Formula Sheet: Foundation Design

Bearing Capacity of Shallow Foundations

Terzaghi's Bearing Capacity Theory

Ultimate Bearing Capacity for Strip Footings (Continuous): \[q_u = c N_c + \gamma D_f N_q + 0.5 \gamma B N_\gamma\]

• \(q_u\) = ultimate bearing capacity (lb/ft² or kN/m²)
• \(c\) = cohesion of soil (lb/ft² or kN/m²)
• \(\gamma\) = unit weight of soil (lb/ft³ or kN/m³)
• \(D_f\) = depth of footing below ground surface (ft or m)
• \(B\) = width of footing (ft or m)
• \(N_c, N_q, N_\gamma\) = Terzaghi's bearing capacity factors (dimensionless)

Ultimate Bearing Capacity for Square Footings: \[q_u = 1.3 c N_c + \gamma D_f N_q + 0.4 \gamma B N_\gamma\] Ultimate Bearing Capacity for Circular Footings: \[q_u = 1.3 c N_c + \gamma D_f N_q + 0.3 \gamma B N_\gamma\] Terzaghi's Bearing Capacity Factors: \[N_q = \frac{e^{2(0.75\pi - \phi/2)\tan\phi}}{2\cos^2(45° + \phi/2)}\] \[N_c = (N_q - 1)\cot\phi\] \[N_\gamma = \frac{1}{2}\left(\frac{K_{p\gamma}}{\cos^2\phi} - 1\right)\tan\phi\]

• \(\phi\) = angle of internal friction (degrees)
• \(K_{p\gamma}\) = passive earth pressure coefficient
• Note: Values are typically obtained from published tables based on \(\phi\)

Meyerhof's Bearing Capacity Theory

General Bearing Capacity Equation: \[q_u = c N_c s_c d_c i_c + q N_q s_q d_q i_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma i_\gamma\]

• \(q\) = effective overburden pressure at foundation level = \(\gamma D_f\) (lb/ft² or kN/m²)
• \(s_c, s_q, s_\gamma\) = shape factors
• \(d_c, d_q, d_\gamma\) = depth factors
• \(i_c, i_q, i_\gamma\) = inclination factors

Meyerhof's Bearing Capacity Factors: \[N_q = e^{\pi\tan\phi}\tan^2\left(45° + \frac{\phi}{2}\right)\] \[N_c = (N_q - 1)\cot\phi\] \[N_\gamma = (N_q - 1)\tan(1.4\phi)\] Shape Factors: For rectangular footings: \[s_c = 1 + 0.2\frac{B}{L}\] \[s_q = 1 + 0.2\frac{B}{L}\] \[s_\gamma = 1 - 0.4\frac{B}{L}\]

• \(L\) = length of footing (ft or m)
• \(B\) = width of footing (ft or m)
• For square footings: \(B/L = 1\)
• For circular footings: use \(B/L = 1\)

Depth Factors (for \(D_f/B \leq 1\)): \[d_c = 1 + 0.2\sqrt{\frac{D_f}{B}}\] \[d_q = 1 + 0.2\sqrt{\frac{D_f}{B}}\] \[d_\gamma = 1.0\] Depth Factors (for \(D_f/B > 1\)): \[d_c = 1 + 0.2\sqrt{\frac{D_f}{B}} \leq 1 + 0.4\tan^{-1}\left(\frac{D_f}{B}\right)\] \[d_q = 1 + 0.2\sqrt{\frac{D_f}{B}} \leq 1 + 2\tan\phi(1-\sin\phi)^2\tan^{-1}\left(\frac{D_f}{B}\right)\]

Hansen's and Vesic's Modifications

Hansen's Shape Factors: For \(\phi = 0\) (purely cohesive): \[s_c = 1 + 0.2\frac{B}{L}\] For \(\phi > 0\): \[s_c = 1 + \frac{B}{L}\frac{N_q}{N_c}\] \[s_q = 1 + \frac{B}{L}\tan\phi\] \[s_\gamma = 1 - 0.4\frac{B}{L} \geq 0.6\] Hansen's Depth Factors: For \(\phi = 0\): \[d_c = 1 + 0.4\frac{D_f}{B}\] For \(\phi > 0\): \[d_c = 1 + 0.4k\] \[d_q = 1 + 2\tan\phi(1-\sin\phi)^2 k\] \[d_\gamma = 1.0\] where: \[k = \frac{D_f}{B} \text{ for } D_f/B \leq 1\] \[k = \tan^{-1}\left(\frac{D_f}{B}\right) \text{ (in radians) for } D_f/B > 1\]

Allowable Bearing Capacity

Factor of Safety Method: \[q_{all} = \frac{q_u}{FS}\]

• \(q_{all}\) = allowable bearing capacity (lb/ft² or kN/m²)
• \(FS\) = factor of safety (typically 2.5 to 3.0 for bearing capacity)

Alternative Formulation: \[q_{all} = \frac{q_u - q}{FS} + q\]

• This formulation considers net ultimate bearing capacity
• \(q = \gamma D_f\) = overburden pressure

