Formula Sheet: Soil Mechanics

Table of Contents
1. Phase Relationships and Index Properties
2. Soil Classification
3. Soil Compaction
4. Effective Stress and Pore Water Pressure
5. Seepage and Flow Nets
6. Stress Distribution in Soil
7. Consolidation Theory
8. Shear Strength of Soils
9. Earth Pressure Theory
10. Slope Stability
11. Bearing Capacity
12. Lateral Earth Support Systems
13. Soil Dynamics and Liquefaction
14. Foundation Settlement
15. Soil Improvement and Ground Modification
16. In-Situ Testing Correlations
17. Plate Load Test
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Phase Relationships and Index Properties

Basic Phase Relationships

• Void ratio (e): \[e = \frac{V_v}{V_s}\] where \(V_v\) = volume of voids, \(V_s\) = volume of solids
• Porosity (n): \[n = \frac{V_v}{V_t} = \frac{e}{1+e}\] where \(V_t\) = total volume
• Degree of saturation (S): \[S = \frac{V_w}{V_v} \times 100\%\] where \(V_w\) = volume of water
• Water content (w): \[w = \frac{W_w}{W_s} \times 100\%\] where \(W_w\) = weight of water, \(W_s\) = weight of solids
• Relationship between w, e, S, and G_s: \[w = \frac{S \cdot e}{G_s}\] where \(G_s\) = specific gravity of solids

Unit Weights

• Total (moist) unit weight: \[\gamma = \frac{W_t}{V_t}\] where \(W_t\) = total weight
• Dry unit weight: \[\gamma_d = \frac{W_s}{V_t} = \frac{\gamma}{1+w}\]
• Saturated unit weight: \[\gamma_{sat} = \frac{(G_s + e)\gamma_w}{1+e}\] where \(\gamma_w\) = unit weight of water (typically 62.4 pcf or 9.81 kN/m³)
• Submerged (buoyant) unit weight: \[\gamma' = \gamma_{sat} - \gamma_w = \frac{(G_s - 1)\gamma_w}{1+e}\]
• Zero air voids unit weight: \[\gamma_{zav} = \frac{G_s \gamma_w}{1 + wG_s}\]
• General unit weight formula: \[\gamma = \frac{(G_s + Se)\gamma_w}{1+e}\]

Relative Density

• Relative density (D_r): \[D_r = \frac{e_{max} - e}{e_{max} - e_{min}} \times 100\%\]
• Alternative form using unit weights: \[D_r = \frac{\gamma_{d,max}}{\gamma_d} \times \frac{\gamma_d - \gamma_{d,min}}{\gamma_{d,max} - \gamma_{d,min}} \times 100\%\]

Soil Classification

Atterberg Limits

• Liquid Limit (LL): Water content at which soil transitions from plastic to liquid state
• Plastic Limit (PL): Water content at which soil transitions from semi-solid to plastic state
• Plasticity Index (PI): \[PI = LL - PL\]
• Liquidity Index (LI): \[LI = \frac{w - PL}{PI}\]
• Consistency Index (CI): \[CI = \frac{LL - w}{PI}\]
• Activity (A): \[A = \frac{PI}{\% \text{ clay fraction}}\] where clay fraction is percent finer than 2 μm

Grain Size Distribution

• Coefficient of uniformity: \[C_u = \frac{D_{60}}{D_{10}}\]
• Coefficient of curvature: \[C_c = \frac{D_{30}^2}{D_{10} \times D_{60}}\]

Well-graded soil criteria: \(C_u > 4\) for gravels or \(C_u > 6\) for sands, and \(1 < c_c=""><>

Soil Compaction

Compaction Relationships

• Dry unit weight from field conditions: \[\gamma_d = \frac{\gamma}{1+w}\]
• Percent compaction: \[\text{Percent Compaction} = \frac{\gamma_d}{\gamma_{d,max}} \times 100\%\]
• Relative compaction: Same as percent compaction

