Pile group: distribution of actions and checks¶

For walls on piles MDC treats the foundation as a rigid pile cap on a group of piles. The actions transmitted by the wall are distributed among the rows through the equilibrium and compatibility of the pile cap; each row is then checked for bearing capacity and bending. This page describes the model; the window in which the data are assigned is described in Piles, the formulas for the single pile (bearing capacity, Matlock and Reese, settlements) in Foundation Pile Computation.

Pile layouts¶

One, two or three rows: the distance of the pile axis from the outer edge of the foundation, the spacing along the wall and, with two rows, the alignment (aligned or staggered piles) are assigned.
Wall on rows of piles: up to 8 rows, each with its distance from the downstream edge, inclination, spacing along the wall and offset. Splayed and A-frame (trestle) foundations can be described.

The pile type can be drilled, driven or micropile. Micropiles are calculated as drilled piles, with the same resistance formulas and the same R3 partial factors; when micropile is selected the reinforcement switches to steel tube (it can be changed).

Conventions: horizontal forces positive towards the downstream side, vertical forces positive downwards, moments positive when they overturn the wall towards the downstream side; the pile inclination α is measured from the vertical and is positive when the tip points towards the downstream side.

Actions at the pile heads¶

The wall calculation (phases I-III of the Computation Process) gives, per metre of wall, the horizontal force H, the vertical force V and the moment. The moment to be assigned to the piles is chosen in the Piles window:

Overturning moment: only the overturning moments about the toe of the foundation, without the stabilizing contribution of the weights. This is the conservative choice.
Resulting moment: equilibrium of all the actions (thrusts, weights of the wall and of the soil on the footing shelves, surcharges) about the toe.

In both cases the moment is transferred to the centroid G of the pile heads:

\[M_G = M_{piede} + V \cdot d\]

where d is the horizontal distance between the toe and G. H, V and M_G are printed in the report.

Percentage of load to the piles (mixed foundation)¶

A percentage p between 70% and 100% of the actions can be assigned to the piles:

V and M are multiplied by p;
H goes entirely to the piles (the pile cap is not considered capable of transmitting shear to the soil);
the remaining part, (1 − p), goes to the soil under the pile cap as in a shallow foundation: the pressures are calculated and printed in the report.

With p < 100% the report and the list of problems flag a mixed foundation: the bearing capacity of the soil under the pile cap is not checked by the program and must be verified separately.

Distribution among the rows: rigid pile cap¶

The pile cap is a rigid body with three unknown displacements: horizontal u, vertical w and the rotation θ about G. For row i, at distance s_i from G and with inclination α_i, the displacement of the head along the pile axis and the one normal to the axis are

\[\delta_{a,i} = u \sin\alpha_i + w \cos\alpha_i + \theta\, s_i \cos\alpha_i\]

\[\delta_{t,i} = u \cos\alpha_i - w \sin\alpha_i - \theta\, s_i \sin\alpha_i\]

and the actions at the pile head are

\[N_i = k_a\, \delta_{a,i} \qquad T_i = k_t\, \delta_{t,i}\]

Writing the equilibrium of the pile cap gives the system

\[\mathbf{K} \begin{Bmatrix} u \\ w \\ \theta \end{Bmatrix} = \begin{Bmatrix} H \\ V \\ M_G \end{Bmatrix} \qquad \mathbf{K} = \sum_i \frac{1}{i_i} \left( k_a\, \mathbf{b}_{a,i} \mathbf{b}_{a,i}^T + k_t\, \mathbf{b}_{t,i} \mathbf{b}_{t,i}^T \right)\]

where b_a,i and b_t,i are the coefficients of u, w, θ in the two previous expressions and i_i is the spacing of the row along the wall (each row counts as 1/i_i piles per metre). This is the rigid pile cap method of Hrennikoff (1949), taken up by Poulos and Davis (1980).

With vertical piles and negligible transverse stiffness (k_t → 0) the system returns the classical distribution

\[N_i = \frac{V}{n} \pm \frac{M_G\, s_i}{\sum s_j^2}\]

which is therefore a special case of the model.

Pile stiffnesses¶

The axial stiffness k_a combines the settlement of the pile in the soil and the elastic shortening of the shaft:

\[k_a = \frac{1}{w_1 + \dfrac{L}{E_p A_p}}\]

where w₁ is the settlement under unit load, calculated with the average elastic modulus of the soil along the shaft (mean of the layers weighted on their thicknesses), L the length, E_p the modulus of the concrete and A_p the pile area. With the average modulus the values remain within about 15% of the Randolph and Wroth (1978) solution; using only the modulus of the layer at the tip the stiffness turned out 1.5-1.7 times higher.