Bearing Capacity for Eccentric Loading

Effective Dimensions Method (Meyerhof): For eccentricity in one direction: \[B' = B - 2e_B\] \[L' = L - 2e_L\]

• \(B'\) = effective width (ft or m)
• \(L'\) = effective length (ft or m)
• \(e_B\) = eccentricity in width direction (ft or m)
• \(e_L\) = eccentricity in length direction (ft or m)
• Use \(B'\) and \(L'\) in bearing capacity equations instead of \(B\) and \(L\)

Eccentricity from Applied Loads: \[e = \frac{M}{P}\]

• \(e\) = eccentricity (ft or m)
• \(M\) = applied moment (lb-ft or kN-m)
• \(P\) = applied vertical load (lb or kN)

Maximum Eccentricity (to avoid tensile stress): For rectangular footings: \[e_{max} = \frac{B}{6}\] For circular footings: \[e_{max} = \frac{D}{8}\]

• \(D\) = diameter of circular footing (ft or m)

Bearing Capacity for Inclined Loading

Hansen's Inclination Factors: For load inclined at angle \(\alpha\) to vertical: \[i_c = i_q - \frac{1 - i_q}{N_c \tan\phi}\] \[i_q = \left(1 - \frac{\alpha}{90°}\right)^2\] \[i_\gamma = \left(1 - \frac{\alpha}{90°}\right)^2\]

• \(\alpha\) = angle of load inclination from vertical (degrees)

Shallow Foundation Settlement

Total Settlement Components

\[\rho_T = \rho_i + \rho_c + \rho_s\]

• \(\rho_T\) = total settlement (in or mm)
• \(\rho_i\) = immediate (elastic) settlement (in or mm)
• \(\rho_c\) = primary consolidation settlement (in or mm)
• \(\rho_s\) = secondary compression settlement (in or mm)

Immediate (Elastic) Settlement

Elastic Settlement Formula: \[\rho_i = q_o B \frac{1 - \nu^2}{E_s} I_s I_f\]

• \(q_o\) = net applied foundation pressure (lb/ft² or kN/m²)
• \(B\) = width or diameter of foundation (ft or m)
• \(\nu\) = Poisson's ratio of soil (dimensionless)
• \(E_s\) = modulus of elasticity of soil (lb/ft² or kN/m²)
• \(I_s\) = shape factor (dimensionless)
• \(I_f\) = depth/rigidity factor (dimensionless)

Simplified Elastic Settlement (Rigid Footing, Center): \[\rho_i = q_o B \frac{1 - \nu^2}{E_s} I\] where \(I\) is an influence factor depending on footing shape and rigidity. Typical Influence Factors for Flexible Footings:

• Center of circular footing: \(I = 1.0\)
• Corner of rectangular footing: depends on \(L/B\) ratio
• Average settlement: typically 0.85 to 0.93 × corner settlement for rigid footings

Primary Consolidation Settlement

For Normally Consolidated Clay: \[\rho_c = \frac{C_c H}{1 + e_0} \log_{10}\left(\frac{\sigma'_0 + \Delta\sigma}{\sigma'_0}\right)\]

• \(C_c\) = compression index (dimensionless)
• \(H\) = thickness of compressible layer (ft or m)
• \(e_0\) = initial void ratio (dimensionless)
• \(\sigma'_0\) = initial effective vertical stress at mid-height of layer (lb/ft² or kN/m²)
• \(\Delta\sigma\) = stress increase at mid-height of layer (lb/ft² or kN/m²)

For Overconsolidated Clay (\(\sigma'_0 + \Delta\sigma \leq \sigma'_c\)): \[\rho_c = \frac{C_r H}{1 + e_0} \log_{10}\left(\frac{\sigma'_0 + \Delta\sigma}{\sigma'_0}\right)\]

• \(C_r\) = recompression index (dimensionless)
• \(\sigma'_c\) = preconsolidation stress (lb/ft² or kN/m²)

For Overconsolidated Clay (\(\sigma'_0 + \Delta\sigma > \sigma'_c\)): \[\rho_c = \frac{C_r H}{1 + e_0} \log_{10}\left(\frac{\sigma'_c}{\sigma'_0}\right) + \frac{C_c H}{1 + e_0} \log_{10}\left(\frac{\sigma'_0 + \Delta\sigma}{\sigma'_c}\right)\] Compression Index Correlations: For undisturbed clays: \[C_c = 0.009(LL - 10)\]

• \(LL\) = liquid limit (%)

For remolded clays: \[C_c = 0.007(LL - 7)\] Relationship between \(C_c\) and \(C_r\): \[C_r = \frac{C_c}{5} \text{ to } \frac{C_c}{10}\] Overconsolidation Ratio: \[OCR = \frac{\sigma'_c}{\sigma'_0}\]

Secondary Compression Settlement

\[\rho_s = \frac{C_\alpha H}{1 + e_p} \log_{10}\left(\frac{t}{t_p}\right)\]

• \(C_\alpha\) = coefficient of secondary compression (dimensionless)
• \(e_p\) = void ratio at end of primary consolidation
• \(t\) = time after end of primary consolidation (years, days, etc.)
• \(t_p\) = time at end of primary consolidation (same units as \(t\))

