Flexible pile caps¶
The feature exclusive to PCX. When the cap does not meet the rigidity condition, the strut does not form in a well-defined way and the real behaviour is bending: the cap works as a beam supported on the piles.
The basis is Silva (2021), published in REEC, which compared the three structural models — strut and tie, simply supported beam and cantilever beam — by reliability analysis.
When to use it¶
The NBR 6118 criterion is \(h \ge (A - a_p)/3\) in the direction considered, or, equivalently, strut angle \(\ge 45^\circ\). A cap that does not meet it is flexible. See Checks.
Designing a flexible cap by Blévot underestimates the reinforcement
It is not a difference of refinement: the strut model assumes a resisting mechanism that does not exist in that cap. The reinforcement it returns is that of a tie which is not what is actually working.
The two models¶
| Model | Assumption |
|---|---|
| Simply supported beam | The piles are the supports; the column applies the load within the span |
| Cantilever beam | Cantilever from the column |
The cantilever beam model allows displacements in the cap — the pile or the cap may settle —, but requires the column to remain undisplaced, or to settle uniformly with the cap and the piles.
With 3 or more piles, it becomes a grillage
The simply supported beam is one-dimensional, and a cap with three or more piles is not. In these cases the program automatically switches to a grillage analysis, which distributes bending in both directions.
Design moment¶
with \(L\) the distance between pile axes and \(b\) the column dimension.
The 15 % distance from the column face
It comes from Alonso (2010), and depends on the column's moment of inertia: \(0.15\,b\) is recommended for columns of large inertia (\(b \ge 60\) cm) and \(0.5\,b\) for columns of small inertia.
It exists because the fixity does not occur at the column face, but a little inside it — and the stiffer the column, the closer to the face.
Bending reinforcement¶
Simple bending design of a rectangular section according to NBR 6118, with \(\alpha_c = 0.85\) and \(\lambda = 0.8\) for \(f_{ck} \le 50\) MPa.
The neutral axis position comes from equilibrium:
and the reinforcement from
The corresponding resisting moment can be written directly as a function of the reinforcement:
Ductility limit¶
Above that the section fails without warning — the concrete crushes before the steel yields. The program warns you and caps it at 0.45, but the warning is meant to be read: the solution is to increase the depth, \(f_{ck}\) or use compression reinforcement.
Moment above what the section can resist
When the discriminant is negative, the section does not resist even with \(x = 0.45d\). The way out is the same: depth, \(f_{ck}\) or compression reinforcement.
For strut angles below 35°, Silva (2021) found that the concrete reaches domain 4 and compression reinforcement becomes necessary.
Minimum reinforcement¶
with \(A_{s,min} = \rho_{min}\,b\,h\). When it governs, the program warns you.
Strut check¶
Even in the beam model, the concrete compression must be checked:
Transverse reinforcement¶
Beam shear criterion. The resistance is the sum of the concrete and steel contributions:
with \(f_{ctd} = 0.15\,f_{ck}^{2/3}\).
If the reaction of the most heavily loaded pile does not exceed the concrete contribution, minimum detailing stirrups are adopted, at 20 cm spacing. If it does, the following is designed
and the adopted spacing lies between 5 and 20 cm.
The most loaded reaction, not the sum
Shear is checked against the reaction of the most heavily loaded pile — it is what defines the critical section, next to the most stressed support.
Flexible cap in tension¶
PCX handles Flexible Cap (Uplift/Tension) explicitly. The mechanism is reversed: the bending reinforcement moves to the opposite face, and the anchorage check gains weight — in uplift, it is the anchorage that holds.
What the reliability analysis showed¶
Silva (2021) applied Monte Carlo with one million simulations to a two-pile cap, varying the strut inclination from 30° to 60°, and compared the three models. The results guide the choice:
| Finding | Practical consequence |
|---|---|
| The simply supported beam produces reinforcement ratios more than 30 % higher than the strut model | It is the least recommended model: bar spacing becomes tight and encourages cracking |
| The cantilever beam results in reinforcement close to the strut-and-tie value — less than 6 % difference for \(\theta > 40^\circ\) | It is the most economical flexible alternative |
| The strut-and-tie model gives the least reinforcement, but only reaches \(\beta \ge 3.8\) with \(\theta > 60^\circ\) (fck 20) or \(\theta > 45^\circ\) (fck 30) | Meeting the rigid cap criterion does not guarantee adequate reliability |
| The strut-and-tie tie equation is independent of \(f_{ck}\) | An inconsistency of the model: increasing \(f_{ck}\) should reduce the reinforcement |
Do not design below 35°
Because of the significant loss of stiffness when reducing the depth of the cap, the guidance is not to design caps with a strut angle below 35°, even using flexible models — because of the increased displacements and the possibility of second-order effects.
Limits of validity¶
Range of application
- Modelling as a beam is a simplification of a solid. It represents the slender cap well; in a cap on the boundary, neither the beam nor the strut describes the behaviour precisely, and the defensible practice is to adopt the envelope of both.
- The grillage analysis distributes bending in both directions assuming linear elastic behaviour. There is no cracking or plastic redistribution.
- \(x/d \le 0.45\) and the minimum reinforcement apply to \(f_{ck} \le 50\) MPa.
- The failure function for excessive deformation was not checked by Silva (2021): there is no estimate in the literature of the maximum allowable deformation for pile caps. Since flexible models are more susceptible to deformation, this is an acknowledged gap.
- The anchorage of the column reinforcement inside the cap is not checked here — and it is one of the main factors that make flexible caps unfeasible in practice.
- Span-to-depth ratios below 2 (angles above 50°) make it a deep beam, and warping of the section is not accounted for.
- Splitting and confinement reinforcement are not checked, here as in the rigid cap.