The transverse stiffness k_t is the inverse of the horizontal displacement of the free head under a unit force, for a pile on a Winkler soil with characteristic length L_b:

\[k_t = \frac{1}{y_1} \qquad y_1 = \frac{L_b^3}{2\, E_p I_p} \;\text{(constant reaction)} \qquad y_1 \simeq \frac{2.5\, L_b^3}{E_p I_p} \;\text{(reaction increasing with depth, Matlock and Reese)}\]

Pile-cap connection: hinged or fixed head¶

With two or more rows the way the pile head is connected to the pile cap is chosen:

Hinged (default): the head can rotate with respect to the pile cap, there is no moment at the head.
Fixed: the head rotates with the pile cap (rotation φ = −θ). For each row the force normal to the axis and the moment at the head follow the head stiffness matrix of Hetényi for the beam on elastic foundation:

\[T_i = k_t \left( 2\,\delta_{t,i} + L_b\, \varphi \right) \qquad M_i = k_t \left( L_b\, \delta_{t,i} + L_b^2\, \varphi \right)\]

These terms enter the matrix K of the pile cap, and the fixed-end moment M_i is passed to the pile check together with N and T.

Which one to choose

Fixed head: pile reinforcement extended into the pile cap for at least the anchorage length, or pile head embedded in the pile cap for at least one diameter. This is the usual case of cast-in-place drilled piles. Hinged head: head embedded in the pile cap by only a few centimetres, without connecting reinforcement (precast or driven piles cut flush, micropiles with a plate). If in doubt, it is advisable to calculate with both: the hinged head gives the higher maximum axial force, the fixed head the higher head moment and reinforcement.

With a single row the moment is passed to the pile anyway and the choice is not active.

Check of all rows¶

For each load combination and for each row the program calculates:

the bearing capacity (tip plus lateral resistance) and the safety factor;
the internal forces along the shaft (displacements, moment, shear) with N, T and, if present, the fixed-end moment of the row;
the tension state: the pile is in tension when N plus its self-weight is negative, and then only the lateral resistance in tension counts.

The row with the minimum safety factor governs the bearing capacity check; the pile reinforcement is designed on the envelope of all the rows. The report contains a table with, for each row, N, T, M, bearing capacity R, safety factor and maximum moment; the details are printed for the row that governs the bearing capacity and, if different, also for the one that governs the reinforcement.

Inclined piles¶

For a pile inclined by α the tip lies at depth L cos α. The bearing capacity takes the true geometry into account:

the layers crossed down to depth L cos α are considered;
the factor Nq is evaluated at the true depth of the tip;
the lateral resistance is integrated along the pile axis: in each layer of thickness h the pile has a length h / cos α.

The distributed actions (N along the axis, T normal to the axis) are those of the rigid pile cap. The horizontal soil reaction on an inclined pile is calculated with the formulas of the vertical pile: the corrections for inclination (Kubo, Awoshika and Reese) are not applied.

Group effect and spacing¶

The group efficiency of Converse-Labarre can be applied to the bearing capacity of each pile. For a wall, with piles continuing along the whole length and m rows:

\[E_g = 1 - \frac{\theta_L + \theta_F\, \dfrac{m-1}{m}}{90} \qquad \theta_L = \arctan\frac{D}{i} \qquad \theta_F = \arctan\frac{D}{s_F}\]

with the angles in degrees, D diameter, i spacing along the wall, s_F distance between the rows (the minimum values are used); E_g does not drop below 0.3. The bearing capacity becomes R · E_g and in the report the formula shows "· Eg". Usually, for drilled piles in sand, E_g = 1 is taken (no reduction); in clays Converse-Labarre is applied.

The spacings are checked when the Piles window is confirmed: a spacing along the wall or a distance between the rows smaller than the diameter is an error; smaller than 3 diameters is a warning, and one can proceed. In the report the spacings are also given in diameters.

Seismic combinations¶

In the seismic combinations the inertial actions of the wall and of the soil are already included in H, V and M, so they reach the piles with the distribution described above. The pile resistance is not reduced as a function of the seismic coefficient k_h.

The kinematic interaction moments (soil deformation during the earthquake) are not calculated. With ground type D or E the report points this out: according to NTC 2018 (§7.2.5) they must be evaluated, among other cases, with ground type D or worse and structures in use class III or IV, in which case the warning also appears in the list of problems.

Global stability¶

When the wall is exported to Slope for the global stability check, all the pile rows are passed, with position, inclination and spacing. With more than 3 rows or different spacings the program warns to check the piles in the Slope project.

References¶

Hrennikoff A. (1949). Analysis of pile foundations with batter piles. Transactions ASCE, 115.
Poulos H. G., Davis E. H. (1980). Pile Foundation Analysis and Design. Wiley.
Randolph M. F., Wroth C. P. (1978). Analysis of deformation of vertically loaded piles. Journal of the Geotechnical Engineering Division ASCE, 104.
Hetényi M. (1946). Beams on Elastic Foundation. University of Michigan Press.
Matlock H., Reese L. C. (1960). Generalized solutions for laterally loaded piles. Journal of the Soil Mechanics and Foundations Division ASCE, 86.
Bowles J. E. Foundation Analysis and Design. McGraw-Hill (Converse-Labarre formula).
D.M. 17 gennaio 2018, Aggiornamento delle Norme tecniche per le costruzioni, §6.4.3 and §7.2.5.

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