Effective Stress in Soils — Transcript Notes

Effective Stress in Soils – Transcript Notes

Context and purpose
- This lecture completes the discussion on stresses and soils by focusing on effective stress, which is central to almost every geotechnical analysis.
- Key idea: total stress in soil is composed of the portion carried by the fluid (pore water) and the portion carried by the solid soil skeleton. The latter is the effective stress, which governs deformation and strength.
- The concept was introduced by Carl Herzog and is foundational for soil mechanics and geotechnical engineering.
Core definitions
- Total vertical stress in a soil body:
  \sigma_V = \gamma \; z
  where \gamma is the unit weight of the soil (which can vary with depth) and z is depth below the ground surface.
- Pore water pressure (static):
  u = \gammaw \; zw
  where \gammaw is the unit weight of water and zw is the depth below the groundwater surface. In saturated soils, the pore pressure exists in all directions and is isotropic:
  uv = uh = u\n
- Effective stress (stress carried by the soil skeleton):
- Vertical: \sigma'V = \sigmaV - u
- General 3D form: \sigma' = \sigma - u (for the corresponding principal stress)
- Horizontal effective stress (when needed): \sigma'H = \sigmaH - u
- Total horizontal stress relation (using effective stresses):
  \sigmaH = \sigma'H + u
- Conceptual takeaway: effective stress is the stress that governs soil deformation and strength; the pore water pressure reduces the effective stress available to the soil skeleton.
Dry vs saturated conditions
- Dry soil: no pore water (or negligible pore pressure). Then u = 0 and \sigma'V = \sigmaV. The solid grains bear the entire load.
- Saturated soil: pore water bears part of the load, reducing the effective stress on the soil skeleton. The effective stress is lower than the total stress by the amount of pore pressure.
- In many geotechnical analyses, we work with the effective stress framework rather than total stress alone because it correlates to settlement, shear strength, and stiffness.
Physical intuition about pore pressure and stress directions
- Pore pressure acts in all directions (isotropic) and reduces the net force transmitted through the soil skeleton.
- The vertical stress often varies with depth due to overburden; pore pressure increases with depth below groundwater, typically linearly with depth:
  u = \gammaw \; zw
- The total vertical stress is a sum of the pore pressure and the vertical effective stress:
  \sigmaV = \sigma'V + u\n
Linking to horizontal stresses and at-rest condition
- Lateral earth pressure coefficient at rest, k0, relates horizontal and vertical effective stresses under undisturbed conditions: k0 = \frac{\sigma'H}{\sigma'V}
- In many problems, \sigma'H = k0 \; \sigma'V and then total horizontal stress is \sigmaH = \sigma'_H + u\n.
- Typical range for k0 in many soils is 0.3 \leq k0 \leq 0.5, though it can be greater than 1 in certain slope situations. When not enough information is available, a common rough assumption is k_0 \approx 0.5 for first-pass calculations.
- Cannot (knot) is the value used to relate horizontal and vertical stresses in the effective stress state at rest, i.e., k0 = \sigma'H / \sigma'_V under in-situ, undisturbed conditions. It is a property of the soil and material state, not directly a property of the water or air.
The flow of the lecture: moving from dry to saturated and to practical calculations
- Start with a simplified dry skeleton: a vertical stack of soil particles and a horizontal plane at depth z. The stress on that plane is due to the weight of soil above it.
- When dry: the pore pressure is zero; the effective stress equals the total vertical stress on that plane.
- When saturated: part of the load is carried by pore water, so the effective stress on the soil skeleton is reduced by the pore pressure.
- The general approach in calculations is to compute the total stress, compute the pore pressure, and then obtain the effective stress by subtraction: \sigma'V = \sigmaV - u (and similarly for horizontal stresses).
Worked example framework (concepts to be applied in the upcoming numerical example)
- The vertical stress increases with depth due to overburden: \sigmaV(z) = \gamma z (dry) or \sigmaV(z) = \gamma{sat} z (saturated; where \gamma{sat} = (W{soil}+W{water})/V).
