Module 6 Notes: Vertical Stresses in Soils

Module 6: Vertical Stresses in Soils

Introduction

• Intergranular or effective pressure:
  • Pressure transmitted through grain-to-grain contact points in a soil mass.
  • Responsible for decreasing void ratio or increasing frictional resistance.
• Pore water pressure or neutral stress:
  • Pressure induced in pore water that tries to separate grains.
  • Increases volume or decreases frictional resistance of the soil mass.
• Terzaghi (1925) developed the effective stress concept, a key concept in modern soil mechanics.
• Effective stress influences:
  • Soil strength and volume change.
  • Capillary rise.
  • Seepage force due to water flow.
  • Quicksand (sand boiling).
  • Heaving at the bottom of excavations.

Topic Outcomes

1. Determine stresses in soil by the action of solid particles and water.
2. Determine the critical hydraulic gradient in soil with upward seepage that will cause heaving of soil.
3. Determine the additional increase in soil stress caused by surface loads.

Stresses in Saturated Soil Without Seepage

• Total stress at a point can be obtained from the saturated unit weight of the soil and the unit weight of water above it.

Total Stress

• The total stress, $\sigma$, can be divided into two parts:

  1. Stress carried by water in the continuous void spaces, acting with equal intensity in all directions.
  2. Stress carried by the soil solids at their points of contact.

• Effective stress,$\sigma'$, is the sum of the vertical components of the forces developed at the points of contact of the solid particles per unit cross-sectional area of the soil mass.

• $\sigma' = \frac{P<em>1(v) + P</em>2(v) + P<em>3(v) + … + P</em>n(v)}{\overline{A}}$

  • Where $P<em>1(v), P</em>2(v), P<em>3(v), …, P</em>n(v)$ are the vertical components of the forces acting at the points of contact.
  • $\overline{A}$ is the cross-sectional area of the soil mass.

• If $a<em>s$ is the cross-sectional area occupied by solid-to-solid contacts ($a</em>s = a<em>1 + a</em>2 + a<em>3 + … + a</em>n$), then the space occupied by water equals $(\overline{A} - a_s)$.

• $\sigma = \frac{\sum<em>{i=1}^{n} P</em>i(v)}{\overline{A}} + u = \frac{\sum<em>{i=1}^{n} P</em>i(v)}{\overline{A} - a_s} + u$

• The value of $a_s$ is extremely small and can be neglected for pressure ranges generally encountered in practical problems.

• $\sigma = \sigma' + u$

  • Where $u$ is the pore water pressure, also referred to as neutral stress.

• $\sigma' = \sigma - u$

• $\sigma' = z\gamma<em>{\text{sat}} - z\gamma</em>w = z(\gamma<em>{\text{sat}} - \gamma</em>w) = z\gamma'$

  • Where $\gamma' = \gamma<em>{\text{sat}} - \gamma</em>w$ equals the submerged unit weight of soil.

• The effective stress at any point A is independent of the depth of water, H, above the submerged soil.

Principle of Effective Stress

• First developed by Terzaghi (1925, 1936).
• Skempton (1960) extended Terzaghi's work.
• Effective stress is approximately the force per unit area carried by the soil skeleton.
• The effective stress in a soil mass controls its volume change and strength.
• Increasing the effective stress induces soil to move into a denser state of packing.
• The effective stress principle is probably the most important concept in geotechnical engineering.
• The compressibility and shearing resistance of a soil depend to a great extent on the effective stress.
• The concept of effective stress is significant in solving geotechnical engineering problems, such as:
  • Lateral earth pressure on retaining structures.
  • Load-bearing capacity and settlement of foundations.
  • Stability of earth slopes.

Example #1

• Calculate the total stress, pore water pressure, and effective stress at points A, B, and C in a soil profile.

