Eccentrically Loaded Foundation - Hydrostatic force analogy

Introduction

Can I use just the sum of stresses ( P/A and M/z) to calculate the maximum gross soil pressure?

Yes, only if the minimum soil pressure on the opposite corner/end is compression. This article provides insight into the calculation of maximum soil pressure when the resultant compression force on the foundation acts outside the kern distance for rectangular, circular, octagon and ring foundations. The fluid mechanic's concept of "Hydrostatic force on plane surfaces" and "Center of Pressure" is used to compute the calculation of maximum pressure for shallow foundations.

Kern Distance

If the rectangular footing plan dimensions ( Length and Width) are designed such that the resultant compressive force applied falls within the middle one-third, the whole footing will remain in contact with the soil.

Kern distance is an extreme position from the section centre of gravity where the force applied will not generate tension.

Kern Distance = Extreme Fiber Elastic Section Modulus / Cross-section Area

Teng Chart

Foundation Design by Wayne C. Teng's book provided a graphical method to calculate maximum soil pressure for cases eccentric force falls outside the middle one-third.

I was curious, What procedure Teng may have used to arrive at a Graphical chart giving coefficient k to compute maximum soil pressure from two ratios ( el/L and eb/B, that is force eccentricity over dimension along that eccentricity)? One thing was sure, the reaction pressure distribution is linear ( f (x,y) = Ax+By+C; with three unknown A, B and C).

Maximum Pressure Coefficient for Rectangular Footing

To solve three unknowns with three equilibrium equations ( Summation Fy, Summation of Moment Mx and Summation of Moment My), I used MathCAD to solve a block, Giving the maximum pressure coefficient (k) for eb/B and eL/ L ratio, as presented in Table 1

Maximum soil Pressure, f = k x P / (LxB)

Loss of soil contact area can be computed using the coefficient k1 presented in Table 2. Foundation contact lost = (k1/100) x L x B

Hydrostatic force on plane surfaces Analogy - Circular Raft

The similarity between soil pressure and hydrostatic force on the plane surface inside the fluid, both are linear. In fluid mechanics, the centre of hydrostatic force is a point where the resultant hydrostatic force acts, it is deeper than the centre of gravity of the plane surface. Figure 3 provides the application of hydrostatic force to compute a maximum soil pressure

The following flow chart describes a process to generate the maximum pressure coefficient and % loss of contact from the ratio of eccentricity and diameter for a circular raft

A graph is plotted e/(2R) versus maximum soil pressure coefficient (k) and percentage soil loss of contact

Octagon Foundation

In Industrial plants, Octagon footing is used for the vertical column or similar heavy equipment. The maximum soil pressure is calculated for bending about a diagonal axis ( Axis B-B) ( Refer to Figure 4) and reinforced concrete design is performed for soil pressure from bending about a flat axis ( Axis A-A)

Ring Foundation

Ring foundation is used for circular shaft supported elevated reservoir or Chimney Structure. Figure 5 provides parameters ( Center line radius R and ring width b ) for the circular ring foundation used for coefficient k.

Coefficient k is presented in Graph 2 for the range of b/(2R) ( 0.05 to 0.3) ring foundations

Graph 3 presents the percentage area contact loss for the ratio e/(2R) for the range of ratio b/(2R)

Conclusion

The coefficient provided in Table form for the eccentrically loaded foundation can be used in Excel or a similar calculation tool/program to interpolate results for intermediate values. The Tables and Graphs provided in this article can be used for manual design. Fluid mechanics concepts ( Hydrostatic force on a plane surface) applied for soil engineering problems.

References

1. Foundation Desing, Wayne C. Teng
2. Fluid Mechanics, Streeter, Victor