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Effective Stresses and Capillary - N. Sivakugan (2004) 1
Chapter 6 Effective Stresses and Capillary
6.1 INTRODUCTION When soils are subjected to external loads due to buildings, embankments or excavations, the state of stress within the soil in the vicinity changes. To study the stability or deformations of the surrounding soil, as a result of the external loads, it is often necessary to know the stresses within the soil mass fairly accurately. Elastic solutions are quite popular in geotechnical engineering. Here, the entire soil mass is assumed to be a continuous elastic media, and the theory of elasticity is applied to determine the state of stress at a point. Some special cases such as the vertical stress increase beneath uniformly loaded square and strip footings are given in most textbooks. Harr (1966), Poulos & Davis (1974), Terzaghi (1943) and several others developed elastic solutions in geotechnical engineering. These developments, with more refinements, have been summarised in design handbooks (Canadian Geotechnical Society 1992; Fang 1991; Winterkorn & Fang 1975). Nevertheless, soils do not deform elastically. Further, they are particulate media. Therefore, the elastic solutions should only be used with caution.
6.2 EFFECTIVE STRESS CONCEPT In saturated soils, the normal stress (σ) at any point within the soil mass is shared by the soil grains and the water held within the pores. The component of the normal stress acting on the soil grains, is called effective stress or intergranular stress, and is generally denoted by σ'. The remainder, the normal stress acting on the pore water, is knows as pore water pressure or neutral stress, and is denoted by u. Thus, the total stress at any point within the soil mass can be written as: (6.1) ' u+=σσ This applies to normal stresses in all directions at any point within the soil mass. In a dry soil, there is no pore water pressure and the total stress is the same as effective stress. Water cannot carry any shear stress, and therefore the shear stress in a soil element is carried by the soil grains only.
6.4 VERTICAL NORMAL STRESSES DUE TO OVERBURDEN GL
Figure 6.1 Overburden stress at a point in a homogeneous soil
Effective Stresses and Capillary - N. Sivakugan (2004) 2
In a dry soil mass having a unit weight of γ (see Fig. 6.1), the normal vertical stress at a depth of h is simply γh. If there is a uniform surcharge q placed at the ground level, this stress becomes γh+q. In a soil mass with three different soil layers as shown in Fig. 6.2, the vertical normal stress at X is γ1h1 + γ2h2 + γ3h3.
Soil 3, γ3
Soil 2, γ2
Soil 1, γ1h1
Figure 6.2. Overburden stress at a point in a layered soil Now let’s see what happens in a saturated soil? For the soil shown in Fig. 6.3, for simplicity we will assume that the water table is at the ground level. Let the saturated unit weight and submerged unit of the soil be γsat and γ′ respectively. The total vertical normal stress at X is given by:
(6.2) hsatv γσ = The pore water pressure at this point is simply,
(6.3) hu wγ= Therefore, the effective vertical normal stress is,
uvv −= σσ '
= γsath - γwh = γ′h
Figure 6.3 Overburden stress at a point in a homogeneous saturated soil When the water table is at some depth below the ground level as shown in Fig. 6.4, the total and effective vertical stresses and the pore water pressure can be written as:
Effective Stresses and Capillary - N. Sivakugan (2004) 3
21 hh satmv γγσ +=
21 ' 'hhmv γγσ +=
Figure 6.4 Overburden stresses at a point when the water table is below the ground level
When computing total vertical stress, use saturated unit weight for soil below the water table and bulk or dry unit weight for soil above water table.
When the soils are partially saturated, the situation is a bit more complex. Here, the normal stress within a soil element is carried by the water, air, and the soil grains. Therefore, the normal stress can be split into three components and written as:
(6.4) )1( ' aw uu χχσσ −++= Here uw and ua are the pore water pressure and pore air pressure respectively, and χ is a constant that can be determined from triaxial test and varies between 0 and 1. In dry soils χ=0 and in saturated soils χ=1.
