ECIV 731 Slope Stability and Earth Retaining Systems — Study Notes

Bloom's Taxonomy

• Produce new or original work
  • verbs: create, design, assemble, construct, conjecture, develop, formulate, author, investigate
• Evaluate
  • verbs: justify a stand or decision, appraise, argue, defend, judge, select, support, value, critique, weigh
• Draw connections among ideas
  • verbs: differentiate, organize, relate, compare, contrast, distinguish, examine, experiment, question, test
• Use information in new situations
  • verbs: execute, implement, solve, use, demonstrate, interpret, operate, schedule, sketch
• Explain ideas or concepts
  • verbs: classify, describe, discuss, explain, identify, locate, recognize, report, select, translate
• Recall facts and basic concepts
  • verbs: define, duplicate, list, memorize, repeat, state

Note: Bloom’s Taxonomy as presented by the lecture places these cognitive processes in a hierarchical framework, often used to frame learning objectives and assessment.

Example

• The slide labeled "Example" appears to illustrate stress orientation and magnitudes on a plane or area with labeled points A, B, D, B', etc.
• Given numerical annotations include values such as 40 psi, 30°, 10 psi, 20 psi, and directional cues (A, B, D, B', etc.).
• The diagram seems to compare stresses under different orientations or loading conditions (e.g., principal-stress directions, rotated coordinate systems).
• Purpose: to show how stress magnitudes change with orientation and how principal directions relate to loading on a surface.

Note: The slide appears schematic; specifics beyond the listed numbers are not fully described in the transcript.

Principal stress planes

• Concept: Principal stress planes are the orientations where shear stress is zero; the normal stresses on these planes are the principal stresses.
• Notation: usually denoted as
  • 
    σ1 = maximum principal stress,
    σ2 = intermediate principal stress,
    σ3 = minimum principal stress.
• Implication: Failure analyses (e.g., Mohr-Coulomb) are often simplified in principal stress space because shear on principal planes is zero.

Geotechnical example: Tank and geostatic stresses (Slide set from Page 5)

• Tank geometry and loading
  • Tank diameter: 153\tfrac{1}{4}\text{ ft} = 153.25\text{ ft}
  • Tank height: 129\tfrac{1}{2}\text{ ft} = 129.5\text{ ft}
  • Dead/structure load or external load parameter (DAS): 5500\ \text{psf}
• Soil properties
  • Soil type: medium to fine sand
  • Unit weight: \gamma = 129\ \text{pcf}
  • In-situ horizontal stress ratio: K_0 = 0.4
• K0/geostatic stress relations
  • Horizontal stress: \sigmah = K0\,\sigma_v
  • Vertical stress: generally approximated as \sigma_v = \gamma z for depth z
  • Given data also references typical stress progressions with depth (0, 10 ft, 20 ft, 30 ft, 40 ft) and corresponding stress increments on a circular/planar area.
• Additional notes observed on the slide
  • A set of curves or data labels (e.g., H 300, 10, 20, 30, 40) indicating depth or load increments and resulting stress magnitudes in kip/ft²
  • The slide title indicates a geostatic analysis context for the circular area under the tank footprint or subjected area.

Stresses under uniform load on circular area (Figure 8.5)

• Topic: Stresses induced by a uniform load on a circular area
• Key takeaways (as depicted by Fig. 8.5):
  • Distribution of stress components under a uniform circular load
  • Variations of the components with distance from center (radial effects)
  • Notation includes KR (likely rheological or reduction factor), NR, XR, and related indicators used in the stress distribution analysis
• Purpose: to relate a uniform circular load to the resulting stress state in the soil or footing, which is essential for stability analysis around circular loading areas

Uniformly loaded square area (flexible) – Plan (Page 7)

