Footing design

1. ANALYSIS OF ECCENTRICALLY LOADED
RECTANGULAR FOOTING RESTING ON SOIL
– A NUMERICAL APPROACH
Jignesh V Chokshi, L&T Sargent & Lundy Limited, Vadodara, India
For analysis of isolated rectangular footings with large bi-axial eccentricity, an accurate and
efficient numerical approach satisfying all equilibrium conditions and suitable on computers is
presented in this paper. Microsoft Excel, a cogent tool globally used by structural engineers,
under its VBA programming environment is chosen for programming the numerical approach
and graphically displaying input and results. A generalized program dealing with any
conditions of eccentricities–zero eccentricity, one-way eccentricity or two-way eccentricity– is
developed for analysis of rectangular footings. Several examples, with different eccentricity
conditions are chosen to investigate accuracy of results and verify performance of the
numerical approach implemented in the program.
Introduction
The bearing pressure distribution for rigid isolated footing resting on soil subjected to axial
load 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 X–axis and Z–Axis respectively,
Ix, Iz = Moment of Inertia of footing about X–axis and Z–axis
respectively and,
x, z = Coordinates of point at which bearing pressure is to be
calculated.
From the above, eccentricity of loading for footing can be
derived as,
ex = Eccentricity along X-axis from center of gravity of footing = Mz / P
ez = 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 is
positive and the soil below footing is said to be in compression. However, if loading point lies
outside the Kern, magnitude of p at few locations in the footing is negative and that portion of
footing is said to be in tension. Since, there exists no mechanism between soil and footing to
Figure 1: Footing Geometry
p
P
A
M x
I x
z+
M z
I z
x+:=

2. resist the tensile stresses, some portion of footing will remain unstressed and the force
equilibrium will occur in the area of footing which remains in contact with soil. Under these
circumstances, bearing pressure at different points of footing will be modified and the line of
zero stresses will shift towards loading point. The portion outside line of zero pressure will be
completely unstressed and is called footing uplift area.
Footings with one-way eccentricity, either ex or ez outside kern, solution to the problem is
simple. However, for footings with two-way eccentricity ex and ez outside Kern, the solution
is not as simple as that for one-way eccentricity. In the available literature, Teng [1] shows
graphical 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 footing
design process on computer, tables or charts are cumbersome to implement and the information
is very brief. Hence, for computer implementation of footing design process, a numerical
approach is the best choice. A numerical approach is described in the paper, which solves this
problem 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 satisfactory
Factor of Safety against overturning. It is also necessary to keep sufficient area of footing
remaining in contact with soil and bearing pressure not exceeding the allowable bearing
pressure 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 footing
2. ex > Lx/6 and ez =0 ; ex outside kern – One-way eccentricity along X axis
3. ex = 0 and ez > Lz/6 ; ez outside kern – One-way eccentricity along Z axis
4. ex > 0 and ez > 0 ; ex, ez outside kern – Two-way eccentricity
It 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 positive
below footing base. The points of zero pressure on footing edges can be obtained by
substituting p = 0 and appropriate coordinate of footing edges in Eq. 1. The initial position of
neutral axis can be obtained by connecting a line between two points having zero stresses on
adjacent or opposite edges. However, for static equilibrium to occur, there will be significant
shift of initial neutral axis to its final position.

3. 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 = 0
Case 3: One end on AB and other end on CD, Pressure at B and C = 0
Case 4: One end on BC and other end on AD, Pressure at C and D = 0
Case 5: One end on AB and other end on AD, Pressure at B, C and D = 0
Case 6: Neutral Axis parallel to Z-axis, Pressure at B & C = 0, Pressure at A= Pressure at D
Case 7: Neutral Axis parallel to X-axis, Pressure at C & D = 0, Pressure at A= Pressure at B
Cases 2 through 5 are for footing with two-way eccentricity and cases 6 and 7 are for footing
with one-way eccentricity.
Numerical Approach
One can imagine that it is almost
impossible to obtain a unified
mathematical equation that solves all of
the above-defined cases. Hence, for
effective solution, the numerical
approach is necessary. For a given size
of footing and loading, the numerical
approach suggested by Peck, et. Al. [3]
is adopted and implemented to obtain
faster and accurate solution. The
numerical procedure essentially works
as follows:
1. Read size of footing and loading.
(P, Mx, Mz, Lx, Lz)
2. Calculate the geometrical
properties of footing.(A, Ix, Iz, ex, ez)
3. Calculate the pressure at corners A,
B, C & D.
4. Obtain initial position of neutral
axis for problems having tension on the
corners.
5. For selected neutral axis, calculate
geometric properties, pressure etc. about neutral axis for portion of footing that remains in
contact 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 detailed
procedure for implementation of the solution technique on digital computer. A systematic
Figure 2: Positions of Neutral Axis

