Analysis of isolated footing subjected to axial load and ...icjonline.com/views/2014.06_POV_Bijay_Sarkar.pdf · Analysis of isolated footing subjected to axial load and high biaxial

60 The Indian Concrete Journal June 2014

POINT OF VIEW POINT OF VIEW

Analysis of isolated footing subjected to axial load and high biaxial moments and numerical

approach for its solution

Bijay Sarkar

In this paper a rigid isolated foundation of square or rectangular shape is analyzed on the action of a vertical load and high biaxial moments at centre. The pressure intensity corresponding to any given set of above loads on footings resting on elastic soils has been found through a general method of analysis and solution is made through a comprehensive numerical procedure. The common assumption of linear contact pressure in footing-soil interface is adopted for the solutions. Special attention has been given where there are inactive parts of foundation, without contact with soil and necessary equations on case to case basis are deduced. Flow Chart for computer procedure is also provided at end. The solution for these cases is not yet given in any Indian Standards.

IntroductIon

When a rectangular/square isolated footing of size L x B is subjected to a set of forces comprising of compressive axial load P and bi-axial moments Mx & My at centre of footing, load P alone may equivalently be considered at eccentricities in X & Y directions from origin at the following location from centre of footing :

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

; &

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

When pressure at any location under the footing is compressive in nature, Centroid of the effective footing

area coincides with the centre of the footing. The Parameters P,

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

,

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

and dimensions of the foundation along with its sectional properties are known and we get pressure distribution under the foundation at any location (X, Y) with respect to centroidal axis of the footing by using following bending equation :

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

;

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

... (1)

whereL = Length of the footingB = Breadth of the footingX = Perpendicular distance from the Centroidal Y-axis of the point on the Cross-Section at which pressure is to be determined.Y = Perpendicular distance from the Centroidal X-axis of the point on the Cross-Section at which pressure is to be determined. P = Vertical Force Acting on the section at centre.

61The Indian Concrete Journal June 2014

POINT OF VIEW

A = Compressive area of the section = BLMy = Bending moment about the Centroidal Y-axis =

xeP×

MxBending moment about the Centroidal X-axis = yeP×Ix = Second moment of area about Centroidal X-axis =

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

Iy = Second moment of area about Centroidal Y-axis =

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

=xe = Eccentricity of Load along X-Axis

=ye Eccentricity of Load along Y-Axis

However, above equation remains valid till whole

cross-sectional area

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

of footing remains

under compression i.e.

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

, or,

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

which can be represented as a general

form of straight line equation

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

, assuming

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

i.e. intersecting the x-axis at

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

and

y-axis at

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

. Considering the opposite quadrant,

we can get the co-ordinates

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

on x-axis and

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

on y-axis. Connecting these co-ordinates, we

get a diamond shaped bounded zone around the centre of the footing which is called “KERN” or “Central Core” of the footing. So as the load is located within this zone, whole cross-sectional area is effective in transferring loads to soil as compression.

ProBlem defInItIon

When the eccentricities are such that the load location crosses the boundary of the “KERN”, equation (1) shows a negative pressure i.e. tensile pressure at some zone under the footing. As underneath soil can not resist such tensile forces, the footing area in that zone gets detached & uplifted from soil and thus the above equations for base pressure calculation do not remain anymore valid.

Due to uplift of the footing area from soil, Neutral Axis as calculated from equation (1), do not remain at the calculated location, however, it gets shifted to other location to adjust the phenomena of uplift redistributing the base pressure under the footing for obtaining an equilibrium.

First we divide the whole footing into four equal quadrants. Assumed, biaxial moments are such that vertical load P alone may be considered acting at

&

= ;

= .

, ,

×±×±=

××−××−=∴

−−=∴

…………….….…….. (1)

×

×

=

=

×

>=

−−

, or, <=

+

=+

, assuming

= ,

=

and yaxis at

.

−

on xaxis and

−

on yaxis.

in the upper rightmost quadrant wrt centre. Thereby, the lower left most corner of the footing will experience the least

pressure. When

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

, the lower left most

corner of the footing starts to uplift i.e. the neutral axis starts to enter into the footing area from lower left most corner intersecting the footing sides. Depending on the intersection of neutral axis with the sides of the footing, the bearing pressure scenario underneath foundation is divided into five different cases :

When NA lies outside the footing area. Effective shape remains rectangular and equation (1) is applicable. Issue is already discussed above (Case – I).

When NA intersects the adjacent two sides of lower left most corner of the footing. Contact area reduces to a pentagonal shape. Issue is discussed under Case – II.

When NA intersects the two opposite shorter sides of the footing. Contact area reduces to a non-rectangular shape. Issue is discussed under Case – III.

When NA intersects the two opposite longer sides of the footing. Contact area reduces to a non-rect-angular shape. Issue is discussed under Case – IV.

When NA intersects the adjacent two sides of up-per right most corner of the footing. Contact area reduces to a triangular shape. Issue is discussed under Case – V.

1.

2.

3.

4.

5.



AnAlySIS of the foundAtIon

For Case – II to Case – V, analysis of footing has been done and general equations on case to case basis are deduced here. In all the cases, aim is to find out the effective area, CG of the effective area, sectional properties of the effective area wrt centroidal axes system i.e. through CG of the effective area. As we are not using principle axis to calculate the sectional properties, we are to calculate the product moment of area of the effective footing also and use all such data in General Form of Bending Equation and assembling all the cases into a graph. Assumed that a set of forces lb MMP ,, are acting at the centre of a rectangular i s o l a t e d footing of size (L x B). P is acting in the vertically downward, bM is acting along B (from down to top of this paper) and lM is acting along L (from left to right of this paper) respectively. When a portion of footing area is lifted, CG of the effective compressed footing area shifts away from the centre and as CG is changed, the equivalent eccentricity of load also changes from bl ee , to say,

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

. Therefore, Revised Moments acting at revised CG location are

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

∴ General Form of Two Dimensional Bending Stress Equation will be :

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

...(2)

where,p = Pressure at co-ordinate ),( nm wrt Centroidal

AxisP = Vertical Load at centre of the footing

rA = Revised Effective area of the foundation

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

= Revised Moment at Revised CG location about Revised Centroidal Axis YY

>

+

,

.

