2 stress in soil

, the

disadvantageous one

is choosen.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Example

The stratum’s conditions and the related physical characteristics parameters of a foundation are shown in Fig below. Calculate the stress due to self-weight at a,b,c. Draw the stress distribution.

w=15.6% e=0.57

γ s =26.6kN/m 3

w

=22%

w

L =32% w p =23% γ s =27.3kN/m 3

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

For silver sand, the

buoyant unit weight

(γ‘)

is used : For saturated clay,  1 '  

s

 1  

e w

 26 .

6  10 1  0 .

67  9 .

9

kN

/

m

3

I L



w



w L



w p w p

 22  23 32  23  0 (

Semisolid

) So, the clay can be considered the watertight; then the

saturated unit weight

(γ sat )

is used :

S r

 

G s w

 1

e e



G s w

 

sat

2  27 .

3 / 10  0 .

22  

s

1 

e



e w

  0 .

6 27 .

3  0 .

6  10 1  0 .

6  20 .

8

kN

/

m

3

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

a σ z =0 b σ z(upper) =γ’ 1 h 1 =9.9

× 2=19.8kPa

σ z(Down) =γ’ 1 h 1 + γ w (h 1 +h w )=9.9

× 2+10 × (2+1.2)=51.8kPa

c σ z =γ’ 1 h 1 + γ w (h 1 +h w )+ γ sat2 h 2 = 9.9

× 2+10 × (2+1.2)+20.8

× 3=114.2kPa

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Exercise 2-2 The stratum’s conditions and the related physical characteristics parameters of a foundation are shown in Fig below. Calculate the stress due to self-weight at 10m depth. Draw the stress distribution.

Note: For saturated clay, both cases (watertight and non-watertight) need to consider.

w=8% e=0.7

γ s =26.5kN/m 3

e=1.5

γ s =27.2kN/m 3

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

§

2.3 Contact stress

♦

2.3.1 Concept of Contact pressure

A foundation is the interface between a structural load and the ground.

The stress p applied by a structure to a foundation is often assumed to be uniform

. The actual pressure then applied by the foundation to the soil is a reaction, called the

contact pressure

p

and its distribution beneath the foundation may be far from uniform.

♦ This distribution depends mainly on: · stiffness of the foundation.

i.e. flexible → stiff → rigid.

· compressibility or stiffness of the soil.

· loading conditions – uniform or point loading.

2.3.2 Contact pressure – uniform loading

The effects of the stiffness of the foundation (flexible or rigid) and the compressibility of the soil (clay or sand) are illustrated in Figure below.

.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Figure The distribution of contact pressure

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

•

2.3.3 Stiffness of foundation

A flexible foundation has no resistance to deflection and will deform or bend into a dish-shaped profile when stresses are applied

. An earth embankment would comprise a flexible structure and foundation.

•

A stiff foundation provides some resistance to bending and well deform into a flatter dish-shape

so that differential settlements are smaller. This forms the basis of design for a raft foundation placed beneath the whole of a structure.

•

A rigid foundation has infinite stiffness and will not deform or bend

, so it moves downwards uniformly. This would apply to a thick, relatively small reinforced concrete pad foundation.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

• •

2.3.4 Stiffness of soil

The stiffness of a clay

will be the same under all parts of the foundation so for a flexible foundation a fairly uniform contact pressure distribution is obtained with a dish-shaped (sagging) settlement profile.

For a rigid foundation

the dish-shaped settlement profile must be flattened out so the contact pressure beneath the centre of the foundation is reduced and beneath the edges of the foundation it is increased. Theoretically, the contact pressure increases to a very high value at the edges although yielding of the soil would occur in practice , leading to some redistribution of stress.

The stiffness of a sand

increase as the confining pressures around it increase so beneath the centre of the foundation the stiffness will be smaller.

A flexible foundation of the sand

will, therefore, produce greater strains at the edges than in the centre so the settlement profile will be dish shaped but upside-down (hogging) with a fairly uniform contact pressure.

For a rigid foundation

profile must be flattened out so the contact pressure beneath the centre would be increased and beneath the edges it would be decreased.

this settlement

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

• •

2.3.5 Contact pressure-point loading

An load analysis

W

for contact pressure beneath a circular raft with a point at its centre resting on the surface of an Borowicka,1939.

incompressible soil (such as clay) has been provided by This shows that the contact pressure distribution is non-uniform irrespective of the stiffness of the raft or foundation.

