Lecture 7 stress distribution in soil

INTERNATIONAL UNIVERSITY FOR SCIENCE & TECHNOLOGY

�م وا�� ا�� ا��و�� ا��

CIVIL ENGINEERING AND

ENVIRONMENTAL DEPARTMENT

303322 - Soil Mechanics

Stress Distribution in Soil

Dr. Abdulmannan Orabi

Lecture

Dr. Abdulmannan Orabi IUST 2

Das, B., M. (2014), “ Principles of geotechnical Engineering ” Eighth Edition, CENGAGE Learning, ISBN-13: 978-0-495-41130-7.

Knappett, J. A. and Craig R. F. (2012), “ Craig’s Soil Mechanics” Eighth Edition, Spon Press, ISBN: 978-0-415-56125-9.

References

� Stress in soil due to self weight

Stress Distribution in Soil

� Stress in soil due to surface load

3Dr. Abdulmannan Orabi IUST

Stress due to self weight

The vertical stress on element A can be determined simply from the mass of the overlying material. If represents the unit weight of the soil, the vertical stress is

Variation of stresses with depth

Ground surface

zz ⋅= γσ

� � ��

��

⋅=⋅++⋅+⋅=n

iinnz hhhh1

2211 ...... γγγγσ

Stresses in a Layered Deposit

The stresses in a deposit consisting of layers of soil having different densities may be determined as

Vertical stress at depth z1

��

�� ∗ ��

�� ∗ �� ∗ ��

��

�� ∗ ��

�� ∗ �� ∗ ��

�� ∗ �� ∗ �� ∗ ��5Dr. Abdulmannan Orabi IUST

With uniform surcharge on infinite land surface

Original land surface

Conversion land surface

� �

�� ∗ � �

� ��

��

�� ∗ �

Vertical stresses due to self weight increase with depth,There are 3 types of geostatic stresses:

a. Total Stress, σtotal

b. Effective Stress, σ'c. Pore Water Pressure, u

Vertical Stresses

Consider a soil mass having a horizontal surface and with the water table at surface level. The total vertical stress at depth z is equal to the weight of all material (solids + water) per unit area above that depth ,i.e

Total vertical stress

��!"!#$ � �%#! ∗ �

The pore water pressure at any depth will be hydrostatic since the void space between the solid particles is continuous, therefore at depth z:

Pore water pressure

� � �& ∗ �

If the pores of a soil mass are filled with water and if a pressure induced into the pore water, tries to separate the grains, this pressure is termed as pore water pressure

Effective vertical stress due to self weight of soil

The difference between the total stress (��!"!#$) and the pore pressure (u) in a saturated soil has been defined by Terzaghi as the effective stress ( ).��

��' � ��!"!#$ − �

The pressure transmitted through grain to grain at the contact points through a soil mass is termed as effective pressure.

Stresses in Saturated Soil

If water is seeping, the effective stress at any point in a soil mass will differ from that in the static case. It will increase or decrease, depending on the direction of seepage.

The increasing in effective pressure due to the flow of water through the pores of the soil is known as seepage pressure.

A column of saturated soil mass with no seepage of water in any direction.

The total stress at the elevation of point A can be obtained from the saturated unit weight of the soil and the unit weight of water above it. Thus,

Stresses in Saturated Soil without Seepage

Solid particle

Pore water

0A Solid particle

Pore water)*

Forces acting at the points of contact of soil particles at the level of point A

�� &) � ,)* − )-�%#!

where �� +��.��+��

�/+�� 0 ��1

�%#! � �+��.+�2��

��

)* � 2��+�34��0 ��

1+�2��+�.�+4�

Valve (closed)

Stress at point A,

• Total stress:• Pore water pressure:• Effective stress:

�* ��&)�

�*' ��* − �* � 0

Stress at point B,• Total stress:

• Pore water pressure• Effective stress:

�: ��&)� �)� ∗ �%#!

�: � ,)��)�-�&

�:' � �: − �:

�:' � )� � �%;<

Stress at point C,• Total stress:

�= ��&)� � � ∗ �%#!