Stress Increase Due to Foundation Loading

Boussinesq Equation (Point Load): \[\Delta\sigma_z = \frac{3Q}{2\pi z^2} \left[\frac{1}{1 + (r/z)^2}\right]^{5/2}\]

• \(Q\) = point load (lb or kN)
• \(z\) = depth below point load (ft or m)
• \(r\) = radial distance from load axis (ft or m)

Stress Increase Below Corner of Uniformly Loaded Rectangular Area: \[\Delta\sigma_z = q I\] where \(I\) is an influence factor based on \(m = L/B\) and \(n = z/B\) \[I = \frac{1}{4\pi}\left[\frac{2mn\sqrt{m^2+n^2+1}}{m^2+n^2+m^2n^2+1}\left(\frac{m^2+n^2+2}{m^2+n^2+1}\right) + \tan^{-1}\left(\frac{2mn\sqrt{m^2+n^2+1}}{m^2+n^2-m^2n^2+1}\right)\right]\]

• \(q\) = uniform load intensity (lb/ft² or kN/m²)
• \(m\) = \(L/B\) ratio
• \(n\) = \(z/B\) ratio
• Note: Values typically obtained from charts

2:1 Method (Approximation): \[\Delta\sigma_z = \frac{q B L}{(B + z)(L + z)}\]

• Assumes stress spreads at 2 vertical to 1 horizontal
• Less accurate than Boussinesq method but simpler

Deep Foundation - Pile Foundations

Single Pile Capacity - Static Methods

Total Axial Capacity: \[Q_u = Q_p + Q_s\]

• \(Q_u\) = ultimate pile capacity (lb or kN)
• \(Q_p\) = end bearing capacity (lb or kN)
• \(Q_s\) = side friction (skin friction) capacity (lb or kN)

End Bearing Capacity: \[Q_p = q_p A_p = A_p (c N_c + \sigma'_v N_q)\]

• \(q_p\) = unit end bearing resistance (lb/ft² or kN/m²)
• \(A_p\) = cross-sectional area of pile tip (ft² or m²)
• \(c\) = cohesion at pile tip (lb/ft² or kN/m²)
• \(\sigma'_v\) = effective vertical stress at pile tip (lb/ft² or kN/m²)
• \(N_c, N_q\) = bearing capacity factors for deep foundations

Side Friction Capacity: \[Q_s = \sum f_s A_s = \sum f_s p \Delta L\]

• \(f_s\) = unit skin friction (lb/ft² or kN/m²)
• \(A_s\) = surface area of pile shaft in contact with soil (ft² or m²)
• \(p\) = pile perimeter (ft or m)
• \(\Delta L\) = incremental length of pile (ft or m)

Pile Capacity in Cohesive Soils (Clay)

α-Method (Total Stress Analysis): End bearing: \[Q_p = 9 c_u A_p\] Side friction: \[Q_s = \alpha c_u A_s\]

• \(c_u\) = undrained shear strength of clay (lb/ft² or kN/m²)
• \(\alpha\) = adhesion factor (dimensionless, 0 < \(\alpha\)="" ≤="">
• Factor of 9 comes from \(N_c = 9\) for deep foundations in clay

Adhesion Factor \(\alpha\) (Tomlinson):

• For \(c_u\) < 24="" kpa="" (500="" lb/ft²):="" \(\alpha="" \approx="">
• For \(c_u\) = 24-72 kPa (500-1500 lb/ft²): \(\alpha\) decreases from 1.0 to 0.5
• For \(c_u\) > 72 kPa (1500 lb/ft²): \(\alpha\) continues to decrease
• Empirical correlations available in charts

β-Method (Effective Stress Analysis): \[Q_s = \beta \sigma'_v A_s\]

• \(\beta\) = empirical coefficient (dimensionless)
• \(\sigma'_v\) = average effective vertical stress along pile shaft (lb/ft² or kN/m²)

λ-Method: \[Q_s = \lambda (\sigma'_v + 2c_u) A_s\]

• \(\lambda\) = empirical coefficient (typically 0.15 to 0.35)

Pile Capacity in Cohesionless Soils (Sand)

End Bearing in Sand: \[Q_p = q_p A_p = \sigma'_v N_q A_p\]

• \(N_q\) = bearing capacity factor (function of \(\phi\))
• For driven piles, \(q_p\) is often limited to 100-400 tsf (10-40 MPa)

Critical Depth Concept: For piles driven deeper than critical depth \(D_c\): \[q_p = \sigma'_v(D_c) N_q\] Typical critical depth: \[D_c = 10B \text{ to } 20B\]

• \(B\) = pile diameter or width (ft or m)

Side Friction in Sand: \[f_s = K \sigma'_v \tan\delta\]

• \(K\) = lateral earth pressure coefficient (dimensionless)
• \(\delta\) = friction angle between pile and soil (degrees)
• Typically \(\delta = 0.5\phi\) to \(0.75\phi\) for driven piles
• Typically \(\delta = 0.67\phi\) to \(\phi\) for drilled shafts

