

Janbu’s general relation is

where σ‘r is a “reference stress = 100 KPa

j > 0

The Janbu Method

9/11/2013 61

Dense Coarse-Grained Soil j = 1

Cohesive Soil j = 0

Sandy or Silty Soils j = 0.5

0

1

'

'ln

1

σ

σε

m=

'1

)''(1

01 σσσε ∆=−=mm

'2

1)''(

2

101 σσσε ∆=−=

mm

KPa

ksf

)''(5

101 σσε −=

m

pm

''(2

1 σσε −=

KPa

ksf

There are direct mathematical conversions

between m and the E and Cc-e0

22

00 69.02lg10ln13.2

110ln

εε==

+=

+=

cc C

e

C

em

9/11/2013 62

For E given in units of KPa (and ksf), the relation between the

modulus number and the E-modulus is

m = E/100 (KPa)

m = E/2 (ksf)

For Cc-e0, the relation to the modulus number is

Typical and Normally Conservative Virgin Modulus Numbers

SOIL TYPE MODULUS NUMBER STRESS EXP.

Till, very dense to dense 1,000 — 300 (j = 1)

Gravel 400 — 40 (j = 0.5)

Sand dense 400 — 250 (j = 0.5)

compact 250 — 150 - " -

loose 150 — 100 - " -

Silt dense 200 — 80 (j = 0.5)

compact 80 — 60 - " -

loose 60 — 40 - " -

Silty clay hard, stiff 60 — 20 (j = 0)

and stiff, firm 20 — 10 - “ -

Clayey silt soft 10 — 5 - “ -

Soft marine clays

and organic clays 20 — 5 (j = 0)

Peat 5 — 1 ( j= 0)

For clays and silts, the recompression modulus, mr, is often five to tentimes greater than the virgin modulus, m, listed in the table

9/11/2013 63

0.40

0.60

0.80

1.00

1.20

10 100 1,000 10,000

Stress (KPa) log scale

Void

Ratio

(- -

)

m = 12

(CR = 0.18)

p'c

0

5

10

15

20

25

10 100 1,000 10,000

Stress (KPa) log scale

Str

ain

(%

)

p 10p

Cc

Cc = 0.37

e0 = 1.01 p'c

p 2.718p

1/m

Slope = m = 12

Evaluation of compressibility from oedometer results

Void-Ratio vs. Stress and Strain vs. Stress — Same test data

9/11/2013 64

Comparison between the Cc/e0 approach

and the Janbu method

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40 0.60 0.80 1.00 1.20

VOID RATIO, e0

CO

MP

RE

SS

ION

IN

DE

X, C

c

0

5

10

15

20

25

30

35

0.400.600.801.001.20

VOID RATIO, e0

VIR

GIN

MO

DU

LU

S N

UM

BE

R,

m

9/11/2013 65

Data from a 20 m thick sedimentary deposit of medium compressibility.

Do these values

indicate a

compressible soil, a

medium compressible

soil, or a non-

compressible soil?

The Cc-e0 approach implies that the the compressibility varies by 30± %.

However, the Janbu methods shows it to vary only by 10± %. The

modulus number, m, ranges from 18 through 22; It would be unusual to

find a clay with less variation.

What about “Immediate Settlement”

and Consolidation?

50

75

100

125

150

0.40 0.60 0.80 1.00 1.20

VOID RATIO, e0

NO

RM

AL

IZE

D

Cc

(%)

50

75

100

125

150

0.400.600.801.001.20

VOID RATIO, e0

NO

RM

AL

IZE

D

m

(%)

9/11/2013 66

CE 5101 - SETTLEMENTS AND 1D Compression Theory

Settlements of Soils - Immediate, Delayed, and Creep Compression

Delayed Consolidation Compression•mv method•e-logP method•Janbu method

Terzaghi’s Theory of 1D ConsolidationEffects of Drainage and Initial Stress Distribution

SPREADSHEET Calculations (UNISETTLE)Finite Element Analysis (PLAXIS)

9/11/2013 67

Foundation Settlement Issues

How Much settlements will occur?