�> �,)��-�&

�>' ��> − �>

�>' � � � �%;<

• Pore water pressure:

• Effective stress:

Total stress

Pore water Pressure, u Effective stress

DepthDepth Depth

Valve (open)

AB-�

Stresses in Saturated Soil with Upward Seepage

Stress at point A,• Total stress:

�* ��&)�

�*' ��* − �* � 0

• Pore water pressure

�: ��&)� � )� ∗ �%#!

�: � ,)��)� � �-�&

�:' ��: − �:

�:' �)� � �%;< − ��&

Stress at point C,

• Total stress:• Pore water pressure:• Effective stress:

�= ��&)� � � ∗ �%#!

�: � ,)��

�-�&

�>' ��> − �>

�>' � � � �%;< −

��&

�>' � � � �%;< − ��&

Note that h/H2 is the hydraulic gradient icaused by the flow, and therefore

Total stressPore water Pressure, u

Effective stress

DepthDepth Depth

At any depth z, is the pressure of the submerged soil acting downward and is the seepage pressure acting upward. The effective pressure reduces to zero when these two pressures balance. This situation generally is referred to as boiling.

�>' � � � �%;< − �>C ��& � 0

�>C ��%;<

�&�.�>C � 3.��3+��D2.+��3�.+2��

For most soils, the value of �>C varies from 0.9 to 1.1

� � �%;<��&

Valve (open)

AB-�

Stresses in Saturated Soil with Downward Seepage

Stress at point A,• Total stress:

�* ��&)�

�*' ��* − �* � 0

• Pore water pressure

�: ��&)� � )� ∗ �%#!

�: � ,)��)� − �-�&

�:' ��: − �:

�:' �)� � �%;< � ��&

Stress at point C,• Total stress:

�= ��&)� � � ∗ �%#!

�: � ,)�� −�

�-�&

�>' ��> − �>

�>' � � � �%;< �

��& �>' � � � �%;< � ��&

Pore water Pressure, uTotal stress Effective stress

DepthDepth Depth

Worked Examples

Example 1

A soil profile is shown in figure below. Calculate total stress, pore water pressure, and effective stress at A, B, C, and D.

A Ground surface

Sand γ= 16.3 kN/m^3

γ= 15.1 kN/m^3

γ= 19.8 kN/m^3

Total stress Effective stress Pore water pressure

DepthDepthDepth

γ1 X H1

γ1 X H1 + γ2 X H2

γ1 X H1 + γ2 X H2 + γ3 X H3

γ1 X H1 + γ2 X H2 + γsub X H3

γw X Hw

To analyze problems such as compressibility of soils, bearing capacity of foundations, stability of embankments, and lateral pressure on earth retaining structures, we need to know the nature of the distribution of stress along a given cross section of the soil profile.

Stress due to surface load

Introduction

When a load is applied to the soil surface, it increases the vertical stresses within the soil mass. The increased stresses are greatest directly under the loaded area, but extend indefinitely in all directions.

Introduction

•Allowable settlement, usually set by building codes, may control the allowable bearing capacity. •The vertical stress increase with depth must be determined to calculate the amount of settlement that a foundation may undergo

Introduction

Foundations and structures placed on the surface of the earth will produce stresses in the soil These stresses will decrease with the distance from the load How these stresses decrease depends upon the nature of the soil bearing the load

Individual column footings or wheel loads may be replaced by equivalent point loads provided that the stresses are to be calculated at points sufficiently far from the point of application of the point load.

Stress Due to a Concentrated Load

Stresses in soil due to surface load

Vertical stress due to a concentrated load • Boussinesq’s Formula • Wastergaard Formula

Boussinesq’s Formula for Point Loads

Joseph Valentin Boussinesq (13 March 1842 – 19 February

1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.

Stresses in soil due to surface load

In 1885, Boussinesq developed the mathematical relationships for determining the normal and shear stresses at any point inside a homogenous, elastic and isotropic mediums due to a concentrated point loads located at the surface

Vertical Stress in Soil

The soil mass is elastic, isotropic (having identical properties in all direction throughout), homogeneous (identical elastic properties) and semi-infinite depth.The soil is weightless.

Assumption:

The distribution of σz in the elastic medium is apparently radially symmetrical. The stress is infinite at the surface directly beneath the point load and decreases with the square of the depth.