Simplified β-Method for Sand: \[f_s = \beta \sigma'_v\] where: \[\beta = K \tan\delta\] Typical values:

• Driven piles: \(\beta = 0.15\) to \(0.35\)
• Drilled shafts: \(\beta = 0.25\) to \(0.40\)

Dynamic Pile Formulas

Engineering News Formula (ENR): For drop hammers: \[Q_u = \frac{2WH}{S + 1.0}\] For steam/air hammers: \[Q_u = \frac{2WH}{S + 0.1}\]

• \(W\) = weight of hammer (lb or kN)
• \(H\) = drop height of hammer (ft or m)
• \(S\) = penetration per blow (in or cm)
• Note: Factor of safety should be applied (typically FS = 6)
• Constants 1.0 and 0.1 are in inches (or 2.54 and 0.254 in cm)

Modified ENR Formula: \[Q_u = \frac{e_h WH}{S + C}\]

• \(e_h\) = hammer efficiency (typically 0.75 to 0.85)
• \(C\) = empirical constant (in or cm)

Gates Formula: \[Q_u = \frac{WH \log(10N)}{0.4}\]

• \(N\) = number of blows per inch of penetration

Allowable Pile Capacity

\[Q_{all} = \frac{Q_u}{FS}\]

• \(Q_{all}\) = allowable pile capacity (lb or kN)
• \(FS\) = factor of safety (typically 2.5 to 3.0 for static analysis, 6 for dynamic formulas)

Pile Group Capacity

Group Capacity in Sand: \[Q_{group} = n Q_{single}\]

• \(n\) = number of piles in group
• \(Q_{single}\) = capacity of single pile (lb or kN)
• Sand groups are typically more efficient than individual pile sum

Group Capacity in Clay (Block Failure): Check both individual pile sum and block failure: \[Q_{group,block} = Q_p + Q_{s,perimeter}\] \[Q_p = 9 c_u A_{group}\] \[Q_{s,perimeter} = \alpha c_u p_{group} L\]

• \(A_{group}\) = area enclosed by pile group perimeter (ft² or m²)
• \(p_{group}\) = perimeter of pile group (ft or m)
• \(L\) = embedded pile length (ft or m)

Use lesser of: \[Q_{group} = \min(n Q_{single}, Q_{group,block})\] Group Efficiency: \[E_g = \frac{Q_{group}}{n Q_{single}}\]

• \(E_g\) = group efficiency (dimensionless)
• Typically \(E_g\) < 1="" in="" clay,="" \(e_g\)="" ≥="" 1="" in="">

Converse-Labarre Equation: \[E_g = 1 - \frac{\theta}{90} \left[\frac{(n-1)m + (m-1)n}{mn}\right]\] where: \[\theta = \tan^{-1}\left(\frac{d}{s}\right)\]

• \(m\) = number of rows
• \(n\) = number of piles per row
• \(d\) = pile diameter (ft or m)
• \(s\) = center-to-center pile spacing (ft or m)
• \(\theta\) = angle in degrees

Pile Group Settlement

Equivalent Footing Method: Treat pile group as an equivalent footing at depth \(L_e\): \[L_e = \frac{2L}{3}\] \[\Delta\sigma = \frac{Q_{group}}{(B_g + L_e)(L_g + L_e)}\]

• \(L\) = embedded pile length (ft or m)
• \(B_g\) = width of pile group (ft or m)
• \(L_g\) = length of pile group (ft or m)
• \(\Delta\sigma\) = stress increase at depth \(L_e\) (lb/ft² or kN/m²)

Negative Skin Friction (Downdrag)

\[Q_{nsf} = \alpha c_u A_s \text{ (in clay)}\] \[Q_{nsf} = \beta \sigma'_v A_s \text{ (in sand)}\]

• \(Q_{nsf}\) = negative skin friction force (downward drag on pile) (lb or kN)
• Reduces effective pile capacity
• Occurs when soil settles relative to pile (e.g., consolidating fill)

Net Available Capacity with Downdrag: \[Q_{net} = Q_u - Q_{nsf}\]

Lateral Pile Capacity

Brom's Method for Short Rigid Piles in Clay: \[H_u = 9 c_u B L\]

• \(H_u\) = ultimate lateral capacity (lb or kN)
• \(B\) = pile width or diameter (ft or m)
• \(L\) = embedded length (ft or m)

Brom's Method for Long Flexible Piles in Clay: \[H_u = 9 c_u B L_f\] where \(L_f\) is the effective depth to point of rotation. Maximum Moment for Laterally Loaded Pile: \[M_{max} = H \times e + M_{applied}\]

• \(e\) = eccentricity or height above ground (ft or m)
• \(M_{applied}\) = applied moment at pile head (lb-ft or kN-m)

Deep Foundation - Drilled Shafts

Drilled Shaft Capacity

Total Capacity (Same form as piles): \[Q_u = Q_p + Q_s\] End Bearing (in Clay): \[Q_p = N_c c_u A_p\]

