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.

The load-bearing capacity of a pile is a combination of tip resistance and shaft friction.

Where:

• $Q_u$: Pile tip resistance ($\text{kN}$).
• $Q_s$: Pile shaft resistance ($\text{kN}$).
• $Q_p$: Pile tip resistance ($\text{kN}$).

In the undrained analysis of cohesive soils, the strength of the piles depends mainly on the adhesion between the soil and the pile surface.

For friction resistance in the shaft, the Alpha method is used:

Where:

• $Q_s$: Pile shaft resistance ($\text{kN}$).
• $\alpha$: Empirical coefficient that varies according to the type of soil (dimensionless).
• $c_u$: Undrained shear strength of the soil ($\text{kN/m}^2$).
• $A_s$: Area of the pile shaft ($\text{m}^2$).

The value of $c_u$ is estimated using the following table:

Where $p_a = \text{atmospheric pressure} \approx 100 \text{kN/m}^2$

The resistance at the tip ($Q_p$) is determined using the following equation:

Where:

• $Q_p$: Pile tip resistance ($\text{kN}$).
• $c_u$: Undrained shear strength of the soil ($\text{kN/m}^2$).
• $A_p$: Cross-sectional area of the pile ($\text{m}^2$).

The frictional resistance of the shaft is defined as:

Where:

• $Q_s$: Pile shaft resistance ($\text{kN}$).
• $p$: Pile perimeter ($\text{m}$).
• $\Delta L$: Length of the section in contact with the ground ($\text{m}$).
• $f$: Friction on the surface of the shaft ($\text{kN/m}^2$).
• $A_s$: Area of the pile shaft ($\text{m}^2$).

In granular soils, the shaft resistance must consider the following aspects:

• The method of installing the pile. Driven piles cause soil densification around the pile.
• The friction of the shaft increases to a depth of 15 times the diameter of the pile ($L' \approx 15 D$) and then remains constant.
• Friction in loose sand is greater for high displacement piles compared to low displacement piles.
• The friction of the shaft is less in drilled piles compared to driven piles.

Thus, the shaft friction is calculated as follows:

For $z = 0$ to $L'$:

For $z = L'$ a $L$, it remains constant:

In these equations:

• $K$: Coefficient of lateral earth pressure (dimensionless).
• $\sigma'_0$: Effective vertical stress, variable with depth ($\text{kN/m}^2$).
• $\delta$: Friction angle between the soil and the pile ($\text{°}$).

The recommended values for $K$ are:

The suggested values for $\delta$ range from $0.5 \phi'$ to $0.8 \phi'$.

The resistance at the tip ($Q_p$) is determined using the Meyerhof equation:

Where:

• $Q_p$: Pile tip resistance ($\text{kN}$).
• $q'$: Effective stress at the tip of the pile ($\text{kN/m}^2$).
• $N^{\ast}_q$: Factor (dimensionalless).
• $p_a$: Atmospheric pressure ($= 100 \text{kN/m}^2$).
• $\phi$: Ground friction angle at the tip ($\text{°}$).
• $A_p$: Cross-sectional area of the pile ($\text{m}^2$).

Pile groups must be analyzed considering the combined behavior of the piles, which means determining whether the failure is governed by the sum of the individual capacities of the piles or by the block failure of the soil.

To determine the failure mode, the group efficiency is calculated as:

Where:

• $\eta$: Efficiency of the pile group (dimensionless).
• $Q_{g(u)}$: Ultimate resistance of the pile group ($\text{kN}$).
• $\sum Q_u$: Sum of the ultimate resistance of each of the individual piles in the group ($\text{kN}$).

When efficiency is greater than or equal to 1, failure occurs when individual piles reach their final strength.

Conversely, when the efficiency is less than 1, block failure occurs before individual piles reach their final strength. In these cases, it is recommended to increase the space between stacks to avoid overlapping friction zones.

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