Vertical Stress in Soil Mass

The vertical stress at any point in a soil mass due to self-weight of the soil may be considered under the follow­ing heads: 1. Total Stress 2. Pore Water Pressure 3. Effective Stress.

1. Total Stress:

The total stress in a soil mass is the stress at any point due to the weight of the soil column above that point. Consider a column of soil of cross-sectional area A at any point P in the soil mass shown in Fig. 7.1. Let h be the height of soil above point P and y be the bulk density of soil. The force acting on the element due to the weight of the soil is given by –

W= γ × A × h

Vertical stress in the element due to the self-weight of the soil will be –

σ = W/A = γAh/A = γh …(7.1)

Equation (7.1) gives the total stress due to the self-weight at any point P, which is at a depth “h” below the ground surface. It is clear from Eq. (7.1) that the total stress in a soil mass increases linearly with the depth as shown in Fig. 7.1(a). Total stress is denoted by the symbol o. Its units are kgf/cm², t/m², or kN/m². It is the initial or original total vertical stress existing at any point in the soil mass before the construction of a foundation and the structure.

It may be noted that the soil mass is fully saturated below the groundwater table (GWT). GWT is the upper sur­face of groundwater in a soil mass. In some cases, the soil mass up to a small height above the GWT may become fully saturated due to the rise of groundwater above GWT by capillary action.

In Eq. (7.1), γ is the bulk density of soil and depends on the degree of saturation of the soil. When the soil is fully saturated, γ = γ_sat, and in this case, Eq. (7.1) becomes –

σ = γ_sat h …(7.2)

When the soil is dry, γ = γ_d, and in this case, Eq. (7.1) becomes –

σ = γ_d h …(7.3)

Figure 7.2 shows a soil mass in which GWT exists at the depth h₁ below the ground surface. If y₁ is the bulk den­sity of soil up to the depth h₁ and γ₂ = γ_sat is the saturated density of the soil below the GWT, then total stress at level 1-1 is given by –

σ₁ = γ₁ h₁

Total stress at level 2-2 is given by –

σ₂ = γ₁ h₁ + γ₂h₂

The distribution of total stress with the depth for the soil mass is shown in Fig. 7.2(a).

2. Pore Water Pressure:

The groundwater exists in voids or pores of the soil mass. Hence, it is also known as pore water. From the know­ledge of hydraulics, it is known that pore water exerts hydrostatic pressure on the soil mass at all points below the GWT. Figure 7.2 shows a soil mass, in which the GWT is at a depth h₂ below the ground level. Consider a point P at a depth h₂ below the GWT. The hydrostatic pressure exerted at any point in a soil mass is known as pore water pressure. We know that the hydrostatic pressure, that is, pore water pressure at point P is given by –

u = γ_wh₂ …(7.4)

Pore water pressure is also known as a neutral pressure and denoted by the symbol u. Its units are kgf/cm², t/m², or kN/m². The pore water pressure at any point on the surface of the groundwater is zero and increases linearly with increase in the depth below GWT, as shown in Fig. 7.2(b). In some cases, the soil above the GWT becomes sat­urated due to the capillary suction as shown in Fig. 7.3. The zone of this capillary water above the GWT is known as capillary fringe. The pore water pressure in the zone of capillary water is negative and proportional to the depth above GWT.

Referring to Fig. 7.3, pore water pressure at level 1-1 at a height h_c above GWT is given by –

u = –γ_w h_c

Pore water pressure at level 2-2 on the groundwater surface is given by –

u = 0

Pore water pressure at level 3-3 at a depth h₂ below GWT is given by –

U = γ_w h₂

The distribution of pore water pressure with the depth in a soil mass with capillary fringe is shown in Fig. 7.3(b).

3. Effective Stress Principle:

It is known from the Archimedes principle that an object immersed in a liquid appears to lose weight and this loss of weight is equal to the weight of the liquid displaced by the object. The actual stress acting at a point in a submerged soil mass, considering the effect of buoyancy, is known as the effective stress. Effective stress at depth h₂ below GWT in a submerged soil mass (as shown in Fig. 7.2) is therefore given by –

σ’ = γ₁h₁ + γ’h₂ = γ₁h₁ + (γ_sat – γ_w)h₂ = γ₁h₁ + γ_sath₂ – γ_wh₂

Þ σ’= σ – u …(7.5)

Equation (7.5) is the mathematical representation of effective stress principle given by Terzaghi in 1946. Thus, as per the effective stress principle, the effective stress acting at any point in a soil mass is equal to the total stress minus the pore water pressure at that point. Effective stress principle, stated by Terzaghi, is a significant develop­ment in the field of soil mechanics.

Effective stress principle has applications in the following areas of geotechnical engineering:

1. Shear strength of a soil is found to be a function of effective normal stress on the failure plane. The shear param­eters, cohesion, and angle of shearing resistance are computed on the basis of effective stress to consider the long-term stability of structures.

2. The process of consolidation is explained by Terzaghi on the basis of development and dissipation of pore water pressure. The process of consolidation is said to be completed when the effective stress increment is equal to the applied stress increment due to the construction, after complete dissipation of pore water pressure.

3. In shallow footings, the safe bearing capacity is computed considering the effective stress principle.

4. In pile foundations, the ultimate load capacity of the pile is computed by considering the effective stresses along the pile shaft and the pile tip.

5. In the analysis of stability of Earth slopes, the factor of safety is determined by considering effective stresses in vertical or inclined direction.

6. In the stability analysis of retaining structures, Rankine’s theory is widely used for the calculation of lateral Earth pressure. In this theory, the lateral Earth pressure is calculated by multiplying the effective stress with a lateral earth pressure coefficient.

Effective Stress in a Soil Mass:

Effective stress at any point in a soil mass is the algebraic difference between the total stress and pore water pressure at that point –

σ ‘ = σ – u …(7.6)

Here effective stress is denoted by the symbol σ’. Its units are kgf/cm², t/m², or kN/m². Referring to Fig. 7.1, effec­tive stress at a depth h below the ground surface is given by –

σ ‘ = σ – u = γh – 0 = γh

The distribution of effective stress with the depth for the soil mass is shown in Fig. 7.1(c). Referring to Fig. 7.2, effec­tive stress at depth h = h₁ + h₂ below ground surface is given by –

σ = σ – u = (γ₁h₁ + γ_sath₂) – γ_wh₂

The distribution of effective stress with depth for the soil mass is shown in Fig. 7.2(c). Referring to Fig. 7.3, effective stress at level 1 – 1 at a height h_c above GWT is given by –

σ’₁ = σ₁ – u₁ = γ₁(h₁ – h_c ) – (–γ_wh_c) = γ₁(h₁ – h_c) + γ_wh_c

Effective stress at level 2 – 2 on the groundwater surface is given by –

σ’₂ = σ₂ – u₂ = [γ₁(h₁ – h_c) + γ_sat h_c] – 0 = γ₁(h₁ – h_c)+ γ_sat h_c

Effective stress at level 3 – 3 at a depth h₂ below GWT is given by –

σ’₃ = σ₃ – u₃= [γ₁(h₁ – h_c) + γ_sat( h_c + h₂)] – (γ_wh₂) = γ₁(h₁ – h_c)+ γ_sat (h_c + h₂) – γ_wh₂

The distribution of effective stress with depth for the soil mass is shown in Fig. 7.3(c).