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Theory & Technical Notes
Based on CE75.05 Geotechnical Engineering for Tall Buildings lecture notes (AIT, Dr. Geoff Chao, 2019)
and standard geotechnical references.
1. Bearing Capacity of Shallow Foundations
1.1 Modes of Bearing Capacity Failure
- General Shear Failure — Most common type. Gradual settlement followed by sudden failure due to loss of interlocking resistance. Occurs in dense soils and rocks.
- Local Shear Failure — Intermediate between general and punching shear. Movement accompanied by sudden jerks. Occurs in medium-density soils.
- Punching Shear Failure — Shear zones never become well-defined, little surface heave. Occurs in very loose sand and weak clay.
1.2 Terzaghi (1943) Bearing Capacity Equation
For a general shear failure of a shallow foundation, assuming homogeneous isotropic soil, horizontal ground surface, and failure zone below the base:
\[ q_u = c'N_cS_c + qN_q + \tfrac{1}{2}\gamma B N_\gamma S_\gamma \]
Where:
- \(q_u\) = ultimate bearing capacity (kPa)
- \(c'\) = effective cohesion (kPa)
- \(q = \gamma D_f\) = surcharge pressure at foundation level (kPa)
- \(\gamma\) = unit weight of soil (kN/m³)
- \(B\) = width of foundation (m)
- \(N_c, N_q, N_\gamma\) = bearing capacity factors (functions of \(\phi'\))
Shape Factors (Terzaghi):
| Foundation Type | \(S_c\) | \(S_\gamma\) |
|---|
| Strip | 1.0 | 1.0 |
| Square | 1.3 | 0.8 |
| Circular | 1.3 | 0.6 |
1.3 Meyerhof (1963) Bearing Capacity Equation
Improved Terzaghi's equation by adding shape, depth, and load inclination factors. Addresses rectangular foundations and inclined loads:
\[ q_u = c'N_cF_{cs}F_{cd}F_{ci} + qN_qF_{qs}F_{qd}F_{qi} + \tfrac{1}{2}\gamma BN_\gamma F_{\gamma s}F_{\gamma d}F_{\gamma i} \]
Shape Factors (De Beer, 1970)
\[ F_{cs} = 1 + \frac{B}{L}\frac{N_q}{N_c}, \quad F_{qs} = 1 + \frac{B}{L}\tan\phi', \quad F_{\gamma s} = 1 - 0.4\frac{B}{L} \]
Depth Factors (Hansen, 1970)
For \(D_f/B \leq 1\):
\[ F_{qd} = 1 + 2\tan\phi'(1-\sin\phi')^2 \frac{D_f}{B} \]
For \(D_f/B > 1\):
\[ F_{qd} = 1 + 2\tan\phi'(1-\sin\phi')^2 \arctan\left(\frac{D_f}{B}\right) \]
\[ F_{cd} = F_{qd} - \frac{1-F_{qd}}{N_q\tan\phi'}, \quad F_{\gamma d} = 1 \]
Load Inclination Factors (Meyerhof, 1963)
\[ F_{ci} = F_{qi} = \left(1 - \frac{\theta}{90^\circ}\right)^2, \quad F_{\gamma i} = \left(1 - \frac{\theta}{\phi'}\right)^2 \]
where \(\theta\) is the angle of load inclination from vertical.
1.4 Drained vs. Undrained Analysis
- Saturated Fine Materials (Clay): Use undrained shear strength (\(c_u\), \(\phi = 0\)). The \(N_\gamma\) term drops out.
- Saturated Coarse Materials (Sand/Gravel): Use effective shear strength (\(c'=0\), \(\phi'\)). The \(N_c\) term drops out.
2. Limit State Design Approach
2.1 Ultimate Limit State (ULS)
The maximum load-carrying capacity of the foundation. Using the LRFD approach:
\[ R_{dg} \geq E_d \]
Where \(R_{dg} = \phi_g \times R_{ug}\) is the design geotechnical strength, \(\phi_g\) is the reduction factor (0.4 to 0.8 depending on site investigation extent), and \(E_d\) is the factored design load.
2.2 Serviceability Limit State (SLS)
Settlement and angular distortion criteria:
\[ \rho_{max} \leq \rho_{all}, \quad \theta_{max} \leq \theta_{all} \]
Common allowable angular distortion for tall buildings: \(\theta_{all} = 1/500 = 0.002\) (20% probability of damage).