= ; = .

−

++

−

++= ……………………..(2)

= Revised Moment at Revised CG location about Revised Centroidal Axis XX

yI = Second Moment of Inertia of the effective area

about Revised Centroidal Axis YYxI = Second Moment of Inertia of the effective area

about Revised Centroidal Axis XXxyI

= Product Moment of Inertia of the effective area m = X-Axis Co-ordinate of the location where pressure is to be found wrt Revised Centroidal Axis n = Y-Axis Co-ordinate of the location where pressure is to be found wrt Revised Centroidal Axis

By assuming,

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

= Constant for given section and

loads

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

= Constant for given section and

loads

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

= Constant for given section and loads

We can write, Am + Bn + C = p. This is the General Form Equation of pressure underneath foundation subjected to high Bi-Axial Eccentricity. This is a straight line equation. Further, it can be observed that at some combinations of m and n, pressure p may remain constant.

Now, we know that pressure at neutral axis is equal to 0. Therefore, for any value of Co-Ordinate (m, n) being on the neutral axis, we get, Am + Bn + C = 0. This is nothing but a straight line equation representing the Neutral Axis.

Substituting the values of

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

, for moments in equation (2), we get

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

...(3)

General equations for Sectional Properties, Eccentricities & co-ordinate of Maximum Pressure Location wrt Revised Centroidal Axis are found out and used in the above equation on case to case basis for finding out the location of Neutral Axis as well as Maximum Bearing Pressure :

cASe - II

When NA cuts AB and AD. Here uplift portion is APQ and effective portion is PQDCBP. This is considered that NA cuts AB at P & AP = yB and NA cuts AD at Q & AQ = xL


POINT OF VIEW

Therefore, uplift fraction in side AB is y & in AD is x and neglecting the ineffective triangular uplifted portion APQ, all the sectional properties of effective portion PQDCBP of the foundation area are calculated as follows :

1. Effective Area of the Section and Revised Centre of Gravity :

Considering an orthogonal axis system origin located at A, co-ordinate of CG at O of the effective foundation area PQDCBP is calculated as follows :

(i) Effective Area (Rectangular PBCS) :

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

Centre of Gravity from AB :

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

Centre of Gravity from AD :

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

(ii) Effective Area (Rectangular QRSD) :

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

where

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

Centre of Gravity from AB:

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

Centre of Gravity from AD:

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

(iii) Effective Area (Triangular PQR) :

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

where

−

+=

−

+=

=

,

−

++++=∴

−

++

−

++=∴

………………………..(3)

( ) =×−= where −=

== where

=

=×+

= where

+=

=×−= where −=

=×+

= where

+=

=×= where

=

=×= where

=

Centre of Gravity from AB:

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=

Putting the value of and simplifying,

( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Centre of Gravity from AD:

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

From (i), (ii) & (iii),

Total Effective Area,

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Therefore, Distance of revised Centre of Gravity from AB :

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Where,

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Similarly, distance of revised Centre of Gravity from AD :

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Where,

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

2. Second Moment of Inertia wrt revised Centroidal Axis XX :

Ix = MI of Area ABCD about XX-axis – MI of Area APQ about XX-axis

Figure 1. Effective area of the section and revised centre of gravity

B

X XO

SPR

A Q Y

Y

D

C

PQ is the location of NAAP = yB ; PB = vB = (1-y)B ; v = 1-yAQ = xL ; QD = uL = (1-x) L ; u = 1-xO is the origin of Centroidal Axes XX & YY



=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Putting the value of

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

and simplifying,

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Where,

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

3. Second Moment of Inertia wrt Centroidal Axis YY :

Iy = MI of Whole Area ABCD about YY-axis – MI of Triangular Area APQ about YY-axis

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

=×= where

=

=×= where

=

++= ×=

−=

Where

−=

++

=

×=×−−

=

−−

=

++

=

×=×−−

=

−−

=

−×−×−

−×+=

−−−

−+=


( )

−−

+−=

×=

−

−×+−=

−××−×−

−×+=

−−−

−+=

Putting the value of gxF and simplifying,


( )

−−

×+−=

×=

Where,

−

−×+−=

Ixy = PMI of ABCD wrt revised CG PMI of APQ wrt revised CG

−

−××+×−−

−

−=

Putting , in above and simplifying,

−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=

and along B and L respectively was calculated based on Centroid being at centre of the footing. As Centroid is relocated due to uplift of some portion of the footing area, eccentricities are required to be recalculated based on the new Centroid to find out the revised moments : (i) Revised Eccentricity along B :

−+=

×−+=

−+=

+=

Where, −=

;

−−

=

Where,


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

4. Product Moment of Inertia (PMI) of the effective section wrt revised Centroidal Axes :

If NA lies outside the section, Centroidal Axes are symmetrical at the centre of the footing. As in such condition, Centroidal axes are the principle axes, Product Moment of Inertia of the section becomes zero. But, when NA lies within the footing area, the Centroidal Axes may not be symmetrical or in other words, the Vertical and Horizontal axes at Centroid are no more the principle axes of the section. Principle axes may be in other orientation than the vertical and horizontal axes. Therefore, when we are considering Vertical and Horizontal axes at centroid of the effective section, we must consider the Product Moment of Inertia of the effective contact area which will take care of as if pressures are being calculated with respect to Principle Axes.