Effective Stress and Pore Water Pressure

Terzaghi's Effective Stress Principle

• Effective stress equation: \[\sigma' = \sigma - u\] where \(\sigma'\) = effective stress, \(\sigma\) = total stress, \(u\) = pore water pressure
• Total vertical stress at depth z: \[\sigma_v = \sum \gamma_i h_i\] where summation is over layers of different unit weights
• Pore water pressure (hydrostatic): \[u = \gamma_w h_w\] where \(h_w\) = height of water above the point
• Effective vertical stress: \[\sigma'_v = \sigma_v - u\]

Capillary Rise

• Height of capillary rise: \[h_c = \frac{C}{eD_{10}}\] where \(C\) ≈ 10 to 50 mm² (empirical constant), \(D_{10}\) in mm
• Simplified form: \[h_c = \frac{C}{D_{10}}\] where \(C\) ≈ 15 mm² for clean sand

Seepage and Flow Nets

Darcy's Law

• Darcy's Law (one-dimensional): \[v = ki\] where \(v\) = discharge velocity (superficial velocity), \(k\) = coefficient of permeability (hydraulic conductivity), \(i\) = hydraulic gradient
• Flow rate: \[q = vA = kiA\] where \(A\) = cross-sectional area
• Hydraulic gradient: \[i = \frac{\Delta h}{L}\] where \(\Delta h\) = head loss, \(L\) = flow length
• Seepage velocity: \[v_s = \frac{v}{n} = \frac{ki}{n}\] where \(n\) = porosity

Permeability Relationships

• Hazen's approximation: \[k = C D_{10}^2\] where \(k\) in cm/s, \(D_{10}\) in mm, \(C\) ≈ 100 for clean sands
• Stratified soil (horizontal flow): \[k_h = \frac{\sum k_i h_i}{\sum h_i}\]
• Stratified soil (vertical flow): \[k_v = \frac{\sum h_i}{\sum \frac{h_i}{k_i}}\]

Flow Net Analysis

• Total seepage flow: \[q = k h \frac{N_f}{N_d}\] where \(h\) = total head loss, \(N_f\) = number of flow channels, \(N_d\) = number of equipotential drops
• Head loss per equipotential drop: \[\Delta h = \frac{h}{N_d}\]
• Hydraulic gradient in flow element: \[i = \frac{\Delta h}{\Delta L}\] where \(\Delta L\) = length of flow element
• Pore pressure at a point: \[u = \gamma_w (z + h_p)\] where \(z\) = elevation head, \(h_p\) = pressure head from flow net

Stress Distribution in Soil

Boussinesq Theory (Point Load)

• Vertical stress increase under point load: \[\Delta \sigma_z = \frac{3Q}{2\pi z^2} \left[\frac{1}{1+(r/z)^2}\right]^{5/2}\] where \(Q\) = point load, \(z\) = depth, \(r\) = radial distance from load axis
• Influence factor form: \[\Delta \sigma_z = \frac{Q}{z^2} I_B\] where \(I_B = \frac{3}{2\pi} \left[\frac{1}{1+(r/z)^2}\right]^{5/2}\)

Uniform Circular Load

• Vertical stress at center (z-axis): \[\Delta \sigma_z = q \left[1 - \frac{1}{(1+(R/z)^2)^{3/2}}\right]\] where \(q\) = uniform pressure, \(R\) = radius of loaded area

Uniform Rectangular Load

• Vertical stress increase: \[\Delta \sigma_z = q I_r\] where \(I_r\) = influence factor (from charts or tables based on \(m = L/z\) and \(n = B/z\))
• Influence factor (corner of rectangle): \[I_r = \frac{1}{4\pi}\left[2mn\sqrt{m^2+n^2+1}\frac{m^2+n^2+2}{m^2+n^2+m^2n^2+1} + \tan^{-1}\frac{2mn\sqrt{m^2+n^2+1}}{m^2+n^2-m^2n^2+1}\right]\]

Newmark Chart Method

• Vertical stress increase: \[\Delta \sigma_z = q \times I \times N\] where \(I\) = influence value of chart, \(N\) = number of blocks covered by loaded area

2:1 Method (Approximate)

• Average vertical stress increase: \[\Delta \sigma_z = \frac{Q}{(B+z)(L+z)}\] where \(B\) and \(L\) = footing dimensions, \(Q\) = total load

