Geotechnical Formulas

Overview

Welcome! In this session, we’ll break down the foundational concepts and formulas underpinning geotechnical engineering calculations. From how soils behave under stress to the crucial ratios and coefficients used in design, you’ll gain a practical understanding of the terms, relationships, and methods that professionals apply daily. By the end, you’ll be equipped to recognize, use, and reason through core geotechnical formulas and definitions with confidence.

Concept-by-Concept Deep Dive

Soil Structure and Phases

Understanding soils begins with recognizing that soil is a three-phase system: solids (minerals), water, and air. The way these phases are arranged and interact greatly influences soil behavior.

Particle Arrangement (Compaction)

• Soil compaction is the process of rearranging soil particles to reduce the volume of air between them, increasing density and strength.
• The arrangement after compaction determines key properties like porosity and permeability.

Volume Relationships

• Total Volume (V): Includes solids, water, and air.
• Void Volume (Vv): Space taken by water and air.
• Solid Volume (Vs): Occupied by soil particles.
• Void Ratio (e): Ratio of void volume to solid volume.
• Porosity (n): Ratio of void volume to total volume.

Recipe: To analyze a soil’s structure, identify the mass and volume of each phase. Use these to compute ratios such as void ratio and degree of saturation.

Misconception: Confusing void ratio (voids/solids) with porosity (voids/total volume). Always double-check which base volume is used in each ratio.

Soil Properties: Strength, Compressibility, and Deformability

Soil mechanics hinges on how soils respond to loading.

Bearing Capacity

• The maximum pressure soil can support without failing.
• Calculated using empirical or theoretical equations, often factoring in cohesion, internal friction angle, and unit weight.

Shear Strength and Modulus

• Shear Strength: Resistance to sliding or shearing.
• Shear Modulus (G): Ratio of shear stress to shear strain—reflects soil stiffness.

Bulk and Elastic Moduli

• Bulk Modulus (K): Resistance to uniform compression; relates volume change to applied pressure.
• Elasticity vs. Plasticity:
  • Elasticity: Ability to return to original shape after stress is removed.
  • Plasticity: Ability to deform permanently without cracking.

Recipe: Identify applied stresses and corresponding strains. Use the appropriate modulus formula to relate them.

Misconception: Assuming soils behave like ideal elastic solids—real soils often exhibit both elastic and plastic behavior.

Water in Soils: Flow and Saturation

Water in soil pores governs many geotechnical phenomena.

Degree of Saturation and Water Content

• Degree of Saturation (S): Percentage of voids filled with water.
• Water Content (w): Ratio of water weight to solid weight.

Permeability and Flow

• Movement of Water: Described by terms like permeability and governed by Darcy's Law.
• Flow direction, rate, and the potential for soil to drain or retain water are critical to stability.

Recipe: Use known masses or volumes to compute water content and saturation. Apply Darcy’s Law to analyze flow.

Misconception: Equating degree of saturation with water content—they measure different things (one is percentage of voids, the other is based on mass).

Soil Density and Specific Gravity

Quantifying how much soil weighs per unit volume is foundational.

Bulk Density (γ)

• Mass per total volume, including solids and any water or air.

Dry Density (γ_d)

• Mass of solids per total volume—excludes water mass.

Specific Gravity (Gs)

• Ratio of density of soil solids to that of water (dimensionless).
• Used to relate soil properties to standard reference materials.

Recipe: Use weighed samples and measured volumes. Always clarify whether you’re including water or considering only the solids.

Misconception: Mixing up dry density and bulk density—be clear which phase(s) the measurement includes.

Earth Pressure and Stress in Soils

Retaining walls and slope stability problems rely on understanding earth pressures.

Earth Pressure Coefficients (Ka, Kp)

• Active (Ka): Soil pushes against a retaining structure (wall moves away from soil).
• Passive (Kp): Structure pushes into the soil (wall moves toward soil).
• Both coefficients depend on the internal friction angle.

Vertical and Lateral Stresses

• Vertical (overburden) stress: Weight of overlying soil per unit area at a given depth.
• Lateral (horizontal) stress: Transmitted sideways, crucial for wall and slope design.

Recipe: Use geometric and soil property relationships (typically involving the friction angle) to compute pressure coefficients.

Misconception: Assuming earth pressure coefficients are the same for all situations—they vary with wall movement and soil properties.

Worked Examples (generic)

Example 1: Calculating Void Ratio

Given: A soil sample has a total volume of V, with a void volume of Vv and a volume of solids Vs.

Process:

1. Find the volume of voids (Vv) and volume of solids (Vs).
2. Compute the void ratio: e = Vv / Vs.
3. Interpret the result: If e > 1, voids exceed solids in volume.

Example 2: Computing Water Content

Given: Mass of wet soil (Mw), mass of dry soil (Md).

Process:

1. Calculate the mass of water: Mw - Md.
2. Water content (w) = (Mw - Md) / Md × 100% (expressed as a percentage).
3. Use in further calculations (e.g., adjusting compaction targets).

Example 3: Determining Dry Density

Given: Mass of dry soil (Ms), total volume of sample (V).

Process:

1. Dry density (γ_d) = Ms / V.
2. Compare with standard values for compaction assessment.

Example 4: Calculating Earth Pressure Coefficient

Given: Soil’s internal friction angle (φ).

Process:

1. Use the relevant formula for Ka or Kp (involving the tangent or sine functions of φ).
2. Substitute φ and solve for the coefficient.

Common Pitfalls and Fixes

• Mixing Up Volume and Mass Ratios: Always double-check whether a formula asks for mass-based or volume-based ratios.
• Confusing Similar Terms: Terms like water content, degree of saturation, and porosity have precise, distinct meanings—review definitions frequently.
• Unit Errors: Many formulas require consistent units (e.g., all in kN, m³, or %). Convert before plugging in.
• Overlooking Soil State: Properties (density, strength) can change dramatically if the soil is wet, dry, loose, or compacted—context matters.
• Improper Application of Modulus: Know when to use bulk modulus vs. shear modulus, and make sure the stress and strain types match.
• Forgetting Phase Relationships: Remember that soil is not just solid—air and water content are equally important in most calculations.

Summary

• Soil is a three-phase system: solids, water, and air; understanding their relationships is key to all geotechnical calculations.
• Ratios like void ratio, porosity, and degree of saturation describe how these phases interact and directly impact soil behavior.
• Mechanical properties (bearing capacity, shear modulus, bulk modulus) define how soils respond to loading and must be calculated using the correct formulas.
• Water movement in soil (permeability) and the resulting stresses are central to stability and design in geotechnical engineering.
• Precise terminology and unit consistency are vital; always clarify what each formula measures and requires before applying it.