Formulas by arrangement¶
The closed-form expressions of the Strut Method for each pile configuration, according to Bastos (2023) and NBR 6118.
In all of them: \(N\) is the column load, \(e\) the distance between pile axes, \(a_p\) the column dimension, \(d\) the effective depth, \(A_p\) the column area and \(A_e\) the pile area.
A rectangular column becomes an equivalent square
The formulations assume a square column, with its centre coinciding with the geometric centre of the cap. For a rectangular column, the following is adopted:
Summary¶
| Piles | Arrangement | \(R_s\) | \(\sigma_{lim}\) at the column | \(\sigma_{lim}\) at the pile |
|---|---|---|---|---|
| 2 | Linear | \(\dfrac{N}{8}\dfrac{2e-a_p}{d}\) | \(1.4\,K_R f_{cd}\) | \(1.4\,K_R f_{cd}\) |
| 3 | Triangle | \(\dfrac{N}{9}\dfrac{e\sqrt3-0.9a_p}{d}\) | \(1.75\,K_R f_{cd}\) | \(1.75\,K_R f_{cd}\) |
| 4 | Square | \(\dfrac{N\sqrt2}{16}\dfrac{2e-a_p}{d}\) | \(2.1\,K_R f_{cd}\) | \(2.1\,K_R f_{cd}\) |
| 5 | Square + centre | \(\dfrac{4}{5}\dfrac{N\sqrt2}{16}\dfrac{2e-a_p}{d}\) | \(2.6\,K_R f_{cd}\) | \(2.1\,K_R f_{cd}\) |
| 5 | Pentagon | \(\dfrac{0.85N}{5d}\left(e-\dfrac{a_p}{3.4}\right)\) | not required | not required |
| 6 | Pentagon + centre | \(\dfrac{0.85N}{6d}\left(e-\dfrac{a_p}{3.4}\right)\) | not required | not required |
| 6 | Hexagon | \(\dfrac{N}{6d}\left(e-\dfrac{a_p}{4}\right)\) | not required | not required |
| 7 | Hexagon + centre | — (see below) | not required | not required |
"Not required" means: if the effective depth is adopted within the interval \(d_{min} \le d \le d_{máx}\), the stress in the struts does not need to be checked. It is the interval itself that ensures a safe inclination.
Watch out for the five-pile cap with one in the centre
It is the only arrangement in which the limit at the column differs from the limit at the pile: \(2.6\,K_R f_{cd}\) versus \(2.1\,K_R f_{cd}\). In the others, the two limits coincide.
Three piles — triangle¶
The column is assumed square, with its centre coinciding with the geometric centre of the cap. The force system is analysed along one of the medians of the triangle formed by the pile centres.
Effective depth¶
| Criterion | Interval |
|---|---|
| Blévot, \(40^\circ \le \alpha \le 55^\circ\) | \(0.485\left(e - 0.52a_p\right) \le d \le 0.825\left(e - 0.52a_p\right)\) |
| Machado (1985), \(45^\circ \le \alpha \le 55^\circ\) | \(0.58\left(e - \dfrac{a_p}{2}\right) \le d \le 0.825\left(e - \dfrac{a_p}{2}\right)\) |
Struts¶
Main reinforcement¶
\(R_s\) acts along the medians. To obtain the component in the direction of the lines between pile axes, by the law of sines:
resulting in the reinforcement parallel to the sides:
Why parallel to the sides, and not along the medians
The arrangement with reinforcement along the medians was widely used in the past, but it has two defects: the three bundles of bars overlap at the centre of the cap, and there is heavy cracking on the side faces caused by the lack of support at the ends of the bars — the so-called "unsupported reinforcement".
Moreover, it does not comply with NBR 6118 (22.7.4.1.1), which requires at least 85 % of the flexural reinforcement within the strips defined by the piles.
The recommended configuration is main reinforcement parallel to the sides, with an orthogonal mesh — the most used in Brazil, with less cracking and greater economy.
Plan dimensions¶
Following the suggestion of Campos (2015), the dimension \(A\) of the triangle flange is approximately \(1.154\,a\), with \(a\) the distance from the pile axis to the face.
Four piles — square¶
\(R_s\) is the tensile force along the diagonals.
Effective depth¶
For \(45^\circ \le \alpha \le 55^\circ\):
Struts¶
Main reinforcement¶
There are four possible detailing layouts, and they are not equivalent:
| Layout | Performance |
|---|---|
| a) Along the diagonals | Excessive side cracking even at low loads |
| b) Parallel to the sides | One of the most efficient — the most usual in practice |
| c) Diagonals + parallel to the sides | — |
| d) Single mesh | Lower failure load than the others, 80 % efficiency; best cracking performance |
Layouts a, c and d do not comply with the NBR 6118 (22.7.4.1.1) requirement that more than 85 % of the reinforcement be within the pile strips.
For layout b, with an added mesh:
Five piles — square with one in the centre¶
The procedure is that of the four-pile cap, replacing \(N\) by \(\frac{4}{5}N\) — the central pile takes its share without generating a tie.