• \(N_c = 9\) for circular base in deep clay

End Bearing (in Sand): \[Q_p = \sigma'_v N_q A_p\] Side Friction (in Clay): \[Q_s = \alpha c_u A_s\]

• \(\alpha\) typically ranges from 0.3 to 1.0 depending on construction method and \(c_u\)

Side Friction (in Sand): \[Q_s = \beta \sigma'_v A_s\]

• \(\beta\) typically 0.25 to 0.40 for drilled shafts (higher than driven piles)

O'Neill and Reese Method for Side Resistance in Clay: \[f_s = \alpha c_u\] where \(\alpha\) varies with \(c_u\):

• For \(c_u\) < 48="" kpa:="" \(\alpha="">
• For 48 kPa ≤ \(c_u\) ≤ 96 kPa: \(\alpha = 0.55 - 0.1(c_u - 48)/48\)
• For \(c_u\) > 96 kPa: \(\alpha = 0.45\)

Belled Drilled Shafts

End Bearing for Bell: \[Q_p = N_c c_u A_{bell}\]

• \(A_{bell}\) = area of bell base (ft² or m²)
• \(N_c = 9\) for standard bells

Bell Geometry: Maximum bell diameter: \[D_{bell} \leq 3D_{shaft}\]

• \(D_{shaft}\) = shaft diameter (ft or m)
• \(D_{bell}\) = bell diameter (ft or m)

Retaining Wall Design and Lateral Earth Pressure

Rankine Earth Pressure Theory

Active Earth Pressure Coefficient: \[K_a = \frac{1 - \sin\phi}{1 + \sin\phi} = \tan^2\left(45° - \frac{\phi}{2}\right)\] Passive Earth Pressure Coefficient: \[K_p = \frac{1 + \sin\phi}{1 - \sin\phi} = \tan^2\left(45° + \frac{\phi}{2}\right)\]

• \(\phi\) = angle of internal friction (degrees)

Active Lateral Pressure at Depth \(z\): \[p_a = K_a \gamma z - 2c\sqrt{K_a}\] For cohesionless soil (\(c = 0\)): \[p_a = K_a \gamma z\] Passive Lateral Pressure at Depth \(z\): \[p_p = K_p \gamma z + 2c\sqrt{K_p}\] For cohesionless soil (\(c = 0\)): \[p_p = K_p \gamma z\]

• \(\gamma\) = unit weight of soil (lb/ft³ or kN/m³)
• \(z\) = depth below ground surface (ft or m)
• \(c\) = cohesion (lb/ft² or kN/m²)

Coulomb Earth Pressure Theory

Active Earth Pressure Coefficient: \[K_a = \frac{\cos^2(\phi - \alpha)}{\cos^2\alpha \cos(\delta + \alpha)\left[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \beta)}{\cos(\delta + \alpha)\cos(\beta - \alpha)}}\right]^2}\] Passive Earth Pressure Coefficient: \[K_p = \frac{\cos^2(\phi + \alpha)}{\cos^2\alpha \cos(\delta - \alpha)\left[1 - \sqrt{\frac{\sin(\phi + \delta)\sin(\phi + \beta)}{\cos(\delta - \alpha)\cos(\beta - \alpha)}}\right]^2}\]

• \(\alpha\) = angle of wall face from vertical (degrees, positive if tilted away from soil)
• \(\beta\) = slope of backfill surface from horizontal (degrees)
• \(\delta\) = friction angle between wall and soil (degrees)
• For vertical wall (\(\alpha = 0\)), horizontal backfill (\(\beta = 0\)), and smooth wall (\(\delta = 0\)), Coulomb reduces to Rankine

Lateral Earth Pressure with Surcharge

Point Load Surcharge: Additional horizontal pressure from point load \(Q\) at distance \(x\) from wall: \[\Delta p = \frac{Q}{\pi H} \left(\frac{n^2}{m^2 + n^2}\right)^2\] where: \[m = \frac{x}{H}, \quad n = \frac{z}{H}\]

• \(H\) = height of wall (ft or m)
• \(z\) = depth from ground surface (ft or m)
• \(x\) = horizontal distance from wall (ft or m)

Uniform Surcharge: Additional lateral pressure: \[\Delta p = K_a q\]

• \(q\) = uniform surcharge load (lb/ft² or kN/m²)
• Applied uniformly over entire height

Line Load Surcharge: Maximum horizontal stress at depth \(z\): \[\Delta p = \frac{2q}{\pi z}\]

• \(q\) = line load per unit length (lb/ft or kN/m)

Lateral Pressure from Water

\[p_w = \gamma_w z\]

• \(\gamma_w\) = unit weight of water (62.4 lb/ft³ or 9.81 kN/m³)
• \(z\) = depth below water surface (ft or m)

Total Lateral Pressure with Water Table: \[p_{total} = K_a \gamma_{sat} z + p_w = K_a \gamma_{sat} z + \gamma_w z\] or equivalently: \[p_{total} = K_a \gamma' z + \gamma_w z\]

