Consolidation and Settlement - Cheat Sheet
Your last-minute revision companion for Consolidation and Settlement. This sheet condenses all formulas, definitions, and board-exam essentials into rapid-fire reference points. Master the compression indices, time-factor calculations, and settlement predictions.
Primary Consolidation Settlement
Formulas
S_c = (C_c / (1 + e_0)) × H × log₁₀((σ'_0 + Δσ) / σ'_0)
Meaning
S_c = primary settlement (mm); C_c = compression index (slope of virgin e–log σ' line); e_0 = initial void ratio; H = layer thickness (mm); σ'_0 = initial effective stress (kPa); Δσ = stress increase (kPa)
Watch out
Use log₁₀ (base 10), NOT natural log. Use C_c ONLY if final stress > preconsolidation stress σ'_c
When to use
Calculate settlement for normally consolidated (NC) clay under load increase
S_c = (C_r / (1 + e_0)) × H × log₁₀((σ'_0 + Δσ) / σ'_0)
Meaning
S_c = recompression settlement (mm); C_r = recompression index (much smaller than C_c, typically 0.1–0.2 × C_c)
Watch out
C_r is much smaller than C_c — using C_c for OC clay OVERESTIMATES settlement significantly
When to use
Calculate settlement for overconsolidated (OC) clay while final stress ≤ preconsolidation stress σ'_c
S_c(total) = S_c(recompression, OC range) + S_c(virgin, NC range)
Meaning
For OC clay loaded beyond σ'_c: split into two stages — recompression (C_r) up to σ'_c, then virgin compression (C_c) beyond
Watch out
Easy to forget the two-stage calculation — check if loading crosses σ'_c
When to use
OC clay with stress path crossing the preconsolidation pressure
Common values
Value
0.20–0.50
Symbol
C_c
Quantity
Compression Index for clay (typical range)
Value
0.01–0.10
Symbol
C_r
Quantity
Recompression Index (typical range)
Value
0.6–1.2
Symbol
e
Quantity
Void ratio for clays (typical range)
Important facts
- Settlement is caused by expulsion of pore water from saturated clay under load — a time-dependent process.
- NC clays settle more than OC clays under the same stress increase because C_c >> C_r.
- The settlement magnitude depends ONLY on C_c (or C_r), e_0, H, and the stress ratio — NOT on c_v or drainage.
- Use H in consistent units with S_c result (mm → use H in mm; m → use H in m).
- The stress ratio log₁₀((σ'_0 + Δσ) / σ'_0) = log₁₀(1 + Δσ/σ'_0) captures the nonlinear compression behavior.
- For clay with c_c ≈ 0.009 × (LL − 10) (empirical correlation), where LL is liquid limit.
- Secondary consolidation (creep) occurs after primary consolidation ceases; usually much smaller than primary settlement.
- Total settlement = primary consolidation + secondary consolidation + immediate (elastic) settlement.
Key definitions
Compression Index (C_c)
Example
C_c = 0.30 means void ratio drops by 0.30 for every 10× increase in effective stress
Slope of the virgin consolidation curve (e vs. log σ') in the e–log σ' diagram for normally consolidated clay.
Recompression Index (C_r)
Example
If C_c = 0.30, then C_r ≈ 0.03–0.06
Slope of the unload–reload curve (flatter than virgin) for overconsolidated clay; typically 0.1–0.2 × C_c.
Normally Consolidated (NC) Clay
Example
Young clay deposits in river deltas; no glacial or tectonic unloading
Clay that has never been subjected to effective stress greater than the current overburden; current stress = maximum historical stress.
Overconsolidated (OC) Clay
Example
Glacially overridden clays; heavily weathered deposits; desiccated surface clay
Clay that has experienced higher effective stress in the past; current stress < preconsolidation stress σ'_c due to erosion or unloading.
Preconsolidation Pressure (σ'_c)
Example
Determined by Casagrande's graphical method or oedometer testing
The maximum effective stress to which clay has ever been subjected; stress at which e–log σ' curve changes from steep (recompression) to gentler (virgin).
