Theory

This section presents the theoretical foundations on which the program is based. The concepts, assumptions, and formulations used in the different analysis methods implemented are explained.

The content is aimed at providing a clear understanding of how the calculation is performed within the software, facilitating both the interpretation of results and the technical validation of the models.

The general bearing capacity equation by Meyerhof is expressed as:

$$q_{u} = c' N_c F_{cs} F_{cd} F_{ci} + \gamma q' N_q F_{qs} F_{qd} F_{qi} + 0.5 \gamma B N_\gamma F_{\gamma s} F_{\gamma d} F_{\gamma i}$$

Where:

• $q\_{u}$: Ultimate bearing capacity ($\text{kN/m}^2$).
• $c'$: Effective soil cohesion (kN/m^2^).
• $q'$: Effective stress at the foundation base ($\text{kN/m}^2$).
• $\gamma$: Unit weight of soil ($\text{kN/m}^3$).
• $B$: Width of foundation ($\text{m}$).
• $N_c$, $N_q$, $N_\gamma$ : Bearing capacity factors (dimensionless).
• $F_{cs}$, $F_{qs}$, $F_{\gamma s}$: Shape factors (dimensionless).
• $F_{cd}, F_{qd}, F_{\gamma d}$: Depth factors (dimensionless).
• $F_{ci}, F_{qi}, F_{\gamma i}$: Inclination factors (dimensionless).

The bearing capacity factors are defined as:

$$N_q = \tan^2 \left( 45° + \frac{\phi'}{2}\right) e^{\pi \tan \phi'}$$ $$N_c = (N_q-1) \cot \phi'$$ $$N_\gamma = 2 (N_q + 1) \tan \phi'$$

Shape factors:

$$F_{cs} = 1 + \left(\frac{B}{L}\right)\left(\frac{N_q}{N_c}\right)$$ $$F_{qs} = 1 + \left(\frac{B}{L}\right)\tan \phi'$$ $$F_{\gamma s} = 1 - 0.4 \left(\frac{B}{L}\right)$$

Depth factors:

Case 1: $\frac{D_f}{B} \leq 1$ and $\phi' = 0$ :

$$F_{cd} = 1 + 0.4 \left(\frac{D_f}{B}\right)$$ $$F_{qd} = 1$$ $$F_{\gamma d} = 1$$

Case 2: $\frac{D_f}{B} \leq 1$ and $\phi' > 0$ :

$$F_{cd} = F_{qd} - \frac{1-F_{qd}}{N_c \tan \phi'}$$ $$F_{qd} = 1 + 2 \tan \phi' (1- \sin \phi')^2 \left( \frac{D_f}{B} \right)$$ $$F_{\gamma d} = 1$$

Case 3: $\frac{D_f}{B} > 1$ and $\phi' = 0$ :

$$F_{cd} = 1 + 0.4 \tan^{-1}\left(\frac{D_f}{B}\right)$$ $$F_{qd} = 1$$ $$F_{\gamma d} = 1$$

Case 4: $\frac{D_f}{B} > 1$ and $\phi' > 0$ :

$$F_{cd} = F_{qd} - \frac{1-F_{qd}}{N_c \tan \phi'}$$ $$F_{qd} = 1 + 2 \tan \phi' (1- \sin \phi')^2 \tan^{-1}\left( \frac{D_f}{B} \right)$$ $$F_{\gamma d} = 1$$

Inclination factors:

$$F_{ci} = F_{qi} = \left( 1 - \frac{\beta°}{90°} \right)^2$$ $$F_{\gamma i} = \left( 1 - \frac{\beta°}{\phi'} \right)^2$$

The influence of the water table on bearing capacity implies modifying the calculation of the parameters $\gamma$ and $q$ in Meyerhof’s General Bearing Capacity Equation. These modifications depend on the saturation condition:

Saturated case $D_w \leq D_f$ :

$$\gamma = \gamma' = \gamma_{sat} - \gamma_w$$ $$q = D_w \gamma + (D_f-D_w) (\gamma_{sat} - \gamma_w)$$

Where:

• $\gamma_{sat}$: Saturated unit weight of soil ($\text{kN/m}^3$).
• $\gamma_{w}$: Unit weight of water ($\text{kN/m}^3$).

Partially saturated case $Df < D_w \leq D_f + B$ :

$$\bar{\gamma} = \gamma' + \frac{D_w - Df}{B} (\gamma - \gamma')$$ $$q = \gamma D_f$$

Dry case $D_w > D_f + B$ :

$$\gamma = \gamma$$ $$q = \gamma D_f$$

Eccentric loads modify the load application area. This calculation is carried out using the Effective Area Method.