At any given non-zero radius, r, from the point of load application, the vertical stress is zero at the surface, increases to a maximum value at a depth where , approximately, and then decreases with depth.

E � 39.25°

According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by

∆��

∆�M ∆�N

According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by

2 2 5/2

2 [1 ( / ) ]z

z r zσ

…… . 7 − 1

∆��.

Equation shows that the vertical stress is

� Directly proportional to the load

� Inversely proportional to the depth squared, and

� Proportional to some function of the ratio ( r/z).

2 [1 ( / ) ]bI

r zπ=

………… . 7 − 2

It should be noted that the expression for z is independent of elastic modulus (E) and

Poisson’s ratio (µ), i.e. stress increase with depth is a function of geometry only.

r/Z IB

0.00 0.4775 0.18 0.4409 0.36 0.3521 0.55 0.2466

0.01 0.4773 0.19 0.4370 0.37 0.3465 0.56 0.2414

0.02 0.477 0.20 0.4329 0.38 0.3408 0.57 0.2363

0.03 0.4764 0.21 0.4286 0.39 0.3351 0.58 0.2313

0.04 0.4756 0.22 0.4242 0.40 0.3294 0.59 0.2263

0.05 0.4745 0.23 0.4197 0.41 0.3238 0.60 0.2214

0.06 0.472 0.24 0.4151 0.42 0.3181 0.61 0.2165

0.07 0.4717 0.25 0.4103 0.43 0.3124 0.62 0.2117

0.08 0.4699 0.26 0.4054 0.44 0.3068 0.63 0.2070

0.09 0.4679 0.27 0.4004 0.45 0.3011 0.64 0.2024

0.1 0.4657 0.28 0.3954 0.46 0.2955 0.65 0.1978

0.11 0.4633 0.29 0.3902 0.47 0.2899 0.66 0.1934

0.12 0.4607 0.30 0.3849 0.48 0.2843 0.67 0.1889

0.13 0.4579 0.31 0.3796 0.49 0.2788 0.68 0.1846

0.14 0.4548 0.32 0.3742 0.50 0.2733 0.69 0.1804

0.15 0.4516 0.33 0.3687 0.51 0.2679 0.70 0.1762

0.16 0.4482 0.34 0.3632 0.52 0.2625 0.71 0.1721

0.17 0.4446 0.35 0.3577 0.53 0.2571 0.72 0.1681

0.54 0.2518 0.73 0.1641

Influence Factor Ib

r/Z IB r/Z IB r/Z IB r/Z IB

0.74 0.1603 0.94 0.0981 1.14 0.0595 1.34 0.0365

0.75 0.1565 0.95 0.0956 1.15 0.0581 1.35 0.0357

0.76 0.1527 0.96 0.0933 1.16 0.0567 1.36 0.0348

0.77 0.1491 0.97 0.0910 1.17 0.0553 1.37 0.0340

0.78 0.1455 0.98 0.0887 1.18 0.0539 1.38 0.0332

0.79 0.1420 0.99 0.0865 1.19 0.0526 1.39 0.0324

0.80 0.1386 1.0 0.0844 1.20 0.0513 1.40 0.0317

0.81 0.1353 1.01 0.0823 1.21 0.0501 1.41 0.0309

0.82 0.1320 1.02 0.0803 1.22 0.0489 1.42 0.0302

0.83 0.1288 1.03 0.0783 1.23 0.0477 1.43 0.0295

0.84 0.1257 1.04 0.0764 1.24 0.0466 1.44 0.0283

0.85 0.1226 1.05 0.0744 1.25 0.0454 1.45 0.0282

0.86 0.1196 1.06 0.0727 1.26 0.0443 1.46 0.0275

0.87 0.1166 1.07 0.0709 1.27 0.0433 1.47 0.0269

0.88 0.1138 1.08 0.0691 1.28 0.0422 1.48 0.0263

0.89 0.1110 1.09 0.0674 1.29 0.0412 1.49 0.0257

0.90 0.1083 1.10 0.0658 1.30 0.0402 1.50 0.0251

0.91 0.1057 1.11 0.0641 1.31 0.0393 1.51 0.0245

0.92 0.1031 1.12 0.0626 1.32 0.0384 1.52 0.0240

0.93 0.1005 1.13 0.0610 1.33 0.0374 1.53 0.0234

Influence Factor Ib

r/Z IB r/Z IB r/Z IB r/Z IB

1.54 0.0229 1.66 0.0175 1.86 0.0114 2.5 0.0034

1.55 0.0224 1.67 0.0171 1.88 0.0109 2.6 0.0029

1.56 0.0219 1.68 0.0167 1.90 0.0105 2.7 0.0024

1.57 0.0214 1.69 0.0163 1.92 0.0101 2.8 0.0021

1.58 0.0209 1.70 0.0160 1.94 0.0097 2.9 0.0017

1.59 0.0204 1.72 0.0153 1.96 0.0093 3.0 0.0015

1.60 0.0200 1.74 0.0147 1.98 0.0089 3.5 0.0007

1.61 0.0195 1.76 0.0141 2.0 0.0085 4.0 0.0004

1.62 0.0191 1.78 0.0135 2.1 0.0070 4.5 0.0002