Effective depth¶
For \(45^\circ \le \alpha \le 55^\circ\), the same as the four-pile cap:
Struts¶
Reinforcement¶
Very elongated column
For very rectangular columns, a rectangular cap on five piles is designed, treated as a four-pile cap with the formulas adapted to the different distances.
Another option is to place one line of three piles and another of two — in which case the calculation resembles that of caps with more than six piles.
Five piles — pentagon¶
The piles are at the vertices of a pentagon, with the centre of the square column coinciding with their geometric centre.
Effective depth¶
For \(45^\circ \le \alpha \le 55^\circ\):
Struts need not be checked
If \(d\) is adopted between \(d_{min}\) and \(d_{máx}\), the compressive stresses in the struts do not need to be checked.
Reinforcement¶
Resolving in the direction parallel to the sides, with \(R'_s = \dfrac{R_s}{2\cos 54^\circ}\):
Six piles — pentagon with one in the centre¶
Proceed as for the five-pile pentagon cap, replacing \(N\) by \(\frac{5N}{6}\):
Effective depth identical to that of the five-pile pentagon, and struts likewise need not be checked within the interval.
By the law of sines, \(R'_s = R_s\dfrac{\sin 54^\circ}{\sin 72^\circ} = 0.85\,R_s\):
Six piles — hexagon¶
The piles are at the vertices of the hexagon, with a centred square column.
Effective depth¶
For \(45^\circ \le \alpha \le 55^\circ\):
Struts need not be checked within the interval.
Reinforcement¶
Here the law of sines gives \(\dfrac{R_s}{\sin 60^\circ} = \dfrac{R'_s}{\sin 60^\circ}\), i.e. \(R'_s = R_s\) — the resolution does not change the force.
Six piles — rectangular¶
Suitable for rectangular, elongated columns. The forces \(R_{sx}\) and \(R_{sy}\) are treated separately in each direction, with the corresponding distances.
Seven piles — hexagon with one in the centre¶
The seventh pile is at the centre of the cap, under the column. For \(45^\circ \le \alpha \le 55^\circ\):
The compression in the struts does not need to be checked if \(d\) is within the interval.
The reinforcement is placed along the diagonals, with ties parallel to the sides:
with
The parameter \(k\) splits the force between diagonals and perimeter ties — it is a design choice within that range.
Complementary reinforcement¶
These apply to any number of piles: the NBR 6118 requirements for mesh and suspension reinforcement are general.
Mesh reinforcement¶
NBR 6118 (22.7.4.1.2) requires, to control cracking, additional bottom reinforcement independent of the main flexural reinforcement, as a mesh uniformly distributed in two orthogonal directions, corresponding to 20 % of the total tensile forces in each of them.
Suspension reinforcement¶
NBR 6118 (22.7.4.1.3) requires suspension reinforcement for the share of load to be balanced if distribution reinforcement is provided for more than 25 % of the total forces or if the spacing between piles is greater than three times the depth of the cap.
In general, regardless of this, the following can be specified:
with \(n_e\) the number of piles. The reinforcement per face is the total divided by the number of faces of the cap.
What it is for
Suspension reinforcement prevents cracks in the regions between the piles. They can appear because compressed struts form that transfer part of the column load to the lower regions of the cap, between the piles, and that bear on the reinforcement parallel to the sides.
This creates tensions that need to be suspended to the upper regions of the cap, from where they travel to the piles.
Top reinforcement¶
Skin reinforcement¶
On each lateral vertical face, as stirrups or horizontal bars:
with \(A_{s,total}\) the total main reinforcement — \(3A_{s,lado}\) in the three-pile cap, \(4A_{s,lado}\) in the four-pile one, and so on.
Spacing: \(s \le \min\left(\dfrac{d}{3};\ 20\ \text{cm}\right)\), and \(s \ge 8\) cm as a practical recommendation.
CEB-70 method¶
An alternative to the Strut Method for rigid caps, similar to the procedure for spread footings. The depth of the cap is limited by
where \(c\) is the distance from the column face to the axis of the farthest pile.
The method calculates the main reinforcement for bending, determined with respect to a reference section \(S_1\) located inside the column, at \(0.15\,a_p\) from the face — and checks the resistance of the cap to shear forces.
PCO does not implement CEB-70
The models offered are Blévot and MBT. CEB-70 is listed here for reference, as one of the methods historically most used in Brazil and accepted by the code.
Limits of validity¶
Range of application
- All expressions assume a centred column and piles equally spaced from the centre of the column. With moments or eccentricity, the pure formulation does not apply.
- They assume a square column — a rectangular one enters through the equivalent \(a_{p,eq} = \sqrt{a_p b_p}\), an approximation that gets worse the more elongated the column.
- The effective depth intervals come from \(45^\circ \le \alpha \le 55^\circ\) (Machado) or \(40^\circ \le \alpha \le 55^\circ\) (Blévot). Outside them, the exemption from checking the struts does not apply, and they must be checked explicitly.
- The limits \(\sigma_{lim}\) are Blévot's, not those of the current NBR 6118. To check against the current code, use MBT.
- Blévot's tests covered caps with up to six piles. The expressions for seven piles and for compound arrangements extrapolate that experimental basis.