ANALYSIS OF ECCENTRICALLY LOADEDRECTANGULAR FOOTING RESTING ON SOIL

A NUMERICAL APPROACHJignesh V Chokshi, L&T Sargent & Lundy Limited, Vadodara, India

For analysis of isolated rectangular footings with large bi-axial eccentricity, an accurate andefficient numerical approach satisfying all equilibrium conditions and suitable on computers ispresented in this paper. Microsoft Excel, a cogent tool globally used by structural engineers,under its VBA programming environment is chosen for programming the numerical approachand graphically displaying input and results. A generalized program dealing with anyconditions of eccentricitieszero eccentricity, one-way eccentricity or two-way eccentricity isdeveloped for analysis of rectangular footings. Several examples, with different eccentricityconditions are chosen to investigate accuracy of results and verify performance of thenumerical approach implemented in the program.

Introduction

The bearing pressure distribution for rigid isolated footing resting on soil subjected to axialload and bending moments can be obtained by,

.........................................................(1)

In the equation 1,p = Bearing pressure under footing base at point (x, z),P = Axial Load; A = Area of Footing,Mx, Mz = Moment about Xaxis and ZAxis respectively,Ix, Iz = Moment of Inertia of footing about Xaxis and Zaxisrespectively and,x, z = Coordinates of point at which bearing pressure is to becalculated. From the above, eccentricity of loading for footing can bederived as,

ex = Eccentricity along X-axis from center of gravity of footing = Mz / Pez = Eccentricity along Z-axis from center of gravity of footing = Mx / P

For isolated rectangular footings, called footings now onwards, when the loading point k(ex,ez) lies in middle third of the footing, called Kern (shaded area in Fig. 1), magnitude of p ispositive and the soil below footing is said to be in compression. However, if loading point liesoutside the Kern, magnitude of p at few locations in the footing is negative and that portion offooting is said to be in tension. Since, there exists no mechanism between soil and footing to

Figure 1: Footing Geometry

pP

A

M x

I xz+

M z

I zx+:=

resist the tensile stresses, some portion of footing will remain unstressed and the forceequilibrium will occur in the area of footing which remains in contact with soil. Under thesecircumstances, bearing pressure at different points of footing will be modified and the line ofzero stresses will shift towards loading point. The portion outside line of zero pressure will becompletely unstressed and is called footing uplift area.

Footings with one-way eccentricity, either ex or ez outside kern, solution to the problem issimple. However, for footings with two-way eccentricity ex and ez outside Kern, the solutionis not as simple as that for one-way eccentricity. In the available literature, Teng [1] showsgraphical method, charts and related equations; Roark [2] provides tables and Peck [3]mentions an iterative method for footing with two-way eccentricity. To automate the footingdesign process on computer, tables or charts are cumbersome to implement and the informationis very brief. Hence, for computer implementation of footing design process, a numericalapproach is the best choice. A numerical approach is described in the paper, which solves thisproblem with tangible accuracy. In this approach, it is assumed that pressure varies linearly,the footing is rigid and the effect of soil displacement has no effect on the pressure distribution.

Equilibrium Conditions

In analysis of eccentrically loaded footings, following equilibrium conditions must comply,1. Volume of bearing pressure envelope shall be equal to the applied load P,2. CG of bearing pressure envelope shall coincide with location of applied load P.

For footings having large eccentricities, large area of footing will remain unstressed and hence,the stability of footing demands special attention. Thus, it is imperative to ensure satisfactoryFactor of Safety against overturning. It is also necessary to keep sufficient area of footingremaining in contact with soil and bearing pressure not exceeding the allowable bearingpressure of the soil.

Eccentricity Conditions

For a footing, possible eccentricity conditions can be enumerated as follows:1. ex = 0 and ez = 0 ; or ex, ez within kern area Compression on entire base of footing2. ex > Lx/6 and ez =0 ; ex outside kern One-way eccentricity along X axis3. ex = 0 and ez > Lz/6 ; ez outside kern One-way eccentricity along Z axis4. ex > 0 and ez > 0 ; ex, ez outside kern Two-way eccentricityIt shall be noted that, conditions 2, 3 and 4 produces tension on some portion of the footing.