Void Ratio (e)
Example
e = 0.80 means 1 m³ of solids corresponds to 0.80 m³ of voids
Ratio of volume of voids to volume of solids; e = V_v / V_s.
Diagrams to know
- e–log σ' (consolidation) curve: virgin line (steep), recompression line (flat), kink at σ'_c
- Oedometer test load increments: typical 50 kPa steps; plot e vs. log(stress); determine C_c and C_r graphically
Time Rate of Consolidation
Formulas
T_v = (c_v × t) / H_dr²
Meaning
T_v = dimensionless time factor (–); c_v = coefficient of consolidation (m²/yr or m²/s); t = elapsed time (yr or s); H_dr = longest drainage path (m)
Watch out
Units must be consistent: if c_v in m²/yr, then t in yr; if c_v in m²/s, then t in s. Do NOT mix.
When to use
Convert real time to time factor for use with U–T_v relationships
H_dr = H / 2 (double drainage, both top and bottom); H_dr = H (single drainage, one end only)
Meaning
H_dr = effective drainage path; double drainage halves the path, speeding consolidation 4×; H = total layer thickness
Watch out
Double drainage (sand layer both top and bottom) ⟹ H_dr = H/2. Single drainage (impermeable base) ⟹ H_dr = H. Confusing these is a common exam error.
When to use
Determine drainage path before calculating time factor
U ≤ 60%: T_v = (π/4) × U² (for fast consolidation approx.)
Meaning
U = degree of consolidation (%) = 0–60%; simplified relation valid in early stages
Watch out
This is an APPROXIMATION. Use only if U ≤ 60%. Beyond 60%, use the exact table or iterative form.
When to use
Quick calculation of T_v from U when U ≤ 60%
U > 60%: T_v = 1.781 − 0.933 × log₁₀(100 − U)
Meaning
U = degree of consolidation (%) = 60–99%; exact empirical relation for later stages
Watch out
Use log₁₀ (base 10). Easy to confuse with the U ≤ 60% formula. Check which range U falls into.
When to use
Calculate T_v when U > 60% or vice versa
t = (T_v × H_dr²) / c_v
Meaning
Solve for real time (yr or s) given T_v, c_v, and H_dr
Watch out
Rearranged form of T_v equation; units must be consistent throughout
When to use
Find elapsed time to reach a target degree of consolidation
U = S / S_∞ (or S_c / S_c∞, if referring to settlement)
Meaning
U = ratio of settlement (or excess pore pressure reduction) at time t to final (100%) settlement; always expressed as decimal or %
Watch out
Ensure numerator and denominator use same units and time reference (e.g., don't mix settlement at different depths)
When to use
Define degree of consolidation; sometimes settlement-based (S/S_c), sometimes pore-pressure based (Δu/Δu_0)
Common values
Value
0.197
Symbol
T_v (U=50%)
Quantity
Time factor at 50% consolidation
Value
0.848
Symbol
T_v (U=90%)
Quantity
Time factor at 90% consolidation
Value
1–5 m²/yr
Symbol
c_v
Quantity
Coefficient of consolidation (typical clay)
Value
5–20 m²/yr
Symbol
c_v
Quantity
Coefficient of consolidation (typical silt)
Important facts
- Settlement MAGNITUDE (S_c) is independent of c_v and time — it depends only on compression index and stress.
- Settlement RATE (how fast S_c is reached) depends on c_v, drainage path, and layer thickness.
- Doubling layer thickness (without changing c_v) increases time to reach same U by 4× (because H_dr² in denominator).
- Double drainage cuts consolidation time to 1/4 compared to single drainage (H_dr halves, but H_dr² quarters the time).
- T_v = 0.197 at U = 50%; T_v = 0.848 at U = 90% — these are anchor points for most board problems.
- Coefficient of consolidation c_v is typically 0.5–10 m²/yr for clays; sandy silts are faster; pure clays are slower.
- In very thick clay, consolidation may take decades; in thin clay (< 1 m), a few years to a decade.
- High permeability ⟹ high c_v ⟹ fast consolidation; high compressibility ⟹ low c_v ⟹ slow consolidation.
- Pore pressure dissipation and settlement are coupled; 50% pore pressure drop ≠ 50% settlement (nonlinear).