• Solution:

  • At Point A:
    • Total stress: $\sigma_A = 0$
    • Pore water pressure: $u_A = 0$
    • Effective stress: $\sigma'_A = 0$
  • At Point B:
    • $\sigma<em>B = 6\gamma</em>{\text{dry(sand)}} = 6 \times 16.5 = 99 \text{ kN/m}^2$
    • $u_B = 0 \text{ kN/m}^2$
    • $\sigma'_B = 99 - 0 = 99 \text{ kN/m}^2$
  • At Point C:
    • $\sigma<em>C = 6\gamma</em>{\text{dry(sand)}} + 13\gamma_{\text{sat(clay)}} = 6 \times 16.5 + 13 \times 19.25 = 99 + 250.25 = 349.25 \text{ kN/m}^2$
    • $u<em>C = 13\gamma</em>w = 13 \times 9.81 = 127.53 \text{ kN/m}^2$
    • $\sigma'_C = 349.25 - 127.53 = 221.72 \text{ kN/m}^2$

Stresses in Saturated Soil With Upward Seepage

• If water is seeping, the effective stress at any point in a soil mass will differ from that in the static case.
• It will increase or decrease, depending on the direction of seepage.
• The effective stress at a point located at a depth z measured from the surface of a soil layer is reduced by an amount $iz\gamma_w$ because of upward seepage of water.
• When the rate of seepage and thereby the hydraulic gradient gradually are increased, a limiting condition will be reached, at which point $\sigma' = 0$.
• $i<em>{cr} = \frac{\gamma'}{\gamma</em>w}$
  * where $i_{cr}$ = critical hydraulic gradient (for zero effective stress).
• Under such a situation, soil stability is lost. This situation generally is referred to as boiling, or a quick condition.
• For most soils, the value of $i_{cr}$ varies from 0.9 to 1.1, with an average of 1.

Example 9.2

• Calculate the maximum depth of cut H that can be made in the clay.

Stresses in Saturated Soil With Downward Seepage

• The hydraulic gradient caused by the downward seepage equals $i = \frac{h}{H_2}$.

Seepage Force

• The effect of seepage is to increase or decrease the effective stress at a point in a layer of soil.

• Often, expressing the seepage force per unit volume of soil is convenient.

• With no seepage, the effective stress at a depth z is equal to $z\gamma'$.

• The effective force on an area A is $P_1' = A z \gamma'$.

• With upward seepage of water, the effective force on an area A at a depth z can be given by $P<em>2' = A(z\gamma' - iz\gamma</em>w)$.

• Hence, the decrease in the total force because of seepage is $P<em>1' - P</em>2' = Aiz\gamma_w$.

• The volume of the soil contributing to the effective force equals zA, so the seepage force per unit volume of soil is $i\gamma_w$.

• The force per unit volume, $i\gamma_w$, for this case acts in the upward direction—that is, in the direction of flow.

• Similarly, for downward seepage, the seepage force in the downward direction per unit volume of soil is $i\gamma_w$.

• The seepage force per unit volume of soil is equal to $i\gamma_w$, and in isotropic soils, the force acts in the same direction as the direction of flow.

• Flow nets can be used to find the hydraulic gradient at any point and, thus, the seepage force per unit volume of soil.

Example 9.3

• Consider the upward flow of water through a layer of sand in a tank.

• Given: void ratio (e) = 0.52 and specific gravity of solids = 2.67.

  • Calculate the total stress, pore water pressure, and effective stress at points A and B.
  • What is the upward seepage force per unit volume of soil?