Position of Neutral Axis

For footings with loading point outside Kern, the pressure will vary from negative to positivebelow footing base. The points of zero pressure on footing edges can be obtained bysubstituting p = 0 and appropriate coordinate of footing edges in Eq. 1. The initial position ofneutral axis can be obtained by connecting a line between two points having zero stresses onadjacent or opposite edges. However, for static equilibrium to occur, there will be significantshift of initial neutral axis to its final position.

As shown in Figures 1 and 2, following positions of neutral axis can be envisaged.Case 1: No neutral axis Compression case (Fig. 1)Case 2: One end on BC and other end on CD, Pressure at C = 0Case 3: One end on AB and other end on CD, Pressure at B and C = 0Case 4: One end on BC and other end on AD, Pressure at C and D = 0Case 5: One end on AB and other end on AD, Pressure at B, C and D = 0Case 6: Neutral Axis parallel to Z-axis, Pressure at B & C = 0, Pressure at A= Pressure at DCase 7: Neutral Axis parallel to X-axis, Pressure at C & D = 0, Pressure at A= Pressure at BCases 2 through 5 are for footing with two-way eccentricity and cases 6 and 7 are for footingwith one-way eccentricity.

Numerical Approach

One can imagine that it is almostimpossible to obtain a unifiedmathematical equation that solves all ofthe above-defined cases. Hence, foreffective solution, the numericalapproach is necessary. For a given sizeof footing and loading, the numericalapproach suggested by Peck, et. Al. [3]is adopted and implemented to obtainfaster and accurate solution. Thenumerical procedure essentially worksas follows:

1. Read size of footing and loading.(P, Mx, Mz, Lx, Lz)2. Calculate the geometricalproperties of footing.(A, Ix, Iz, ex, ez)3. Calculate the pressure at corners A,B, C & D.4. Obtain initial position of neutralaxis for problems having tension on thecorners.5. For selected neutral axis, calculate

geometric properties, pressure etc. about neutral axis for portion of footing that remains incontact with soil.

6. Calculate the volume of pressure diagram envelope.7. Calculate the center of gravity of pressure diagram envelope.8. Compare values of P, ex, and ez obtained in step 6 and 7 with input parameters. If

difference is too large, alter the position of neutral axis and repeat step 5 to step 8.

Programming Strategy

The solution methods suggested in the literature are very brief and do not explain a detailedprocedure for implementation of the solution technique on digital computer. A systematic

Figure 2: Positions of Neutral Axis

numerical procedure is described here demonstrating each component of the programmingimplemented for the solution of the problem. Microsoft Excel with its powerful VBA supportis selected for implementing the numerical procedure on computer. The strategy describedhere is for case 2. For other cases, necessary changes are taken care in the generalizedprogram.

1. Read size of footing and applied forces.2. Establish the acceptable numerical error in results and limit of number of iterations.3. Calculate geometrical properties, eccentricities and pressures at each corner of footing.4. Identify the pressure case of footing from Fig. 2 to know initial position of neutral axis.5. For cases 1, 6 and 7, simply solve the problem using known method. For cases 2 to 5, find

out the position of points G and H on appropriate edges of footing where p=0.6. Extend point G on edge AB to locate point E and extend point H on edge AD to locate

point J. Now, the problem is restricted to triangle EAJ, triangle EBG and triangle HDJ.7. In this method, the iterations are performed in two phases. In the first phase, line EJ - the

neutral axis, will be moved, parallel to EJ, towards point K in subsequent iterations. Selectappropriate step for iteration.

8. For each position of neutral axis EJ, calculate distance Z of loading point K, distance b1for corner A, b2 for corner B and b4 for corner D normal to neutral axis EJ.