Key definitions
Degree of Consolidation (U)
Example
U = 50% means half the total settlement has occurred; U = 90% means 90% complete, only 10% remaining
Ratio of consolidation (settlement or excess pore pressure dissipation) achieved at time t to final consolidation; ranges 0–100%.
Time Factor (T_v)
Example
T_v = 0.197 corresponds to U = 50%; T_v = 0.848 corresponds to U = 90%
Dimensionless group T_v = c_v t / H_dr² relating real time, soil properties, and drainage path; used to find U from charts/tables.
Coefficient of Consolidation (c_v)
Example
High c_v (e.g., 10 m²/yr) ⟹ fast consolidation; low c_v (e.g., 0.5 m²/yr) ⟹ slow consolidation
Measure of soil's ability to drain and consolidate under load; c_v = k / (γ_w × m_v), where k = permeability, m_v = volume compressibility.
Drainage Path (H_dr)
Example
2 m clay layer with sand above and below: H_dr = 1 m. Same layer with rock below: H_dr = 2 m
Longest distance water must travel to escape the clay layer; H/2 for double drainage (two permeable boundaries), H for single drainage (one permeable boundary).
Primary Consolidation
Example
Takes months to years; 90% completion typically defines end of primary consolidation
Time-dependent settlement caused by gradual dissipation of excess pore water pressure and volume change of soil skeleton.
Diagrams to know
- U vs. T_v curve: S-shaped, slow start, rapid middle, asymptotic tail approaching 100%
- T_v vs. time graphs for different H, c_v, and drainage configurations; importance of H_dr²
NC vs. OC Clay — Settlement Calculation Strategy
Formulas
σ'_c > σ'_0 + Δσ ⟹ ENTIRE loading range is OC; use C_r only
Meaning
If final stress stays below preconsolidation, clay behaves as OC throughout; only recompression occurs
Watch out
Forgetting to check OCR (overconsolidation ratio) = σ'_c / σ'_0 before choosing index
When to use
Check if clay is overconsolidated and whether loading stays in OC range
σ'_0 < σ'_c < σ'_0 + Δσ ⟹ TWO-STAGE: recompression (C_r) to σ'_c, then virgin (C_c) beyond
Meaning
Loading path crosses preconsolidation; split settlement into OC stage (C_r) and NC stage (C_c)
Watch out
Must calculate TWO logarithmic terms, not one; easily confused with single-stage NC
When to use
Most realistic OC clay scenario in practice
σ'_c ≤ σ'_0 ⟹ ENTIRE loading is NC (virgin compression); use C_c only
Meaning
Preconsolidation stress is already exceeded; all loading is on virgin curve
Watch out
Easy to assume OC, but if σ'_c ≤ σ'_0, then it's effectively NC for this loading event
When to use
NC clay or highly loaded OC clay; stress path entirely on steep portion of e–log σ' curve
Common values
Value
1–2
Symbol
OCR
Quantity
OCR for lightly OC clay
Value
2–10
Symbol
OCR
Quantity
OCR for heavily OC clay (glaciated)
Important facts
- NC clay: C_c ~ 0.30, settles ~150–300 mm per 100 kPa load over ~1 m layer.
- OC clay: C_r ~ 0.03–0.06, settles ~15–30 mm per 100 kPa load over same layer (10× less).
- Two-stage OC settlements are often split: ~5–10% from recompression, ~90–95% from virgin compression.
- OCR decreases with depth (deeper clays are often more NC due to higher overburden history).
- Determining C_c and C_r requires lab oedometer test or empirical correlations (C_c ≈ 0.009(LL − 10)).
- OCR can be estimated from cone penetration or torvane tests; lab is most accurate.
Key definitions
Overconsolidation Ratio (OCR)
Example
OCR = 2 ⟹ clay was historically subjected to 2× current stress; settling slowly
Ratio OCR = σ'_c / σ'_0; OCR > 1 means clay is OC; OCR ≈ 1 means NC.