• \(\gamma_{sat}\) = saturated unit weight of soil (lb/ft³ or kN/m³)
• \(\gamma'\) = submerged (buoyant) unit weight = \(\gamma_{sat} - \gamma_w\) (lb/ft³ or kN/m³)

Resultant Force and Location

Total Resultant Force (Triangular Pressure Distribution): \[P_a = \frac{1}{2}K_a \gamma H^2\]

• \(P_a\) = total active force per unit length of wall (lb/ft or kN/m)
• \(H\) = height of wall (ft or m)

Location of Resultant (from base): \[\bar{z} = \frac{H}{3}\]

• For triangular distribution, resultant acts at one-third height from base

With Uniform Surcharge: \[P_a = \frac{1}{2}K_a \gamma H^2 + K_a q H\]

At-Rest Earth Pressure

At-Rest Coefficient (Jaky's Formula): \[K_0 = 1 - \sin\phi\] For Overconsolidated Soils: \[K_0(OC) = K_0(NC) \times OCR^{\sin\phi}\]

• \(OCR\) = overconsolidation ratio
• \(K_0(NC)\) = at-rest coefficient for normally consolidated soil

At-Rest Pressure: \[p_0 = K_0 \gamma z\]

Retaining Wall Stability Analysis

Overturning Stability: \[FS_{OT} = \frac{M_R}{M_O}\]

• \(FS_{OT}\) = factor of safety against overturning (typically ≥ 2.0)
• \(M_R\) = sum of resisting moments (stabilizing) (lb-ft/ft or kN-m/m)
• \(M_O\) = sum of overturning moments (lb-ft/ft or kN-m/m)

Sliding Stability: \[FS_S = \frac{F_R}{F_D}\]

• \(FS_S\) = factor of safety against sliding (typically ≥ 1.5)
• \(F_R\) = sum of resisting forces (lb/ft or kN/m)
• \(F_D\) = sum of driving (lateral) forces (lb/ft or kN/m)

Resisting Force (Base Friction): \[F_R = \mu \sum V + c_b B\]

• \(\mu\) = coefficient of friction between base and soil = \(\tan\delta_b\)
• \(\sum V\) = sum of vertical forces (lb/ft or kN/m)
• \(c_b\) = adhesion/cohesion at base (lb/ft² or kN/m²)
• \(B\) = base width of wall (ft or m)

Bearing Capacity Check: \[q_{max} = \frac{\sum V}{B}\left(1 + \frac{6e}{B}\right)\] \[q_{min} = \frac{\sum V}{B}\left(1 - \frac{6e}{B}\right)\]

• \(q_{max}, q_{min}\) = maximum and minimum bearing pressure (lb/ft² or kN/m²)
• \(e\) = eccentricity of resultant from centerline of base (ft or m)

Eccentricity: \[e = \frac{B}{2} - \bar{x}\] where: \[\bar{x} = \frac{M_R - M_O}{\sum V}\]

• \(\bar{x}\) = location of resultant from toe (ft or m)
• For stability, \(e \leq B/6\) (middle third rule)

Reinforced Soil Walls

Required Reinforcement Length: \[L_e = \frac{L - L_a}{FS}\] where: \[L_a = \frac{H - z}{\tan(45° - \phi/2)}\]

• \(L_e\) = embedment length in resistant zone (ft or m)
• \(L\) = total reinforcement length (ft or m)
• \(L_a\) = length in active zone (ft or m)
• \(H\) = wall height (ft or m)
• \(z\) = depth to reinforcement layer (ft or m)
• \(FS\) = factor of safety (typically 1.5 for pullout)

Tensile Force in Reinforcement: \[T = K_a \gamma z S_v\]

• \(T\) = tensile force per unit width (lb/ft or kN/m)
• \(S_v\) = vertical spacing of reinforcement (ft or m)

Pullout Resistance: \[P_R = 2L_e B \sigma'_v \tan\phi_r\]

• \(P_R\) = pullout resistance (lb or kN)
• \(B\) = width of reinforcement strip (ft or m)
• \(\sigma'_v\) = effective vertical stress at reinforcement depth (lb/ft² or kN/m²)
• \(\phi_r\) = interface friction angle (degrees)
• Factor 2 accounts for both faces of reinforcement

Shallow Foundation on Soil Slopes

Bearing Capacity Reduction on Slopes

Reduction Factor for Foundations Near Slopes: \[q_u(slope) = q_u(level) \times N_{cq} \times N_{\gamma q}\]

• \(N_{cq}\) = reduction factor for cohesion term
• \(N_{\gamma q}\) = reduction factor for surcharge and unit weight terms
• Values depend on slope angle \(\beta\), setback distance \(b\), and foundation depth
• Factors are less than 1.0, obtained from charts

Minimum Setback Distance

For foundation near slope crest: \[b_{min} = H \tan(45° - \phi/2)\]

• \(b_{min}\) = minimum setback from slope crest (ft or m)
• \(H\) = slope height (ft or m)

Mat Foundations

Mat Foundation Bearing Pressure

Uniform Bearing Pressure (Rigid Mat): \[q = \frac{P}{A}\]