Product Moment of Inertia (PMI) of the Revised Effective Footing Area,

Ixy = PMI of ABCD wrt revised CG - PMI of APQ wrt revised CG


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

Putting xG , yG in above and simplifying,


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

Where,


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


POINT OF VIEW

and ;


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

5. Revised Eccentricities of Load wrt. revised Centroidal Axes :

Earlier eccentricities be and le along B and L respectively was calculated based o n Centroid being at centre of the footing. As Centroid is relocated due to uplift of some portion of the footing area, eccentricities are required to be re-calculated based on the new Centroid to find out the revised moments :

(i) Revised Eccentricity along B :


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

Where, ;


( )

−−

×+−=

×=

Where,

−

−×+−=


−

−××+×−−

−

−=


−

−−+

−

−=

×=

−

−−+

−

−=

and

−−

= ;

−−

=


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

(ii) Revised Eccentricity along L : (ii) Revised Eccentricity along L :

−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&

(i) For == for NA coordinate and as ≠

, substituting all the terms in eqn (3), we simplify to get,

( ) ( )

=

−

−−

++

−

−+−

++∴

(ii) Revised Eccentricity along L :

−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

Where,


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

;


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

6. Co-ordinate of points P & Q of NA intersecting the footing side AB & AD wrt revised

Centroidal Co-ordinate System :

(i) Co-ordinate of point P where NA is intersecting the footing side AB :


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

where


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

where


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

(ii) Co-ordinate of point Q where NA is intersecting the footing side AD :


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

where


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

where


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

7. Co-ordinate of maximum pressure location C :


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

RELATIONSHIP BETWEEN KNOWN ECCENTRICITIES


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

&


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

AND FOOTING LIFTED FRACTIONS x & y :

As points


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

and


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

lie on NA, Pressure p is 0 at these locations. Now in general form of two dimensional bending pressure equation (3), we put the co-ordinates of NA points


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

&


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

one by one and other sectional properties of the effective area of footing and equate the same to zero.

(i) For


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

For NA co-ordinate and as


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

substituting all the terms in eqn (3), we simplify to get,


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴


−+=

×−+=

−+=

+=

Where, −=

;

−−

=

×−=−= where

−−

=

( ) −×=−= where

−−

=

( ) −×=−= where

−−

=

×−=−= where

−−

=

( ) −=−=

( ) −=−=

and

&



( ) ( )

=

−

−−

++

−

−+−

++∴

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=

(ii) Similarly, for NA location at ==

( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

... (4)



Where

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

;

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

(ii) Similarly for NA location at

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

and substituting all the terms in eqn (3), we get

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

...(5)

Where

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

;

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

Solving (4) & (5),

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

... (6)

In equation (6), it may be seen that all right hand side terms are in x & y only.

cASe – III

When NA cuts AB and DC. Here uplift portion is APQDA and effective portion is PQCBP. This is considered that NA cuts AB at P where, AP = yB and NA cuts DC at Q where, DQ = xB

Therefore, considering effective fraction in side AB is v & in DC is u, and uplift part in AB is y & in CD is x, we have

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

, where x, u and y, v are fraction parts of B & B respectively. We will now involve only the effective fractions u & v to ease formation of the equations as follows which can be transferred in terms of x & y from above.


Considering an orthogonal axis system origin located at A, co-ordinate of CG at O of the effective foundation area PQCBP is calculated as follows :


POINT OF VIEW

(i) Effective Area (Rectangular PBCS) :

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

where,

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

(ii) Effective Area (Triangular PQS)

=+×+×∴

......…….……(4)

Where ( )

−

−+−= ;

( )

−

−−=

( ) ( )

+

−

−−+

−

−+−=


( ) ( )

=+

−

−+−

++

−

−−

+∴

=+×+×∴

…….…….…(5)

Where ( )

−

−−= ;

( )

−

−+−=

( ) ( )

+

−

−+−+

−

−−=

−−

= ;

−−

= ……………….(6)

( ) −= ; ( ) −= , == where, =

== where,

= ; =

−= where,

−=

=

−

= where,

−

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

From (i) & (ii),

Total Effective Area

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=


== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

3. Second Moment of Inertia wrt revised Centroidal Axis YY :

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=


== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

5. Revised Eccentricities of Load with respect to revised Centroidal Axes :


== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

== where,

= ;

=−−

= where,

−−

=

=+

=+=

where,

+=

=

+=

+=

where,

+

=

=

+=

+=

where,

+

=

( ) ( )

=

−+

−+−+=

( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=


−+=

×−+=

−+=

+=

Where, −=

;

+

=

(ii) Revised Eccentricity along L : (ii) Revised Eccentricity along L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&

== for NA coordinate and as ≠

,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

Where,


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

;


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)



6. Co-ordinate of points P & Q of NA intersecting the footing side AB & DC wrt revised Centroidal Co-ordinate System :

(i) Co-ordinate of point P where NA is intersecting the footing side AB :


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

(ii) Co-ordinate of point Q where NA is intersecting the footing side CD :


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)



−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

RELATIONSHIP BETWEEN DIMENSIONLESS ECCENTRICITY RATIOS


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

AND DIMENLESS FOOTING LIFTED FRACTIONS x & y :

As points


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

and


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)



−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

&


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

one by one and all other sectional properties of the effective area of footing and equate the same to zero.

(i) For


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

for NA co-ordinate and as


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)



−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

Wherefrom, we get


−+=

×−+=

−+=

+=

Where, −=

;

+

=

−=

( ) −−=

( ) −=

( ) −−= ( ) −=

( ) −=

and

&


,

( ) ( )

=

−

−−−

++

−

−−+−

++∴

=+×+×∴

......…….……(7)

…(7)

Where Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

;

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

(ii) Similarly, for NA location at

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =


Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Wherefrom, we get

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

...(8)

Where

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

;

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =


POINT OF VIEW

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

Solving (7) & (8),

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

... (9)

In equation (9), see that all right hand side terms are in x & y.

cASe – IV

When NA cuts BC and AD. Here uplift portion is ABPQA and effective portion is PQDCP. This is considered that NA cuts BC at P where, BP = yL and NA cuts AD at Q where, AQ = xL

Therefore, considering effective fraction in side BC is v & in AD is u, and uplift part in BC is y & in AD is x, we have

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

, where x, u and y, v are fraction parts of L & L respectively. We will now involve only the effective fractions u & v to ease formation of the equations as follows which can be easily transferred in terms of x & y from above.