• Solution:

  • Part a
    • The saturated unit weight of sand is calculated as follows:
      • $\gamma<em>{sat} = \frac{(G</em>s + e)\gamma_w}{1+e} = \frac{(2.67 + 0.52)9.81}{1 + 0.52} = 20.59 \text{ kN/m}^3$
    • The stresses at point A:
      • Total stress, $\sigma$ = 0.7$\gamma<em>w$ + 1$\gamma</em>{sat}$ = (0.7)(9.81) + (1)(20.59) = 27.46 kN/m²
      • Pore water pressure, u = [(1 + 0.7) + (1.5/2) (1)] $\gamma_w$ = (2.45)(9.81) = 24.03 kN/m²
      • Effective stress, $\sigma'$ = 3.43 kN/m²
    • The stresses at point B:
      • Total stress, $\sigma$ = 0.7$\gamma<em>w$ + 2$\gamma</em>{sat}$ = (0.7)(9.81) + (2)(20.59) = 48.05 kN/m²
      • Pore water pressure, u = (4.2)(9.81) = 41.2 kN/m²
      • Effective stress, $\sigma'$ = 6.85 kN/m²
  • Part b
    • Hydraulic gradient (i) = 1.5/2 = 0.75.
    • Seepage force per unit volume = i$\gamma_w$ = (0.75)(9.81) = 7.36 kN/m³

Stress Distribution in Soils Due to Surface Loads

• Estimation of vertical stresses at any point in a soil-mass due to external vertical loadings are of great significance in the prediction of settlements of buildings, bridges, embankments, and many other structures.

• Equations have been developed to compute stresses at any point in a soil mass on the basis of the theory of elasticity.

• According to elastic theory, constant ratios exist between stresses and strains.

• For the theory to be applicable, the real requirement is not that the material necessarily be elastic, but there must be constant ratios between stresses and the corresponding strains.

• Therefore, in non-elastic soil masses, the elastic theory may be assumed to hold so long as the stresses induced in the soil mass are relatively small.

• Since the stresses in the subsoil of a structure having adequate factor of safety against shear failure are relatively small in comparison with the ultimate strength of the material, the soil may be assumed to behave elastically under such stresses.

• When a load is applied to the soil surface, it increases the vertical stresses within the soil mass.

• The increased stresses are greatest directly under the loaded area but extend indefinitely in all directions.

• Many formulas based on the theory of elasticity have been used to compute stresses in soils.

• They are all similar and differ only in the assumptions made to represent the elastic conditions of the soil mass.

• The formulas that are most widely used are the Boussinesq and Westergaard formulas.

• These formulas were first developed for point loads acting at the surface.

• These formulas have been integrated to give stresses below uniform strip loads and rectangular loads.

• The extent of the elastic layer below the surface loadings may be any one of the following:

  1. Infinite in the vertical and horizontal directions.

  2. Limited thickness in the vertical direction underlain with a rough rigid base such as a rocky bed

• The loads at the surface may act on flexible or rigid footings.

• The stress conditions in the elastic layer below vary according to the rigidity of the footings and the thickness of the elastic layer.

• All the external loads considered are vertical loads only as the vertical loads are of practical importance for computing settlements of foundations.

Boussinesq’s Formula for Point Load

• A semi-infinite solid is the one bounded on one side by a horizontal surface, here the surface of the earth, and infinite in all the other directions.

• The problem of determining stresses at any point P at a depth z as a result of a surface point load was solved by Boussinesq (1885) on the following assumptions.

  1. The soil mass is elastic, isotropic, homogeneous and semi-infinite.

  2. The soil is weightless.

  3. The load is a point load acting on the surface.

• The soil is said to be isotropic if there are identical elastic properties throughout the mass and in every direction through any point of it.

• The soil is said to be homogeneous if there are identical elastic properties at every point of the mass in identical directions.

• The expression obtained by Boussinesq for computing vertical stress $\sigma_z$, at point P due to a point load Q is

• $\Delta\sigma<em>z = \frac{3Q}{2\pi z^2} \left[ \frac{1}{1 + (r/z)^2} \right]^{5/2} = \frac{Q}{z^2}I</em>B$

  • where,

    • r = the horizontal distance between an arbitrary point P below the surface and the vertical axis through the point load Q .
    • z = the vertical depth of the point P from the surface .