• \(q\) = uniform bearing pressure (lb/ft² or kN/m²)
• \(P\) = total load (lb or kN)
• \(A\) = mat area (ft² or m²)

With Moment (Linear Distribution): \[q_{max/min} = \frac{P}{A} \pm \frac{M_x c_y}{I_x} \pm \frac{M_y c_x}{I_y}\]

• \(M_x, M_y\) = moments about x and y axes (lb-ft or kN-m)
• \(c_x, c_y\) = distances from neutral axis to edge (ft or m)
• \(I_x, I_y\) = moment of inertia about x and y axes (ft⁴ or m⁴)

For Rectangular Mat: \[I_x = \frac{BL^3}{12}, \quad I_y = \frac{LB^3}{12}\]

• \(B\) = mat width (ft or m)
• \(L\) = mat length (ft or m)

Compensated Foundation

Net Foundation Pressure: \[q_{net} = q_{applied} - \gamma D_f\]

• \(q_{net}\) = net pressure increase on soil (lb/ft² or kN/m²)
• \(q_{applied}\) = applied foundation pressure (lb/ft² or kN/m²)
• \(\gamma D_f\) = weight of soil excavated (lb/ft² or kN/m²)

Fully Compensated Foundation: \[q_{net} = 0\] Achieved when: \[\gamma D_f = \frac{P}{A}\]

• Foundation weight equals excavated soil weight
• Minimizes settlement

Foundation Vibration and Dynamics

Natural Frequency

Vertical Vibration: \[\omega_n = \sqrt{\frac{k_z}{m}}\] \[f_n = \frac{\omega_n}{2\pi}\]

• \(\omega_n\) = natural circular frequency (rad/s)
• \(f_n\) = natural frequency (Hz or cycles/s)
• \(k_z\) = vertical spring constant (lb/ft or kN/m)
• \(m\) = mass of foundation and supported equipment (lb-s²/ft or kg)

Vertical Spring Constant (Circular Foundation): \[k_z = \frac{4Gr_0}{1-\nu}\]

• \(G\) = shear modulus of soil (lb/ft² or kN/m²)
• \(r_0\) = equivalent radius of foundation (ft or m)
• \(\nu\) = Poisson's ratio of soil (dimensionless)

Equivalent Radius: \[r_0 = \sqrt{\frac{A}{\pi}}\]

• \(A\) = foundation contact area (ft² or m²)

Amplitude of Vibration

Resonance Condition: Avoid when: \[\frac{f_{operating}}{f_n} \approx 1.0\] Typical acceptable ranges:

• \(f_{operating}/f_n < 0.5\)="" or="" \(f_{operating}/f_n=""> 1.5\)

Soil Improvement Methods

Compaction

Relative Compaction: \[RC = \frac{\gamma_d}{\gamma_{d,max}} \times 100\%\]

• \(RC\) = relative compaction (%)
• \(\gamma_d\) = field dry unit weight (lb/ft³ or kN/m³)
• \(\gamma_{d,max}\) = maximum dry unit weight from Proctor test (lb/ft³ or kN/m³)

Dry Unit Weight: \[\gamma_d = \frac{\gamma_{wet}}{1 + w}\]

• \(\gamma_{wet}\) = wet (moist) unit weight (lb/ft³ or kN/m³)
• \(w\) = water content (decimal)

Zero Air Voids Line: \[\gamma_d = \frac{G_s \gamma_w}{1 + w G_s}\]

• \(G_s\) = specific gravity of soil solids (dimensionless)
• \(\gamma_w\) = unit weight of water (62.4 lb/ft³ or 9.81 kN/m³)

Preloading and Surcharging

Degree of Consolidation: \[U = \frac{\rho_t}{\rho_f} \times 100\%\]

• \(U\) = degree of consolidation (%)
• \(\rho_t\) = settlement at time \(t\) (in or mm)
• \(\rho_f\) = final settlement (in or mm)

Time Factor: \[T_v = \frac{c_v t}{H_{dr}^2}\]

• \(T_v\) = time factor (dimensionless)
• \(c_v\) = coefficient of consolidation (ft²/day or m²/year)
• \(t\) = time (days or years)
• \(H_{dr}\) = drainage path length (ft or m)

For double drainage: \(H_{dr} = H/2\)
For single drainage: \(H_{dr} = H\) Relationship between \(T_v\) and \(U\): For \(U < 60\%\):="" \[t_v="\frac{\pi}{4}\left(\frac{U}{100}\right)^2\]" for="" \(u=""> 60\%\): \[T_v = 1.781 - 0.933\log_{10}(100 - U)\]

Stone Columns

Improvement Factor: \[n = \frac{s}{d}\]

• \(n\) = spacing ratio
• \(s\) = center-to-center spacing of stone columns (ft or m)
• \(d\) = diameter of stone column (ft or m)

Area Replacement Ratio: For triangular pattern: \[a_r = \frac{A_s}{A} = \frac{2\pi}{3\sqrt{3}n^2}\] For square pattern: \[a_r = \frac{A_s}{A} = \frac{\pi}{4n^2}\]