9. Calculate moment of inertia of polygon ABGHDA about its base GH using, Igh = I(DEAJ) I(DEBG) I(DHDJ).10. Calculate pressure at points A, B and D using pi = ( P x Z x bi ) / ( Igh ). In cases 2 to 7,

pressure at C = 0.11. Calculate volume and CG of pressure envelop of polygon ABGHD using properties of

triangle and tetrahedron.12. Compare volume of polygon with applied load P, and center of gravity of pressure envelop

with ex and ez. Calculate percentage error in the achieved solution. If the numerical erroris more than acceptable limit, select another axis EJ at next step and repeat step 8 to 12.

13. Store the positions of neutral axis whenindividual error for P, ex and ez is withinacceptable limit. This results in storage of threepositions of line EJ. This means that, at any ofthese three positions, error for only one of P, ex orez will be within acceptable limits.14. Terminate further iterations when these threepositions of line EJ are traced. Figure 3 shows thelocation of line EJ where individual error for P, exand ez is found within acceptable limits. Thiscompletes the first phase of iterations where line EJis moved parallel to initial neutral axis. It can beinferred that the true solution, the unique positionof line EJ where numerical error for P, ex and ez is

simultaneously within acceptable limits, lies within the band bounded by three positions ofneutral axis. To extract the solution band limits, find out the lower-most and upper-mostposition of EJ. As shown in Fig. 3, the solution band is bounded by a polygon connectedbetween points E1, E2, J1 and J2.

Figure 3: Solution Band

15. It was observed that to achieve a tangible accuracy of 99 percentage or better, a slightlylarger band shall be used than originally extracted. The same is implemented inprogramming by slightly shifting point E1 on left, E2 on right; J1 downward and J2upward before initiating second phase of iterations. In the second phase, the objective is tofind the position of EJ where error for P, ex and ez is within limits simultaneously.

16. The second phase of iterations within the newly formulated solution band is initiated byassuming the neutral axis as a line joining points E1 and J2 (see Fig. 3). Here, the point E1is pivoted first and second point of neutral axis is altered from J2 to J1 with appropriatestep size. At every position of neutral axis during the iterations, all steps to find outvolume of pressure diagram and CG of pressure envelope are repeated as explained earlier.Also, the numerical errors for P, ex and ez are calculated to monitor the convergence andlimit on number of iterations is also verified at each step. If the solution is not convergedwith the selected pivot, then pivot E1 is shifted at the next step towards E2. The entirerange from E1 to E2 will be pivoted during these iterations, with other end from J2 to J1until the true solution is found. While iterating within J2 to J1, if the solution diverges, theprogram abandons further iterations within J2 and J1 and new pivot point within E1 andE2 is selected.

17. It shall be noted that, during iterations, the position of line EJ may get changed from onecase to another. For example, at the beginning of the iterations, the position of line EJmay be representing case 2. However, during subsequent iterations, the position of line EJmay represent case 3, 4 or 5. The program constantly monitors the case of current neutralaxis and calculates required properties accordingly.

18. For true solution to occur, it is imperative that for a particular position of neutral axiswithin solution band, the numerical errors for P, ex and ez, all simultaneously, shall bewithin allowable limits. The very first instance of such convergence is reported andfurther iterations are abandoned. At this point, essential results such as pressures at A, Band D, uplift area, position of final neutral axis are reported by the program.

19. Since, solution search is an iterative process; it is expected that there may be otherpositions of final neutral axis. It is found that the results of other positions do not varymuch for the desired accuracy, and hence, the accuracy of the first instance of solution isacceptable for all practical purposes.

Results and Graphics Interface

After successful execution of the program, the following output is generated:1) The input parameters, 2) position of initial neutral axis, 3) position of final neutral axis, 4)effective compression area, 5) load and loading point coordinates recovered, 6) maximumpressures at corners and 7) numerical difference in recovering P, ex and ez.