• The values of the Boussinesq coefficient $I_B$ can be determined for a number of values of r/z.

• $I_B$ has a maximum value of 0.48 a t r/ z = 0, i.e., indicating thereby that the stress is a maximum below the point load .

• Or in three-dimension,
  $\Delta\sigma_z = \frac{3P}{2\pi} \frac{z^3}{(r^2 + z^2)^{5/2}}$

• The relationship for $\Delta\sigma_z$ can be rewritten as

• The variation of $I_1$ for various values of r/z is given in Table 14

Example #1:

• Consider a point load P = 5 kN. Calculate the vertical stress increase $\Delta\sigma_z$ at z = 0, 2 m, 4 m, 6 m, 10 m, and 20 m. Given x = 3 m and y = 4 m.

Example #2:

• A concentrated load of 1000 kN is applied at the ground surface. Compute the vertical pressure (i) at a depth of 4m below the load, (ii) at a distance of 3m at the same depth . Use Boussinesq's equation

Vertical Stress Caused by a Vertical Line Load

• The vertical stress increase, $\Delta\sigma_z$, inside the soil mass can be determined by using the principles of the theory of elasticity, or

• $\Delta\sigma_z = \frac{2q z}{\pi (x^2 + z^2)}$

• The value of $\Delta\sigma_z$ is the additional stress on soil caused by the line load.

• The value of $\Delta\sigma_z$ does not include the overburden pressure of the soil above point A.

Example #3:

• Determine the increase of stress at point A.

Vertical Stress Caused by a Horizontal Line Load

• The vertical stress increase at point A in the soil mass can be given as

• $\Delta\sigma_z = \frac{2q}{\pi} \frac{x z^2}{(x^2 + z^2)^2}$

Example #4:

• An inclined line load with a magnitude of 1000 lb/ft is shown in Figure below. Determine the increase of vertical stress $\Delta\sigma_z$ at point A due to the line load.
• Solution:
  • $\Delta\sigma_z$ = 23 lb/ft2
  • $\Delta\sigma_z$ = 10.7 lb/ft2

Vertical Stress Below the Center of a Uniformly Loaded Circular Area

• Using Boussinesq’s solution for vertical stress $\Delta\sigma_z$ caused by a point load, one also can develop an expression for the vertical stress below the center of a uniformly loaded flexible circular area.
  The increase in the stress at point A caused by the entire loaded area can be found by integration:
• $\Delta\sigma_z = q\left[1 - \frac{1}{\left[1 + \left(\frac{R}{z}\right)^2\right]^{3/2}}\right]$

Vertical Stress Caused by a Rectangularly Loaded Area

• Boussinesq’s solution also can be used to calculate the vertical stress increase below a flexible rectangular loaded area, as shown in Figure 90.

• To determine the increase in the vertical stress ($\Delta\sigma_z$) at point A, which is located at depth z below the corner of the rectangular area, we need to consider a small elemental area dx dy of the rectangle.

• The increase in the stress, at point A caused by the entire loaded area can now be determined by integrating the preceding equation. We obtain

• The increase in the stress at any point below a rectangularly loaded area can be found by using the above equation

Example #5:

• The plan of a uniformly loaded rectangular area is shown in the figure below. Determine the vertical stress increase $\Delta\sigma_z$ below point A’ at a depth of z = 4 m.

Solution:

• A$\sigma$ = A$\sigma$(1) A$\sigma$(2)

• A$\sigma$ (1) = stress increase due to the loaded area

• A$\sigma$ (2) = stress increase due to the loaded area

• For the loaded area:

• m = B/Z = 2/4 = 0.5

• n = L/Z = 4/4 = 1

• From Figure 10.21 for m = 0.5 and n = 1, the value of 13 = 0.1225. So

• A$\sigma$(1) = q13 = (150) (0.1225) = 18.38 kN/m²

• Similarly, for the loaded area :

• m = B/Z = 1/4 = 0.25

• n = L/Z = 2/4 = 0.5

• Thus, 13 = 0.0473. Hence

• $\Delta\sigma$(2) = (150)(0.0473) = 7.1 kN/m²

• $\Delta\sigma$ = $\Delta\sigma$(1) - $\Delta\sigma$(2) = 18.38 - 7.1 = 11.28 kN/m²

2:1 Approximate Slope Method

• Vertical stress $\sigma_{v,0}$, on the ground surface is P/(B × L).