Considering an orthogonal axis system origin located at A, co-ordinate of CG at O of the effective foundation area PQDCP is calculated as follows :

(i) Effective Area (Rectangular QSCD) :

Where ( )

−

−−+−= ;

( )

−

−−−=

( ) ( )

+

−

−−−+

−

−−+−=

==

( ) ( ) ( ) ( )

=+

−

−+−−

++

−

−−+−

+∴

..(10)

=+×+×∴

…….…….…(8)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

−−

= ;

−−

= ……………….(9)

( ) −= ; ( ) −= ,

( ) == where, =

CG of Area A1 :

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

where,

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

(ii) Effective Area (Triangular PQS) :

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

where,

CG of Area A2 :

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

From (i) & (ii) above,

Total Effective Area

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

CG of total Effective Area :

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=




=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=


=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

4. Product Moment of Inertia (PMI) of the effective section wrt Centroidal Axes :

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=



=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

=

−= where,

−=

;

== where,

=

=

−

= where,

−

=

=−−

= where,

−−

= ;

== where,

=

=+

=+= where

+=

=

+=

+= where,

+=

=

+=

+= where,

+=

( ) ( )

=

−+

−+−+=

( ) ( ) ( )

=

−+

−+−+=

( )( ) ( ) ( )( )

=

−−+

−−−−=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

6. Co-ordinate of points P & Q of NA intersecting the footing side BC & AD wrt revised Centroidal Co-ordinate System :

(i) Co-ordinate of point P where NA is intersecting the footing side BC :

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


As points

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

and

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

one by one and all other sectional properties of the effective area of footing and equate the same to zero.


POINT OF VIEW

(i) For

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

for NA co-ordinate and as

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

Wherefrom we get,

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)

...(10)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,


Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

and substituting all the terms in equ (3), we get

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Wherefrom we get,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

…(11)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Solving (10) & (11),

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

...(12)

In equation (12), see that all right hand side terms are in x & y.

cASe – V

When NA cuts BC and DC. Here uplift portion is ABPQDA and effective portion is PQC. This is considered that NA cuts BC at P where, BP = yL and NA cuts DC at Q where, DQ = xL

Therefore, considering effective fraction in side BC is v & in DC is u, and uplift part in BC is y & in DC is x, we have

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

where x, u and y, v are fraction parts of B & L respectively. We will now involve only the effective fractions u & v to ease formation of the equations as follows which can be easily transferred in terms of x & y from above.


Considering an orthogonal axis system origin located at A, co-ordinate of CG at O of the effective foundation area PQC is calculated as follows :



Total Effective Area :

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( )

=+

−

−−+−

++

−

−−−

+∴

=+×+×∴

…….…….…(11)

Where ( )

−

−−−= ;

( )

−

−−+−=

( ) ( )

+

−

−−+−+

−

−−−=

−−

= ;

−−

= ……………….(12)

( ) −= ; ( ) −= ,

==

where, =

=

−=

where,

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=



−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=

6. Co-ordinate of points P & Q of NA intersecting the footing side BC & DC wrt revised Centroidal Co-ordinate System :

(i) Co-ordinate of point P where NA is intersecting the footing side BC :

−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


−= ; =

−=

where,

−=

== where,

=

== where,

=

== where,

=

B :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −=

( ) −−=


POINT OF VIEW


+=

Where, −=

;

+

=

L :

−+=

×−+=

−+=

+=

Where, −=

;

+

=

( ) −−=

( ) −=

( ) −−=

−= ( ) −=

( ) −=

and

&

==

≠

,

( ) ( ) ( ) ( )

=+

−

−−+−

++

−

−+−−

+∴

=+×+×∴

......…….……(10)


( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=


As points

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=


( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

one by one and all other sectional properties of the effective area of footing and equate the same to zero. (i) For

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

for NA

co-ordinate and as

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

, substituting all the terms in

eqn (3), we simplify to get,

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

Wherefrom, we get,

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

…(13)

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=


( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=


( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

…(14)

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

( ) −=

( ) −=

and

&

== for NA

≠

,

( ) ( ) ( ) ( )

=

−

−−+−

++

−

−+−−

++∴

=+×+×∴

......…….……(13)

Where ( ) ( )

−

−+−−= ;

( ) ( )

−

−−+−=

( ) ( ) ( ) ( )

+

−

−−+−+

−

−+−−=

==

( ) ( ) ( ) ( )

=

−

−+−−

++

−

−−+−

++∴

=+×+×∴

…….…….…(14)

Where ( ) ( )

−

−−+−= ;

( ) ( )

−

−+−−=

( ) ( ) ( ) ( )

+

−

−+−−+

−

−−+−=

Solving (13) & (14),

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

...(15)

In equation (15), see that all right hand side terms are in x & y

ProBlem SolutIon

(A) Method for generating the whole graph for all cases :

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

are all functions of x and y which

in turn are the fractions of L or B.

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

are known



parameters. Equations involving known

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

on the right

hand side and terms involving x, y on the right hand

side are deduced for all the cases (Case – II to Case – V).

Equations involving known

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

on the left hand side

and terms involving x, y on the right hand side are also deduced for all the cases (Case – II to Case – V). These two equations with two unknowns x & y for one particular case are non-linear simultaneous equations and can be used

for generating the graph as a whole by assuming various

combination of x & y and finding the corresponding

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

. However, for a particular known

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

values, efficient trial and error method shall be adopted to decrease the number of iterations. From the equations,

we can see that though it is an iterative process to find

out the values of x and y for known values of

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

, however, in reverse, if we think that x and y are known

parameters and unknowns are

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

, the problem

gets much easier. For several such known values of x and

y, we can find out corresponding

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

by using

equation (6) for Case-II, (9) for Case-III, (12) for Case-

IV, (15) for Case-V. Knowing x, y,

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

, we can

calculate corresponding maximum pressure on the base corner C of the footing by putting all known parameters along with coordinates of C in pressure equation (3) and simplifying :

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

...(16)

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

−−

= ;

−−

= ……………….(15)

L or B. and

&

&

and

,

and

,

and

Knowing x, y, and

,

( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=∴ .(16)

×=∴

Where ( ) ( ) ( ) ( )

+

−

−+−

++

−

−+−

+=

Right hand side parameters shall be calculated on case to case basis depending on the NA location. After generating numerous sets of dimensionless parameters

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.

