Notes on Total Stress, Pore Pressure, and Effective Stress
Overview: foundation for geomechanics rests on understanding how stresses are applied in soils, including total stress, pore pressure, and effective stress. The course will build from static in-situ conditions to more complex scenarios (varying vertical vs horizontal stresses, depth variation, and groundwater effects).
Key goals: define and evaluate stresses within a soil profile and grasp the principle of effective stress (to be covered in depth in the next lecture).
Notation to memorize:
Total vertical stress: \sigma_V (V = vertical)
Pore water pressure: u
Vertical effective stress: \sigma'V (often written with a dash, i.e., \sigma'V). In some older literature this is shown as a bar over the symbol.
Effective stress relation (in many geomechanics problems): \sigma'V = \sigmaV - u
Unit weight of water: \gamma_w = 9.81\ \text{kN/m}^3
Soil unit weight: \gamma (or \gamma1, \gamma2 for layered soils)
Depth notation: depth increases positively downward from the ground surface (z).
Units: stresses are in kilopascals (kPa) and weights in kilonewtons per meter cubed (kN/m^3). 1 kPa = 1 kN/m^2.
Why this matters: practical calculations require knowing how stress changes with depth and how groundwater (pore pressure) modifies the stress state that the soil grains 'feel' (the effective stress). This governs strength, stability, and deformation.
Visualization metaphor: a gnome under the soil “feels” the stress from the soil above him, not from what lies below him. This helps illustrate that total stress depends on what’s above the point of interest, while pore pressure and effective stress quantify how that stress translates into soil reaction and strength.
Capillary and unsaturated effects (brief preview): water can rise into voids from the phreatic surface due to capillary forces, producing suction (negative pore pressure) which can temporarily increase soil strength. In many geotechnical analyses, these unsaturated/capillary effects are neglected (u ≈ 0 above groundwater table) for conservative design, but they are important for slope stability during rainfall and for understanding capillary rise. Capillary rise height depends on grain size and void structure: sands/gravels have only a few centimeters of rise; silts can rise ~1 m; clays can have tens of meters of rise. The discussion of capillary effects and seepage is reserved for later lectures.
Foundational references mentioned: Mechanics sections 3.1–3.2 (textbooks can be used to cross-check derivations) and Introductions to Geotechnical Engineering (Kovacs). The lecturer notes a historical error in Kovacs section 6.9, reminding students that even well-known textbooks can contain mistakes.
Summary of two core “stresses” plots you’ll encounter:
Water-column case (pure water): total stress and pore pressure are the same at a given depth (in a liquid column without soils present). The stress distribution has a constant gradient set by the unit weight of water.
Soil column with layers: total stress increases with depth, but the slope (gradient) changes when you cross a layer boundary if the layer has a different unit weight. The pore pressure distribution depends on groundwater position (water table) and can alter the effective stress profile.
Practical takeaway: to evaluate stresses at a depth z in a soil profile, you need (a) the layerwise unit weights, (b) the depth to groundwater (water table), and (c) the geometry of layers. Then you can compute \sigmaV(z), u(z), and \sigma'V(z) = \sigma_V(z) - u(z).
Next topics to expect in the course: Tuesday will cover effective stress in detail (the most fundamental concept), including how vertical and horizontal stresses relate to each other at rest and how effective stress governs failure and deformation in soils.
Quick notation refresher (for quick reference in your notes):
Vertical total stress: \sigma_V
Vertical effective stress: \sigma'_V
Pore water pressure: u
Unit weight of water: \gamma_w
Dry unit weight: \gamma_{dry}
Saturated unit weight: \gamma_{sat}
Depth to groundwater table (phreatic surface): often denoted as the depth where pore pressure resets to atmospheric (zero gauge pressure) at the surface of the groundwater, with u increasing with depth below it.
Important caveats mentioned:
In older literature you may see a bar notation on the dash (e.g., \sigma_V with a bar) indicating the same thing as a dashed notation, i.e., the effective stress.
There can be mistakes in derivations in published textbooks; cross-check with lecture notes and other references.
The instructor emphasizes the intuition behind the math: think about what the soil is “feeling” from the soil above and how water in voids changes the effective stress state.
Two core physical setups discussed (for intuition):
Static water column: stress increases linearly with depth, with a single gradient equal to the unit weight of water; pore pressure equals total stress in this pure-water scenario; useful for reviewing basic hydrostatics.
Layered soil with and without groundwater: total stress is piecewise linear with depth, changing slope at layer boundaries; pore pressure depends on whether you are above or below the groundwater table; effective stress is the difference between total stress and pore pressure, and will be discussed in the next lecture.
Key formulas to memorize (LaTeX):
Total vertical stress in a single homogeneous soil layer (depth z from ground surface): \sigma_V(z) = \gamma \, z
Two-layer vertical stress (with layer 1 of depth up to z1 and layer 2 from z1 downward):
For z \le z1: \sigmaV(z) = \gamma_1 \, z
For z > z1: \sigmaV(z) = \gamma1 \, z1 + \gamma2 \, (z - z1)
Pore water pressure distribution (with groundwater table at depth z_w):
If z \le z_w (above groundwater): u(z) = 0
If z > zw (below groundwater): u(z) = \gammaw \,(z - z_w)
Effective vertical stress: \sigma'V(z) = \sigmaV(z) - u(z)
Unit weight of water: \gamma_w = 9.81\ \text{kN/m}^3
Capillary rise concept (qualitative): capillary rise height depends on soil void size; give ranges for sands, silts, clays; capillary suction can temporarily increase soil strength above the groundwater table.