Diagrams to know
- Overlay of OC and NC curves on e–log σ' plot: showing kink at σ'_c and two-stage loading path
Quick Reference: Key Equations & Lookup Tables
Formulas
Settlement equation summary: S_c = (C_* / (1 + e_0)) × H × log₁₀(σ'_f / σ'_0)
Meaning
C_* = C_c for NC or OC loading beyond σ'_c; C_* = C_r for OC loading below σ'_c; σ'_f = σ'_0 + Δσ
Watch out
Numerator is C_*, NOT (C_c or C_r) — be clear on which applies
When to use
Universal form; always start here and substitute appropriate C
Time to consolidation: t (yr) = (T_v × H_dr²) / c_v, where H_dr in m, c_v in m²/yr
Meaning
Key for all time-rate problems; T_v from U–T_v table or formulas
Watch out
H_dr² is critical; single vs. double drainage changes answer by 4×
When to use
Any problem asking 'how long until settlement reaches X% or how much settlement after Y years'
U lookup table: (50%, T_v=0.197), (60%, T_v≈0.286), (70%, T_v≈0.403), (80%, T_v≈0.567), (90%, T_v=0.848)
Meaning
Standard values from Terzaghi consolidation theory; memorize 50% and 90%
Watch out
T_v is NOT linear with U; nonlinear curve; small errors at boundary (60%) can compound
When to use
Exam problems often use these standard U values; saves time vs. solving formula
Common values
Value
40–100 %
Symbol
LL
Quantity
Liquid limit (LL) for clay (typical)
Value
15–40 %
Symbol
PL
Quantity
Plastic limit (PL) for clay (typical)
Important facts
- All formulas use log₁₀ (base 10 logarithm) unless explicitly stated otherwise.
- Settlement is typically expressed in mm; thickness H should be in mm for consistency.
- Stress always in kPa in board problems (SI standard); watch for kN/m² (same as kPa).
- Water unit weight γ_w = 9.81 kN/m³ ≈ 10 kN/m³ (if problem gives 'unit weight', check if total or effective).
- Coefficient of consolidation c_v can be estimated from c_v = k / (γ_w × m_v) where m_v = compression modulus inverse.
- Practical settlement limit: typical allowable = 50–100 mm; differential settlement = 25–50 mm (depends on structure type and RA 544 / NSCP 2015).
Secondary Consolidation & Total Settlement
Formulas
S_secondary = C_α × H × log₁₀(t_2 / t_1)
Meaning
S_secondary = secondary (creep) settlement (mm); C_α = secondary compression index; H = layer thickness (mm); t_1, t_2 = times before and after interval
Watch out
C_α is MUCH smaller than C_c (typically 0.01–0.05 × C_c); often overlooked in exams but can be significant over decades
When to use
Estimate long-term creep settlement after primary consolidation (t > t_100% primary)
S_total = S_immediate + S_primary + S_secondary
Meaning
Total settlement = elastic (immediate) + consolidation (primary) + creep (secondary)
Watch out
Secondary consolidation often neglected but can reach 20–30% of primary in organic clays; critical for long-term projects
When to use
Final settlement prediction for design; not all three always significant
S_immediate ≈ (Δσ / E_s) × H (rough elastic estimate)
Meaning
Immediate settlement using modulus E_s or E_0; order-of-magnitude only
Watch out
This is VERY approximate for saturated clays; more relevant for sands. Do not use without noting 'order-of-magnitude only'
When to use
Quick check; not rigorous; more accurate methods exist (Schmertmann, Boussinesq)
Common values
Value
0.001–0.010
Symbol
C_α
Quantity
Secondary compression index (inorganic clay)
Value
0.010–0.050
Symbol
C_α
Quantity
Secondary compression index (organic clay, peat)
Important facts
- Primary consolidation: excess pore pressure reduces to ~zero; typically 90–95% complete in 1–5 years for typical clay.
- Secondary consolidation: excess pore pressure zero; soil continues to compress very slowly, rate proportional to log(time).
- Secondary settlement often negligible in inorganic clays but significant (30–50% of primary) in organic clays and peats.
- Total settlement includes immediate (elastic) + primary (consolidation) + secondary (creep); rarely all three are large.
- Settlement rate slows dramatically after primary consolidation; most damage occurs during primary phase.
