Theory

This section presents the theoretical foundations on which the program is based. It explains 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 results and the technical validation of the models.

Load Capacity of Individual Piles

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

Where:

• $Q_u$: Ultimate resistance of the pile ($\text{kN}$).
• $Q_s$: Shaft resistance of the pile ($\text{kN}$).
• $Q_p$: Tip resistance of the pile ($\text{kN}$).

Cohesive Soils: Undrained Analysis

In the undrained analysis for cohesive soils, the pile resistance mainly depends on the adhesion between the soil and the pile surface.

For shaft friction resistance, the Alpha method is used:

Where:

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

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

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

Tip resistance ($Q_p$) is determined using the following equation:

Where:

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

Granular Soils

Shaft friction resistance is defined as:

Where:

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

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

• The pile installation method. Driven piles cause densification of the soil around the pile.
• Shaft friction increases with depth up to approximately 15 times the pile diameter ($L' \approx 15 D$) and then remains constant.
• Friction in loose sand is higher for high-displacement piles compared to low-displacement piles.
• Shaft friction is lower in bored piles compared to driven piles.

Thus, shaft friction is calculated as follows:

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

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

In these equations:

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

Recommended values for $K$ are:

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

Tip resistance ($Q_p$) is determined using Meyerhof’s equation:

Where:

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

Pile Group Analysis

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

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

Where:

• $\eta$: Pile group efficiency (dimensionless).
• $Q_{g(u)}$: Ultimate resistance of the pile group ($\text{kN}$).
• $\sum Q_u$: Sum of the ultimate resistance 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 ultimate resistance.

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

References

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