1.63 0.0187 1.80 0.0129 2.2 0.0058 5.0 0.0001

1.64 0.0183 1.82 0.0124 2.3 0.0048

1.65 0.0179 1.84 0.0119 2.4 0.0040

Influence Factor Ib

Equation may be used to draw three types of pressure distribution diagram. They are:

� The vertical stress distribution on a horizontal plane at depth of z below the ground surface

� The vertical stress distribution on a vertical plane at a distance of r from the load point, and

� The stress isobar.

Pressure Distribution Diagram

� The vertical stress distribution on a horizontal plane at depth of z below the ground surface

Distribution on a horizontal plane

�The vertical stress distribution on a vertical plane at a distance of rfrom the point load

.��

Distribution on a vertical plane O

��

Stress isobars

An isobar is a line which connects all points of equal stress below the ground surface. In other words, an isobar is a stress contour.

What is the vertical stress at point A of figure below for the two loads, P1 and P2 ?

P1 = 350 kNP2 = 470 kNZ

2.3 m 1.1 m

Worked Examples

Example 2

A four concentrated forces are located at corners of a rectangular area with dimensions 8 m by 6 m as shown in figure in the next slide. Compute the vertical stress at points A and B, which are located on the lines A – A’ , B – B’ at depth of 4 m below the ground surface.

Worked Examples

Example 3

700 kN700 kN

Worked Examples

Example 3

Dr. Abdulmannan Orabi IUST 51

Westergaard Formula

Westergaard proposed a formula for the computation of vertical stress �� by a point load, P at the surface as

�� O+

2V�� +� �.�

� �/�

In which µ is Poisson’s ratio

+ � 1 − 2X /,2 − 2X-

… . 7 − 3

Stress below a Line Load

The vertical stress increase due to line load , , inside the soil mass can be determined by using the principles of the theory of elasticity, or

��

�� 2 ��

V P� � ��

This equation can be rewritten as��

/��

V 1 �P�

� � 1

… . 7 − 4

Vertical Stress caused by a horizontal line load

The vertical stress increase (��) at point A in the soil mass caused by a horizontal line load can be given as :

�� 2 P��

V P� � ��

/��

… . 7 − 5

Vertical Stress caused by a strip load

The fundamental equation for the vertical stress increase at a point in a soil mass as the result of a line load can be used to determine the vertical stress at a point caused by a flexible strip load of width B.

The term strip loading will be used to indicate a loading that has a finite width along the x axis but an infinite length along the y axis.

Vertical Stress caused by a strip load

Vertical stress at point A can be determined by equation:

[ sin cos ( 2 ) ]oz

qσ α α α β

π= + +

… . 7 − 6

� � 0.25\

��0.5\

� � \

0.5\0.25\

Worked Examples

Example 4

Refer to figure below, The magnitude of the strip load is 120 kPa. Calculate the vertical stress at points, a , b, and c.

1 2 2[( ) ( ) ( )]oz

q a b b

a aσ α α α

+= + −

Vertical Stress Due to Embankment Loading

The vertical stress increase in the soil mass due to an embankment of height H may be expressed as

" � � � )where:

� � �� `4+�a`��

) � �� `4+�a`��

… . 7 − 7

120aO+

Refer to figure below. The magnitude of the load is 120 kPa. Calculate the vertical stress at points,

A , B, and C.