For a flexible foundation the contact pressure is concentrated beneath the point load which is to be expected and for a stiff foundation it is more uniform.

For a rigid foundation the stresses beneath the edges are very pressure distribution similar to the distribution produced by a uniform pressure on a clay the is centre obtained.

of uniform pressure.

a This suggests that a point load at rigid foundation is comparable to a considerably increased and a

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

• •

2.3.6 Stress distribution

The stresses that already exist within the ground due to self-weight of the soil are discussed in Chapter 4.

When a load or pressure from a foundation or structure is applied at the surface of the soil this pressure is distributes throughout the soil and the original normal stresses and shear stresses ate altered. For most civil engineering applications the changes in vertical stress are required so the methods given below are for increases in vertical stress only.

-Central Loading

P

=P/F

B P p=P/F

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

-Eccentric Loading

p p

2 1 

P F



Pe W



P

( 1 

F

6

e

)

B

(0

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

p

'  3 (

B

(e>B/6) 2 2

P



e

)

L

e P e P p2 e>B/6 p1 p1 B

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

2.3.7

Additional pressure on the bottom of foundation ¦ Ãh B

p a =p γh

Where

p a

-additional pressure; p-contact pressure;

γ-

natural unit weight of soil( bouyant unit weight if below water surface);

h-

buried depth of the foundation.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Exercise 2-3 A load of 1200kN as well as a moment of 1500 kNm is carried on a rectangular foundation (axb=6x4m) at a shallow depth in a soil mass as shown in Fig below. (1) Determine the maximum & minimum contact stresses on the bottom of foundation.

(2) If the stress is tensional, Where does the width of foundation extend assuming the stress at right tip is zero and the position & value of loading, the length of the foundation a do not change?

Calculate the contact stress after the extension.

M=1500kNm N=1200kN x 4m

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

§

2.4 Stress due to loading

2.4.1 Stresses beneath point load

（点荷载） • • • •

Boussinesq

published in 1885 a solution for the stresses beneath a point load on the surface of a material which had the following properties: semi-infinite – this means infinite below the surface therefore providing no boundaries of the material apart from the surface homogeneous isotropic – the same properties at all locations –the same properties in all directions • elastic –a linear stress-strain relationship.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Linear elastic assumption

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Y P X Z β x r R y σz z M σy (x,y,z) τzy τyx τyz τzx τxz τx y σx

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2



z

 3

P

2  

Z

3

R

5 Substitute

Z/R

for

Cos β

, 

z

 3

p

2   cos 3

R

2  

z

 3

P

2 

Z

2     1  1

r Z

2    5 2  

P Z

2   2     1  3

r Z

2    5 2 Properties

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

p x

along horizontal surface

z

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

P σz Bulbs of pressure( 压力泡 ) Isoline( 等值线 ) ——Lines or Contours of equal stress increase

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

P 1 P 2 σz 1 σz 2 Z σz 1 + σz 2 Z

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.2 Stresses due to line load

（线荷载） ８ pdy p X Y X Z ８ σx τxz τzx M(X,Z) Z

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

d



z

 3

Z

3

pdy

2 

R

5  3

pdy

2   

X

2 

Y

2

Z

3 

Z

2  5 2 

z

    

d



z

  

X

2 2

pZ



Z

3 2  2   

p Z

  2        1  1

X Z

2       2

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

-polar coordinates

R

0 2 

X

2 

Z

2 , cos  

Z R

0 

z

 2

p



R

0 cos 3  

x

 2

p



R

0 cos  sin 2  

xz

 2

p



R

0 sin  cos 2 

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.3 Stresses due to uniform vertical loading on an infinite strip

（条形荷载） ８ Po b/2 Z b/2 Po d ξ ξ d ξ X σx τxz σz τzx Z M(X,Z) X( ξ)

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

d



z

   

X

2

Z

 3 

p

0  2

d

 

Z

2 2  

z

  

b b

2 / 2

d



z

  

b b

2 / 2   

X

2

Z

 3 

p

0  2

d

 

Z

2  2 

p

 0      

arctg

  

p

0

X



b

2

Z



arctg X



b

2

Z



Z Z

2 

b

2

b

2

Z

2 

Z

2 

b

2

b

2 2      

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

-polar coordinates  b dx P 0 A B β 1 0 β β 2 R 0

dx



R

0

d

 cos  Z dβ β σz Z M

d



z

 2

p

0 

R

0

dx

cos 3    2 

p

0  

R

0

R

0

d

 cos    cos 3   2 

p

0 cos 2 

d

 X

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2



z



p

0      1 2

d



z

    1  1 2 sin 2 

p

0    2 1 cos 2 

d

 2  1     2   1 2 sin   2  2    

x



p

 0    1  1 2 sin 2  1     2   1 2 sin  2  2    

xz



p

0 2   cos 2  2  cos 2  1  -Principal stress  1  3  

X

 