Key definitions
Secondary Compression Index (C_α)
Example
C_α ≈ 0.005 for typical clay; very small but cumulative over years
Slope of e vs. log t curve during secondary consolidation; measure of creep rate in clay after excess pore pressure dissipates.
Secondary Consolidation (Creep)
Example
Organic clays and peats exhibit significant creep; inorganic clays show minor creep
Time-dependent volume change at constant effective stress; occurs after primary consolidation (excess pore pressure = 0).
Diagrams to know
- Settlement vs. log(time) curve: showing immediate jump, primary consolidation (curve), then secondary creep (linear on semi-log plot)
Must remember
- Settlement formula is S_c = (C_* / (1 + e_0)) × H × log₁₀(σ'_f / σ'_0); use C_c for NC/virgin, C_r for OC recompression.
- Drainage path H_dr = H/2 (double) or H (single); halving H_dr cuts consolidation time by 4×; confusing these is a common exam error.
- Time factor T_v = c_v × t / H_dr²; key pairs: T_v = 0.197 at U = 50%, T_v = 0.848 at U = 90%.
- For U ≤ 60% use T_v = (π/4)U²; for U > 60% use T_v = 1.781 − 0.933 log₁₀(100 − U); pick the right branch.
- NC clay has C_c ~ 0.20–0.50 (large); OC clay has C_r ~ 0.01–0.10 (small, ~0.1–0.2 × C_c); using wrong index gives 10× error.
- Settlement MAGNITUDE depends ONLY on compression index, e_0, H, and stress ratio — NOT on c_v or time.
- Settlement RATE depends on c_v, H_dr, and thickness; magnitude is independent of rate.
- Two-stage OC settlement: recompression (C_r) from σ'_0 to σ'_c, then virgin (C_c) from σ'_c to σ'_f; do not forget either stage.
- All logarithms in consolidation equations are log₁₀ (base 10), not natural log; easy mistake.
- Effective stress σ' = total stress σ − pore pressure u; always use effective stress in settlement and time calculations, NOT total.
Last minute tips
- In double-drainage problems, time reduces by 4×, not 2×. The drainage path is H/2, so (H/2)² = H²/4. Many students forget the square.
- When loading an OC clay and the final stress exceeds σ'_c, split the calculation: first leg uses C_r, second leg uses C_c. Forgetting this is an instant 50% error.
- T_v ≈ 0.197 at U = 50% and T_v ≈ 0.848 at U = 90%. If your problem uses different values, you chose the wrong formula or made a calculation error.
- Always check units: c_v in m²/yr matches t in yr; c_v in m²/s matches t in s. Mixing units is a silent killer in time calculations.
- If a problem does not explicitly state 'double drainage', assume SINGLE drainage (H_dr = H). Double drainage is special and usually mentioned.
Comparison tables
NC vs. OC Clay Settlement Behavior
| Property | Normally Consolidated (NC) | Overconsolidated (OC) |
|---|---|---|
| Compression Index Used | C_c (larger, ~0.20–0.50) | C_r (smaller, ~0.01–0.10) — until σ'_c is exceeded |
| Settlement per 100 kPa Load | Large (~100–300 mm per m) | Small (~10–30 mm per m) in OC range; increases sharply if σ'_c exceeded |
| e–log σ' Curve Slope | Steep (virgin line); C_c is slope | Flat (recompression line); C_r is slope until kink at σ'_c |
| Consolidation Time | Time factor same for given c_v and H_dr; no difference from OC in TIME RATE | Same TIME RATE (T_v) as NC; settlement MAGNITUDE is difference |
| History | σ'_c ≈ σ'_0 (current stress is max historical stress) | σ'_c > σ'_0 (clay experienced higher stress; now unloaded/eroded) |
| Typical Examples | Young river deltas; marine clays; recent deposits | Glaciated regions; heavily weathered profiles; desiccated clays |
Time Factor (T_v) vs. Degree of Consolidation (U) — Key Pairs
| Degree of Consolidation (U, %) | Time Factor (T_v) | Quick Approximation Formula | Settlement Progress |