Pmax. ∆x & ∆y ∆n = a xprev & yprev

against different sets of x and y, and

considering of

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


in X-axis and

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


in Y-axis, we can

plot corresponding x values, y values and Kmax values.

After plotting many of such sets, we get the x-curves, y-

curves & Kmax-curves plotted against

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


. 8 (Eight)

example hand calculations for each case using the above

equations on case to case basis are placed in a tabular form (Annexure Table 1 for Case – II, Table 2 for Case – III, Table 3 for Case – IV, Table 4 for Case – V) at Annexure-A. Here, the curves for whole graph showing NA locations

and Pressure contours in respect of

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


are drawn

by preparing a computer program and is presented in

Figure. 5 above.

(B) To find out the maximum pressure intensity for a given set of

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


:

1. MANNUAL METHOD : Once a full graph is ready, we can use this graph for our future design purpose by reading off values of maximum pressure coefficient Kmax and NA uplift quantity

against given set of

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


for design purpose

and multiply this Kmax with

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


to get the maximum pressure under footing.

2. COMPUTER PROCEDURE : A computer program may be created for solution of the non-linear

equations for finding out the uplift fractions of the

footing against the known values of

,

and

in Xaxis and

Kmax Kmax

,

,

,

with

,

.


Once

the uplift fractions i.e. the neutral axis location is known, we can calculate the Kmax and maximum pressure under the footing. A Flow Chart for framing a program in any computer language is


POINT OF VIEW

shown later in Figure 6. Following notes may be read in conjunction with the flow chart :

Sets of Equations used in the Flow Chart are all taken from the above discussion in the paper and are provided at the end of the flow chart. Different variable names have been used in the Equation Sets with that of the paper for use in VB Macro.

1.

As the final equations are dimensionless (as discussed above), L or B of the footing are not necessary to use in a program and hence, equa-tions in the given Equation Sets are provided as independent of L or B except for calculation of the Maximum Pressure Pmax.

For each of the Equation Sets, different function may be created and called as necessary by passing

2.

3.



Read M , M , P, B, LCalculate E /B = M /(P x B)Calculate E /L = M / (P x L)Assume x = 1 ; Y=0.1

FLOW CHART

STARTx

x

y

y

y

yyb

b

11

1

1

1

Case 2 : Calculate Eff.Sectional Propertiesfrom Eqn Set (2)

Calculate E B B andE B L from Eqn. Set (A)

Initialise p=0, q=0, r=0, s=0, x & Y=a small valueConsider a less value of x & y for each new iteration

Y = Y

Yes

No

Equal

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Increase y = y + yp = p + 1

Decrease y = y - yq = q + 1

If p> 0 and q = 0or

If p = 0 and q >0

If x < = 1 and y < = 1

y = y

If x > 1 and y < = 1

If x <= 1 and y >1

If x > 1 and y > 1

Case = 2x = x - 0y = y - 0

Case = 3u = 2 - xv = 1 - y

Case = 4u = 1 - xv = 2 - y

Case = 5u = 2 - xv = 2 - y

Use Eqn. Set (2)To Calculate Eff.

Sectional Properties







if E B L < E / L

prev

yCalculate E B L and

E B B from Eqn. Set (A)bMark - B

ERROR

Mark -A

prev


POINT OF VIEW

Yes

Yes

Yes

Yes

Yes

Yes

Equal

Mark -A

Mark - B









1y

yCalculate E B L andE B B from Eqn. Set (A)

Calculate P from eqn. Set (B)Print Case No, x, y, K P

b

END

Case = 2x = x - 0y = y - 0

Case = 3u = 2 - xv = 1 - y

If x < = 1 and y < = 1

If x > 1 and y < = 1

If x <= 1 and y >1 Case = 4u = 1 - xv = 2 - y

Case = 5u = 2 - xv = 2 - y

If x > 1 and y > 1

If E B L - E /L < nand

If E B B - E /B < nx = x

No

No

No

No

No

maxmax, max

Increase x = x + xr = r + 1

Decrease x = x - xs = s + 1

If r > 0 and s = 0or

If r = 0 and s > 0

If E B B < E /B

x = xprev

b by

1

b by

1yprev

No



parameters for necessary calculations and getting the result.

The values of fractional part x & y have been var-ied from 0 to 2 (considering x= 0 to 1 for edge AD & x= 1 to 2 for edge DC and y= 0 to 1 for edge AB & y= 1 to 2 for edge BC).

A check is to be kept on x & y so that calculated eccentricity always falls on the first quadrant (+ve) i.e. upper rightmost quadrant of the footing.

Different values of ∆x & ∆y may be chosen by calling a function for the same. For each new itera-tion, ∆x & ∆y shall be less than the ∆x & ∆y values of the previous iteration to gradually converge into the solution.

Mark-A and Mark-B of one page of the Flow Chart shall be considered merged with the same Mark of other page to study the Flow Chart.

Abbreviations Used in Flow Chart : P = Axial force, Mx = Moment about x-axis, My = Moment about y-axis, Eb/B = (Actual ecc. / fdn. Dimn.) along y-axis, El/L = (Actual ecc. / fdn. Dimn.) along x-axis, ElByL and EbByB = Ecc./fdn dimn based on assumed x & y i.e. NA location to com-pare the same with the actual ones to reach at a solution for x & y. ∆n = a negligible predefined number to verify whether a solution is reached or not. p, q, r, s = variables keeping record whether y or x is increasing or decreasing. xprev & yprev are keeping the values of x & y which are being changed in the later part of the program.

