SOIL MECHANICS - CHAPTER 4 ppt

The pressure at the bottom in each case is Only in case of a container with vertical sides this is equal to the total weight of the water in the container.. cross section in the center,

STRESSES IN SOILS

As in other materials, stresses may act in soils as a result of an external load and the volumetric weight of the material itself Soils, however, have a number of properties that distinguish it from other materials Firstly, a special property is that soils can only transfer compressive normal stresses, and no tensile stresses Secondly, shear stresses can only be transmitted if they are relatively small, compared to the normal stresses Furthermore

. .

.

.

.

.

x

y

Figure 4.1: Stresses

it is characteristic of soils that part of the stresses is transferred by the water in the pores This will

be considered in detail in this chapter

Because the normal stresses in soils usually are compressive stresses only, it is standard practice to use a sign convention for the stresses that is just opposite to the sign convention of classical continuum mechanics, namely such that compressive stresses are considered positive, and tensile stresses are negative The stress tensor will be denoted by σ The sign convention for the stress components is illustrated in Figure 4.1 Its definition is that a stress component when it acts in a positive coordinate direction on a plane with its outward normal in a negative coordinate direction, or when it acts in negative direction on a plane in positive direction This means that the sign of all stress components

is just opposite to the sign that they would have in most books on continuum mechanics or in applied mechanics

stress is acting, and the second index denotes the direction of the stress itself This means, for instance,

sign convention for forces is the same as in mechanics in general

Soil is a porous material, consisting of particles that together constitute the grain skeleton In the pores of the grain skeleton a fluid may be present: usually water The pore structure of all normal soils is such that the pores are mutually connected The water fills a space of very complex form, but it constitutes a single continuous body In this water body a pressure may be transmitted, and the water may also flow

25

through the pores The pressure in the pore water is denoted as the pore pressure.

In a fluid at rest no shear stresses can be transmitted This means that the pressure is the same in all directions This can be proved

by considering the equilibrium conditions of a small triangular element, see Figure 4.2, bounded by a vertical plane, a horizontal plane and a

.

.

.

.

.

. pA pA Figure 4.2: Pascal sloping plane at an angle of 45◦ If the pressure on the vertical plane at the right is p, the force on that plane is pA, where A is the area of that plane Because there is no shear stress on the lower horizontal plane, the horizontal force pA must be equilibrated by a force component on the sloping plane That component must therefore also be pA Because on this plane also the shear stress is zero, it follows that there must also be a vertical force pA, so that the resulting force on the plane is perpendicular to it This vertical force must be in equilibrium with the vertical force on the lower horizontal plane of the element Because the area of that element is also A, the pressure on that plane is p, equal to the pressure on the vertical plane Using a little geometry it can be shown that this pressure p acts on every plane through the same point This is often denoted as Pascal’s principle If the water is at rest (i.e when there is no flow of the water), the pressure in the water is determined by the location of the point considered with respect to the water surface As shown by Stevin the magnitude of the water pressure on the bottom of a container filled with water,

d .

.

.

.

.

.

.

.

.

.

.

.

.

Figure 4.3: Hydrostatic water pressure depends upon depth only

depends only upon the height of the column of water and the volumetric weight of the water, and not upon the shape of the container, see Figure 4.3 The pressure at the bottom in each case is

Only in case of a container with vertical sides this is equal to the total weight of the water in the container Stevin showed that for the other

of the water body, taking into account that the pressure in every point on the walls must always be perpendicular to the wall The container at

the extreme right in Figure 4.3 resembles a soil body, with its pore space It can be concluded that the water in a soil satisfies the principles of hydrostatics, provided that the water in the pore space forms a continuous body

On an element of soil normal stresses as well as shear stresses may act The simplest case, however, is the case of an isotropic normal stress, see Figure 4.4 It is assumed that the magnitude of this stress, acting in all directions, is σ In the interior of the soil, for instance at a

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

..

.

.

.

..

.

.

.

.

.

.

.

...

Figure 4.4: Isotropic stress

cross section in the center, this stress is transmitted by a pore pressure p in the water, and by stresses in the particles The stresses in the particles are generated partly by the concentrated forces acting in the contact points between the particles, and partly by the pressure in the water, that almost completely surrounds the particles It can be expected that the deformations of the particle skeleton are almost completely determined by the concentrated forces in the contact points, because the structure can deform only by sliding and rolling in these contact points The pressure in the water results in an equal pressure in all the grains It follows that this pressure acts on the entire surface of a cross section, and that by subtracting p from the total stress σ a measure for the contact forces is obtained It can also be argued that when there are no contact forces between the particles, and a pressure p acts in the pore water, this same pressure p will also act in all the particles, because they are completely surrounded

by the pore fluid The deformations in this case are the compression of the particles and the water caused by this pressure p Quartz and water are very stiff materials, having an elastic modulus about 1/10 of the elastic modulus os steel, so that the deformations in this case are very small

These considerations indicate that it seems meaningful to introduce the difference of the total stress σ and the pore pressure p,

not mean that the stresses in the grains are zero in that case, because there will always be a stress in the particles equal to the pressure in the surrounding water The basic idea is, as stated above, that the deformations of a granular material are almost completely determined by changes of the concentrated forces in the contact points of the grains, which cause rolling and sliding in the contact points These are described (on the average) by the effective stress, a concept introduced by Terzaghi Eq (4.2) can, of course, also be written as