• This $\sigma_{v,0}$ is redistributed over a wider loading area with increasing depth z.

• A slope with 2 in vertical to 1 in horizontal defines spread loading areas within the soil mass.

• Stress is spread over an area of (B + z) × (L + z) at depth z.

• Accordingly, the vertical stress increment $\Delta\sigma_v$ at depth z can be calculated from

• $\Delta\sigma_v = \frac{P}{(B+z)(L+z)}$

• Vertical stress decreases with increasing depth z with increased distributed area

• It is assumed that the stress is uniformly distributed over (B + z) × (L + z) area and it suddenly becomes zero beyond the zone defined by 2:1 slope.

Example #6:

• A 5 kN point load is applied at the center of 1 m × 1 m square footing on the ground surface. Compute and plot the magnitudes of a vertical stress increment under the center of the footing at the depths 2, 4, 6, 8, and 10 m from the ground surface. Use the 2:1 approximate slope method.

End of Module Exercises:

1. Point loads of magnitude 2000, 4000, and 6000 lb act at A, B, and C, respectively. Determine the increase in vertical stress at a depth of 10 ft below point D. Use Boussinesq’s equation.
2. Refer to the figure below. Determine the vertical stress increase, $\Delta\sigma_z$, at point A with the following values:
   • q1 = 75 kN/m
   • x1 = 2 m
   • q2 = 300 kN/m
   • x2 = 3 m
   • z = 2 m
3. Refer to the figure. Due to the application of line loads q1 and q2, the vertical stress increase, $\Delta\sigma_z$, at A is 30 kN/m2 . Determine the magnitude of q2.
4. The plan of a flexible rectangular loaded area is shown in the figure below. The uniformly distributed load on the flexible area, q, is 100 kN/m2 . Determine the increase in the vertical stress, $\Delta\sigma_z$, at a depth of z = 2 m below
   • Point A
   • Point B
   • Point C
5. Three footings are placed at locations forming an equilateral triangle of 13 ft sides. Each of the footings carries a vertical load of 112.4 kips. Estimate the vertical pressure increase by means of the Boussinesq equation at a depth of 9 ft at the following locations :
   • Vertically below the centers of the footings,
   • Below the center of the triangle.
6. Three concentrated loads Ql = 255 kips, Q2 = 450 kips and Q3 = 675 kips act in one vertical plane and they are placed in the order Ql-Q2-Q3. Their spacings are 13 ft and 10 ft respectively. Determine the vertical pressure at a depth of 5 ft along the center line of footings.
7. A square footing of size 13 x 13 ft founded on the surface carries a distributed load of 2089 lb/ft2. Determine the increase in pressure at a depth of 10 ft by the 2:1 method
8. A loaded footing ABCD with q = 200 kN/m2 on the ground is shown in the figure below. Compute $\Delta\sigma_z$ under Points E, F, B, and G at a depth of 5 m
9. 50, 100, and 150 kN point loads are applied at Points A, B, and C, respectively, on the ground surface as seen in the figure. Compute the vertical stress increment under Point D down to the depth z = 20 m.

Bibliography

• Principles of Geotechnical Engineering; 7th Edition; Braja M. Das
• Geotechnical Engineering (Principles and Practices of Soil Mechanics and Foundation Engineering); V.N.S. Murthy
• Soil Mechanics Fundamentals and Applications; 2nd Edition; Isao Ishibashi, Hemanta Hazarika