Water tower on three columns

Assumptions

Each column and footing transmits one-third of the total load as a point load: $P = 10^6/3 = 3.333\times10^5$ kN (as printed, $10^6$ kN). The columns are at the corners of an equilateral triangle of side 10 m. Boussinesq's point load equation is used, with $z = 5$ m:

$$\Delta\sigma_z = \frac{3P}{2\pi z^2}\left[\frac{1}{1+(r/z)^2}\right]^{5/2}$$

        C1
       /  \
   10 m    10 m          centre O at 5.774 m from
     /  O   \            each footing
   C2 ------- C3
        10 m

Distance from the centre O to each column: $r = \dfrac{10}{\sqrt3} = 5.774$ m.

(a) At the centre of the tank (z = 5 m)

$r/z = 1.155$, $1 + (r/z)^2 = 2.333$

For one column: $\Delta\sigma = \dfrac{3\times333333}{2\pi\times25}\,(2.333)^{-2.5} = 6366.2\times0.1202 = 765.5$ kPa

For three columns:

$$\Delta\sigma_z = 3\times765.5 = 2296\ \text{kPa}$$

(b) Below each footing (z = 5 m)

• Own column ($r = 0$): $\dfrac{3P}{2\pi z^2}\times1 = 6366.2$ kPa
• Each of the other two columns ($r = 10$ m, $r/z = 2$, $(1+4)^{-2.5} = 0.01789$): $6366.2\times0.01789 = 113.9$ kPa

$$\Delta\sigma_z = 6366.2 + 2\times113.9 = 6594\ \text{kPa}$$

Answer: $\Delta\sigma_z \approx 2296$ kPa at the centre and $\approx 6594$ kPa below each footing, 5 m below the foundation level.

The values are very large because the load is taken as $10^6$ kN; for $10^6$ N the stresses are 1000 times smaller (2.30 kPa and 6.59 kPa).