4. numerical procedure is described here demonstrating each component of the programming
implemented for the solution of the problem. Microsoft Excel with its powerful VBA support
is selected for implementing the numerical procedure on computer. The strategy described
here is for case 2. For other cases, necessary changes are taken care in the generalized
program.
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. Select
appropriate step for iteration.
8. For each position of neutral axis EJ, calculate distance Z of loading point K, distance b1
for 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(∆EAJ) – I(∆EBG) – I(∆HDJ).
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 error
is more than acceptable limit, select another axis EJ at next step and repeat step 8 to 12.
13. Store the positions of neutral axis when
individual error for P, ex and ez is within
acceptable limit. This results in storage of three
positions of line EJ. This means that, at any of
these three positions, error for only one of P, ex or
ez will be within acceptable limits.
14. Terminate further iterations when these three
positions of line EJ are traced. Figure 3 shows the
location of line EJ where individual error for P, ex
and ez is found within acceptable limits. This
completes the first phase of iterations where line EJ
is moved parallel to initial neutral axis. It can be
inferred that the true solution, the unique position
of line EJ where numerical error for P, ex and ez is
simultaneously within acceptable limits, lies within the band bounded by three positions of
neutral axis. To extract the solution band limits, find out the lower-most and upper-most
position of EJ. As shown in Fig. 3, the solution band is bounded by a polygon connected
between points E1, E2, J1 and J2.
Figure 3: Solution Band

5. 15. It was observed that to achieve a tangible accuracy of 99 percentage or better, a slightly
larger band shall be used than originally extracted. The same is implemented in
programming by slightly shifting point E1 on left, E2 on right; J1 downward and J2
upward before initiating second phase of iterations. In the second phase, the objective is to
find 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 by
assuming the neutral axis as a line joining points E1 and J2 (see Fig. 3). Here, the point E1
is pivoted first and second point of neutral axis is altered from J2 to J1 with appropriate
step size. At every position of neutral axis during the iterations, all steps to find out
volume 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 and
limit on number of iterations is also verified at each step. If the solution is not converged
with the selected pivot, then pivot E1 is shifted at the next step towards E2. The entire
range from E1 to E2 will be pivoted during these iterations, with other end from J2 to J1
until the true solution is found. While iterating within J2 to J1, if the solution diverges, the
program abandons further iterations within J2 and J1 and new pivot point within E1 and
E2 is selected.
17. It shall be noted that, during iterations, the position of line EJ may get changed from one
case to another. For example, at the beginning of the iterations, the position of line EJ
may be representing case 2. However, during subsequent iterations, the position of line EJ
may represent case 3, 4 or 5. The program constantly monitors the case of current neutral
axis and calculates required properties accordingly.
18. For true solution to occur, it is imperative that for a particular position of neutral axis
within solution band, the numerical errors for P, ex and ez, all simultaneously, shall be
within allowable limits. The very first instance of such convergence is reported and
further iterations are abandoned. At this point, essential results such as pressures at A, B
and 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 other
positions of final neutral axis. It is found that the results of other positions do not vary
much for the desired accuracy, and hence, the accuracy of the first instance of solution is
acceptable 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) maximum
pressures at corners and 7) numerical difference in recovering P, ex and ez.
Extensive effort is put on the graphical presentation of input and results. Extraordinary
features 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 area
is divided into many small parts to produce refined bearing pressure diagram.