Set Of Equations for Use in Flow Chart :: Equations given below may be directly used in computer program in excel sheet (VB macro) with the same variable names and suitable changes of operators are required for other languages ::

Case – II : Equation Set (2) : Effective Sectional PropertiesArea = 1 - x * y / 2 ; CGX = (3 - x ^ 2 * y) / (6 - 3 * x * y) ; CGY = (3 - x * y ^ 2) / (6 - 3 * x * y)Icgx = 1 / 12 * (1 - (x * y ^ 3) / 3 + x * y / 3 * (3 - 2 * y) ^ 2 / (x * y - 2))Icgy = 1 / 12 * (1 - (x ^ 3 * y) / 3 + x * y / 3 * (3 - 2 * x) ^ 2 / (x * y - 2))Icgxy = Abs((CGX - 0.5) * (CGY - 0.5) + ((x ^ 2 * y ^ 2 / 72) - (x * y / 2 * (CGX - x / 3) * (CGY - y / 3))))X1 = -CGX ; Y1 = (y - CGY) ; X2 = (x - CGX) ; Y2 = -CGY

4.

5.

6.

7.

8.

Case – III : Equation Set (3) : Effective Sectional PropertiesArea1 = v ; cgx1 = 1 / 2 ; cgy1 = 1 - v / 2 ; Area2 = (u - v) / 2 ; cgx2 = 2 / 3 ; cgy2 = (3 - 2 * v - u) / 3 ; Area = Area1 + Area2 ; CGX = (Area1 * cgx1 + Area2 * cgx2) / Area ; CGY = (Area1 * cgy1 + Area2 * cgy2) / AreaIcgx = v ^ 3 / 12 + Area1 * (CGY - cgy1) ^ 2 + (u - v) ^ 3 / 36 + Area2 * (CGY – cgy2) ^ 2Icgy = v / 12 + Area1 * (CGX - cgx1) ^ 2 + (u - v) / 36 + Area2 * (CGX – cgx2) ^ 2Icgxy = Abs(Area1 * (CGX - cgx1) * (CGY - cgy1) - (u - v) ̂ 2 / 72 + Area2 * (CGX - cgx2) * (CGY - cgy2))X1 = -CGXY1 = 1 - v - CGYX2 = 1 - CGXY2 = 1 - u - CGY

Case – IV : Equation Set (4) : Effective Sectional PropertiesArea1 = u ; cgx1 = 1 - u / 2 ; cgy1 = 1 / 2 ; Area2 = 1 / 2 * (v - u) ; cgx2 = (3 - 2 * u - v) / 3 ; cgy2 = 2 / 3Area = Area1 + Area2CGX = (Area1 * cgx1 + Area2 * cgx2) / Area ; CGY = (Area1 * cgy1 + Area2 * cgy2) / AreaIcgx = u / 12 + Area1 * (CGY - cgy1) ^ 2 + (v - u) / 36 + Area2 * (CGY - cgy2) ^ 2Icgy = u ^ 3 / 12 + Area1 * (CGX - cgx1) ^ 2 + (v - u) ^ 3 / 36 + Area2 * (CGX – cgx2) ^ 2Icgxy = Abs(Area1 * (CGX - cgx1) * (CGY - cgy1) - (v - u) ̂ 2 / 72 + Area2 * (CGX - cgx2) * (CGY - cgy2))X1 = 1 - v – CGX ; Y1 = 1 – CGY ; X2 = 1 - u – CGX ; Y2 = -CGY

Case – V : Equation Set (5) : Effective Sectional PropertiesArea = v * u / 2 ; CGX = 1 - v / 3 ; CGY = 1 - u / 3 Icgx = v * u ^ 3 / 36 ; Icgy = v ^ 3 * u / 36 ; Icgxy = Abs(v ^ 2 * u ^ 2 / 72)X1 = 1 - v – CGX ; Y1 = 1 – CGY ; X2 = 1 – CGX ; Y2 = 1 - u - CGY

Equation Set (A) : Stress Equation on NA for calculating EbByB and ElByL : Feccb = 0.5 – CGY ; Feccl = 0.5 – CGX ; a = 1 / AreaP1 = (Icgx * X1 + Icgxy * Y1) / (Icgx * Icgy - Icgxy ^ 2) ; Q1 = (Icgy * Y1 + Icgxy * X1) / (Icgx * Icgy - Icgxy ^ 2) R1 = (a + Feccl * P1 + Feccb * Q1)P2 = (Icgx * X2 + Icgxy * Y2) / (Icgx * Icgy - Icgxy ^ 2) ; Q2 = (Icgy * Y2 + Icgxy * X2) / (Icgx * Icgy - Icgxy ^ 2) R2 = (a + Feccl * P2 + Feccb * Q2)ElByL = (Q2 * R1 - Q1 * R2) / (P2 * Q1 - Q2 * P1)EbByB = (P2 * R1 - P1 * R2) / (P1 * Q2 - Q1 * P2)

Equation Set (B) : Max Pressure Equation at Corner of footing : X3 = 1 – CGX ; Y3 = 1 - CGYa3 = (Icgy * Y3 + Icgxy * X3) / (Icgx * Icgy - Icgxy ^ 2) ; b3 = (Icgx * X3 + Icgxy * Y3) / (Icgx * Icgy - Icgxy ^ 2)Max Pressure Coefficient Kmax = a + ((EbByB + Feccb) * a3 + (ElByL + Feccl) * b3)Max Pressure = Kmax * P / (B * L)

References “Foundation Design” by Wayne C. Teng published by Prentice Hall of India Private Limited, New Delhi-110001 Published in : 1979 pp 130-133

1.