- Pore pressure distribution depends on the groundwater table location. If the groundwater table is at the surface, the pore pressure at depth z is simply u = \gammaw z. If the groundwater table is below the surface, use the depth below the groundwater surface for u = \gammaw z_w.
- The effective stress distribution is then obtained by subtracting pore pressure from total stress and can be plotted as a function of depth to compare the three stress components:
- Total vertical stress: \sigma_V(z)
- Pore pressure: u(z)
  - Isotropic in all directions: uv = uh = u
- Vertical effective stress: \sigma'V(z) = \sigmaV(z) - u(z)
- If needed, horizontal stresses: \sigma'H(z) = k0 \; \sigma'V(z) and total horizontal: \sigmaH(z) = \sigma'_H(z) + u(z)
Key points about groundwater level effects
- If the groundwater level changes below the ground surface, the effective stress below changes (because both total stress and pore pressure change in a way that can alter the difference \sigma'V = \sigmaV - u).
- If the groundwater level changes above the ground surface (e.g., a surface lake or river rising/falling), the effective stress below the ground does not change through the direct mechanism of pore pressure because the pore pressure at depth is governed by the depth below the groundwater surface, not the surface water level above.
- The total stress always increases with depth; the pore pressure increases with depth below the groundwater surface; the resulting change in effective stress is what governs deformation and strength.
Summary of the three key stresses
- Total stress: \sigma (vector magnitude in a given direction; vertical often emphasized as \sigma_V)
- Pore pressure: u (isotropic, directionally equal in all principal directions)
- Effective stress: \sigma' = \sigma - u
- The principal effective stresses are what control soil behavior under loading and drainage conditions.
Concrete example walkthrough (case scenarios and numbers)
- Setup: a common in-class example uses dry sand over saturated sand with a groundwater table at a certain depth. This allows illustration of how the stress components vary with depth and groundwater position.
- Case where the groundwater table is at the ground surface (u acts from the surface downward):
- At depth 4 m in saturated soil (gamma_sat = 18 kN/m^3):
  - Total vertical stress: \sigmaV = \gamma{sat} \; z = 18 \times 4 = 72\ \text{kPa}
  - Pore water pressure: (surface groundwater) u = \gamma_w \; z = 9.81 \times 4 = 39.2\ \text{kPa}
  - Vertical effective stress: \sigma'V = \sigmaV - u = 72 - 39.2 = 32.8\ \text{kPa}
- This demonstrates how pore pressure reduces the effective stress carried by the soil skeleton.
- Case A (A is 2 m below water surface; i.e., two metres of water above the point):
- Pore pressure: u = \gamma_w \times 2 = 9.81 \times 2 = 19.6\ \text{kPa}
- Total vertical stress: \sigma_V = u = 19.6\ \text{kPa} (since only the water column contributes here in this simplified view)
- Vertical effective stress: \sigma'V = \sigmaV - u = 0\ \text{kPa}
- Case C (two metres of water above ground plus four metres of saturated soil below):
- Total vertical stress from soils and water:
  \sigma_V = (\text{water head}) + (\text{soil head}) = (2 \text{ m} \times 9.81) + (4 \text{ m} \times 18) = 19.6 + 72 = 91.6\ \text{kPa}
- Pore pressure at depth z = 6 m below the groundwater surface (two metres above ground plus four metres of soil):
  u = \gamma_w \times 6 = 9.81 \times 6 = 58.9\ \text{kPa}
- Vertical effective stress: \sigma'V = \sigmaV - u = 91.6 - 58.9 = 32.7\ \text{kPa} \approx 32.8\ \text{kPa}
- Note: This yields the same vertical effective stress as the single-case scenario with 4 m soil depth and surface groundwater, illustrating consistency in the effective-stress framework.