6. Verification Examples
Many practical examples were
selected to validate results
produced by the program and
monitor accuracy of the numerical
approach presented here. The
results were compared with input
data and not with solution
obtained from any other reference.
Table 1 shows input data and true
solution for selected problems.
Note that in all problems a
tangible accuracy of 99.9% is
achieved. The table also
demonstrates number of iterations
performed to solve the problem
and run time taken on PC with P4
-1.5GHz processor and 512MB
RAM. Graphical representation
of footing geometry and pressure
distribution diagrams for
examples 1, 2, 3 and 5 are shown
in Fig. 4.1 to 4.8.
Observations and Conclusions
The numerical approach suggested
in this paper produces impressive
results having a tangible accuracy
of 99.9 percentage or better for all
problems under investigation. The
time taken for finding the solution
is computationally economical for
incredible accuracy achieved.
Hence, the numerical approach
presented here can be effectively
implemented to automate the
footing analysis and design.
The use of Excel with its VBA
environment is phenomenally user
friendly and endorses the structural engineers’ acceptance of Excel as a cogent tool for
automating structural design work processes. Even for such a complex problem like footings
with two-way eccentricity, use of Excel is found highly efficient.
Table 1
Verification Problems and Comparison of Results
Problem No
Item
1 2 3 4 5 6
Geometry and Load Data (Units kN and m)
P 278.00 1300.0 1250.0 333.00 2000.0 2000.0
Mx 278.00 162.50 2813.0 150.00 1500.0 1500.0
Mz 250.00 1800.0 750.00 400.00 4000.0 3000.0
Lx 6.00 5.00 6.00 4.00 5.00 5.00
Lz 5.00 2.50 5.00 3.00 2.50 2.50
ex 0.899 1.385 0.600 1.201 2.000 1.500
ez 1.000 0.125 2.250 0.450 0.750 0.750
ex/Lx 0.150 0.277 0.100 0.300 0.400 0.300
ez/Lz 0.200 0.050 0.450 0.150 0.300 0.300
Bearing Pressure at Corners
(Before Modification of Pressure)
PA 28.72 308.00 179.18 102.75 832.00 736.00
PB 12.05 -37.60 129.18 2.75 64.00 160.00
PC -10.18 -100.00 -95.85 -47.25 -512.00 -416.00
PD 6.48 245.60 -45.85 52.75 256.00 160.00
Results obtained by Numerical Method
Case 2 3 4 3 5 5
Step 0.0030 0.0020 0.0030 0.0020 0.0020 0.0010
P’A 32.41 360.24 749.89 146.10 3000.0 1500.0
P’B 11.14 0.00 395.56 0.00 0.00 0.00
P’C 0.00 0.00 0.00 0.00 0.00 0.00
P’D 4.88 265.47 0.00 50.57 0.00 0.00
c 4.624 2.200 4.077 2.923 - -
d 2.976 1.200 4.513 0.888 - -
as % of (Lx x Lz)Contact
Area 77.07 66.01 14.09 52.36 16.00 32.00
Comparison of Results
Precovered 277.97 1300.8 1249.8 333.27 2000.0 2000.0
exrecovered 0.8984 1.3831 0.5996 1.2000 2.0000 1.5000
ezrecovered 1.0008 0.1251 2.2505 0.4503 0.7500 0.7500
(%) Error in
P 0.0088 0.0652 0.0126 0.0804 0.0000 0.0000
ex 0.0647 0.0640 0.0733 0.0838 0.0000 0.0000
ez 0.0822 0.0876 0.0219 0.0737 0.0000 0.0000
Run Time Data
Iterations 1245 640 2081 673 946 1067
Time
(Sec)
7 3 9 3 3 3

7. Example Problem No. 1 ( Case 2 )
Figure 4.1 : Footing Geometry Figure 4.2: Bearing Presure Diagram
Example Problem No. 2 ( Case 3 )
Figure 4.3 : Footing Geometry Figure 4.4: Bearing Presure Diagram
Example 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-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA
-3.00
-2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00-
2.50
-2.00
-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
Points along X Axis
PointsalongZAxis
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-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA
-2.50
-2.25
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-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
-0.25
0.00
0.25
0.50
0.75
1.00
1.25
Points along X Axis
PointsalongZAxis
Base Pressure Distribution Diagram - 2D
-20.000-20.000 20.000-60.000 60.000-100.000 100.000-140.000
140.000-180.000 180.000-220.000 220.000-260.000 260.000-300.000
300.000-340.000 340.000-380.000
-3.00
-2.50
-2.00
-1.50
-1.00
-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-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA

8. 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 the
support, encouragement and providing computational facilities for this programming work.
References
1. Foundation Design, Teng W. C., Prentice-Hall Inc., Englewood cliffs, New Jersey.
2. Roark’s 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.
-2.50
-2.25
-2.00
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-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-Axis:WidthofFooting(Lz)
Footing Load Point Original_NA Final NA