Theory

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

The content is intended to provide a clear understanding of how the calculation is performed within the software, facilitating both the interpretation of the results and the technical validation of the models.

Meyerhof’s general bearing capacity equation 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:

• $q_{u}$: Bearing capacity ($\text{kN/m}^2$).
• $c'$: Effective soil cohesion (kN/m^2^).
• $q'$: Effective stress at the base of the foundation ($\text{kN/m}^2$).
• $\gamma$: Unit weight of soil ($\text{kN/m}^3$).
• $B$: Foundation width ($\text{m}$).
• $N_c, N_q, N_\gamma$: Bearing capacity factors (dimensionless).
• $F_{cs}, F_{qs}, F_{\gamma s}$: (dimensional) form factors.
• $F_{cd}, F_{qd}, F_{\gamma d}$: depth factors (dimensionless).
• $F_{ci}, F_{qi}, F_{\gamma i}$: tilt factors (dimensionless).

Bearing capacity factors are defined as:

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

Form 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 factor:

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 the carrying capacity involves modifying the calculation of the parameters $\gamma$ and $q$ in the Meyerhof 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 area of load application. This calculation is performed using the effective area method.

First, the maximum and minimum pressure at the base of the foundation 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 at 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 ground.

The safety factor is evaluated as:

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

Where:

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

To calculate the load capacity $Q_u$, Meyerhof suggests using the following effective dimensions in the general load capacity equation:

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

This modifies the load 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 $F_{\gamma d}$ must still be computed with $B$ and $L$, not $B'$ and $L'$.

Furthermore, 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 the theory of elasticity. The 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 the soil (dimensionless).
• $E$: Elastic modulus of the soil ($\text{kN/m}^2$).
• $I_s$: Influence factor depending on the depth of rigid layer (dimensionless).
• $I_f$: Influence factor depending on the depth of the foundation (dimensionless).

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

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

Where:

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

Being $A_0$, $A_1$ and $A_2$:

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

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:

• $D_f$: Foundation depth.
• $B$: Foundation width.

When there is more than one soil layer, the weighted average of the elastic modulus of the soil layers 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 smallest value between $H$ and $5 B$.

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