Interested in Ultimate settlements in fully drained state, as well as long-term creep settlements

How Fast and how long will it take for most of settlements to occur?

Involved Consolidation and Secondary Compression theories to estimate rate of settlements, and

Methods to accelerate settlements and minimise long-term settlements

9/11/2013 68

Types of Ground Movements and Causes of Settlements

•Compaction - due to vibrations, pile driving, earthquake

•Elastic Volumetric Settlement in OC Clay in recompression,

use E’ and νννν’ or recompression index, Cr or κκκκ

•Immediate or Undrained Settlement - Distortion without

volume change, use Eu and ννννu

•Moisture changes - Expansive soils, high LL and PI, high

swelling and shrinkage

Swell Potential (%) = 0.1(PI-10) log (s/p’)

•Effects of vegetation - related to moisture changes by root

system

•Effects of GWT lowering - shrinkage ad consolidation

9/11/2013 69

Types of Ground Movements and Causes of Settlements

•Effects of temperatures - Frost heaving,

drying by furnace and boilers

•Effects of seepage and scouring - Erosion by

piping, scouring and wind action, mineral

cement dissolved by GW eg limestone, rock

salts and chalk areas

•Loss of lateral support - Footings beside

unsupported excavation, movement of natural

slopes and cuttings

•Effects of mining subsidence - collapse of

ground cavities

•Filled ground - settlements of the fill soils,

compaction, consolidation, and creep

9/11/2013 70

9/11/2013 71

Shrink/ swell potential is function of Clay Activity = %clay fraction/ PI

Swell Potential (%) = 0.1(PI-10) log (s/p’)

Where s=suction before construction and p’=final bearing pressure

1D Settlements

Soil deformations are of two types:• Distortion (change of shape 2D effects)•Compression (change of volume)

Components of Settlements:

ncompressiosecondarys

ncompressiosettlementionconsolidats

distortionsettlementimmediateswhere

ssss

s

c

i

scit

=

=

=

++=

,

,

9/11/2013 72

Undrained Immediate Settlements

Distortion of clay layer:•Calculate by elastic theory eg Janbu Chart

layerclay of ratio sPoisson'

layerclay of modulus Undrained E

B dthfootong wi on loadAverage q

by JanbufactorsessdimensionlIandI

arealoadedflexibleforsettlementimmediateswhere

E

qBIIs

10

i

i

=

=

=

=

=

−=

υυυυ

νννν )1( 2

10

9/11/2013 73

Immediate Settlements in Clays by Janbu

9/11/2013 74

Example on Janbu Chart: Foundation 4x2m with q=150kPa,

located at 1m in Clay layer 5m thick with Eu=40MN/m2. Below

is second clay layer of 8m thickness and Eu=75 MN/m2.What

is average settlement under foundation? Assume νννν=0.5

H

D

B(((( )))) mm5.35.01

40

2*150*7.0*9.0s

7.0 2,4/2L/B and 24/2H/B

:MN/m40E withlayer,clay upper Consider (1)

9.0 0.5, 1/2D/B 2;4/2L/B Now

2

1i

1

2

u

0

====−−−−====

====⇒⇒⇒⇒================

====

====⇒⇒⇒⇒================

µµµµ

µµµµ

(((( )))) mm3.25.0175

2*150*85.0*9.0s

85.0 2,4/2L/B and 612/2H/B

:MN/m75E withcombined, layers two Consider (2)

2

2i

1

2

u

====−−−−====

====⇒⇒⇒⇒================

====

µµµµ

(((( )))) mm9.15.0175

2*150*7.0*9.0s

7.0 2,4/2L/B and 24/2H/B

:MN/m75E withlayer, upper Consider (1)

