Distribution of actions on a pile group

The program allows defining the stiffness to be attributed to the foundation structure, so as to consider it as a deformable constraint, rather than a rigid one, which can determine significant changes both in terms of stresses (hyperstatic structures) and for the verifications of deformation compatibility. Furthermore, the “group effects” of the pile foundation, which each pile has on the other, are considered.

The application operates under the assumption of elastic behavior, so the nonlinear behaviors that occur in piles at higher stress levels are not simulated. Therefore, the results can be reliable only if they are sufficiently far from the ultimate load.

It is divided into three calculation sections: Axial Actions, Horizontal and Bending Actions, and Pile Distribution.

Axial Actions

Figure 1 – Schematic of Axial Actions on a single pile

The first section, called “Axial Actions“, allows calculating the stiffness of a single pile subjected to axial action. The input data to be entered are:

• Thickness of the layers in m (coincides with the pile length);
• Transverse deformation modulus Gsup at the top of the layer in kN/m²;
• Transverse deformation modulus Ginf at the bottom elevation in kN/m²;
• Transverse deformation modulus Gbase at the base (for end-bearing piles) in kN/m²;
• Poisson’s ratio ν;
• Pile diameter D in m;
• Concrete Elastic Modulus E_cls in MPa.

The pile length L, expressed in m, will be derived from the sum of the layer thicknesses. G_base is considered in the algorithm only for the deepest layer. The elastic modulus values must be secant corresponding to the expected stress levels, and the most used classes are proposed:

• C20/25;
• C25/30;
• C28/35;
• C40/50.

The proposed elastic modulus value derives from relation (1):

(1)

Where:

The user can also enter an elastic modulus value if the proposed options are not relevant to the case under consideration.

The output results will be:

• Pile moment of inertia J_pile, expressed in m⁴, obtained from relation (2):

(2)

• Radius of influence or extinction r_m, expressed in m, through formula (3):

(3)

Where:

• • ;
  • in which G_L and G_b are respectively G_inf and G_base of the i-th layer;
  • For stratified soils equal to the average between G_inf e G_base of the i-th layer.
• Stiffness under axial actions expressed in kN/m.

Horizontal and Bending Actions

Figure 2 – Schematic of Horizontal and Bending Actions on a single pile

In the second calculation section, called “Horizontal and Bending Actions“, it is possible to determine the stiffness of the pile subjected to shear and bending actions by treating the pile as a beam on a bed of springs and imposing unit moments at the head. As for the input data, the scheme is analogous to the one previously described in Axial Actions.

The calculation of stiffnesses for lateral actions reports values for a single pile, but the application also calculates them for reduced subgrade reaction coefficients, necessary to take into account the interaction between the piles:

(4)

Where P_b/P_t is the fraction of the total load absorbed at the base.

It is also necessary to enter a value for the discretization step of the elements Dim_elem, with which the structural element is discretized.

Pile Distribution

Figure 3 – Schematic of pile arrangement and sign convention

In the third and final calculation section called “Pile Distribution“, the geometry of the pile foundation is defined. In particular, the arrangement is done by entering the planimetric coordinates x and y, expressed in meters, of the piles, up to a maximum of 30. To establish the number of piles to be considered in the calculation, simply select it from the drop-down menu.

Furthermore, it is possible to establish the number of load conditions, as for the piles, through the drop-down menu. A specific table is available to enter the load values. The stiffnesses calculated with the axial force, shear, and bending analyses are reported in the latter and cannot be modified.

Another input data is the point of application of the resulting loads (Actions): you can choose to refer the actions to the center of gravity of the pile foundation or by setting specific coordinates.

Once the calculation is performed, the stresses on all piles under the various load conditions will be available in table format. In addition, the most significant values are reported, corresponding to:

• Maximum and minimum axial force;
• Maximum moment;
• Maximum shear.

The values of M and T are the vector sums of the components along x and y and are always positive. For each pile, the displacement components u, v, w along the three directions x, y, and z are also calculated.

Depending on the position of the piles, the planimetric layout is generated; it is possible to save the input data as a text file and load it later. Finally, a complete calculation report is generated with the data used, the respective calculation results, and theoretical background.