Extensive effort is put on the graphical presentation of input and results. Extraordinaryfeatures of Excel chart options are explored and the graphical features of the program includes:1. Footing Geometry: Size of footing, origin, loading point, Kern, initial neutral axis and

final neutral axis.2. Bearing Pressure Diagrams: 2D and 3D presentation of contours showing variation of

pressure, after equilibrium conditions are met, over the footing surface. The footing areais divided into many small parts to produce refined bearing pressure diagram.

Verification Examples

Many practical examples wereselected to validate resultsproduced by the program andmonitor accuracy of the numericalapproach presented here. Theresults were compared with inputdata and not with solutionobtained from any other reference.

Table 1 shows input data and truesolution for selected problems.Note that in all problems atangible accuracy of 99.9% isachieved. The table alsodemonstrates number of iterationsperformed to solve the problemand run time taken on PC with P4-1.5GHz processor and 512MBRAM. Graphical representationof footing geometry and pressuredistribution diagrams forexamples 1, 2, 3 and 5 are shownin Fig. 4.1 to 4.8.

Observations and Conclusions

The numerical approach suggestedin this paper produces impressiveresults having a tangible accuracyof 99.9 percentage or better for allproblems under investigation. Thetime taken for finding the solutionis computationally economical forincredible accuracy achieved.Hence, the numerical approachpresented here can be effectivelyimplemented to automate thefooting analysis and design.

The use of Excel with its VBAenvironment is phenomenally userfriendly and endorses the structural engineers acceptance of Excel as a cogent tool forautomating structural design work processes. Even for such a complex problem like footingswith two-way eccentricity, use of Excel is found highly efficient.

Table 1Verification Problems and Comparison of Results

Problem NoItem1 2 3 4 5 6

Geometry and Load Data (Units kN and m)P 278.00 1300.0 1250.0 333.00 2000.0 2000.0Mx 278.00 162.50 2813.0 150.00 1500.0 1500.0Mz 250.00 1800.0 750.00 400.00 4000.0 3000.0Lx 6.00 5.00 6.00 4.00 5.00 5.00Lz 5.00 2.50 5.00 3.00 2.50 2.50ex 0.899 1.385 0.600 1.201 2.000 1.500ez 1.000 0.125 2.250 0.450 0.750 0.750ex/Lx 0.150 0.277 0.100 0.300 0.400 0.300ez/Lz 0.200 0.050 0.450 0.150 0.300 0.300Bearing Pressure at Corners(Before Modification of Pressure)PA 28.72 308.00 179.18 102.75 832.00 736.00PB 12.05 -37.60 129.18 2.75 64.00 160.00PC -10.18 -100.00 -95.85 -47.25 -512.00 -416.00PD 6.48 245.60 -45.85 52.75 256.00 160.00Results obtained by Numerical MethodCase 2 3 4 3 5 5Step 0.0030 0.0020 0.0030 0.0020 0.0020 0.0010PA 32.41 360.24 749.89 146.10 3000.0 1500.0PB 11.14 0.00 395.56 0.00 0.00 0.00PC 0.00 0.00 0.00 0.00 0.00 0.00PD 4.88 265.47 0.00 50.57 0.00 0.00c 4.624 2.200 4.077 2.923 - -d 2.976 1.200 4.513 0.888 - -

as % of (Lx x Lz)ContactArea 77.07 66.01 14.09 52.36 16.00 32.00Comparison of ResultsPrecovered 277.97 1300.8 1249.8 333.27 2000.0 2000.0exrecovered 0.8984 1.3831 0.5996 1.2000 2.0000 1.5000ezrecovered 1.0008 0.1251 2.2505 0.4503 0.7500 0.7500(%) Error inP 0.0088 0.0652 0.0126 0.0804 0.0000 0.0000ex 0.0647 0.0640 0.0733 0.0838 0.0000 0.0000ez 0.0822 0.0876 0.0219 0.0737 0.0000 0.0000Run Time DataIterations 1245 640 2081 673 946 1067Time(Sec) 7 3 9 3 3 3

Example Problem No. 1 ( Case 2 )