Worked Examples

Example 4

1- Under the center: The increase in the vertical

stress (��) at depth z ( point A)under the center

of a circular area of diameter D = 2R carrying a uniform pressure q is given by

Vertical Stress due to a uniformly loaded circular area

�� 1 −1

Q/� � � 1 �/�

… . 7 − 8

2- At any point: The increase in the vertical stress (��) at any point located at a depth z at any distance r from the center of the loaded area can be given

where and are functions of z/R and r/R.

�� 1' � \'

… . 7 − 9

Variation of with z/R and r/R. 1'

Variation of with z/R and r/R. \'

Vertical Stress in SoilVariation of with z/R and r/R. \'

Vertical Stress Caused by a Rectangular loaded area

The increase in the vertical stress (��) at depth z under a corner of a rectangular area of dimensions B = m z and L = n z carrying a uniform pressure q is given by:

z o zq Iσ =

c� � ��3�+3� .20�2�� .+�� d

�+�2

where :

… . 7 − 10

c� �1

2`� `� � �� 1

`� � �� `�� 1

`� � �� 2

`� � �� 1� �+�e�

2`� `� � �� 1

`� � �� −`�� 1

The influence factor

can be expressed as

` �d

�+�2� �

�where :

… . 7 − 11

The increase in the stress at any point below a rectangular loaded area can be found by dividing the area into four rectangles. The point A’ is the corner common to all four rectangles.

6'��* � �� f

��g � c�g

−h5�

�h5i−h5�

+ h5� h5

��* � �� − �� − �� f

Variation of with m and nc�

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.1 0.0047 0.0092 0.0132 0.0168 0.0198 0.0222 0.0242 0.0258

0.2 0.0092 0.0179 0.0259 0.0328 0.0387 0.0435 0.0474 0.0504

0.3 0.0132 0.0259 0.0374 0.0474 0.0559 0.0629 0.0686 0.0731

0.4 0.0168 0.0328 0.0474 0.0602 0.0711 0.0801 0.0873 0.0931

0.5 0.0198 0.0387 0.0559 0.0711 0.0840 0.0947 0.1034 0.1104

0.6 0.0222 0.0435 0.0629 0.0801 0.0947 0.1069 0.1168 0.1247

0.7 0.0242 0.0474 0.0686 0.0873 0.1034 0.1169 0.1277 0.1365

0.8 0.0258 0.0504 0.0731 0.0931 0.1104 0.1247 0.1365 0.1461

0.9 0.0270 0.0528 0.0766 0.0977 0.1158 0.1311 0.1436 0.1537

1.0 0.0279 0.0547 0.0794 0.1013 0.1202 0.1361 0.1491 0.1598

0.9 1 1.2 1.4 1.6 1.8 2.0 2.5

0.1 0.0270 0.0279 0.0293 0.0301 0.0306 0.0309 0.0311 0.0314

0.2 0.0528 0.0547 0.0573 0.0589 0.0599 0.0606 0.0610 0.0616

0.3 0.0766 0.0794 0.0832 0.0856 0.0871 0.0880 0.0887 0.0895

0.4 0.0977 0.1013 0.1063 0.1094 0.1114 0.1126 0.1134 0.1145

0.5 0.1158 0.1202 0.1263 0.1300 0.1324 0.1340 0.1350 0.1363

0.6 0.1311 0.1361 0.1431 0.1475 0.1503 0.1521 0.1533 0.1548

0.7 0.1436 0.1491 0.1570 0.1620 0.1652 0.1672 0.1686 0.1704

0.8 0.1537 0.1598 0.1684 0.1739 0.1774 0.1797 0.1812 0.1832

0.9 0.1619 0.1684 0.1777 0.1836 0.1875 0.1899 0.1915 0.1938

1.0 0.1684 0.1752 0.1851 0.1914 0.1955 0.1981 0.1999 0.2024

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.2 0.0293 0.0573 0.0832 0.1063 0.1263 0.1431 0.1570 0.1684

