Design of Pile cap using Bending theory – Worked Example

Table of Contents

This article presents a worked example on the design of a pile cap using bending theory. The pile cap shall be proportioned, designed for bending, vertical shear, as well as verified for punching shear both at column perimeter and control perimeter all according to EN 1992-1-1-2004 (Eurocode 2).

The pile cap to be designed is to transmit column loads to piles. A geotechnical test has been carried out on the site located at Greenwich, South-East London. Due to the occurrence of London clay, which is highly unstable, on the upper 6m layer on the site, it is recommended that a deep foundation in the form of piles to a depth of 9m is adopted. The recommended pile size is 600mm with safe working load of 400KN. The characteristics strength of concrete and reinforcement for the pile cap are 25MPa and 500MPa respectively. The dimension of the column, and the load acting on it are given below.

Column: Size = 300 x 300mm

Dead Load = 1205KN

Live Load = 291KN

Estimate the number of piles required

Calculate the number of piles to support the column load.

Total service load on the column = 1205 + 291 = 1496KN

No of piles = Total service load/capacity of a pile = 1496/400 = 3.74

Adopt 4 number of piles

Determine the size and Dimension of the Pile cap

The distance between each pile shall be kØ

K =3,

Ø = 600mm

kØ = 3 x 600 = 1800

Edge distance = 150mm

Since the piles are four then they will be arranged in a square array so that they can be symmetrically distributed under the column.

The length of each side of pile cap = Pile dist + 2 x Ø/2 + 2 x edge distance

= 1800 + 2 x 300 + 2 x 150 = 2700mm

The height of this pile cap was taken as 1350mm when this pile cap was design using strut-and-tie method. However, the height of the pile cap shall be drastically reduced here so that the pile cap shall be designed using bending theory.

We shall assume the height of the pile cap to 900mm

The layout of the pile cap is shown below

Design the pile cap for flexure

Since the pile cap is to be designed using bending theory, we shall analyse the pile cap for maximum bending moment. This shall govern the main reinforcement in the pile cap.

Determine the moment in pile cap

The critical moment in the pile cap shall be taken from the centre of a pile to the face of the column.

N = Ultimate axial load = 1.35 x 1205 + 1.5 x 291 = 2063.25KN

Pile Reaction = 2063.25KN/4 = 515.81KN

L = distance from pile centre to column face = 1800/2 = 900mm

Max moment = Pile Reaction x L = 515.81 x 0.9 = 464.23KNm

Design for the computed Moment

Calculate the effective depth

Assumptions

Cover = 50mm

Main reinforcement diameter = 20mm

Diameter of links = 10mm

Effective depth = h-c-ᴓ-ᴓ/2

= 900-50-10-20/2

= 830mm

2) Check whether section is to be designed as singly or doubly reinforced beam

$ K\,=\,\,\frac{M}{bd^2f_{ck}} $

$ K\,=\,\,\frac{464.23×10^6}{2700×830^2×25} $

= 0.01

Since K (0.01) < K’ (0.168); design as singly reinforced.

3) Calculate the lever arm (Z)

$ Z\,\,=\,\,d\left( 0.5+\sqrt{\text{0.25}-\,\,\frac{K}{1.134}} \right) $

$ Z\,\,=\,\,830\left( 0.5+\sqrt{\text{0.25}-\,\,\frac{0.01}{1.134}} \right) $ = 822.63

Since 822.63 > 0.95d (788.5): use Z = 788.5

4. Calculate the area of steel

$ A_{st\,\,=\,\,\frac{M_{Ed}}{0.87f_{yk}Z}} $

$ A_{st\,\,=\,\,\frac{464.23×10^6} {0.87x500x788.5}} $

Ast = 1353mm²

Check minimum area of reinforcement

Asmin = 0.26 x f_ctm/f_yk x b x d ≥ 0.0013bd

Fctm = 0.3 x fck^(2/3) = 0.3 x 25^(2/3) = 2.57MPa

Asmin = 0.26 x 2.57/500 x 2700 x 830

Asmin = 2989mm²

Since the minimum area of reinforcement (2989mm²) is greater than area of steel required (1353mm²), then provide minimum area of steel.

Provide 12Y20 @ 225c/c (3770.4mm²) at the bottom along the two directions.

Top Face Reinforcement

Although EC2 allow for the top of a pile cap to be left unreinforced. It is however a good practice to provide them with minimum reinforcement against secondary stresses.

Since minimum reinforcement is already calculated for the bottom face, hence:

Provide 12Y20 @ 180c/c (2989mm²) at the top face along the two directions.

Check the Pile Cap for Vertical Shear

Vertical shear shall be provided at Ф/5 within the pile diameter (Click overview of pile cap design for more explanation on concept)

Shear at critical section = Reaction of two piles within the critical section

Shear at critical section = 2063/2 = 1031.5

Calculate a_v to allow for shear enhancement. a_v is the distance from the face of the column to one-fifth of the diameter of the pile.

a_v = 1800/2 – 600/2 – 300/2 + 600/5 = 570

Shear enhancement =a_v/2d

Effective shear = V_EDx a_v/2d = $ \frac{\text{1031.5}\times \,\,570}{\text{2}\times \,\,830} $ =257.875KN

Calculate the shear resistance without shear reinforcement

v_{Rdc x}b_wd = (0.12K(100ρ_Lf_ck_)^1/3)b_wd ≥ v_min b_wd

K = (1 +√200/d) ≤ 2.0

K = (1 +√200/830) = 1.49

ρ_L = Asl/bwd ≤ 0.02

ρ_L = (1353) / (2700 × 830) = 0.0013

v_Rdc₌_{{0.12 x 1.49 (100 x 0.0013 x 25)^1/3} ≥ vmin}

v_Rdc= 0.597 N/mm²

Calculate the minimum resistance without reinforcement.

_{vmin = {0.035K³/2fck^1/2}}

_{vmin = 0.035 x 1.49^3/2 x 25^1/2} = 0.319 N/mm²

Since v_Rdc (0.597N/mm²) > v_min(0.319 N/mm²); hence v_Rdcpasses minimum reinforcement requirement

Convert the shear capacity to resistance and compare against design shear force.

V_Rdc = v_Rdc x bd

V_Rdc = 0.597 x 2700 x 830 x 10^ (-3) = 1336.9KN

Since V_ED < V_Rdc the section is adequate for one way shear

Check the Pile Cap for Punching Shear

The punching shall be checked at the column perimeter and at perimeter 2d from the column face

Punching shear at column perimeter

Shear force acting on column = 2063KN

Calculate the maximum resistance of the compressive struct

v_RDmax = 0.3 (1 -f_ck/250)f_cd

f_cd = fck/1.5 = 25/1.5 = 16.67 N/mm2

v_RDmax = 0.3 (1 -25_/250)16.67

v_RDmax=4.5 N/mm²

Shear resistance (V_RDmax) = v_RDmaxud

where u is the perimeter of the column Shear resistance

(v_RDmax) = 4.5 x 4(300) x x 10 ^ (-3)

V_{RDmax =}4482KN

V_RDmax > V_Ed, the section passes punching check

Check Punching shear at basic control perimeter

Since the piles at spaced at not more than 3 x pile diameter, then punching shear check at basic control perimeter is not necessary.

Circumferential Reinforcement

The circumferential reinforcement should be provided in accordance with surface reinforcement for deep beams by providing 0.2% of reinforcement according to the UK national annex. This will however result to large area of reinforcement. We shall instead provide the circumferential reinforcement following the guidance of IStructE EC2 design manual. Hence, we shall Y12@150mm c/c