CEGB 333 GEOTECHNICAL ENGINEERING CHAPTER 3: STRESS AND DISPLACEMENT

FIGURE 3.1: Surface stress (load) cause settlement (Pisa Tower, Italy)

FIGURE 3.2: Changes in stress caused distress to pavement and drainage (signs of settlement)

STRESS & DISPLACEMENTINTRODUCTION

◦ All structures are constructed on soil or rock.

◦ These structures apply load on the soil. When soil has to carry additional load, the soil particles have to adjust themselves and regroup themselves in closer packing to withstand that load.

P

This adjustment and rearranging soil settlement

Therefore, there’s necessity to estimate or predict settlement to ensure the functionality of the structure is not affected.

settlement

STRESSESSOURCES OF STRESS IN THE GROUND◦ GEOSTATIC STRESSES (BODY STRESSES)◦ are those that occur due to the weight of the soil above the point being

evaluated. Geostatic stresses are naturally present in the ground. Human activities such as filling and excavation can cause them to change.

◦ Are caused by gravity acting on the soil or rock vertical normal stress sig. impact on eng behavior of soil and indirectly produces horizontal normal stresses and shear stresses.

• INDUCED STRESSES

◦ External load (structural foundation, vehicles, tanks, stockpile etc)

◦ These stresses can be the source of excessive settlement, shear failure and other problems.

TYPE OF LOADINGShape of external load:

1 . Point load on concentrated load applied from column, wheel of machine, load called point load, because it is effect in point.

2 .Line load which the load is effect on line as the load on Rail way and Dimension kN/m'.

3 .Uniform load which the load effect on area, whereas the load uniform and dimension kN/m2

4 .Triangular load which applied from embankment, so from the dam, the load dimension is kN/m

VERTICAL STRESS INCREASE DUE TO LOADING

1. STRESS CAUSED BY POINT LOAD

2. VERTICAL STRESS CAUSED BY LINE LOAD

3. VERTICAL STRESS DUE TO STRIP LOAD

4. VERTICAL STRESS BELOW A UNIFORMLY LOADED CIRCULAR AREA

5. VERTICAL STRESS CAUSED BY A RECTANGULARLY LOADED AREA

STRESS CAUSED BY POINT LOAD

Assumptions made based in Boussinesq Theory (1883) are:◦ Soil is homogeneous

◦ Soil is considered as elastic material

◦ Stress produced is isotropic

◦ The area of applied load is an infinite large half space

POINT LOAD

FIGURE 3.3: stresses due to point load

Vertical Horizontal

POINT LOAD

FIGURE 3.4: Stress in elastic medium caused by a point load

POINT LOAD

(3.1)

(3.2)

(3.3)

According to Figure 3.4, Boussinesq’s solution for normal stresses at a point A caused by the point load P is given by:

POINT LOAD

(3.4)

(3.5)

Note that Eqs 3.1 and 3.2 which are the expressions for horizontal normal stresses, are dependent on Poisson’s ratio of the medium. However, the relationship for the vertical normal stress, Δσz, as given by Eq (3.3), is independent of Poisson’s ratio. The relationship for Δσz can be written in the following form:

POINT LOADTABLE 3.1: Variation of IlThe variation of Il for

various value of r/z is given in Table 3.1:

TABLE 3.2: Representative values of Poisson’s ratio

Example 3.1 (question 8.10)Point loads of 9, 18, 27 kN act at A, B and C, respectively (Figure 3.5). Determine the increase in vertical stress at a depth of 3m below point D. Use Boussinesq’s equation.

3mAB

C D

3m

1.5m

FIGURE 3.5

VERTICAL STRESS CAUSED BY A LINE LOAD

FIGURE 3.6

TABLE 3.3

EXAMPLE 3.2 (QUESTION 8.13)Refer to Figure 8.20 in DAS 4th Edition. Determine the vertical stress increase,

VERTICAL STRESS CAUSED BY STRIP LOAD

FIGURE 3.7

TABLE 3.4

VERTICAL STRESS BELOW UNIFORMLY LOADED CIRCULAR AREA

FIGURE 3.8

TABLE 3.5

VERTICAL STRESS BELOW UNIFORMLY LOADED RECTANGULAR AREA

FIGURE 3.9

VERTICAL STRESS BELOW UNIFORMLY LOADED RECTANGULAR AREA

Variation of I2 with m and nFIGURE 3.10:

SUPERPOSITION THEORY

FIGURE 3.11

Superposition Theory :To estimate the stress under foundation in the soil.