2.3 Settlement Components
\[ S_{TF} = S_e + S_c + S_s \]
- \(S_e\) = Immediate (elastic) settlement — shear deformation under constant volume
- \(S_c\) = Primary consolidation settlement — excess pore pressure dissipation
- \(S_s\) = Secondary consolidation (creep) — time-related plastic adjustment
Immediate Settlement (Bowles, 1987)
\[ S_{e} = q_o(\alpha B') \frac{1-\mu_s^2}{E_s} I_s I_f \]
Rigid foundation: \(S_{e(rigid)} \approx 0.93 \times S_{e(flexible, center)}\)
Primary Consolidation Settlement
Normally consolidated clay:
\[ S_c = \frac{C_c H_c}{1+e_o} \log\frac{\sigma'_{vo} + \Delta\sigma}{\sigma'_{vo}} \]
Overconsolidated clay (\(\sigma'_{vo} + \Delta\sigma \leq \sigma'_c\)):
\[ S_c = \frac{C_s H_c}{1+e_o} \log\frac{\sigma'_{vo} + \Delta\sigma}{\sigma'_{vo}} \]
3. Mononobe-Okabe Seismic Earth Pressure Theory
3.1 Static Earth Pressure Coefficients (Coulomb)
Active earth pressure coefficient:
\[ K_a = \frac{\sin^2(\beta + \phi)}{\sin\beta\sin(\beta - \delta)\left[1 + \sqrt{\frac{\sin(\phi+\delta)\sin(\phi-\alpha)}{\sin(\beta-\delta)\sin(\beta+\alpha)}}\right]^2} \]
Passive earth pressure coefficient:
\[ K_p = \frac{\sin^2(\beta - \phi)}{\sin\beta\sin(\beta + \delta)\left[1 - \sqrt{\frac{\sin(\phi+\delta)\sin(\phi+\alpha)}{\sin(\beta+\delta)\sin(\beta+\alpha)}}\right]^2} \]
3.2 Seismic Active Earth Pressure (Mononobe-Okabe)
The Mononobe-Okabe method extends Coulomb's earth pressure theory to include seismic effects using pseudo-static horizontal and vertical accelerations:
\[ K_{ae} = \frac{\sin^2(\beta + \phi - \theta)}{\cos\theta\sin\beta\sin(\beta - \delta - \theta)\left[1 + \sqrt{\frac{\sin(\phi+\delta)\sin(\phi-\alpha-\theta)}{\sin(\beta-\delta-\theta)\sin(\beta+\alpha)}}\right]^2} \]
Where the seismic inclination angle:
\[ \theta = \arctan\left(\frac{k_h}{1 - k_v}\right) \]
and \(k_h\), \(k_v\) are the horizontal and vertical seismic coefficients.
3.3 At-Rest Earth Pressure
\[ K_0 = 1 - \sin\phi \]
3.4 Parameters
| Symbol | Description |
|---|
| \(\phi\) | Angle of internal friction of backfill soil |
| \(\beta\) | Angle of wall face from horizontal |
| \(\delta\) | Wall-soil friction angle |
| \(\alpha\) | Angle of backfill slope from horizontal |
| \(k_h, k_v\) | Horizontal and vertical seismic coefficients |
| \(\theta\) | Seismic inclination angle |
4. Retaining Wall Stability
4.1 Rankine Earth Pressure
\[ K_a = \frac{1 - \sin\phi}{1 + \sin\phi}, \quad K_p = \frac{1 + \sin\phi}{1 - \sin\phi} \]
\[ P_a = \tfrac{1}{2}\gamma H^2 K_a \quad \text{(acting at } H/3 \text{ from base)} \]
4.2 Sliding Safety Factor
\[ FS_{sliding} = \frac{\mu W + cB + P_p}{P_a} \geq 1.5 \]
4.3 Overturning Safety Factor
\[ FS_{overturning} = \frac{\sum M_{resisting}}{\sum M_{overturning}} \geq 2.0 \]
4.4 Eccentricity Check
\[ e = \frac{B}{2} - \frac{M_R - M_O}{W} \leq \frac{B}{6} \]
Base pressure distribution:
\[ q_{max,min} = \frac{W}{B}\left(1 \pm \frac{6e}{B}\right) \]
References
- Terzaghi, K. (1943). Theoretical Soil Mechanics. Wiley.
- Meyerhof, G.G. (1963). Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, 1(1), 16-26.
- Das, B., Sivakugan, N. (2017). Principles of Foundation Engineering (9th ed.). Cengage.
- Poulos, H. (2017). Tall Building Foundation Design. CRC Press.
- Bowles, J.E. (1987). Elastic foundation settlements on sand deposits. J. Geotech. Eng., ASCE, 113(8), 846-860.
- Vesic, A.S. (1973). Analysis of ultimate loads of shallow foundations. J. Soil Mech. Found. Div., ASCE, 99(SM1), 45-73.
- Mononobe, N. & Matsuo, H. (1929). On the determination of earth pressures during earthquakes. Proc. World Engineering Congress, Tokyo, Vol. 9.
- Okabe, S. (1926). General theory of earth pressures. J. Japan Society of Civil Engineers, 12(1).
- Chao, G. (2019). CE75.05 Geotechnical Engineering for Tall Buildings Lecture Notes, Asian Institute of Technology, Thailand.