Terzaghi’s effective stress principle is often quoted as “total stress equals effective stress plus pore pressure”, but it should be noted that this applies only to the normal stresses Shear stresses can be transmitted by the grain skeleton only

It may be noted that the concept is based upon the assumption that the particles are very stiff compared to the soil as a whole, and also upon the assumption that the contact areas of the particles are very small These are reasonable assumptions for a normal soil, but for porous rock they may not be valid For rock the compressibility of the rock must be taken into account, which leads to a small correction in the formula

To generalize the subdivision of total stress into effective stress and pore pressure it may be noted that the water in the pores can not contribute to the transmission of shear stresses, as the pore pressure is mainly isotropic Even though in a flowing fluid viscous shear stresses may be developed, these are several orders of magnitude smaller than the pore pressure, and than the shear stresses than may occur in a soil This suggests that the generalization of (4.3) is

This is usually called the principle of effective stress It is one of the basic principles of soil mechanics The notation, with the effective stresses

Even though the equations (4.4) are very simple, and may seem almost trivial, different expressions may be found in some publications

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

. .

.

. .

.

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

p

Figure 4.5: Effective stress

this is that the pore water pressure acts in the pores only, and that therefore a quantity np must be subtracted from the total stress σ to obtain a measure for the stresses in the particle skeleton That seems to make sense, and it may even give a correct value for the average stress

in the particles, but it ignores that soil deformations are not in the first place determined by deformations of the individual particles, but mainly by changes in the geometry of the grain skeleton This average granular stress might be useful if one wishes to study the effect of stresses

on the properties of the grains themselves (for instance a photo-elastic or a piezo-electric effect), but in order to study the deformation of soils it is not useful Terzaghi’s notion that the soil deformations are mainly determined by the contact forces only leads directly to the concept of

are no contact forces The pore pressure must be considered to act over the entire surface to obtain a good measure for the contact forces, see Figure 4.5

The equations (4.4) can be written in matrix notation as



Calculating the effective stresses in soils is one of the main problems of soil mechanics The effective stresses are important because they determine the deformations In the next chapter the procedure for the determination of the effective stress will be illustrated for the simplest case, of one-dimensional deformation In later chapters more general cases will be considered, including the effect of flowing groundwater

The concept of effective stress is so important for soil mechanics that it deserves careful consideration It may be illuminating, for instance, to note that the concept of effective stress is in agreement with the principle of Archimedes for the upward force on a submerged body

The remaining force is

which must be transmitted to the bottom on which the particles rest If the area of the volume is denoted by A, and the height by h, then the

Following Terzaghi the effective stresses must be determined as the difference of the total stress and the pore pressure The total stress is

Because for a saturated soil the volumetric weight is

this can also be written as

This is identical to the expression (4.7) Terzaghi’s principle of effective stress appears to be in agreement with the principle of Archimedes, which is a fundamental principle of physics It may be noted that in the two methods it has been assumed that the determining factor is the force transmitted between the particles and an eventual rigid surface, or the force transmittance between the grains This is another basic aspect

of the concept of effective stress, and it cannot be concluded that Archimedes’ principle automatically leads to the principle of effective stress

. . . . . . .

Figure 4.6: Archimedes

Terzaghi’s approach, leading to the expression (4.8), is somewhat more direct, and especially more easy to generalize In this method the porosity n is not needed, and hence it is not necessary to determine the porosity to calculate the effective stress On the other hand, the porosity is hidden in the volumetric weight

flowing groundwater, is relatively simple For a non-saturated soil the total stresses will be smaller, because the soil is lighter The pore pressure remains hydrostatic, and hence the effective stresses will be smaller, even though there are just as many particles as in the saturated case The effective principle can also be applied in cases involving different fluids (oil and water, or fresh water and salt water) In the case of flowing groundwater the pore pressures must be calculated separately, using the basic laws of groundwater flow Once these pore pressures are known they can be subtracted from the total stresses to obtain the effective stresses The procedure for the determination of the effective stresses usually is that first the total stresses are determined, on the basis of the total weight of the soil and all possible loads Then the pore pressures are determined, from the conditions on the groundwater Then finally the effective stresses are determined by subtracting the pore pressures from the total stresses

Problems

4.1 A rubber balloon is filled with dry sand The pressure in the pores is reduced by 5 kPa with the aid of a vacuum pump Then what is the change of the total stress, and the change of the effective stress?

4.2 An astronaut carries a packet of vacuum packed coffee into space What is the rigidity of the pack (as determined by the effective stresses) in a spaceship? And after landing on the moon, where gravity is about one sixth of gravity on earth?

4.3 A packet of vacuum coffee is dropped in water, and it sinks to a depth of 10 meter Is it harder now?

4.4 A treasure hunter wants to remove a collection of antique Chinese plates from a sunken ship Under water the divers must lift the plates very carefully,

of course, to avoid damage Is it important for this damage to know the depth below water of the ship?

4.5 The bottom of a lake consists of sand The water level in the lake rises, so that the water pressure at the bottom is increased Will the bottom of the lake subside by deformation of the sand?