Note on depth convention used in the examples:
Depth z increases downward from the ground surface.
The water table is the phreatic surface; above it, pores are unsaturated (u ≈ 0 in many problems); below it, pores are saturated (u increases with depth).
Quick study prompts you can test yourself with:
Given a two-layer soil with γ1 and γ2 (γ2 > γ1) and a groundwater table at depth zw, compute σV and u at a point depth z. Then obtain σ'V = σV - u. How does σ'_V change when the groundwater table rises or falls?
For a purely water column, verify that σV(z) = u(z) and hence σ'V(z) = 0 at all depths.
Two worked example outlines (as in the lecture, to help you study):
Case A (groundwater table located at z_w = 4 m): two soil layers with γ1 = 15 kN/m^3 (layer 1) and γ2 = 18 kN/m^3 (layer 2).
A: ground surface point => \sigma_V = 0\ \text{kPa},\ u = 0\,\text{kPa}
B: at z = 2 m (within layer 1): \sigma_V = 2\cdot 15 = 30\ \text{kPa},\ u = 0\ \text{kPa}
C: at z = 4 m (2 m in layer 1, 2 m in layer 2): lecture notes show \sigmaV = 60\ \text{kPa} (note: if you strictly use the piecewise form with z1 = 2 m, you would get \sigma_V = 2\cdot 15 + 2\cdot 18 = 66\ \text{kPa}; the key idea is the gradient changes across the layer boundary). Here, u = 0\ \text{kPa}
D: at z = 14 m (4 m in layer 1, 10 m in layer 2): \sigma_V = 4\cdot 15 + 10\cdot 18 = 240\ \text{kPa},\ u = 10\text{ m} \cdot 9.81 = 98.1\ \text{kPa}
Resulting effective stress at D: \sigma'_V = 240 - 98.1 = 141.9\ \text{kPa}
Case B (groundwater table rises to z_w = 2 m): same depths and layer properties; compute anew.
A: still 0 and 0
B: 2 m depth in layer 1: \sigma_V = 30\ \text{kPa}, u = 0\ \text{kPa}
C: z = 4 m with two layers: \sigmaV = 2\cdot 15 + 2\cdot 18 = 66\ \text{kPa}; u = (z - zw)\gamma_w = (4-2)\cdot 9.81 = 19.6\ \text{kPa}
D: z = 14 m: \sigma_V = 4\cdot 15 + 10\cdot 18 = 240\ \text{kPa}; u = (14-2)\cdot 9.81 = 117.7\ \text{kPa}
Resulting effective stress at C: \sigma'_V = 66 - 19.6 = 46.4\ \text{kPa}
Resulting effective stress at D: \sigma'_V = 240 - 117.7 = 122.3\ \text{kPa}
Takeaways from these examples:
Elevating the groundwater table (increasing zw) generally increases u at depths below zw, which lowers the effective stress \sigma'_V at those points (all else equal).
The presence of multiple layers changes the gradient of the total stress with depth when crossing layer boundaries.
The same depth with different groundwater conditions yields different u and thus different \sigma'_V; this directly impacts soil strength and stability analyses.
Quick “how to” for exams or problem sets:
Step 1: identify layer boundaries and unit weights for each layer (i.e., determine γ in each interval).
Step 2: decide whether the point of interest is above or below the groundwater table to assign u.
Step 3: compute σ_V(z) using the appropriate piecewise formula for the depth relative to layer boundaries.
Step 4: compute u(z) from the groundwater position: if z > zw, use u(z) = \gammaw\,(z - z_w), otherwise u(z) = 0 (above water table).
Step 5: obtain \sigma'V(z) = \sigmaV(z) - u(z).
Ethical/philosophical/practical note: while capillary suction and unsaturated effects can increase soil strength in dry or partially dry conditions, engineers often adopt conservative assumptions (e.g., ignoring suction above the groundwater table) for safety. However, understanding when and where suction can occur is important for risk assessment (e.g., rainfall-induced slope failures). Always check whether the problem context warrants considering unsaturated soil mechanics or capillary effects.
Visual aids to reinforce learning (suggested in lectures):
Draw a depth vs. unit weight plot for a two-layer system to visualize the kink at z_1 where the gradient changes.
Sketch a phreatic surface with the water table and shade unsaturated vs saturated zones; annotate u accordingly.
Use the gnome metaphor to annotate what each point “feels” above it (the portion of soil above that point).
Final reminder: practice these steps with the actual numbers from your problem sets and Canvas exercises. The exact numeric results will depend on the precise layer thicknesses and the depth to groundwater assumed for each case, but the methodology (compute σV, then u, then σ'V) stays constant.
If you want, I can convert the Case A and Case B numbers into a compact table or plot the stress vs depth diagrams to accompany your notes for quick reference on exam day.