X

X

X

A

DC

A

DC

BBA

DC

B

Stress at X= XECH-XEDG-

XFAH+XFBG

E

F

GH

E

F

G

H

Stress at X=

XHAG+XFBG+XECH+XEDF

Stress at X= XGCE+XHDE-

XFAG-XHBF

E

F

G H

*** to use superposition, all shape MUST share the point in question under

one of their corners.

FIGURE 3.12

(a) (b) (c)

EXERCISE 1

A

B

F

ED

C

JH

G The plan of a uniformly loaded rectangular area is shown in Figure E1. Determine the vertical stress increase, Δσ below point A at a depth z = 5m.

Figure E1

100kN/m2

EXERCISE 2

A

B

F

ED

C

JH

G The plan of a uniformly loaded rectangular area is shown in Figure E2. Determine the vertical stress increase, Δσ below point A at a depth z = 5m.

Figure E2

150kN/m2

EXERCISE 3

A

B

F

ED

C

JH

G The plan of a uniformly loaded rectangular area is shown in Figure E3. Determine the vertical stress increase, Δσ below point A at a depth z = 10m.

Figure E3

100kN/m2

Newmark’s ChartBased on Boussinesq’s Theory

Vertical stress can be determined at any point below an area of any shape carrying a uniform pressure, q.

Using tracing paper.

Loaded area drawn to scale.

Z depth = AB scale

Figure 3.13

Figure 3.14: Settlement of soil

Settlement from elastic theoryIs called elastic settlement or immediate settlement.

Settlement that occurs immediately after load is being placed.

Elastic settlement is caused by the elastic deformation of dry soil and of moist and saturated soils without any change in the moisture content.

The vertical displacement (s) under an area carrying a uniform pressure, q on the surface of a semi infinite, homogeneous, isotropic mass, with a linear stress-strain relationship, can be expressed as

IsvE

qBs )1( 2

Where E is the Young’s Modulus of the soil and Is is a influence factor depending on the shape of the loaded area. In the case a rectangular area, B is lesser dimension (the greater dimension being L); in the case of a circular area, B is the diameter.

Value of influence factors are given in Table 3.6 as follows for displacement under the centre and a corner (the edge in the case of circle) of a flexible loaded area (i.e. having negligible bending stiffness) and also for the average displacement under the area as a whole.

Table 3.6Shape Flexible* Rigid^

Centre Corner Average

Circle 1.00 0.64 0.85 0.79

Rectangle

𝐿

𝐵

1.0 1.122 0.561 0.946 0.82

1.5 1.358 0.679 1.148 1.06

2.0 1.532 0.766 1.300 1.20

3.0 1.783 0.892 1.527 1.42

4.0 1.964 0.982 1.694 1.58

5.0 2.105 1.052 1.826 1.70

10.0 2.540 1.270 2.246 2.10

100.0 4.010 2.005 3.693 3.47

* After Giroud (1968)^ after Skempton (19510

In the case of an extensive, homogeneous deposit of saturated clay, it is a reasonable approximation to assume that E is constant throughout the deposit and the distribution of Figure 3.15 (a) applies.

In the case of sand, the value of E varies with confining pressure and therefore will vary across width of the loaded area, being greater under the centre of the area than the edges as in Figure 3.15 (b); the contact pressure will again be uniform if the area is flexible.

Figure 3.15: settlement behaviour of flexible foundation

(a) clay (b) sand

If loaded area is rigid (infinitely stiff in bending), the vertical displacement will be uniform across the width of the area and its magnitude will be only slightly less than the average displacement under a corresponding flexible area. Influence factor for rigid foundation are also given in Table 3.6.

Under a rigid area, the contact pressure distribution is not uniform. For a circular area the forms of the distributions of contact pressure on clay and sand respectively are shown in Figure 3.16 (a) and 3.16(b).

Figure 3.16: settlement behaviour of rigid foundation

(a) clay (b) sand

In most cases, in practice the soil deposit will be of limited thickness and will be underlain by hard stratum (e.g. bedrock).

For this case, the average displacement under a flexible area carrying a uniform pressure q is given by

Se = μ0μ1𝑞𝐵

𝐸Where μ0 depends on the depth of embedment and μ1 depends n the layer thickness and the shape of loaded area.

ELASTIC SETTLEMENTIn predicting settlement, these factors must be taken into account:◦The magnitude of stresses applied,◦The geometry of the plan area which is stressed,◦The depth at which the stress is applied in relation to the magnitude of the width of the stresses area as well as the thickness of the compressible soil stratum,

◦The compressibility of the soil stratum, and◦The rigidity of foundation element.

Exercise Chapter 3

Sem 1 2011/2012 T2 : Question 2

Sem 1 2011/2012 FE : Question 3

Sem 2 2012/2013 T1: Question 3

Sem 2 2011/2012 FE : Question 2