6 stresses in soil mass

University of Anbar College of Engineering Civil Engineering Department Iraq-Ramadi

Asst. Prof. Khalid R. Mahmood (PhD.)

248

Stresses in a Soil Mass

Topics Normal and Shear Stresses on a Plane Stress distribution in soils Stress Caused by a Point Load Vertical Stress Caused by a Line Load Vertical Stress Caused by a Strip Load Vertical Stress Due to Embankment Loading Vertical Stress below the Center of a uniformly Loaded Circular Area Vertical Stress at any Point below a uniformly Loaded Circular Area Vertical Stress Caused by a Rectangularly Loaded Area Influence Chart for Vertical Pressure (Newmark Chart) Approximate methods

Added Stress

Geostatic Stress



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Normal and Shear Stresses on a Plane

n, - n

M

x xy2

2

n, - n

N

T

M

n

n, - n

yx

yxM

xy

x

y

xy

xy

A B

C D

E

F

F

E B

Y > X



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From geometry for the free body diagram EBF

sin

cos

EFFB

EFEB

Summing forces in N and T direction, we have cossin)(2cos)(sin)()( 22 EFEFEFEF xyyxn

2sin2cos22 xy

xyxyn …………………………..……….(1)

Again 22 sin)(cos)(cossin)(cossin)()( EFEFEFEFEF xyxyyxn

2cos2sin2 xy

xyn ……………………………………(2)



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If n = 0 then

xy

xy22tan …………………………………………….(3)

This eq. gives 2 values of that are 90o apart, this means that there are 2 planes that are right angles to each other on which shear stress = 0, such planes are called principle planes and the normal stress that act on the principle planes are to as principle stresses.



252

To find the principle stress substitute eq.3 into eq.1, we get

stressprincipleor

stressprinciplemajor

xyxyxy

n

xyxyxy

n

min22

22

22

3

22

1

These stresses on any plane can be found using Mohr’s circle



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Mohr’s circle



254

Refer to the element shown in Fig. above



255

Pole Method

x, +

x

Pole

x

n 1 3

-

y

y,- xy

y

+

n, n

y

y

x x

xy

yx

yx

xy

A B

C

E



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Geostatic stresses

Total Stress Effective Stress

Pore Water Pressure

Total Stress= Effective Stress+ Pore Water Pressure

Bossinisque Equations

•Point Load

•Line Load

•Strip Load

•Triangular Load

•Circular Load

•Rectangular Load

Added Stresses (Point, line, strip, triangular, circular, rectangular)

Stress Distribution in Soils

Influence Charts Newmark Charts Stress Bulbs

Westergaard’s Method

Approximate Method

A

Added

Geostatic

x

y

xy



257

Geostatic stresses The vertical geostatic stress at point X will be computed as following

hV homogenous soils

n

iiV h1

stratified soils h

v dz0

density varies continuously with depth



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The horizontal geostatic stress can be computed as following

vh K where K is the coefficient of lateral stress or lateral stress

ratio

v

hK 11 K

Geostatic stress are principle stresses ( 1, 2 and 3 major, intermediate and minor principle stresses) and hence the horizontal and vertical planes through any point are principle planes.



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1K 1v 3h h32 1K 321hv Isotropic 1K 1h 3v h12

The largest shear stress will found on plane lying at 45o to the horizontal

1K )1(2max Kv

1K 0max

1K )1(2max Kv



260

Stress Caused by a Point Load

GROUND

SURFACE

Z

Y

X

h

= v

General

Stresses Principal

Stresses

z = (3P/2 (Z3/L5) Boussinesq’s Equation

L=(x2+y2+z2)1/2=(r2+z2)1/2

z = 1

Using Influence Factor Tablebelow

z = (P/Z2) Ip

P

r=(x2+y2)1/2



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262

Vertical Stress Caused by a Line Load

Z

X

Z

h

z

General

Stresses Principal

Stresses

X

Z

WALL

q (Load/Unit Length) q (Load/Unit Length)

Y

z = {2 q Z3/ XUsing Influence Factor Table below

z = (q/Z) IL



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Vertical Stress Caused by a Strip Load

Z

X

Z

h

z

General

Stresses Principal

Stresses

Strip footing

Y

Using Influence Factor Tablebelow

z = (q/Z) IB

B q (Load/Unit Area)



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Vertical Stress Due to Embankment Loading

Z

X

Z

h

v

General Highway embankment

EMBANKMENT

z = q/ 2)]

B2 B1 B1 B2

H

q= H

1(radians)=tan-1{(B1+B2)/z}-tan-1(B1/z)

2=tan-1(B1/z)

Using Influence Factor fig.(9.11-pp238)

z = qI2

Principal

1 2



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Vertical Stress below the Center of a uniformly Loaded Circular Area

Qo X

Z

GROUND

SURFACE

Z

Y

Z

Principle

q

X

z= q { 1- 1/[(R/Z)2 + 1 ]3/2 }

z = qIc

Using Influence Factorfrom table.