Consolidation Theory

One-Dimensional Consolidation

• Void ratio change: \[\Delta e = \frac{\Delta H}{H_0/(1+e_0)}\] where \(\Delta H\) = settlement, \(H_0\) = initial thickness, \(e_0\) = initial void ratio
• Compression index (C_c): \[C_c = \frac{\Delta e}{\log(\sigma'_2/\sigma'_1)}\] for normally consolidated soil (virgin compression curve)
• Recompression index (C_r) or Swell index (C_s): \[C_r = C_s = \frac{\Delta e}{\log(\sigma'_2/\sigma'_1)}\] for overconsolidated soil (recompression curve)
• Coefficient of volume compressibility: \[m_v = \frac{\Delta e}{(1+e_0)\Delta \sigma'} = -\frac{\Delta \varepsilon_v}{\Delta \sigma'}\]
• Coefficient of compressibility: \[a_v = \frac{\Delta e}{\Delta \sigma'}\]

Settlement Calculations

• Primary consolidation settlement (normally consolidated): \[S_c = \frac{C_c H_0}{1+e_0} \log\left(\frac{\sigma'_0 + \Delta \sigma}{\sigma'_0}\right)\] where \(\sigma'_0\) = initial effective stress, \(\Delta \sigma\) = stress increase
• Primary consolidation settlement (overconsolidated, \(\sigma'_0 + \Delta\sigma \leq \sigma'_p\)): \[S_c = \frac{C_r H_0}{1+e_0} \log\left(\frac{\sigma'_0 + \Delta \sigma}{\sigma'_0}\right)\] where \(\sigma'_p\) = preconsolidation pressure
• Primary consolidation settlement (overconsolidated, \(\sigma'_0 + \Delta\sigma > \sigma'_p\)): \[S_c = \frac{C_r H_0}{1+e_0} \log\left(\frac{\sigma'_p}{\sigma'_0}\right) + \frac{C_c H_0}{1+e_0} \log\left(\frac{\sigma'_0 + \Delta \sigma}{\sigma'_p}\right)\]
• Settlement using coefficient of volume compressibility: \[S_c = m_v H_0 \Delta \sigma'\]
• Overconsolidation ratio (OCR): \[OCR = \frac{\sigma'_p}{\sigma'_0}\]

Empirical Correlations for C_c

• Terzaghi and Peck: \[C_c = 0.009(LL - 10)\]
• Skempton: \[C_c = 0.007(LL - 7)\] for remolded clays
• Relationship with natural water content: \[C_c \approx 0.30(e_0 - 0.27)\] for undisturbed clays
• Recompression index approximation: \[C_r \approx \frac{C_c}{5} \text{ to } \frac{C_c}{10}\]

Time Rate of Consolidation

• Coefficient of consolidation: \[c_v = \frac{k}{m_v \gamma_w}\] where \(k\) = permeability
• Time factor (one-way drainage): \[T_v = \frac{c_v t}{H_{dr}^2}\] where \(H_{dr}\) = drainage path length (= \(H/2\) for two-way drainage, = \(H\) for one-way drainage)
• Average degree of consolidation (U <> \[U = \sqrt{\frac{4T_v}{\pi}} = 2\sqrt{\frac{T_v}{\pi}}\]
• Average degree of consolidation (U > 60%): \[U = 1 - 10^{-\frac{T_v}{\pi/4}}\]
• Settlement at time t: \[S_t = U \times S_c\]

Secondary Compression

• Secondary compression settlement: \[S_s = C_\alpha H_0 \log\left(\frac{t}{t_p}\right)\] where \(C_\alpha\) = coefficient of secondary compression, \(t_p\) = time at end of primary consolidation, \(t\) = total time
• Alternative form: \[S_s = \frac{C_\alpha}{1+e_p} H_0 \log\left(\frac{t}{t_p}\right)\] where \(e_p\) = void ratio at end of primary consolidation

Shear Strength of Soils

Mohr-Coulomb Failure Criterion

• Shear strength equation: \[\tau_f = c + \sigma \tan\phi\] where \(\tau_f\) = shear stress at failure, \(c\) = cohesion, \(\sigma\) = normal stress on failure plane, \(\phi\) = friction angle
• Effective stress shear strength: \[\tau_f = c' + \sigma' \tan\phi'\] where \(c'\) = effective cohesion, \(\sigma'\) = effective normal stress, \(\phi'\) = effective friction angle
• Undrained shear strength (total stress, saturated clay): \[\tau_f = c = s_u\] where \(s_u\) = undrained shear strength (\(\phi = 0\) concept)