POINT OF VIEWAnnexure Table 1. Calculation of pressure co-efficients CASE - II (Short Side & Long Side Intersected) - A Triangular Part is uplifted

Example No → 1 2 3 4 5 6 7 8Notations used

in paper ↓ Assumed x, y are +ve fractions ; x<1 in AD portion and y<1 in AB portion

x 0.000 0.000 1.000 1.000 0.900 0.800 0.700 0.300y 0.000 1.000 0.000 1.000 0.700 0.600 0.500 0.400a1 1.000 0.000 1.000 0.000 0.300 0.400 0.500 0.600c1 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500d1 0.500 1.000 0.500 1.000 0.850 0.800 0.750 0.700a2 0.000 1.000 0.000 0.000 0.070 0.120 0.150 0.280c2 0.500 0.500 1.000 1.000 0.950 0.900 0.850 0.650d2 0.000 0.500 0.000 0.500 0.350 0.300 0.250 0.200a3 0.000 0.000 0.000 0.500 0.315 0.240 0.175 0.060c3 0.000 0.000 0.667 0.667 0.600 0.533 0.467 0.200d3 0.000 0.667 0.000 0.667 0.467 0.400 0.333 0.267ar 1.000 1.000 1.000 0.500 0.685 0.760 0.825 0.940Fgx 0.500 0.500 0.500 0.667 0.592 0.574 0.557 0.526Fgy 0.500 0.500 0.500 0.667 0.623 0.595 0.571 0.523Fix 0.083 0.083 0.083 0.028 0.042 0.050 0.057 0.074Fiy 0.083 0.083 0.083 0.028 0.051 0.058 0.063 0.073Fixy 0.000 0.000 0.000 0.014 0.019 0.019 0.017 0.009Feccl 0.000 0.000 0.000 -0.167 -0.092 -0.074 -0.057 -0.026Feccb 0.000 0.000 0.000 -0.167 -0.123 -0.095 -0.071 -0.023x1 -0.500 -0.500 -0.500 -0.667 -0.592 -0.574 -0.557 -0.526y1 -0.500 0.500 -0.500 0.333 0.077 0.005 -0.071 -0.123x2 -0.500 -0.500 0.500 0.333 0.308 0.226 0.143 -0.226y2 -0.500 -0.500 -0.500 -0.667 -0.623 -0.595 -0.571 -0.523x3 0.500 0.500 0.500 0.333 0.408 0.426 0.443 0.474y3 0.500 0.500 0.500 0.333 0.377 0.405 0.429 0.477P1 -6.000 -6.000 -6.000 -24.000 -13.209 -11.326 -9.900 -7.543Q1 -6.000 6.000 -6.000 0.000 -4.132 -4.168 -4.195 -2.594R1 1.000 1.000 1.000 6.000 3.181 2.545 2.069 1.317P2 -6.000 -6.000 6.000 0.000 0.630 0.039 -0.468 -4.047Q2 -6.000 -6.000 -6.000 -24.000 -14.519 -11.854 -10.094 -7.552R2 1.000 1.000 1.000 6.000 3.182 2.436 1.952 1.344

el/L 0.083 0.167 0.000 0.250 0.170 0.149 0.130 0.139eb/B 0.083 0.000 0.167 0.250 0.227 0.206 0.187 0.103Kmax 2.000 2.000 2.000 6.000 4.108 3.560 3.143 2.496

Max Pressure p = Kmax * P/BL

Annexure Table 2 . calculation of pressure co-efficients CASE - III (both the short sides intersected by na)

Example No 1 2 3 4 5 6 7 8Assumed x, y are +ve fractions and x<1 for DC portion, y<1 in AB portion, x<y to keep the

eccentricity in first quadrantx 0.000 0.200 0.300 0.400 0.500 0.600 0.700 0.800y 1.000 0.300 0.400 0.500 0.600 0.700 0.800 0.900u 1.000 0.800 0.700 0.600 0.500 0.400 0.300 0.200v 0.000 0.700 0.600 0.500 0.400 0.300 0.200 0.100a1 0.000 0.700 0.600 0.500 0.400 0.300 0.200 0.100c1 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500d1 1.000 0.650 0.700 0.750 0.800 0.850 0.900 0.950a2 0.500 0.050 0.050 0.050 0.050 0.050 0.050 0.050c2 0.667 0.667 0.667 0.667 0.667 0.667 0.667 0.667d2 0.667 0.267 0.367 0.467 0.567 0.667 0.767 0.867ar 0.500 0.750 0.650 0.550 0.450 0.350 0.250 0.150Fgx 0.667 0.511 0.513 0.515 0.519 0.524 0.533 0.556Fgy 0.667 0.624 0.674 0.724 0.774 0.824 0.873 0.922Fix 0.028 0.035 0.023 0.014 0.008 0.004 0.001 0.0003426Fiy 0.028 0.062 0.054 0.046 0.037 0.029 0.021 0.012Fixy 0.014 0.003 0.003 0.002 0.002 0.001 0.001 0.001Feccl -0.167 -0.011 -0.013 -0.015 -0.019 -0.024 -0.033 -0.056Feccb -0.167 -0.124 -0.174 -0.224 -0.274 -0.324 -0.373 -0.422x1 -0.667 -0.511 -0.513 -0.515 -0.519 -0.524 -0.533 -0.556y1 0.333 -0.324 -0.274 -0.224 -0.174 -0.124 -0.073 -0.022x2 0.333 0.489 0.487 0.485 0.481 0.476 0.467 0.444y2 -0.667 -0.424 -0.374 -0.324 -0.274 -0.224 -0.173 -0.122x3 0.333 0.489 0.487 0.485 0.481 0.476 0.467 0.444y3 0.333 0.376 0.326 0.276 0.226 0.176 0.127 0.078P1 -24.000 -8.685 -10.138 -12.165 -15.185 -20.139 -29.638 -54.154Q1 0.000 -9.912 -13.032 -17.884 -26.016 -41.143 -73.846 -160.000R1 6.000 2.663 3.941 6.013 9.634 16.659 32.557 77.231P2 0.000 7.267 8.252 9.535 11.267 13.695 17.164 20.923Q2 -24.000 -11.327 -15.204 -21.461 -32.520 -54.857 -110.769 -320.000R2 6.000 2.662 4.084 6.486 10.927 20.294 44.782 140.615

el/L 0.250 0.022 0.026 0.030 0.037 0.047 0.066 0.10714eb/B 0.250 0.249 0.282 0.316 0.349 0.382 0.414 0.44643Kmax 6.000 2.840 3.307 3.956 4.918 6.486 9.474 17.143