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Vertical Stress at any Point below a uniformly Loaded Circular Area

Or we can use the stress bulb charts

Qo X

Z

GROUND

SURFACE

Z

Y

Z

h

z v

General

Stresses

q

X

z= q (A/+B/)

r

Where (A/&B/) are functions of z/R and r/R

from tables



270

x

z

x

xy

2

1

1

2

X/R

Z/R

0.95

1

2

3

4

1 2 3 0

0.90

0.80

0.70

0.30

0.50

0.40

0.10

0.20

0.15

qs

1/qs

0.60

x x

xy

Circular Load: (Major Principal Stress)/(Surface Stress)

z =

z =

x = x =



271

X/R

Z/R

0.40

1

2

3

4

1 2 3 0

0.55

0.60

0.55

0.30

0.45

0.40

0.10

0.20

0.15

1 3)/qs

0.50

0.50

0.60

0.35

0.25

qs

z = 1

z = 1

x = 2 x = 2



272

X/R

Z/R

1

2

3

4

1 2 3 0

0.90

0.80

0.70

0.30

0.50

0.40

0.10

0.20

qs

z/qs

0.60

Circular Load: (Vertical Stress)/(Surface

0.05

z =

z =

x = 2 x =

0.15



273

Vertical Stress Caused by a Rectangularly Loaded Area

from tables or one can use the charts below

h

z v

Z

L

B q = Load /Area

IR = f (m,n)

m = B/Z

n = L/Z m & n from Charts or tables

Loaded Area Foundation Level

IR = 1/4 {[ (2. m.n (m2 + n2 + 1)1/2 ) / (m2 + n2 + m2. n2 + 1) ] [(m2 + n2 + 2)/(m2 + n2 + 1)] +tan-1 (2.m .n (m2 + n2 + 1)1/2 / (m2 + n2 - m2 . n2 + 1)}

B

P = q IR

Corner of the Building X

L



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Calculation of Stress below an interior point of the loaded area

)()()()([ OTDXIOZCTIOYBZIOXAYIqz

O P o in t o f in t e r e s t

O

A B

D C

X

Y

Z

T

F i g . 1 0 s t r e s s i n c r e a s e a t a p o i n t b e l o w a l o a d e d r e c t a n g u l a r r e g i o n

z

Q = q x Area

Plan

Elevation



276

Calculation of Stress below a point outside of the loaded area

)()()()([ OZCTIXZCDITYBZIABCDIqz

O P o in t o f in te r e s t

O

A B

D C

X

Y

Z

T

F ig . 1 0 s t r e s s in c r e a s e a t a p o in t b e lo w a l o a d e d r e c t a n g u la r r e g io n

z

z



277

Influence Chart for Vertical Pressure (Newmark Chart) 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’s chart is given in the data sheets and is reproduced in part in Fig 15. The procedure for its use is outlined below 1.The scale for this procedure is determined by the depth z at

which the stress is to be evaluated, thus z is equal to the distance OQ shown on the chart.



278

2.Draw the loaded area to scale so that the point of interest (more correctly its vertical projection on the surface) is at the origin of the chart, the orientation of the drawing does not matter

3.Count the number of squares (N) within the loaded area, if more than half the square is in count the square otherwise neglect it.

4.The vertical stress increase z = N [scale factor(0.001)] [surface stress (p)]



279

O Q4m

LoadedArea



280

Approximate Methods Equivalent Point Load Method

In dividing the loaded area into smaller units, we have to remember to do it such that z/B 3; that is to say, in relation to the specified depth, the size of any unit area should not be greater than one-third of the depth.

pii

z IzQ

2



281

Each Q is the resultant of the uniform load on the unit area acts at the center of it and treated as a point load



282

2:1 Method

))(( ZLZBQ

z Rectangular area

2)( ZBQ

z Square area

2)(4

ZD

Qz Circular area



283

Examples (1-3)



284



285



286

Example 4 A rectangular foundation 6 x 3m carries a uniform pressure of 300 kN/m2 near the surface of a soil mass. Determine the vertical stress at a depth of 3m below a point (A) on the centre line 1.5m outside a long edge of the foundation using influence factors

m = 1 m=1

n =1.5 n=0.5

I = 0.193 I=0.120



287

Example 5 Determine the stress increase under the embankment at points A1 and A2



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6 stresses in soil mass

Education

normal stress

soils stress

point load vertical

strip load vertical

line load vertical stress

xy y n

mahmood phd

x pole x n