Mohr's Circle and Principal Stresses

• Principal stresses: \[\sigma_{1,3} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}\] where \(\sigma_1\) = major principal stress, \(\sigma_3\) = minor principal stress
• Maximum shear stress: \[\tau_{max} = \frac{\sigma_1 - \sigma_3}{2}\]
• Angle of failure plane: \[\theta_f = 45° + \frac{\phi}{2}\] measured from major principal plane
• Relationship at failure (Mohr-Coulomb): \[\sigma_1 = \sigma_3 \tan^2\left(45° + \frac{\phi}{2}\right) + 2c\tan\left(45° + \frac{\phi}{2}\right)\]
• Alternative form: \[\sigma_1 = \sigma_3 K_p + 2c\sqrt{K_p}\] where \(K_p = \tan^2(45° + \phi/2)\) = passive earth pressure coefficient

Undrained Shear Strength Correlations

• Unconfined compression test: \[s_u = \frac{q_u}{2}\] where \(q_u\) = unconfined compressive strength
• Unconsolidated undrained (UU) triaxial test: \[s_u = \frac{\sigma_1 - \sigma_3}{2}\] at failure
• SPT correlation (Terzaghi and Peck): \[s_u \text{ (tsf)} \approx \frac{N_{60}}{8}\] for clays
• Consistency vs. s_u:
  Very soft: \(s_u\) < 0.25="" tsf="" (12="">
  Soft: 0.25-0.5 tsf (12-25 kPa)
  Medium: 0.5-1.0 tsf (25-50 kPa)
  Stiff: 1.0-2.0 tsf (50-100 kPa)
  Very stiff: 2.0-4.0 tsf (100-200 kPa)
  Hard: > 4.0 tsf (> 200 kPa)

Pore Pressure Parameters

• Skempton's pore pressure parameter: \[\Delta u = B[\Delta \sigma_3 + A(\Delta \sigma_1 - \Delta \sigma_3)]\] where \(B\) and \(A\) are pore pressure parameters
• For saturated soils: \(B \approx 1\)
• At failure: \[\Delta u_f = \Delta \sigma_3 + A_f(\Delta \sigma_1 - \Delta \sigma_3)\]

Critical State Soil Mechanics

• Stress ratio at critical state: \[M = \frac{q}{p'} = \frac{6\sin\phi'}{3-\sin\phi'}\] where \(q = \sigma_1 - \sigma_3\), \(p' = (\sigma'_1 + 2\sigma'_3)/3\)

Earth Pressure Theory

Rankine Earth Pressure Theory

• Coefficient of active earth pressure (horizontal backfill): \[K_a = \tan^2\left(45° - \frac{\phi}{2}\right) = \frac{1-\sin\phi}{1+\sin\phi}\]
• Coefficient of passive earth pressure (horizontal backfill): \[K_p = \tan^2\left(45° + \frac{\phi}{2}\right) = \frac{1+\sin\phi}{1-\sin\phi}\]
• Coefficient of at-rest earth pressure: \[K_0 = 1 - \sin\phi\] (Jaky's formula for normally consolidated soil)
• At-rest for overconsolidated soil: \[K_0 = (1-\sin\phi)(OCR)^{\sin\phi}\]
• Active earth pressure (cohesionless soil): \[\sigma_a = K_a \sigma'_v = K_a \gamma z\]
• Passive earth pressure (cohesionless soil): \[\sigma_p = K_p \sigma'_v = K_p \gamma z\]
• Active earth pressure (c-φ soil): \[\sigma_a = K_a \sigma'_v - 2c\sqrt{K_a}\]
• Passive earth pressure (c-φ soil): \[\sigma_p = K_p \sigma'_v + 2c\sqrt{K_p}\]
• Total active force (cohesionless, horizontal surface): \[P_a = \frac{1}{2}K_a \gamma H^2\] per unit width, acting at \(H/3\) from base
• Total passive force (cohesionless, horizontal surface): \[P_p = \frac{1}{2}K_p \gamma H^2\] per unit width, acting at \(H/3\) from base