2

3i

1

2

u

====−−−−====

====⇒⇒⇒⇒================

====

µµµµ

mm 3.9 1.9 -2.3 5.3s

ssss

;inciplePrionSuperposit By

i

3i2i1ii

====++++====

−−−−++++====

9/11/2013 75

1D Primary Consolidation Settlements

Time delayed Primary Consolidation Compression:

water

solids

0e

1

e∆

0H

H∆

)()(

∆

00

0

volumeverticalc

0

volumevertical

HHHs

e1

e

H

H

εεεεεεεε

εεεεεεεε

==∆=

+==

∆=

'vv P where P), vs(e methodm σσσσ=

e

0e

fe

fP0P P

P

e -ility compressib of coeffnav

∆

∆==

0

v

0

ve1

a

P

1

)e(1

e -change volume of coeffnm

++++====

++++========

∆∆∆∆

∆∆∆∆

0

0

vc

0v0

0

c

HPe1

a s

HPm He1

e sH

∆∆∆∆

∆∆∆∆∆∆∆∆

∆∆∆∆

++++====

====++++

========

9/11/2013 76

mv and Constrained Modulus D

For wide loaded area, get 1D compression under Ko condition, elastic modulus to apply is called the Constrained Modulus D defined by:

(((( ))))(((( ))))(((( ))))

0.35)( ratio Poisson drained Soil

elasticity of modulus drained Soil E

changevolume of tCoefficien mwhere

211

1E

m

1D

v

v

<<<<====

====

====

−−−−++++

−−−−========

υυυυ

υυυυυυυυ

υυυυ

9/11/2013 77

Sc by (e vs logP) Method

Normally Consolidated Clays - non-linear stress strain for soil,

but linear in logP

e

0e

fe

fP0P Plog

indexncompressioCc =

(((( ))))0

f

c0fcP

PlogCPlogPlogCe ====−−−−====∆∆∆∆

0

f

0

c0c

P

Plog

e1

CHHs

+== ∆∆∆∆

0

cc

e1

CC CRRatio, nCompressio

++++======== εεεε

9/11/2013 78

Sc by (e vs logP) Method

Over-Consolidated Clays (Pf < Pc), Preconsolidation Pressure

e

0e

fe

fP0P Plog

indexncompressioCc =

(((( ))))0

f

r0frP

PlogCPlogPlogCe ====−−−−====∆∆∆∆

cr

0

f

0

r0c

C than smaller times10 to5 is C as

ssettlementsmall

P

Plog

e1

CHHs

++++======== ∆∆∆∆

0

rr

e1

CC RR, Ratio ncompressioRe

++++======== εεεε

cP

indexswellingC

indexionrecompressC

s

r

=

=

9/11/2013 79

Sc by (e vs logP) Method

Over-Consolidated Clays (Pf > Pc), Preconsolidation Pressure

e

0e

ce

fP0P Plog

cC

(((( ))))0

cr0cr1

P

PlogCPlogPlogCe ====−−−−====∆∆∆∆

ssettlementlarge mean willThis

P

PlogC

P

PlogC

e1

H

e1

eeHHs

c

f

c

0

cr

0

0

0

210c

++

=

+

+==

∆∆∆∆∆∆∆∆∆∆∆∆

cP

sr CorC

fe(((( ))))

c

f

ccfc2P

PlogCPlogPlogCe ====−−−−====∆∆∆∆

9/11/2013 80

Sc by Janbu Tangent Modulus MethodCanadian Foundation Manual 1985

Janbu (1963, 1965,1967,1998); For Cohesionless Soils; j>0

Theory is used in Program UniSettle (by UNISOFT, Canada)