6.3 CAPILLARY EFFECTS IN SOILS
Capillary tube of inner diameter d
α α T T
Figure 6.5 Capillary tube in water
Effective Stresses and Capillary - N. Sivakugan (2004) 4
Let’s look at some simple physics on capillary. A capillary tube is placed in a dish containing water as shown in Fig. 6.5. Immediately, water rises to a height of hc within the tube. The water column is held by the surface tension T at the top (see inset), which acts at an angle of α to vertical. For equilibrium of the water column, the weight of the water column is balanced by the vertical components of the surface tension. This can be written as:
(6.5) d cos 4
παγπ Thd wc = Therefore,
(6.6) cos 4
wd Thc γ
Using typical values of T = 0.073 N/m, α = 0° and γw = 9810 N/m3 in Eq. 6.6, it can be shown that:
What do these have to do with soils? The interconnected voids within the soil can act like capillary tubes (not straight though) and allow the water to rise well above the water table. The “capillary tube” diamater of a soil is approximately 1/5 of D10. Therefore, the capillary rise within a soil can be written as:
15.0)( 10 mmD
As you would expect, finer the soil, smaller the capillary tube diameter, and larger the capillary rise. This can be also inferred from Eq. 6.8, which works well for sands and silts. Gravels are so coarse that there will be negligible capillary effects. Clays have the most capillary rise. Capillary rise can be few milli metres in sands to several metres in clays. Capillary pressure is a pore water pressure that is always negative. Since this occurs while there is no change in total stress, it increases the effective stresses significantly.
Capillary pressure is always negative and gives a suction effect, and increases the effective stress.
You have to be quite clear with the effective stress principle, which will come in any time we have to compute stresses when there is water in the soil.
1. Plot the variation of total and effective vertical stresses, and pore water pressure with depth for the soil profile shown below in Fig. 6.6.
Effective Stresses and Capillary - N. Sivakugan (2004) 5
Sand γsat = 19.5 kN/m3
Sandy gravel γsat = 19.0 kN/m3
4 m Gravely sand γsat = 18.5 kN/m3; γm = 17.8 kN/m3
Figure 6.6 Soil profile for Problem 6.1 Solution: Within a soil layer, the unit weight is constant, and therefore the stresses vary linearly. Therefore, it is adequate if we compute the values at the layer interfaces and water table location, and join them by straight lines. At the ground level, σv = 0 ; σv’ = 0; and u=0 At 4 m depth,
σv = (4)(17.8) = 71.2 kPa; u = 0 ∴σv’ = 71.2 kPa
At 6 m depth,
σv = (4)(17.8) + (2)(18.5) = 108.2 kPa u = (2)(9.81) = 19.6 kPa
∴σv’ = 108.2 – 19.6 = 88.6 kPa At 10 m depth,
σv = (4)(17.8) + (2)(18.5) + (4)(19.5) = 186.2 kPa u = (6)(9.81) = 58.9 kPa
∴σv’ = 186.2 – 58.9 = 127.3 kPa
At 15 m depth, σv = (4)(17.8) + (2)(18.5) + (4)(19.5) + (5)(19.0) = 281.2 kPa
u = (11)(9.81) = 107.9 kPa ∴σv’ = 281.2 – 107.9 = 173.3 kPa
The values of σv, u and σv’ computed above are summarized in Table 6.1.
Table 6.1 Values of σv, u and σv’ in Ex. 1 depth (m) σv (kPa) u (kPa) σv' (kPa)
0 0 0 0 4 71.2 0 71.2 6 108.2 19.6 88.6
10 186.2 58.9 127.3 15 281.2 107.9 173.3
Effective Stresses and Capillary - N. Sivakugan (2004) 6
The plot is shown below in Fig. 6.7.
0 50 100 150 200 250 300
Stress or Pressure (kPa)
Total stress Pore water pressure Effective stress
Figure 6.7 Variation of σv, u and σv’ with depth
REFERENCES Canadian Geotechnical Society (1992). Canadian foundation engineering manual, 3rd edition. Fang, H-Y. (1991). Foundation engineering handbook, van Nostrand Reinhold, New York. Harr, M.E. (1962). Foundations of theoretical soil mechanics, McGraw-Hill, New York. Poulos, H.G. and Davis, E.H. (1974). Elastic solutions for soil and rock mechanics, John
Wiley & Sons. Terzaghi , K. (1943). Theoretical soil mechanics, John Wiley & Sons, New York. Winterkorn, H.F. and Fang, H-Y. (1975). Foundation engineering handbook, van Nostrand