Rankine with Sloping Backfill

• Active earth pressure coefficient (sloping backfill): \[K_a = \cos\beta \frac{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi}}{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi}}\] where \(\beta\) = backfill slope angle
• Passive earth pressure coefficient (sloping backfill): \[K_p = \cos\beta \frac{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi}}{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi}}\]

Coulomb Earth Pressure Theory

• Coulomb active earth pressure coefficient: \[K_a = \frac{\sin^2(\alpha + \phi)}{\sin^2\alpha \sin(\alpha - \delta)\left[1+\sqrt{\frac{\sin(\phi+\delta)\sin(\phi-\beta)}{\sin(\alpha-\delta)\sin(\alpha+\beta)}}\right]^2}\] where \(\alpha\) = wall angle from horizontal, \(\delta\) = wall friction angle, \(\beta\) = backfill slope
• Coulomb passive earth pressure coefficient: \[K_p = \frac{\sin^2(\alpha - \phi)}{\sin^2\alpha \sin(\alpha + \delta)\left[1-\sqrt{\frac{\sin(\phi+\delta)\sin(\phi+\beta)}{\sin(\alpha+\delta)\sin(\alpha+\beta)}}\right]^2}\]
• Total active force (Coulomb): \[P_a = \frac{1}{2}K_a \gamma H^2\] acting at angle \(\delta\) to normal to wall

Earth Pressure with Surcharge

• Uniform surcharge pressure: \[\Delta \sigma_a = K_a q\] where \(q\) = surcharge pressure
• Total active force with surcharge: \[P_a = \frac{1}{2}K_a \gamma H^2 + K_a q H\]

Slope Stability

Infinite Slope Analysis

• Factor of safety (dry slope, parallel seepage): \[FS = \frac{c'}{\gamma z \sin\beta \cos\beta} + \frac{\tan\phi'}{\tan\beta}\] where \(\beta\) = slope angle, \(z\) = depth
• Factor of safety (submerged slope): \[FS = \frac{c'}{\gamma' z \sin\beta \cos\beta} + \frac{\tan\phi'}{\tan\beta}\] where \(\gamma'\) = submerged unit weight
• Factor of safety (seepage parallel to slope): \[FS = \frac{c'}{\gamma_{sat} z \sin\beta \cos\beta} + \frac{\gamma'}{\gamma_{sat}} \frac{\tan\phi'}{\tan\beta}\]

Finite Slope - Method of Slices (Fellenius/Ordinary Method)

• Factor of safety: \[FS = \frac{\sum[c'b + (W\cos\alpha - ub)\tan\phi']}{\sum W\sin\alpha}\] where \(W\) = weight of slice, \(b\) = width of slice base, \(\alpha\) = base angle, \(u\) = pore pressure

Simplified Bishop Method

• Factor of safety: \[FS = \frac{\sum\frac{1}{m_\alpha}[c'b + W\tan\phi' - ub\tan\phi']}{\sum W\sin\alpha}\] where \[m_\alpha = \cos\alpha + \frac{\sin\alpha \tan\phi'}{FS}\]
• This equation requires iterative solution for FS

Total Stress Analysis (φ = 0)

• Factor of safety (circular failure): \[FS = \frac{c \times L}{\sum W \times d}\] where \(L\) = arc length, \(d\) = moment arm
• Factor of safety (simple circular slip): \[FS = \frac{s_u L_c}{W d_w}\] where \(s_u\) = undrained shear strength, \(L_c\) = chord length, \(d_w\) = horizontal distance from center to centroid

Stability Number Method

• Stability number: \[N_s = \frac{\gamma H}{s_u}\] where \(H\) = slope height
• Critical height: \[H_c = \frac{N_s \times s_u}{\gamma}\]
• \(N_s\) values depend on slope angle and depth factor (obtained from charts)