(((( ))))

kPa100 stresseffective verticalReference

test field or lab from obtained number, modulus Janbu m

exponent stress Janbuj

P stresseffective vertical final

P stresseffective vertical initial

stresseffective vertical inchange by induced strain vertical where

mj

1 get to Integrate m

'M

,

r

f

,

f

0

,

0

v

j

,

r

,

0

j

,

r

,

f

v

j1

'

r

''

rt

========

====

====

========

========

====

−−−−

====

====

∂∂∂∂

∂∂∂∂====

−−−−

σσσσ

σσσσ

σσσσ

σσσσσσσσ

σσσσσσσσ

εεεε

εεεεσσσσ

σσσσσσσσ

εεεε

σσσσ

9/11/2013 81

Cohesionless Sands and Silts; j=0.5

Normally Consolidated Cohesionless Soils

−−−−====

========

σσσσσσσσεεεε,

0

,

fv

,

r

5m

1

kPa100 σ0.5jSOIL TYPE m

SAND DENSE 400-250

MEDIUM 250-150

LOOSE 150-100

SILT DENSE 200-80

MEDIUM 80-60

LOOSE 60-40

9/11/2013 82

Cohesionless Sands and Silts; j=0.5

Over-Consolidated Cohesionless Soils

m than larger times10 to5 usually is m number, modulus OC

kPa inpressure dationpreconsoli

where

5m

1

5m

1

kPa100 'P

'f

r

,

P

,

P

,

f

,

0

,

Pr

v

,

r0.5j; for CASE

====

−−−−++++

−−−−====

========>>>>

σσσσ

σσσσσσσσσσσσσσσσ

σσσσσσσσ

εεεε

σ

−−−−====

====≤≤≤≤ ====

σσσσσσσσεεεε

σσσσσσσσ

,

0

,

fr

v

,

r

'

p

'

f

5m

1

kPa,100 ; for CASE ,0.5j σ

9/11/2013 83

Cohesive Soils; j=0

Normally Consolidated Clays

++++====

++++====

++++====

====

========

c

0

c

0

,

0

,

f

0

c

,

0

,

f

v

,

r

C

e13.2

C

e110lnm

loge1

Cln

m

1

kPa100 0j

σσσσσσσσ

σσσσσσσσ

εεεε

σ

SOIL TYPE m

SILTYCLAYS

HARD,STIFF

60-20

AND STIFF,FIRM

20-10

CLAYEYSILT

FIRM TOSOFT

10-5

SOFTMARINECLAYS

20-5

ORGANICCLAYS

10-3

PEAT 5-1

9/11/2013 84

Over-consolidated Clays; j=0

Over- Consolidated Clays

++++====

++++====

++++====

++++====

++++++++

++++====

++++

====

r

0

r

0r

c

0

c

0

,

P

,

f

0

c

,

0

,

P

0

rv

,

P

,

f

,

0

,

P

r

v

C

e13.2

C

e110lnmand

C

e13.2

C

e110lnm

loge1

Clog

e1

C

lnm

1ln

m

1

σσσσσσσσ

σσσσσσσσ

σσσσσσσσ

σσσσσσσσ

εεεε

εεεε

9/11/2013 85

Linear Elastic Soil; j=1

(((( ))))

100

Em

m100E,therefore

m100

1

mj

1

,

0

,

f

j

,

r

,

0

j

,

r

,

f

v

====

====

−−−−====

−−−−

==== σσσσσσσσ

σσσσσσσσ

σσσσσσσσ

εεεε

9/11/2013 86

Janbu compared to e-logP

9/11/2013 87

Oedometer Results in linear scale by Janbu

9/11/2013 88

Calculation of Settlement

•Determine soil profile to get initial effective stresses

•Determine soil compressibility parameters, e-logP, Cc,

Cr and Pc OR Janbu modulus number, m and mr

•Determine final effective stresses due to imposed loads,

excavations, fills, GWT changes etc

•Divide each soil layer into sublayers, calculate strain

caused by change from initial to final effective stresses

in each sublayer

•Calculate the settlement for each sublayer and the

accumulated settlement

9/11/2013 89

UNISETTLE Calculation of Settlement

9/11/2013 90

Hand Calculation

9/11/2013 91

UniSettle Input

9/11/2013 92

InputLoading and Excavation

9/11/2013 93

Results

9/11/2013 94

Settlements Distribution

9/11/2013 95

Alternative Conditions

9/11/2013 96