Posted in Geotechnical Engineering

Going Around in Circles for Rigid and Flexible Foundations

In an earlier post Analytical Boussinesq Solutions for Strip, Square and Rectangular Loads, we discussed elastic solutions for these types of foundations. Most of the results shown were for perfectly elastic foundations. In this post we will concentrate on a) circular foundations and b) the difference between rigid and elastic foundations, and those which find themselves in between.

Circular Foundations

Engineers have been familiar with charts such as this, from NAVFAC DM 7.01:

Chart for influence coefficients for uniformly loaded circular foundations, from NAVFAC DM 7.01 (1986). The chart is redrawn from Foster and Ahlvin (1954)

The influence coefficients (see diagram above) for the vertical, horizontal and shear stresses respectively directly under the centre of the load (x = 0) are not difficult to compute, being

I_z = 1 - \frac{z^3}{(z^2+r^2)^\frac{3}{2}}
I_r = (1+\nu)\frac{z}{\sqrt{z^2+b^2}} - \frac{1}{2}(1 - \frac{z^3}{(z^2+r^2)^\frac{3}{2}})
I_{zr} = 0

For just about everywhere else (except for the edge) closed form solutions are hard to come by. Why is this? Because, for most every other point, the solution for the influence coefficients involves the use of elliptical integrals. An illustration of the values of these is shown below.

Examples of elliptical integrals (from Jahnke and Emde (1945))

Although mathematical packages such as Maple and Matlab are certainly capable of evaluating these, they are still not the common companions of engineers. Fortunately most of the questions about the stresses under circular foundations centre (sorry!) around the point under x = 0, so the above formulae are useful.

Foundations Rigid and Flexible

Up to this point, we’ve considered for the most part the response of the soil–both stress and deflection–to a purely flexible foundation. For these foundations at the soil-foundation interface the pressure exerted on the foundation and the pressure the foundation exerts on the soil is the same. If there is any rigidity in the foundation–and virtually any foundation has some–then both deflections and stresses in the foundation are redistributed.

It’s probably useful to note that, for deflections in general, the formula we use is

s = \frac{\omega p b (1-\nu^2)}{E}

where

  • s = settlement of the foundation at the point of interest
  • \omega = I = influence factor
  • p = uniform pressure on the foundation
  • b = B = smaller dimension of rectangle or dimension of square side
  • \nu = Poisson’s Ratio of the soil
  • E = Modulus of elasticity of the soil

The values for \omega are given below.

Values for the influence coefficients omega (from Tsytovich (1976))

Turning to the flexible circular foundation, the value for \omega for all of the radii can be computed using the formula (Timoshenko and Goodier (1951)):

\omega=\frac{2}{\pi}\,\int_{0}^{1/2\,\pi}\!\sqrt{1-{\frac{{x}^{2}\left(\sin(\psi)\right)^{2}}{{r}^{2}}}}{d\psi}

This still has a complete elliptical integral of the second kind, but it more manageable. It can be solved by applying the trapezoidal rule and using small integration increments over the interval. Values for \omega for ratios of various values of x (see diagram above) to the actual radius of the circle are shown below.

x/r\omega
01.000x/r\omega
0.011.0000.510.931
0.021.0000.520.929
0.031.0000.530.926
0.041.0000.540.923
0.050.9990.550.919
0.060.9990.560.916
0.070.9990.570.913
0.080.9980.580.910
0.090.9980.590.906
0.10.9970.60.903
0.110.9970.610.899
0.120.9960.620.896
0.130.9960.630.892
0.140.9950.640.888
0.150.9940.650.884
0.160.9940.660.880
0.170.9930.670.876
0.180.9920.680.872
0.190.9910.690.867
0.20.9900.70.863
0.210.9890.710.859
0.220.9880.720.854
0.230.9870.730.849
0.240.9850.740.844
0.250.9840.750.839
0.260.9830.760.834
0.270.9820.770.829
0.280.9800.780.824
0.290.9790.790.818
0.30.9770.80.813
0.310.9760.810.807
0.320.9740.820.801
0.330.9720.830.795
0.340.9700.840.788
0.350.9690.850.782
0.360.9670.860.775
0.370.9650.870.768
0.380.9630.880.761
0.390.9610.890.754
0.40.9590.90.746
0.410.9570.910.738
0.420.9540.920.730
0.430.9520.930.721
0.440.9500.940.712
0.450.9470.950.702
0.460.9450.960.692
0.470.9420.970.681
0.480.9400.980.669
0.490.9370.990.655
0.50.93410.637
Values of the influence factor omega vs. the ratio of the radius of a particular point to the entire radius of the circular foundation.

Although many elastic calculations assume the flexible foundation, as noted earlier in reality foundations have rigidity. For a perfectly rigid foundation, the deflection of the entire foundation under a concentric load is uniform. The effect of this on the stresses can be seen below.

Diagrams of contact pressures a) under an absolutely rigid foundation, b) under foundations of various flexibilities. (From Tsytovich (1976))

For the foundation in (a), if it is a circular foundation, the vertical stresses at the base can be computed by the formula

\sigma_{z}=\frac{p}{2\sqrt{1-\left(\frac{x}{r}\right)^{2}}}

At the corners of the foundation, the stresses are theoretically infinite. This means that the lower bound solution for such a foundation is zero stress. In reality it is reasonable to assume that a) no foundation is perfectly rigid and b) the soil will proceed into plastic deformation, which will redistribute the stresses.

It is interesting to note that, for the strip loads we discussed previously, the distribution for a rigid strip is similar to the circle, thus

\sigma_{z}=\frac{p}{2\sqrt{1-\left(\frac{2y}{b}\right)^{2}}}

where the notation is as it was in that discussion.

The figure (b) above shows different vertical stress distributions for different flexibility ratios, given by the variable \Gamma . This can be approximated by the formula (Tsytovich (1976))

\Gamma \cong 10\frac{E_s l^3}{E_f h_1^3}

where

  • E_s = modulus of elasticity of the soil
  • l = half-width of the foundation, shown above
  • E_f = modulus of elasticity of the foundation
  • h_1 = height of the foundation

To get a better approximation would require plate theory, where the mathematics are very involved; an example of this, using a pile toe where the circle is loaded around the edge, can be found here.

The rigidity of the foundation also influences the distribution of stresses under the foundation, as shown below.

Isobars in Soil under Foundations a)absolutely rigid foundation, b)flexible foundation (from Tsytovich (1976))

There’s certainly life beyond elastic theory. In reality the type of soil affects the distribution of contact pressure on the foundation, even with a uniformly loaded foundation, as shown below.

Contact pressure under a)rigid footings and b)elastic foundation on an elastic half-space (from NAVFAC DM 7.01 (1986))

The clay distribution most resembles that of an elastic response of a soil with a rigid footing, as discussed earlier. The purely flexible foundation will have the same uniform reaction as the load, as noted earlier.

Other Sources

  • Foster, C.R. and Ahlvin, P.G. (1954) “Stresses and Deflections Induced by Uniform Circular Load.” Highway Research Board Proceedings, Highway Research Board, Washington, DC.
  • Jahnke, E. and Emde, F. (1945) Tables of Functions with Formulae and Curves. New York: Dover Publishers.
  • Timoshenko, S., and Goodier, J.N. (1951) Theory of Elasticity. New York: McGraww-Hill Book Company, Inc.

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