Bearing Capacity

Terzaghi Bearing Capacity Theory

• Ultimate bearing capacity (strip footing): \[q_{ult} = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma\] where \(c\) = cohesion, \(\gamma\) = unit weight, \(D_f\) = depth of footing, \(B\) = footing width, \(N_c, N_q, N_\gamma\) = bearing capacity factors
• Ultimate bearing capacity (square footing): \[q_{ult} = 1.3 c N_c + \gamma D_f N_q + 0.4 \gamma B N_\gamma\]
• Ultimate bearing capacity (circular footing): \[q_{ult} = 1.3 c N_c + \gamma D_f N_q + 0.3 \gamma B N_\gamma\]

Bearing Capacity Factors (Terzaghi)

• N_q: \[N_q = \frac{e^{2(0.75\pi - \phi/2)\tan\phi}}{2\cos^2(45° + \phi/2)}\]
• N_c: \[N_c = (N_q - 1)\cot\phi\]
• \(N_\gamma\) values are obtained from tables or charts

Meyerhof Bearing Capacity Theory

• General bearing capacity equation: \[q_{ult} = c N_c s_c d_c i_c + q N_q s_q d_q i_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma i_\gamma\] where \(q = \gamma D_f\), \(s\) = shape factors, \(d\) = depth factors, \(i\) = inclination factors

Shape Factors (Meyerhof)

• For φ > 0:
  \(s_c = 1 + 0.2K_p(B/L)\)
  \(s_q = s_\gamma = 1 + 0.1K_p(B/L)\)
  where \(K_p = \tan^2(45° + \phi/2)\)
• For φ = 0:
  \(s_c = 1 + 0.2(B/L)\)
  \(s_q = s_\gamma = 1.0\)

Depth Factors (Meyerhof)

• For φ > 0:
  \(d_c = 1 + 0.2\sqrt{K_p}(D_f/B)\)
  \(d_q = d_\gamma = 1 + 0.1\sqrt{K_p}(D_f/B)\)
• For φ = 0:
  \(d_c = 1 + 0.2(D_f/B)\)
  \(d_q = d_\gamma = 1.0\)

Inclination Factors (Load Inclination)

• For horizontal load at angle θ from vertical:
  \(i_c = i_q = (1 - \theta/90°)^2\)
  \(i_\gamma = (1 - \theta/\phi)^2\)

Hansen Bearing Capacity Factors

• N_c: \[N_c = (N_q - 1)\cot\phi\]
• N_q: \[N_q = e^{\pi\tan\phi}\tan^2(45° + \phi/2)\]
• N_γ: \[N_\gamma = 1.5(N_q - 1)\tan\phi\]

Allowable Bearing Capacity

• Allowable bearing pressure: \[q_{allow} = \frac{q_{ult}}{FS}\] where FS typically = 2.5 to 3.0
• Net ultimate bearing capacity: \[q_{net,ult} = q_{ult} - \gamma D_f\]
• Net allowable bearing pressure: \[q_{net,allow} = \frac{q_{net,ult}}{FS}\]

Bearing Capacity for Special Cases

• Pure clay (φ = 0): \[q_{ult} = c N_c s_c d_c + q\] where \(N_c = 5.14\) for strip, modified by shape and depth factors
• Skempton's formula for saturated clay (φ = 0):
  For strip: \[q_{net,ult} = 5.14 s_u (1 + 0.2 D_f/B)\] for \(D_f/B \leq 2.5\)
  For circular/square: \[q_{net,ult} = 6.2 s_u (1 + 0.2 D_f/B)\]
• For D_f/B > 2.5: Use \(D_f/B = 2.5\) in above equations

Eccentric Loading

• Effective dimensions method:
  \(B' = B - 2e_B\)
  \(L' = L - 2e_L\)
  where \(e_B\), \(e_L\) = eccentricities in B and L directions
• Effective area: \[A' = B' \times L'\]
• Use \(B'\) and \(L'\) in bearing capacity equations
• Maximum contact pressure (one-way eccentricity): \[q_{max} = \frac{P}{A}\left(1 + \frac{6e}{B}\right)\] for \(e \leq B/6\)

Lateral Earth Support Systems

Braced Excavations

• Apparent pressure diagram (Peck) - Sands:
  Rectangular distribution: \(p_a = 0.65 K_a \gamma H\)
• Apparent pressure diagram - Soft to medium clay:
  Trapezoidal: \(p_a = \gamma H K_a\) but not less than \(0.3\gamma H\) or greater than \(0.4\gamma H\) to \(1.0\gamma H\)
• Apparent pressure diagram - Stiff fissured clay:
  Top region: \(p_a = 0.2\gamma H\) to \(0.4\gamma H\)
• Strut loads: Calculated from tributary area method using apparent pressure diagram