First, the maximum and minimum pressure at the foundation base must be calculated:

$$e = \frac{M}{Q}$$ $$q_{max} = \frac{Q}{BL} \left(1+\frac{6e}{B}\right)$$ $$q_{min} = \frac{Q}{BL} \left(1-\frac{6e}{B}\right)$$

Where:

• $Q$: Total vertical load ($\text{kN}$).
• $M$: Acting moment on the foundation ($\text{kN m}$).

The load eccentricity must not exceed $e > B/6$ since from this value, $q\_{min}$ becomes negative and a gap will occur between the plate and the soil.

The safety factor is evaluated as:

$$FS = \frac{Q_u}{Q}$$

Where:

• $Q_u$: Bearing capacity ($\text{kN}$).
• $Q$: Applied vertical load ($\text{kN}$).

To compute the bearing capacity $Q_u$, Meyerhof suggests using the following effective dimensions in the General Bearing Capacity Equation:

$$B' = \text{effective width} = B - 2 e$$ $$L' = \text{effective length} = L$$

This modifies the bearing capacity equation:

$$q'_{u} = c' N_c F_{cs} F_{cd} F_{ci} + \gamma q' N_q F_{qs} F_{qd} F_{qi} + 0.5 \gamma B' N_\gamma F_{\gamma s} F_{\gamma d} F_{\gamma i}$$

To calculate the factors $F_{cs}$, $F_{qs}$ and $F_{\gamma s}$, replace the values of $B$ and $L$ with $B'$ and $L'$ respectively.

However, $F_{cd}$, $F_{qd}$ and $F_{\gamma d}$ should still be calculated with $B$ and $L$, not $B'$ and $L'$.

Also, the load application area must be changed to the effective area:

$$A' = B' L'$$

Therefore:

$$Q_u = q'_u A'$$

This method is based on elasticity theory. Elastic settlement ($S_e$) is calculated with the following equation:

$$S_e = \frac{qB}{E} (1 - \mu^2) I_s I_f$$

Where:

• $S_e$: Elastic settlement ($\text{m}$).
• $q$: Load applied by the foundation ($\text{kN/m}^2$).
• $B$: Foundation width ($\text{m}$).
• $\mu$: Poisson’s ratio of soil (dimensionless).
• $E$: Elastic modulus of soil ($\text{kN/m}^2$).
• $I_s$: Influence factor depending on rigid layer depth (dimensionless).
• $I_f$: Influence factor depending on foundation depth (dimensionless).

When a rigid layer underlies the soil, an influence factor ($I_s$) must be considered to reduce the calculated settlement.

$$I_s = F_1 + \frac{1-2 \mu_s}{1-\mu_s} F_2$$

Where:

$$F_1 = \frac{1}{\pi}(A_0 + A_1)$$ $$F_2 = \frac{n'}{2 \pi}\tan^{-1} A_2$$

With $A_0$, $A_1$ and $A_2$ being:

$$A_0 = m' \ln \frac{\left(1+\sqrt{m'^2+1}\right) \sqrt{m'^2+n'^2}}{m' \left( 1+\sqrt{m'^2+n'^2+1} \right)}$$ $$A_1 = \ln \frac{\left(m'+\sqrt{m'^2+1}\right) \sqrt{1+n'^2}}{\left(m'+\sqrt{m'^2+n'^2+1} \right)}$$ $$A_2 = \frac{m'}{n' + \sqrt{m'^2+n'^2+1}}$$

And finally:

$$m' = \frac{L}{B}$$ $$n' = \frac{H}{B}$$

Where:

• $L$: Foundation length.
• $B$: Foundation width.
• $H$: Depth to the rigid layer.

The influence factor $I_f$ depends on the depth of the foundation:

$$I_f = \frac{1}{1 + \left(\frac{D_f}{B}\right)^2}$$

Where:

• $D_f$: Depth of the foundation.
• $B$: Width of the foundation.

When there is more than one soil layer, the weighted average of the elastic modulus of the soil deposits must be calculated:

$$E_s = \frac{\sum E_{s(i)}\Delta z}{\bar z}$$

Where:

• $E_s$: average elastic modulus.
• $E_{s(i)}$: elastic modulus for a layer of thickness $\Delta z$.
• $\bar z$: the lesser value between $H$ and $5 B$.

• Das, B. M., & Sivakugan, N. (2018). Principles of Foundation Engineering.