Z

2    

X

 

Z

2   2   2

xz

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

 1  3

ψ

 

β 1 p

0    

β 2



sin

 

(Out)

or ψ



β 1



β 2

(In)

tg

2  

tg

(  1   2 ),

then

   1   2 2  max  1 2   1   3  

p

0 

Sin

 Properties

(1) Isoline( 等值线 ) (2) Direction of principal stress (3) The value of maximum shear stress

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

A β 1 b β 2 B P 0 θ ψ /2 ψ σ 3 Μ σ1 Χ Ζ

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.4 Stresses due to Linearly increasing vertical loading on an infinite Strip

-X Y Ο ξ b ξP 0 b dξ P 0 dξ Χ ξ σ Ζ X Z Ζ ∝ X

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

d



z

   

X

2

Z

3    2 

Z

2  2 

p

0 

d



b



z

 2

Z

3 

b p

0 

p

0    

X b

  

p

0  0

b

 

X

 

arctg X Z

  

d

  2 

Z

2  2 

arctg X Z



b



Z

2

Z

 

X



Z

 

b b

  2     

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 2.4.5 Stresses due to a uniform loaded rectangular area

-central point method （中点法） b/2 b/2 Y a/2 o r ξ dξ n dn a/2 R z σ Ζ z m( x.y.z ) x

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

R

   2   2 

Z

2  1 2 

z

  2 

p

0 

a a

2   2 

b b

2 2 3

Z

3

p

0 2 

R d



d

 5   

arctg

2

Z a

2

ab



b

2 

a

0

p

0  4

Z

2  

a

2  4

Z

2 2

abZ



b

2 

a

2  4

Z

 2

b

 2  8

Z

2 

a

2 

b

2  4

Z

2   

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

-Corner method （角点法） b/2 b/2 Y a/2 z o POd ξ dξ n dn R C p o z σ Ζ M( x.y.z ) a/2 p o x



Z



p

0 2   

c

    / / 2 

a

 

a



b b

2 / 

a

2  / 2 2 3

Z Z

2

abZ



b

2 3

p

0

d

2 

R

5 

d

 

a

2  

Z

2

b

2  

a

2 2

Z

 2

b

 2 

p

0 

Z

2 

arctg Z ab a

2 

b

2 

Z

2    

c

 1 2     (

m

2

mn

(

m

2  2

n

2 

n

2 )( 1 

n

2 )

m

 1 ) 2 

n

2  1 

arctg n Where

,

m



a b

,

n



z b m

2

m



n

2  1   

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Principle of superposition

For stresses beneath points other than the corner of the loaded area the principle of superposition should be used, as described in Fig below.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2 Stresses beneath flexible area of any shape



Stresses beneath flexible area of any shape

(Figure 5.9)  Newmark (1942) devised charts to obtain the vertical stress at any depth, beneath any point (inside or outside) of an irregular shape. Use of the charts is explained in Figure 2.9.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Stresses beneath a flexible rectangle

♦ 

Stresses beneath a flexible rectangle (figure 2.7

) The vertical stressσ y at a depth

z

beneath the corner of a flexible rectangle supporting a uniform pressure σ v =

qI q

has been determined using: and influence factors

I

. given by Giroud (1970) are presented in Figure 5.7. they are for an infinite soil thickness.

 These curves are equivalent to the commonly used charts of Fadum (1948) but are easier to use.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Stresses beneath a rigid rectangle

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Exercise 2-4 A point load of P is applied at the ground surface.

Calculate the stress and plot the variation: (1) Vertical stress due to load P along vertical line(r=0) at the different depths Z=1,2,3,4,5m.

(2) Vertical stress due to load P from different distances (r=0,1,2,3,4,5m, respectively) along horizontal surface (at the depth Z=2m). (3) Vertical stress due to load P along vertical line(r=1.5m) at the different depths Z=0,0.5,1,2,3,4,5,6m.

Civil Engineering Department of Shanghai University Soil Mechanics Chapter 2

Exercise 2-5 A uniform load of p(=20kpa) is applied at the ground surface as shown in Fig below.

Calculate the stress due to load p beneath point A at 5m depth.