Sheet Pile Walls

• Free earth support (cantilever): Depth determined by moment equilibrium and passive resistance
• Fixed earth support (anchored): Additional embedment for fixity, typically \(D = 1.2 D_0\) where \(D_0\) = depth for zero moment
• Factor of safety on embedment depth: Typically 1.5 to 2.0
• Reduction factor for passive pressure: Often reduce \(K_p\) by dividing by FS = 1.5 to 2.0

Soil Dynamics and Liquefaction

Dynamic Soil Properties

• Shear wave velocity: \[V_s = \sqrt{\frac{G}{\rho}}\] where \(G\) = shear modulus, \(\rho\) = mass density
• Small-strain shear modulus: \[G_{max} = \rho V_s^2\]
• Mass density: \[\rho = \frac{\gamma}{g}\] where \(g\) = gravitational acceleration

Liquefaction Potential

• Cyclic stress ratio (CSR): \[CSR = 0.65 \frac{a_{max}}{g} \frac{\sigma_v}{\sigma'_v} r_d\] where \(a_{max}\) = peak ground acceleration, \(\sigma_v\) = total vertical stress, \(\sigma'_v\) = effective vertical stress, \(r_d\) = stress reduction factor
• Stress reduction factor (Liao and Whitman): \[r_d = 1.0 - 0.00765z\] for \(z \leq 9.15\) m
  \[r_d = 1.174 - 0.0267z\] for 9.15 m < \(z\)="">< 23="">
  where \(z\) = depth in meters
• Simplified stress reduction (Seed and Idriss): \(r_d \approx 1.0 - 0.012z\) for shallow depths
• Cyclic resistance ratio (CRR): Obtained from SPT or CPT correlations
• Factor of safety against liquefaction: \[FS_L = \frac{CRR}{CSR}\]
• Liquefaction potential exists when \(FS_L < 1.0\);="" typically="" require="" \(fs_l="" \geq="" 1.1\)="" to="">

SPT Corrections

• Corrected SPT blow count: \[N_{60} = N_{measured} \times \frac{ER}{60}\] where ER = energy ratio (percent)
• Overburden correction: \[(N_1)_{60} = N_{60} \times C_N\] where \[C_N = \sqrt{\frac{P_a}{\sigma'_v}}\] or \[C_N = \left(\frac{P_a}{\sigma'_v}\right)^{0.5}\]
• \(P_a\) = atmospheric pressure (≈ 2000 psf or 100 kPa)
• Clean sand equivalent: \[(N_1)_{60,cs} = (N_1)_{60} + \Delta N_{1,60}\] where correction depends on fines content

Foundation Settlement

Elastic (Immediate) Settlement

• Settlement on elastic half-space: \[S_e = q B \frac{(1-\nu^2)}{E} I_f\] where \(q\) = applied pressure, \(B\) = footing width, \(\nu\) = Poisson's ratio, \(E\) = elastic modulus, \(I_f\) = influence factor (shape and rigidity)
• Steinbrenner/Schmertmann influence factor approach: \[S_e = C_1 C_2 (q - \sigma'_v) \sum \frac{I_z \Delta z}{E_s}\] where \(C_1\) = depth correction factor, \(C_2\) = creep factor, \(I_z\) = strain influence factor, \(E_s\) = soil modulus
• Depth correction factor: \[C_1 = 1 - 0.5 \frac{\sigma'_v}{q - \sigma'_v}\]
• Creep correction factor: \[C_2 = 1 + 0.2 \log\left(\frac{t}{0.1}\right)\] where \(t\) = time in years

Schmertmann's Method Strain Influence Factor

• Peak strain influence (square/circular): \[I_{zp} = 0.5 + 0.1\sqrt{\frac{q - \sigma'_v}{\sigma'_v}}\]
• Peak strain influence (strip): \[I_{zp} = 0.2 + 0.3\sqrt{\frac{q - \sigma'_v}{\sigma'_v}}\]
• Peak occurs at depth \(z_p = 0.5B\) for square, \(z_p = B\) for strip
• \(I_z\) varies triangularly from 0 at surface to \(I_{zp}\) at \(z_p\), then linearly to 0 at depth \(z = 2B\) to \(4B\)

