Why E-footing shows that “one small shift of the pile coordinate will make the system unstable” in one line pile group.E-Footing uses a standard equation:

( )( )∑ ∑ ∑

∑ ∑

∑ ∑ ∑∑∑

−

−=

−−

=

±±=

222

2

222

2

nnnn

nnxxnyy

nnnn

nnyynxx

nnn

yxyx

yxeyeNB

AndyxyxyxexeN

A

Where

BxAynNP

)()(

For the calculation of pile stress. In a one-line pile group system, the dominator of A and B will tend to zero because either nx or ny is zero for all n .To prove this let’s us create a simple pile group as follow

200 200

500 450

225 x 225 stump with 300kN axial force

125 x 125 pile with 12 tones capacity

Failure pile with 0 contributions

kNnN

nosn

kNN

10033003

300

=

=

=∴

=pilefailureis1pile Since

x)the tosymetric is group pile (

..

x

centre stump about moments take group pile the of centroid the determine

Themm

ny

y

mmm

nx

To

030

083303383

3450500200

=

=

=

−=−=

+−−=

=

∑

∑

The same centroid calculation is shown also in E-Footing software.

mmeme

xx

yy

008330

== .

0.116m

eyy

= 0.083m

0.416m 0.533m

0

0416001160053300

0000

4700

533041601160

2

2222

2

2222

=

=−+−++=

=++=

=

++=

∑

∑

∑

∑

)(

).(*).(*).(*

.

...

nn

nn

n

n

yx

yx

y

m

x

( )

ndefined

B

ndefined

A

PofstressPile

u000

)0(0*47.0)0*00*0833.0(300

u000

00*47.00*083.047.0*0300

3

2

=

−=

−−=

=

−=

−−=

For one-line pile group, the A and B are undefined and the above equation breaks down. However there is a catch: read on.

Modified formula of E-footing to cater case like “one line pile group” For piles that are exactly aligned along a line where the equivalent load point is falling on, the above equation is still applicable with some modification. The idea is that we get the limit of the term A and B as both their numerator and dominator approaches 0 using L' Hospital's Rule (Calculus: An Intuitive and Physical Approach (Second Edition), Morris Kline ). The derivation is shown below:

( )( )

axes xx aboutgroup pile the of area ofmoment secondIWhere

)(

)(

,

)()(

know, we As

=

•=

=

−

−=

=

−

−=

−−

=

=−+−+=

±±=

∑

∑ ∑∑∑

∑ ∑ ∑∑ ∑

∑ ∑ ∑∑∑

∑

xx

xx

xx

n

xx

nn

nxx

n

nnnn

nnxxnyy

nnnn

nnyynxx

n

nnn

IeNy

Neyx

XeNA

xeqFrom

eqyxyx

yxeyeNB

And

eqyxyxyxexeN

A

QQqqxWhere

BxAynNP

2

22

2

222

2

222

2

0

0

01

2

1

0

qq

QQ

axesyy about group pile the of area of moment secondI Where

)(

)(

,

yy =

•=

=

−

−=

=

∑

∑ ∑∑∑

yy

yy

n

yy

nn

nyy

n

IeNx

Neyx

yeNB

xeqFrom

2

22

2

0

0

02

nyy

yyn

xx

xxn X

INe

yINe

nNP ±±=

belowasmodifyisgrouppilelineoneforformulafinal theconclusionAs