Footing With Biaxial Moments

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, p :=

P

+

Mx

z+

Mz

x

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

Figure 1: Footing Geometry

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,

A

Ix

Iz

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

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.

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:

6. 7. 8.

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. Figure 2: Positions of Neutral Axis 5. For selected neutral axis, calculate geometric properties, pressure etc. about neutral axis for portion of footing that remains in contact with soil. Calculate the volume of pressure diagram envelope. Calculate the center of gravity of pressure diagram envelope. 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

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. 2. 3. 4. 5.

Read size of footing and applied forces. Establish the acceptable numerical error in results and limit of number of iterations. Calculate geometrical properties, eccentricities and pressures at each corner of footing. Identify the pressure case of footing from Fig. 2 to know initial position of neutral axis. 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 Figure 3: Solution Band 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.

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.

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.

Table 1 Verification Problems and Comparison of Results Problem No Item 1 2 3 4 5 6 Geometry and Load Data P 278.00 1300.0 1250.0 333.00 Mx 278.00 162.50 2813.0 150.00 Mz 250.00 1800.0 750.00 400.00 Lx 6.00 5.00 6.00 4.00 Lz 5.00 2.50 5.00 3.00 ex 0.899 1.385 0.600 1.201 ez 1.000 0.125 2.250 0.450 ex/Lx 0.150 0.277 0.100 0.300 ez/Lz 0.200 0.050 0.450 0.150 Bearing Pressure at Corners (Before Modification of Pressure) PA 28.72 308.00 179.18 102.75 PB 12.05 -37.60 129.18 2.75 PC -10.18 -100.00 -95.85 -47.25 PD 6.48 245.60 -45.85 52.75 Results obtained by Numerical Method Case 2 3 4 3 Step 0.0030 0.0020 0.0030 0.0020 P’A 32.41 360.24 749.89 146.10 P’B 11.14 0.00 395.56 0.00 P’C 0.00 0.00 0.00 0.00 4.88 265.47 0.00 50.57 P’D 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 Comparison of Results Precovered 277.97 1300.8 1249.8 333.27 exrecovered 0.8984 1.3831 0.5996 1.2000 ezrecovered 1.0008 0.1251 2.2505 0.4503 (%) Error in P 0.0088 0.0652 0.0126 0.0804 ex 0.0647 0.0640 0.0733 0.0838 ez 0.0822 0.0876 0.0219 0.0737 Run Time Data Iterations 1245 640 2081 673 Time 7 3 9 3 (Sec)

(Units kN and m) 2000.0 2000.0 1500.0 1500.0 4000.0 3000.0 5.00 5.00 2.50 2.50 2.000 1.500 0.750 0.750 0.400 0.300 0.300 0.300

832.00 64.00 -512.00 256.00

736.00 160.00 -416.00 160.00

5 5 0.0020 0.0010 3000.0 1500.0 0.00 0.00 0.00 0.00 0.00 0.00 16.00

32.00

2000.0 2000.0 2.0000 1.5000 0.7500 0.7500 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 946

1067

3

3

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.

Example Problem No. 1 ( Case 2 ) Base Pressure Distribution Diagram - 2D

Footings with Two-Way Eccentricity

1.00 0.00 -1.00

-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-

Y-Axis: Width of Footing (Lz)

2.00

Points along Z Axis

2.50 2.00 1.50 1.00 0.50 0.00 -0.50 -1.00 -1.50 -2.00 2.50

3.00

-2.00

Points along X Axis -3.00 -2.00

-1.00

0.00

1.00

2.00

3.00

X-Axis: Length of Footing (Lx) Footing

Load Point

Original_NA

-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

Final NA

Figure 4.1 : Footing Geometry Example Problem No. 2 ( Case 3 )

Figure 4.2: Bearing Presure Diagram Base Pressure Distribution Diagram - 2D

Footings with Two-Way Eccentricity

1.25 1.00 0.75 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 1.25

2.00 1.00 0.00 -1.00

-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-

Y-Axis: Width of Footing (Lz)

3.00

-2.00

Points along X Axis

-3.00 -3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

X-Axis: Length of Footing (Lx) Footing

Load Point

Original_NA

Final NA

Figure 4.3 : Footing Geometry Example Problem No. 3 (Case 4 )

-20.000-20.000 140.000-180.000 300.000-340.000

Footings with Two-Way Eccentricit y

60.000-100.000 220.000-260.000

100.000-140.000 260.000-300.000

698.000798.000

Base Pressure Distribution Diagram - 3D

598.000698.000

2.00

798.0

498.000598.000

698.0

1.00

598.0

398.000498.000

498.0

0.00

398.0

298.000398.000

298.0

-1.00

198.0 98.0

-2.00

-2.00

-1.00

0.00

1.00

2.00

3.00

X-Axis: Length of Footing (Lx) Footing

2.00

-2.0

Load Point

Original_NA

Final NA

Figure 4.5 : Footing Geometry

-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

Y-Axis: Width of Footing (Lz)

20.000-60.000 180.000-220.000 340.000-380.000

Figure 4.4: Bearing Presure Diagram

3.00

-3.00 -3.00

Points along Z Axis

-3.00

0.50 -1.00 -2.50

198.000298.000 98.000198.000 -2.00098.000

Figure 4.6: Bearing Presure Diagram

Example Problem No. 5 (Case 5 ) Footings with Two-Way Eccentricit y

Y-Axis: Width of Footing (Lz)

2700.0003100.000

Base Pressure Distribution Diagram - 3D

3.00

2300.0002700.000

2.00

1900.0002300.000

3100.0 1.00

2700.0

1500.0001900.000

2300.0 1900.0

0.00

1100.0001500.000

1500.0 1100.0

-1.00

-2.00

700.0

700.000-

300.0

1.001100.000

-3.00 -3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

X-Axis: Length of Footing (Lx) Footing

Load Point

Original_NA

Final NA

Figure 4.7 : Footing Geometry

-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

-100.0

0.25 300.000-0.50 -1.25

700.000 -100.000300.000

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. 2. 3.

Foundation Design, Teng W. C., Prentice-Hall Inc., Englewood cliffs, New Jersey. Roark’s Formulas for Stress and Strain, 7th Edition, Young W. C. and Budynas R. G., McGraw Hill, Englewood cliffs, New Jersey. Foundation Engineering, 2nd Edition, Peck R. B., Hanson W. E., and Thornburn W. H., John Wiley and Sons, New York.