## Tuesday, January 8, 2013

### Foundation Design: Calculation of Applied Bearing Pressures.

Figure 10.15 shows the typical example where the loading is axial and there is no variation in ground level or surcharge. While this simple example will cover a large proportion of foundations constructed, the more general situation needs to be considered, ﬁrstly for calculating the total and net bearing pressures with variations in surcharge and/or ground levels and then for the effects of introducing asymmetrical loading.

While on good bearing soils modest surcharges and/or changes in ground levels will have little effect on the bearing capacity of the soils, in poor soil conditions or where the load changes are signiﬁcant they can have a dramatic effect. For a general case therefore the  net increase in load, N, is given by the formula

This is shown diagrammatically in Fig. 10.16.

It should be noted that where the soil level has been signiﬁcantly reduced by a major regrading of the site or
by construction of basements and the like, consideration should be given to the effects of heave particularly in clays or where there are artesian groundwater pressures.

It is almost always sufﬁciently accurate to take the weight of the new foundation and backﬁll as equal to the weight of soil displaced, i.e. FB ~ SB. Thus the equations for net increase in load and net increase in soil pressure simplify to:

When the ground levels and surcharge pressures are only nominally changed, FS ~ SS , and so the formulae reduce to
i.e. the net increase in soil load is equal to the load from the superstructure as mentioned previously.

In the examples above, the foundations have been axially loaded such that the total bearing pressure is given by

While this is the most common situation, and it is clearly an efﬁcient design principle to create a foundation which uses the maximum available bearing pressure over its entire base, there are many occasions when this is not practical and non-uniform foundation pressures have to be con- sidered. This non-uniformity is typically caused by:

(1) The applied superstructure load  P not being on the centroid of the foundation.
(2) The superstructure being ﬁxed to the foundations such that moments are transferred into the foundation (e.g ﬁxed bases of rigid sway frames).
(3) The application of horizontal loads.

(4) Variations in relative loads on combined bases (e.g. bases carrying two or more columns).

Thus in a general case the total pressure under a base with a small out-of-balance moment is

The moment MT is calculated by taking moments about the centroid at the underside of the foundation. In these cases it is usually beneﬁcial to consider the  total bearing pressure which allows for the balancing effect of the resultant force due to eccentric loads and/or applied moments.

As with simple beam design if

the pressure will be negative and tension, theoretically,  will be developed. However, for most foundations it is impossible to reliably develop tension, and the foundation pressure is either compressive or zero.

For a simple rectangular foundation

where  eT is the resulting eccentricity of the foundation.

Therefore if  eT is less than  L/6, the foundation will be fully in compression. This is known as the middle third rule which is illustrated previously.

Where eT is greater than L/6, a triangular stress distribution is generated under part of the base and zero under the remainder, and the maximum bearing pressure is calculated using the shortened base theory, which, for a rectangular base is

(see Fig. 10.17 (c)).

Again beneﬁts can be made by considering the total bearing pressure, thus utilizing the foundation loads which reduce the overturning and increase the effective length of the pressure diagram. Consideration should also be given to the positioning of the base so that the vertical loads P and  F are used to counteract the effects of any moment or  horizontal loads. In the example shown in Fig. 10.17, the load P should be to the left of the centreline such that the formula for calculating the total eccentricity becomes

The ideal situation is that eT should be zero or

While it is appropriate to compare the existing  load with  the new load on the ground when designing axially loaded foundations, in the more general case where the loads are eccentric, it is necessary to consider the allowable bearing pressure (net or total) with the applied foundation pressure (net or total) and it is recommended that pressures are compared rather than loads in all cases to maintain consistency
and avoid confusion.

Eccentrically loaded rectangular pad or strip foundations are generally designed on the middle third rule where this applies. For other shapes and conditions a trial and error basis is adopted. A base size is selected and the resulting bearing pressures compared with the allowable; the base size is adjusted up or down and the calculations repeated until the maximum bearing pressure is close to the allowable. Experience will soon enable the engineer to make a fairly accurate ﬁrst guess on the size of base required and reduce the number of iterations necessary.

Fig. 10.16 Deﬁnition of loads and pressures – general case.

Fig. 10.17 Foundation in bending about single axis.