Max pressure p = Kmax * P/BL

Annexure Table 3. calculation of pressure co-efficients CASE - IV (both the long sides intersected by na)Example No 1 2 3 4 5 6 7 8

Assumed x, y are +ve fractions ; x<1 in AD portion, y<1 in BC portion, x>y to keep the eccentricity in first quadrant.

x 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000y 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900u 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000v 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100a1 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000c1 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000d1 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500a2 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050c2 0.267 0.367 0.467 0.567 0.667 0.767 0.867 0.967d2 0.667 0.667 0.667 0.667 0.667 0.667 0.667 0.667ar 0.750 0.650 0.550 0.450 0.350 0.250 0.150 0.050Fgx 0.624 0.674 0.724 0.774 0.824 0.873 0.922 0.967Fgy 0.511 0.513 0.515 0.519 0.524 0.533 0.556 0.667Fix 0.062 0.054 0.046 0.037 0.029 0.021 0.012 0.003Fiy 0.035 0.023 0.014 0.008 0.004 0.001 0.000 0.000Fixy 0.003 0.003 0.002 0.002 0.001 0.001 0.001 0.000Feccl -0.124 -0.174 -0.224 -0.274 -0.324 -0.373 -0.422 -0.467Feccb -0.011 -0.013 -0.015 -0.019 -0.024 -0.033 -0.056 -0.167x1 -0.424 -0.374 -0.324 -0.274 -0.224 -0.173 -0.122 -0.067y1 0.489 0.487 0.485 0.481 0.476 0.467 0.444 0.333x2 -0.324 -0.274 -0.224 -0.174 -0.124 -0.073 -0.022 0.033y2 -0.511 -0.513 -0.515 -0.519 -0.524 -0.533 -0.556 -0.667x3 0.376 0.326 0.276 0.226 0.176 0.127 0.078 0.033y3 0.489 0.487 0.485 0.481 0.476 0.467 0.444 0.333P1 -11.327 -15.204 -21.461 -32.520 -54.857 -110.769 -320.000 -2400.000Q1 7.267 8.252 9.535 11.267 13.695 17.164 20.923 0.000R1 2.662 4.084 6.486 10.927 20.294 44.782 140.615 1140.000P2 -9.912 -13.032 -17.884 -26.016 -41.143 -73.846 -160.000 0.000Q2 -8.685 -10.138 -12.165 -15.185 -20.139 -29.638 -54.154 -240.000R2 2.663 3.941 6.013 9.634 16.659 32.557 77.231 60.000

el/L 0.249 0.282 0.316 0.349 0.382 0.414 0.446 0.475eb/B 0.022 0.026 0.030 0.037 0.047 0.066 0.107 0.250Kmax 2.840 3.307 3.956 4.918 6.486 9.474 17.143 60.000

Max pressure p = Kmax * P/BL

Annexure Table 4. calculation of pressure co-efficients CASE - V (Long Side & Short Side Intersected) - A Triangular Part remains effectiveExample No 1 2 3 4 5 6 7 8

Assumed x, y are +ve fractions ; x<1 in DC portion, y<1 in BC portion

x 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.800y 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900u 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.200v 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100a1 0.280 0.210 0.150 0.100 0.060 0.030 0.010 0.010c1 0.733 0.767 0.800 0.833 0.867 0.900 0.933 0.967d1 0.767 0.800 0.833 0.867 0.900 0.933 0.967 0.933a2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000c2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000d2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000ar 0.280 0.210 0.150 0.100 0.060 0.030 0.010 0.010Fgx 0.733 0.767 0.800 0.833 0.867 0.900 0.933 0.967Fgy 0.767 0.800 0.833 0.867 0.900 0.933 0.967 0.933Fix 0.008 0.004 0.002 0.001 0.000 0.000 0.000 0.000Fiy 0.010 0.006 0.003 0.001 0.001 0.000 0.000 0.000Fixy 0.004 0.002 0.001 0.001 0.000 0.000 0.000 0.000Feccl -0.233 -0.267 -0.300 -0.333 -0.367 -0.400 -0.433 -0.467Feccb -0.267 -0.300 -0.333 -0.367 -0.400 -0.433 -0.467 -0.433x1 -0.533 -0.467 -0.400 -0.333 -0.267 -0.200 -0.133 -0.067y1 0.233 0.200 0.167 0.133 0.100 0.067 0.033 0.067x2 0.267 0.233 0.200 0.167 0.133 0.100 0.067 0.033y2 -0.467 -0.400 -0.333 -0.267 -0.200 -0.133 -0.067 -0.133x3 0.267 0.233 0.200 0.167 0.133 0.100 0.067 0.033y3 0.233 0.200 0.167 0.133 0.100 0.067 0.033 0.067P1 -53.571 -81.633 -133.333 -240 -500 -1333.333 -6000 -12000Q1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000R1 16.071 26.531 46.667 90.000 200.000 566.667 2700 5700P2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Q2 -61.224 -95.238 -160.000 -300.000 -666.667 -2000 -12000 -6000R2 19.898 33.333 60.000 120.000 283.333 900 5700 2700

el/L 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475eb/B 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.450Kmax 10.714 14.286 20.000 30.000 50.000 100.000 300 300

Max Pressure p = Kmax * P/BL


POINT OF VIEW

Bijay Sarkar holds a degree in Civil Engineering from Jadavpur University, Kolkata. He is a Superintending Engineer (Civil) in Engineering & Planning Cell of Project Department at DVC Head Quarters, Kolkata. He has a long experience in civil construction and design for more than last 25 years in Damodar Valley Corporation (DVC), a PSU under Ministry of Power, Govt of India. He is experienced in structural design works for power house & boiler building structures, mill & bunker structures, coal conveying structures of power plants for 500MW capacity & above owned by DVC. His keen interest is on preparing software modules in respect of civil engineering aspects.

Documents

Analysis of isolated footing subjected to axial load and ...icjonline.com/views/2014.06_POV_Bijay_Sarkar.pdf · Analysis of isolated footing subjected to axial load and high biaxial