• Topic: Comparison between loading shapes and their effect on stress distribution
• Key elements shown:
  • Uniformly loaded square area with a flexible boundary condition
  • Labels such as 5B, 2.5B, 2B indicating dimensionless or relative sizes used in the analysis
  • A graph or schematic showing the relationship between the applied load and resulting shear/normal stress components, with a reference to a divisor 0.9 = Δστ (change in shear stress)
• Takeaway:
  • The shape of the loaded area (square vs circular) affects the stress distribution beneath the area
  • The slide contrasts rigid vs flexible boundaries or shapes in planning footing/retaining systems

Instrumentation: Sensor layout and tactile pressure (Pages 8–9)

• Physical setup
  • Rigid or shallow footing with a shallow contact surface
  • Sensor array configuration: shallow footing with tactile pressure sensor sheets
  • Sensor sizes displayed: 10 cm × 10 cm sensors, 25 cm × 25 cm footprint, and a larger 91 cm reference (likely spacing or placement dimension)
• Sensor type
  • Tactile Pressure Sensor (Sheet)
• Data capture (in-situ measurements)
  • Ready A I-Scan HS – Sensor 1 through Sensor 4
  • Example readings (Force):
  • Sensor 1: F_1 = 2.3027\ \text{kN}
  • Sensor 2: F_2 = 1.8667\ \text{kN}
  • Sensor 3: F_3 = 1.6796\ \text{kN}
  • Sensor 4: F_4 = 0.9401\ \text{kN}
  • Time stamps: around 388–389 seconds into the measurement sequence
  • Derived quantities displayed:
  • Net Area for each frame: e.g., A_{net} = 0.0381\ \text{m}^2, 0.0577\ \text{m}^2, 0.0593\ \text{m}^2, 0.0582\ \text{m}^2
  • Net Force corresponding to these areas
• Pressure data across the array
  • Pressure readings vs distance across columnar sensors
  • Sensor 4: P = 4.370\ \text{kPa} at distance 0.005588\ \text{m}
  • Sensor 3: P = -5.599\ \text{kPa} at distance 0.011176\ \text{m}
  • Sensor 2: P = 2.319\ \text{kPa} at distance 0.011176\ \text{m}
  • Sensor 1: P = -1.039\ \text{kPa} at distance 0.011176\ \text{m}
• Data interpretation
  • Time-series force data for each sensor show how load redistributes over the surface during testing
  • Pressure distribution maps suggest local variations in contact pressure under the footing

Failure criteria: purpose and theoretical failure plane (Page 10)

• Questions addressed:
  • What is the failure criterion?
  • Why is it needed?
  • What is the theoretical failure plane?
• Context: Establishes the framework for judging soil/rock stability and predicting failure planes under given stress states

Mohr–Coulomb Failure Criteria (Page 11)

• Governing relation (shear strength on a plane):
  • \tau_f = c + \sigma \tan\phi
  • where:
  • (\tau_f) = shear strength on the considered plane
  • (c) = cohesion
  • (\sigma) = normal effective stress on the plane
  • (\phi) = friction angle
• Meaning:
  • Describes the shear strength of soil/rock as a function of confining pressure (via \sigma) and intrinsic material properties (c and \phi)
  • Used as a criterion to predict failure under complex stress states by transforming to a Mohr circle envelope in the shear-stress/normal-stress space

Mohr envelope and parameter interpretation (Pages 12–13)

• Mohr envelope concept
  • A straight-line envelope that fits the failure data from Mohr-Coulomb relationships
  • Two representative fits are shown:
  • Fit 1: c = 5.6\ \text{psi}, \; \phi = 37^{\circ}
  • Fit 2: c = 0, \; \phi = 46^{\circ}
• Graphical interpretation (as depicted)
  • The envelope (actual) is plotted with data points; a straight-line best fit is overlaid to estimate material parameters (c and \phi)
  • The x-axis denotes normal stress (usually (\sigma_n) or equivalent), the y-axis denotes shear stress ((\tau))
  • The intercept on the shear axis corresponds to cohesion (for some plots under a certain convention)
• Purpose:
  • Extract material strength parameters (c and \phi) from test data and apply in stability analyses

Mohr envelope visualizations (Page 13)