• \(a_r\) = area replacement ratio (dimensionless)
• \(A_s\) = area of stone column (ft² or m²)
• \(A\) = total tributary area (ft² or m²)

Special Foundation Considerations

Expansive Soils

Swell Potential: \[S_p = \frac{\Delta H}{H} \times 100\%\]

• \(S_p\) = swell potential (%)
• \(\Delta H\) = vertical swell (in or mm)
• \(H\) = initial height of specimen (in or mm)

Plasticity Index Correlation: Rough classification:

• PI < 15:="" low="" swell="">
• PI = 15-35: Medium swell potential
• PI > 35: High swell potential

Collapsible Soils

Collapse Potential: \[CP = \frac{\Delta e}{1 + e_0} \times 100\%\]

• \(CP\) = collapse potential (%)
• \(\Delta e\) = change in void ratio upon wetting (dimensionless)
• \(e_0\) = initial void ratio (dimensionless)

Classification:

• CP < 1%:="" no="">
• CP = 1-5%: Moderate problem
• CP = 5-10%: Moderately severe
• CP > 10%: Severe problem

Frost Depth and Frost Heave

Modified Berggren Equation: \[X = \sqrt{\frac{48k}{\lambda L}F}\]

• \(X\) = depth of frost penetration (in or cm)
• \(k\) = thermal conductivity of soil (Btu/hr-ft-°F or W/m-K)
• \(\lambda\) = volumetric latent heat of fusion (Btu/ft³ or kJ/m³)
• \(L\) = correction factor for soil properties (dimensionless)
• \(F\) = freezing index (degree-days °F or degree-days °C)

Minimum Foundation Depth: Foundation depth should be below frost depth to prevent frost heave: \[D_{min} = X + \text{safety margin}\]

Foundation Load Transfer and Distribution

Combined Footings

Location of Resultant: \[\bar{x} = \frac{\sum P_i x_i}{\sum P_i}\]

• \(\bar{x}\) = location of resultant from reference point (ft or m)
• \(P_i\) = individual column loads (lb or kN)
• \(x_i\) = distance of each load from reference point (ft or m)

Centroid of Footing: For uniform soil pressure under rigid footing, centroid of footing should coincide with resultant of loads. Required Footing Length: \[L = 2(\bar{x} - x_1) + \text{edge distance}\]

• \(x_1\) = distance to first column from reference (ft or m)

Strap (Cantilever) Footings

Equilibrium for Exterior Footing: Taking moments about interior column: \[P_1 e = T d\]

• \(P_1\) = load on exterior column (lb or kN)
• \(e\) = eccentricity of exterior footing (ft or m)
• \(T\) = tension/compression in strap (lb or kN)
• \(d\) = distance between columns (ft or m)

Design for Zero Eccentricity: Size exterior footing such that: \[\bar{x}_{footing} = \bar{x}_{resultant}\]

Seismic Considerations for Foundations

Seismic Earth Pressure

Mononobe-Okabe Active Seismic Coefficient: \[K_{ae} = \frac{\cos^2(\phi - \theta - \alpha)}{\cos\theta\cos^2\alpha\cos(\delta + \alpha + \theta)\left[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \beta - \theta)}{\cos(\delta + \alpha + \theta)\cos(\beta - \alpha)}}\right]^2}\] where: \[\theta = \tan^{-1}\left(\frac{k_h}{1 - k_v}\right)\]

• \(K_{ae}\) = active seismic earth pressure coefficient
• \(k_h\) = horizontal seismic coefficient (dimensionless)
• \(k_v\) = vertical seismic coefficient (dimensionless)
• \(\theta\) = seismic angle (degrees)

Dynamic Increment of Earth Pressure: \[\Delta P_{ae} = \frac{1}{2}(K_{ae} - K_a)\gamma H^2\]

• \(\Delta P_{ae}\) = additional lateral force due to seismic effects (lb/ft or kN/m)
• Applied at 0.6H from base for dynamic increment

Liquefaction Potential

Factor of Safety Against Liquefaction: \[FS_L = \frac{CRR}{CSR}\]

• \(FS_L\) = factor of safety against liquefaction (typically require ≥ 1.25 to 1.5)
• \(CRR\) = cyclic resistance ratio (dimensionless)
• \(CSR\) = cyclic stress ratio (dimensionless)

Cyclic Stress Ratio: \[CSR = 0.65\frac{a_{max}}{g}\frac{\sigma_v}{\sigma'_v}r_d\]

• \(a_{max}\) = peak ground acceleration (ft/s² or m/s²)
• \(g\) = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• \(\sigma_v\) = total vertical stress (lb/ft² or kN/m²)
• \(\sigma'_v\) = effective vertical stress (lb/ft² or kN/m²)
• \(r_d\) = stress reduction factor (dimensionless, function of depth)

Stress Reduction Factor (simplified): \[r_d = 1.0 - 0.00765z \text{ for } z \leq 9.15 \text{ m}\] \[r_d = 1.174 - 0.0267z \text{ for } 9.15 \text{ m} < z="" \leq="" 23="" \text{="" m}\]="">

• \(z\) = depth (m)