Total Settlement

• Total settlement: \[S_{total} = S_e + S_c + S_s\] where \(S_e\) = elastic (immediate), \(S_c\) = primary consolidation, \(S_s\) = secondary compression

Soil Improvement and Ground Modification

Relative Density and Compaction

• Relationship between SPT and relative density: \[D_r(\%) \approx 15\sqrt{N_{60}}\] (rough approximation for normally consolidated clean sand)
• Meyerhof correlation: \[D_r(\%) \approx \sqrt{\frac{N_{60}}{0.23\sigma'_v/P_a + 0.06}}\] where \(\sigma'_v\) and \(P_a\) in same units

Vibrocompaction and Stone Columns

• Unit cell concept - stress concentration ratio: \[n = \frac{\sigma_c}{\sigma_s}\] where \(\sigma_c\) = stress on column, \(\sigma_s\) = stress on soil
• Area replacement ratio: \[a_s = \frac{A_c}{A_{cell}}\] where \(A_c\) = column area, \(A_{cell}\) = tributary area
• Settlement reduction factor: \[\beta = \frac{S_{treated}}{S_{untreated}} = \frac{1}{1 + a_s(n-1)}\]

In-Situ Testing Correlations

Standard Penetration Test (SPT)

• Relative density of sand (Skempton): \[D_r^2 = \frac{N_{60}}{A + B\sigma'_v/P_a}\] where \(A \approx 60\), \(B \approx 25\) for normally consolidated sand
• Friction angle (Peck et al.): \[\phi \approx 27.1° + 0.3 N_{60} - 0.00054(N_{60})^2\] (approximate, for normally consolidated clean sand)
• Elastic modulus (granular soils): \[E_s \approx (5 \text{ to } 10) N_{60}\] in tsf or \(\approx 500N_{60}\) in kPa

Cone Penetration Test (CPT)

• Soil behavior type index: \[I_c = \sqrt{(3.47 - \log Q_t)^2 + (\log F_r + 1.22)^2}\] where \(Q_t\) = normalized cone resistance, \(F_r\) = friction ratio
• Normalized cone resistance: \[Q_t = \frac{q_t - \sigma_v}{\sigma'_v}\] where \(q_t\) = corrected cone resistance
• Friction ratio: \[F_r = \frac{f_s}{q_t - \sigma_v} \times 100\%\] where \(f_s\) = sleeve friction
• Undrained shear strength (clay): \[s_u = \frac{q_t - \sigma_v}{N_k}\] where \(N_k\) = cone factor (typically 10-20, often 15)
• Friction angle (Robertson and Campanella): \[\phi \approx \tan^{-1}\left[0.1 + 0.38\log\left(\frac{q_c}{\sigma'_v}\right)\right]\] for clean quartz sand

Field Vane Shear Test

• Undrained shear strength: \[s_u = \frac{T}{K}\] where \(T\) = applied torque at failure
• Vane constant (rectangular vane): \[K = \frac{\pi D^2}{2}\left(\frac{H}{2} + \frac{D}{3}\right)\] where \(D\) = vane diameter, \(H\) = vane height
• Corrected undrained strength (Bjerrum): \[s_{u,corrected} = \mu \times s_{u,measured}\] where \(\mu\) = correction factor (function of PI, typically 0.7-1.0)

Plate Load Test

• Bearing capacity extrapolation (clay): \[\frac{q_{ult,foundation}}{q_{ult,plate}} = 1.0\]
• Bearing capacity extrapolation (sand): \[\frac{q_{ult,foundation}}{q_{ult,plate}} = \frac{B_f}{B_p}\] where \(B_f\) = foundation width, \(B_p\) = plate width
• Settlement extrapolation (clay): \[\frac{S_f}{S_p} = \frac{B_f}{B_p}\]
• Settlement extrapolation (sand): \[\frac{S_f}{S_p} = \left(\frac{2B_f}{B_p + B_f}\right)^2\] (Terzaghi and Peck)