• Title indicates a Mohr envelope plot (labeled “Mohr envelope”) with annotations such as (03) and (03/11)
• Use:
  • To illustrate the relationship between normal and shear stresses at failure
  • To show how the envelope aligns with the failure data points
• Note: The exact numerical details are not fully legible in the transcript, but the key idea is the same as above: derive c and \phi from the envelope

Stress path and its significance (Pages 14–15)

• Stress path concept
  • A graphical representation of how a soil element’s stress state evolves under loading, unloading, or sampling in field/lab tests
  • The path in p–q space (mean stress vs. deviatoric stress) provides a rational framework to study field and laboratory behavior
• Definitions (from the slide):
  • p = \frac{\sigma1 + \sigma3}{2}
  • q = \pm \frac{\sigma1 - \sigma3}{2}
• Orientation note
  • The sign of q depends on the orientation of the principal stress increment relative to the axes:
  • "+" if, for example, (\sigma_1) is inclined equal to or less than ±45° to the vertical
  • "-" if (\sigma_1) is inclined less than ±45° to the horizontal
• Significance:
  • Provides a rational approach to interpret how loading paths affect the development of failure or yielding in soils

Example: Find p and q (Page 16)

• Example setup: Given a state of stress with principal stresses (\sigma1) and (\sigma3), compute the mean and deviatoric stresses:
  • p = \frac{\sigma1 + \sigma3}{2}
  • q = \pm \frac{\sigma1 - \sigma3}{2}
• Purpose: Practice the conversion from principal stresses to p–q space to analyze the stress path and potential yielding/failure under Mohr-C Coulomb criteria
• Note: The slide title indicates this is an exercise to reinforce the p–q framework; actual numerical values are not provided in the transcript

Connections to foundational principles and real-world relevance

• Tie-ins to soil mechanics fundamentals
  • Stress transformation concepts (principal stresses, normal and shear stresses on planes)
  • Geostatic stress concepts (vertical vs horizontal stresses, K0 condition)
  • Shear strength and failure criteria (Mohr–Coulomb)
• Real-world relevance
  • Slope stability and earth retaining systems rely on understanding how loads induce stress states in soil, how to interpret p–q paths, and how to estimate strength parameters from envelope plots
  • Instrumentation (sensor sheets, tactile pressure sensors, and in-situ tests) provide data to validate analytical models and observe actual stress redistributions

Formulas and key numerical references (LaTeX)

• Mohr–Coulomb failure criterion
  • \tau_f = c + \sigma \tan\phi
• Mean and deviatoric stresses (stress path context)
  • p = \frac{\sigma1 + \sigma3}{2}
  • q = \pm \frac{\sigma1 - \sigma3}{2}
• Geostatic stress relation (example context)
  • \sigmah = K0 \sigmav, \quad K0 = 0.4
  • \sigma_v = \gamma z (typical vertical stress relation with depth)

Quick glossary of symbols used in the slides

• (\sigma_1): maximum principal stress
• (\sigma_3): minimum principal stress
• (\sigma_h): horizontal stress
• (\sigma_v): vertical stress
• (K_0): coefficient of earth pressure at rest (horizontal-to-vertical stress ratio)
• (c): cohesion
• (\phi): friction angle
• (p): mean effective stress
• (q): deviatoric stress

Summary takeaways

• Bloom’s Taxonomy provides a framework for designing learning objectives and assessments.
• Principal stress planes simplify analysis by aligning with stress directions where shear is zero.
• Geostatic stresses in soil can be represented with a horizontal-to-vertical ratio (K0), and vertical stress generally scales with depth via soil unit weight.
• Mohr–Coulomb criteria combine cohesion and friction to define shear strength as a function of normal stress, forming the basis for predicting failure.
• Mohr envelopes extracted from data yield the parameters (c, φ) that characterize material strength.
• Stress-path analyses (p–q) offer a robust approach to understanding the evolution of soil stress states under loading.
• Instrumentation and in-situ tests (e.g., tactile sensor sheets) provide empirical data to validate theoretical models and inform design decisions for slope stability and retaining structures.