1.4 0.0301 0.0589 0.0856 0.1094 0.1300 0.1475 0.1620 0.1739

1.6 0.0306 0.0599 0.0871 0.1114 0.1324 0.1503 0.1652 0.1774

1.8 0.0309 0.0606 0.0880 0.1126 0.1340 0.1521 0.1672 0.1797

2.0 0.0311 0.0610 0.0887 0.1134 0.1350 0.1533 0.1686 0.1812

2.5 0.0314 0.0616 0.0895 0.1145 0.1363 0.1548 0.1704 0.1832

3.0 0.0315 0.0618 0.0898 0.1150 0.1368 0.1555 0.1711 0.1841

4.0 0.0316 0.0619 0.0901 0.1153 0.1372 0.1560 0.1717 0.1847

5.0 0.0316 0.0620 0.0901 0.1154 0.1374 0.1561 0.1719 0.1849

6.0 0.0316 0.0620 0.0902 0.1154 0.1374 0.1562 0.1719 0.1850

0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5

1.2 0.1777 0.1851 0.1958 0.2028 0.2073 0.2103 0.2124 0.2151

1.4 0.1836 0.1914 0.2028 0.2102 0.2151 0.2184 0.2206 0.2236

1.6 0.1874 0.1955 0.2073 0.2151 0.2203 0.2237 0.2261 0.2294

1.8 0.1899 0.1981 0.2103 0.2183 0.2237 0.2274 0.2299 0.2333

2.0 0.1915 0.1999 0.2124 0.2206 0.2261 0.2299 0.2325 0.2361

2.5 0.1938 0.2024 0.2151 0.2236 0.2294 0.2333 0.2361 0.2401

3.0 0.1947 0.2034 0.2163 0.2250 0.2309 0.2350 0.2378 0.2420

4.0 0.1954 0.2042 0.2172 0.2260 0.2320 0.2362 0.2391 0.2434

5.0 0.1956 0.2044 0.2175 0.2263 0.2324 0.2366 0.2395 0.2439

6.0 0.1957 0.2045 0.2176 0.2264 0.2325 0.2367 0.2397 0.2441

Approximate Method

z��

2V:1H method

A simple but approximate method is sometimes used for calculating the stress change at various depths as a result of the application of a pressure at the ground surface.

The transmission of stress is assumed to follow outward fanning lines at a slope of 1 horizontal to 2 vertical.

Approximate Method

For uniform footing (B x L) we can estimate the change in vertical stress with depth using the Boston Rule. Assumes stress at depth is constant below foundation influence area

�� "d\

,d � �-,\ � �-

��

" �O

d � \

… . 7 − 12

2V:1H method

Approximate Method

Stress on this plane " �j

d ∗ \

Stress on this plane at depth z, �� "d\

,d � �-,\ � �-

Rectangular footing

2V:1H method

Newmark Method

• Stresses due to foundation loads of arbitrary shape applied at the ground surface

• Newmark’s chart provides a graphical method for calculating the stress increase due to a uniformly loaded region, of arbitrary shape resting on a deep homogeneous isotropic elastic region.

Newmark Method

• The Newmark’s Influence Chart method consists of concentric circles drawn to scale, each square contributes a fraction of the stress.

• In most charts each square contributes 1/200 (or 0.005) units of stress. (influence value, I)

Newmark Method

The use of the chart is based on a factor termed the influence value, determined from the number of units into which the chart is subdivided.

Influence value 0.005

1 unit

Newmark Method

A BInfluence value = 0.005

Total number of block on chart = 200 and influence value = 1/200

The influence chart may be used to compute the pressure on an element of soil beneath a footing, or from pattern of footings, and for any depth z below the footing. It is only necessary to draw the footing pattern to a scale of z = length AB of the chart. (If z= 6m and AB = 30mm, the scale is 1/200).

Newmark Method

The footing plan will be placed on the influence chart with the point for which the stress is desired at the center of the circles.

Newmark Method

The units (segments or partial segments) enclosed by the footing are counted, and the increase in stress at the depth z is computed as

�� "cj

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Soil Mechanics: Stress and Strain - PDH Courses Online ... Mechanics- Stress and... · Soil Mechanics: Stress and Strain ... Soil mass is generally a three phase system that consists

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