Figure 4.1 : Footing Geometry Figure 4.2: Bearing Presure DiagramExample Problem No. 2 ( Case 3 )

Figure 4.3 : Footing Geometry Figure 4.4: Bearing Presure DiagramExample Problem No. 3 (Case 4 )

Figure 4.5 : Footing Geometry Figure 4.6: Bearing Presure Diagram

Footings with Two-Way Eccentricity

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00

X-Axis: Length of Footing (Lx)

Y-A

xis

: W

idth

of

Foo

tin

g (

Lz)

Footing Load Point Original_NA Final NA

-3.0

0-2

.50

-2.0

0-1

.50

-1.0

0-0

.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-2.50-2.00-1.50-1.00-0.500.000.501.001.502.002.50

Points along X Axis

Po

ints

alo

ng

Z A

xis

Base Pressure Distribution Diagram - 2D

-2.000-2.000 2.000-6.000 6.000-10.000 10.000-14.000 14.000-18.000

18.000-22.000 22.000-26.000 26.000-30.000 30.000-34.000 34.000-38.000

Footings with Two-Way Eccentricity

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00

X-Axis: Length of Footing (Lx)

Y-A

xis

: W

idth

of

Foo

tin

g (

Lz)

Footing Load Point Original_NA Final NA

-2.5

0-2

.25

-2.0

0-1

.75

-1.5

0-1

.25

-1.0

0-0

.75

-0.5

0-0

.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.25

Points along X Axis

Po

ints

alo

ng

Z A

xis

Base Pressure Distribution Diagram - 2D

-20.000-20.000 20.000-60.000 60.000-100.000 100.000-140.000140.000-180.000 180.000-220.000 220.000-260.000 260.000-300.000300.000-340.000 340.000-380.000

-3.0

0-2

.50

-2.0

0-1

.50

-1.0

0-0

.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

-2.50

-1.00

0.50

2.00-2.0

98.0

198.0

298.0

398.0

498.0

598.0

698.0

798.0

Base Pressure Distribution Diagram - 3D 698.000-798.000

598.000-698.000

498.000-598.000

398.000-498.000

298.000-398.000

198.000-298.000

98.000-198.000

-2.000-98.000

Footings with Two-Way Eccentricity

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00

X-Axis: Length of Footing (Lx)

Y-A

xis:

Wid

th o

f Fo

oti

ng

(Lz

)

Footing Load Point Original_NA Final NA

Example Problem No. 5 (Case 5 )

Figure 4.7 : Footing Geometry Figure 4.8: Bearing Presure Diagram

Acknowledgement

I thank my company M/s. L&T Sargent and Lundy Limited, Vadodara, Gujarat, India, for thesupport, encouragement and providing computational facilities for this programming work.

References

1. Foundation Design, Teng W. C., Prentice-Hall Inc., Englewood cliffs, New Jersey.2. Roarks Formulas for Stress and Strain, 7th Edition, Young W. C. and Budynas R. G.,

McGraw Hill, Englewood cliffs, New Jersey.3. Foundation Engineering, 2nd Edition, Peck R. B., Hanson W. E., and Thornburn W. H.,

John Wiley and Sons, New York.

-2.5

0-2

.25

-2.0

0-1

.75

-1.5

0-1

.25

-1.0

0-0

.75

-0.5

0-0

.25

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

-1.25

-0.50

0.25

1.00

-100.0

300.0

700.0

1100.0

1500.0

1900.0

2300.0

2700.0

3100.0

Base Pressure Distribution Diagram - 3D 2700.000-3100.000

2300.000-2700.000

1900.000-2300.000

1500.000-1900.000

1100.000-1500.000

700.000-1100.000

300.000-700.000

-100.000-300.000

Footings with Two-Way Eccentricity

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

-3.00 -2.00 -1.00 0.00 1.00 2.00 3.00

X-Axis: Length of Footing (Lx)

Y-A

xis

: W

idth

of

Foo

tin

g (

Lz)

Footing Load Point Original_NA Final NA