- Case D (deeper depth with more extreme loading):
- At depth D (e.g., 14 m below ground surface) with a certain total vertical stress (example given):
  - Total vertical stress: \sigma_V = 246\ \text{kPa}
  - Pore pressure: u = 117.7\ \text{kPa}
  - Vertical effective stress: \sigma'V = \sigmaV - u = 128.3\ \text{kPa}
  - If we assume k0 = 0.5 (case where horizontal effective stress is a function of vertical):
    \sigma'H = k0 \; \sigma'_V = 0.5 \times 128.3 = 64.1\ \text{kPa}
  - Total horizontal stress: \sigmaH = \sigma'H + u = 64.1 + 117.7 = 181.8\ \text{kPa} (≈ 181.9 kPa in lecture notes)
- Alternate (slightly different groundwater table) scenario (Case B):
- If the groundwater table is 2 m higher (i.e., different pore-pressure distribution), then the vertical effective stress at a given depth can be different (e.g., at point C, it might be 46.4 kPa instead of 60 kPa in the example’s variation). This illustrates how groundwater fluctuations alter the effective-stress distribution through changes to u and the resultant difference with the total stress.
- Summary of the calculations in this example family:
- Always compute: \sigmaV, u, then \sigma'V = \sigma_V - u.
- If horizontal stresses are needed under in-situ (undisturbed) conditions: use \sigma'H = k0 \; \sigma'V and then \sigmaH = \sigma'_H + u.
- The key takeaway is that effective stress is controlled by the difference between the total stress and the pore pressure, and it governs soil behavior rather than the total stress alone.
Practical implications and takeaways
- The effective stress principle is the cornerstone of soil mechanics; most deformation and strength behavior in saturated soils are dictated by changes in effective stress.
- Fluctuations in the groundwater table can modify the effective stress distribution below the surface. If the groundwater table moves, total stress and pore pressure change in a way that alters the effective stress; if the surface water level changes, the effective stress below can remain the same depending on the geometry of the pore-pressure distribution.
- When performing hand calculations, a common approach for a first approximation is to assume a reasonable knot (often around 0.5) for the lateral earth-pressure coefficient unless more detailed data are available. Then compute the horizontal effective stress and total horizontal stress accordingly.
- Always distinguish between: (i) total stress, (ii) pore pressure, and (iii) effective stress; and remember the isotropy of pore pressure in all directions when computing principal stresses.
Quick recap of key equations to memorize
- Total vertical stress: \sigmaV = \gamma \; z or \sigmaV = \gamma_{sat} \; z (saturated region)
- Pore pressure in general: u = \gammaw \; zw
- Effective stress (vertical): \sigma'V = \sigmaV - u
- Effective stress (general): \sigma' = \sigma - u
- Horizontal effective stress (in many problems): \sigma'H = k0 \; \sigma'_V
- Total horizontal stress: \sigmaH = \sigma'H + u
Quick definitions to keep straight
- Dry soil: u = 0, \sigma'V = \sigmaV
- Saturated soil: u > 0, \sigma'V = \sigmaV - u
- In situ (at-rest) condition: k0 = \sigma'H / \sigma'_V and it relates horizontal to vertical effective stresses.
Final note
- The lecture emphasizes that effective stress is the single most important concept in soil mechanics for understanding how saturated soils behave, including strength and deformation, and it lays the foundation for many of the subsequent topics in geotechnical engineering (including the upcoming discussion on seepage).
Quick reminder for instructors and students
- Always check whether the scenario is dry, partially saturated, or fully saturated, and whether the groundwater table is at the surface or at some depth below the surface.
- Use the correct unit weights: dry unit weight, saturated unit weight, and water unit weight (often ~9.81 kN/m^3 for water) depending on the depth and soil condition.
- When possible, relate changes in groundwater level to their effect on sigma'_V and discuss practical consequences (e.g., potential settlement or strength reduction when effective stress decreases).
References to future topics
- Tomorrow’s topic will be seepage and how water movement through soil interacts with the effective-stress framework.
- The discussion also lays groundwork for understanding vertical versus horizontal stress distributions, including the at-rest state and the role of k0 in static earth pressures.