|---|---|---|---|
| 20% | ~0.031 | T_v ≈ (π/4)U² = 0.785 × 0.04 ≈ 0.031 | Very early, rapid consolidation rate |
| 50% | 0.197 | T_v = (π/4) × 0.25 = 0.196 ≈ 0.197 ✓ | HALF settlement reached; exam staple |
| 60% | ~0.286 | T_v ≈ 0.286 (transition; both formulas valid) | Boundary between approximation regimes |
| 70% | ~0.403 | T_v = 1.781 − 0.933 log(30) ≈ 0.404 ✓ | 70% done; 30% remaining |
| 90% | 0.848 | T_v = 1.781 − 0.933 log(10) ≈ 0.849 ✓ | PRACTICAL completion; exam staple |
| 99% | ~1.781 | T_v = 1.781 − 0.933 log(1) = 1.781 ✓ | Near complete; takes very long time |
Drainage Path (H_dr) & Consolidation Time Scaling
| Drainage Configuration | H_dr | Time Factor Change | Example |
|---|---|---|---|
| Double drainage (sand above & below) | H/2 | Consolidation 4× faster (H_dr² in denominator) | 2 m clay layer → H_dr = 1 m; t(50%) halves vs. single |
| Single drainage (impermeable base) | H | Reference; baseline time | 2 m clay layer → H_dr = 2 m; baseline |
| Triple drainage (sand both sides + internal sand layer) | H/3 (or H/4 depending on layer configuration) | Consolidation 9× faster (for H/3) | Rarely considered; mostly academic |
Common Board-Exam Formulas: When to Use
| Scenario | Use This Formula | Key Check | Typical Units |
|---|---|---|---|
| Find settlement amount for NC clay | S_c = (C_c / (1 + e_0)) H log(σ'_f / σ'_0) | Is final stress > σ'_c? If yes, NC applies. | S_c in mm, H in mm, stresses in kPa |
| Find settlement for OC clay staying below σ'_c | S_c = (C_r / (1 + e_0)) H log(σ'_f / σ'_0) | Final stress ≤ σ'_c? Use C_r only. | S_c in mm, H in mm, stresses in kPa |
| OC clay loading crosses σ'_c | Split: S = S_c(recomp) + S_c(virgin); calculate each separately | σ'_0 < σ'_c < σ'_f? Calculate two log terms. | Two settlement calculations combined |
| Find time for degree U at single drainage | t = (T_v × H²) / c_v | Is it double drainage? If no, use H full thickness. | t in yr (if c_v in m²/yr); H in m |
| Find time for degree U at double drainage | t = (T_v × (H/2)²) / c_v = (T_v × H²) / (4 c_v) | Is it double drainage? If yes, use H/2. | Consolidation 4× faster; t in yr |
| Find U given time and c_v | Calculate T_v = c_v t / H_dr²; look up U from T_v–U table or formula | Which U formula applies? Check T_v vs. 0.286 | T_v dimensionless; then find U from table |
Diagrams
Settlement Calculation Decision Tree
Decision flowchart for determining which compression index (Cc or Cr) to use and whether a single- or two-stage settlement calculation is required. Critical for avoiding the most common NC vs. OC errors.
Time-Rate of Consolidation: From Time to Degree
Flowchart for calculating degree of consolidation (U) from elapsed time. Highlights the critical decision point at Tv ≈ 0.286 (U = 60%) where the T_v–U formula changes form.
Consolidation & Settlement Concept Map
Hierarchical concept map showing the relationships between settlement magnitude (NC vs. OC), consolidation rate (time factor, drainage, cv), and secondary effects. Helps distinguish between magnitude and rate concepts that students often confuse.
Consolidation Settlement Phases Over Time
State transition diagram showing the three phases of total settlement: immediate (elastic), primary consolidation (pore pressure driven), and secondary consolidation (creep). Illustrates the time scales and dominant mechanisms in each phase.
Typical Settlement Composition (NC Clay, 1 m Layer)
Pie chart showing typical distribution of total settlement components for a normally consolidated clay layer. Primary consolidation dominates; secondary creep and immediate settlement are usually minor except in organic clays or over very long timescales.