In the previous chapter it has been seen that under certain conditions the effective stresses in the soil may be reduced to zero, so that the soil looses its coherence, and a structure may fail. Even a small additional load, if it has to be supported by shear stresses, can lead to a calamity. Many examples of failures of this type can be given : the bursting of the bottom of excavation pits, and the uplift or floatation of basements, tunnels and pipelines. The conditions for uplift or floatation of structures are discussed in this chapter.

## Archimedes

The basic principle of the uplift force on a body submerged in a fluid is due to Archimedes. This principle can best be explained by first considering a small rectangular element, which is at rest in a fluid, see Figure 9.1. The material of the block is irrelevant, but it must be given to be at rest, perhaps by the action of some external forces. The pressure in the fluid is a function of depth only, and in a homogeneous fluid the pressure distribution is

p = Pgz

where P is the density of the fluid, g the acceleration of gravity, and z the depth below the fluid surface. The pressures on the left hand side and the right hand side are equal, but act in opposite direction, and therefore are in equilibrium. The pressure below the element is greater than the pressure above it. The resultant force is equal to the difference in pressure, multiplied by the area of the upper and lower surfaces. Because the pressure difference is just Pgh, where h is the height of the element, the upward force equals g times the volume of the element. That is just the volumetric weight of the water multiplied by the volume of the element. Because any body can be constructed from a number of such elementary blocks, the general applicability of Archimedes’ principle (a submerged body experiences an upward force equal to the weight of the displaced fluid) follows.

A different argument, that immediately applies to a body of arbitrary shape, is that in a state of equilibrium the precise composition of a body is irrelevant for the force acting upon it. This means that the force on a body of water must be the same as the force on a body of some other substance, that then perhaps must be kept in equilibrium by some additional force. Because the body when composed of water is in equilibrium it follows that the upward force must be equal to the weight of the water in the volume. On a body of some other substance the resultant force of the water pressures must be the same, i.e. an upward force equal to the weight of the water in the volume. This is the proof that is given in most textbooks on elementary physics. The upward force is often denoted as the buoyant force, and the effect is denoted as buoyancy.

The buoyancy force on a body in a fluid may have as a result that the body floats on the water, if the weight of the body is smaller than the upward force. Floatation will happen if the body on the average is lighter than water. More generally, floatation may occur if the buoyancy force is larger than the sum of all downward forces together. This may happen in the case of basements, tunnels, or pipelines. In principle floatation can easily be prevented: the body must be heavy enough, and may have to be ballasted. The problem of possible floatation of a foundation is that care must be taken that the effective stresses are always positive, taking into account a certain margin of safety. In practice this may be more difficult than imagined, because perhaps not all conditions have been foreseen. Some examples may illustrate the analysis.

## A concrete floor under water

As a first example the concrete floor of an excavation is considered. Such structures are often used as foundations of basements, or as the pavement of the access road of a tunnel. One of the functions of the concrete plate is to give additional weight to the soil, so that it will not float. Care must be taken that the water table is lowered only after the construction of the concrete plate. Therefore a convenient procedure is to build the concrete plate under water, before lowering the water table, see Figure 9.2. After excavation of the building pit, under water,

perhaps using dredging equipment, the concrete floor can be constructed, taking great care of the continuity of the floor and the vertical walls of the excavation. When the concrete structure has been finished, the water level can be lowered. In this stage the weight of the concrete is needed to prevent floatation. There are two possible methods to perform the stability analysis. The best method is to determine the effective stresses just below the concrete floor. If these are always positive, in every stage of the building process, a compressive stress is being transferred in all stages, and the structure is safe. Whenever tensile stresses are obtained, even in a situation that is only temporary, the design must be modified, because the structure is not always in equilibrium, and will float or break. It is assumed that in the case shown in Figure 9.2

the groundwater level is at a depth d = 1 m below the soil surface, and that the depth of the top of the concrete floor should be located at a depth h = 5 m below the soil surface. Furthermore the thickness of the concrete layer (which is to be determined) is denoted as D. The total stress just below the concrete floor nowis

The effective stress will be positive if the thickness of the concrete floor is larger than the critical value. In the example, with h − d = 4 m and the concrete being a factor 2.5 heavier than water, it follows that the thickness of the floor must be at least 2.67 m.

It may be noted that the required thickness of the concrete floor should be even larger if the groundwater level may also coincide with the soil surface, namely 3.33 m. One must be very certain that this condition cannot occur if the concrete plate is thinner than 3.33 m.

It may also be noted that in time of danger, perhaps when the groundwater pressures rise beyond the design level because of some emergency or because of some human error, the foundation can often be saved by submerging it with water. The damage to a basement or a tunnel due to a temporary layer of water on it is usually less than the damage if the concrete floor is cracked and has to be replaced.

The analysis can be done somewhat faster by directly requiring that the weight of the concrete must be sufficient to balance the upward force acting upon it from below. This leads to the same result. The analysis using the somewhat elaborate process of calculating the effective stresses may take some more time, but it can more easily be generalized, for instance in case of a groundwater flow, when the groundwater pressures are not hydrostatic.

The concrete floor in a structure as shown in Figure 9.2 may have to be rather thick, which requires a deep excavation and large amounts of concrete. In engineering practice more advanced solutions have been developed, such as a thin concrete floor, combined with tension piles. It should be noted that this requires a careful (and safe) determination of the tensile capacity of the piles. A heavy concrete floor may be expensive, its weight is always acting.

## Floatation of a pipe

The second example is concerned with a pipeline in the bottom of the sea (or a circular tunnel under a river), see Figure 9.3. The pipeline is supposed to consist of steel, with a concrete lining, having a diameter 2R and a total weight (above water) G, in kN/m. This weight consists of the weight of the steel and the concrete lining, per unit length of the pipe.

For the risk of floatation the most dangerous situation will be when the pipe is empty.

For the analysis of the stability of the pipeline it is convenient to express its weight as an average volumetric weight p, defined as the total weight of the pipeline divided by its volume. In the most critical case of an empty pipeline this is

where Yw is the volumetric weight of water. If the upward force F is smaller than the weight G there will be no risk of floatation. The pipeline then sinks in open water. This will be the case if Yp > Yw. For a pipeline on the bottom of the sea this is a very practical criterion. If one would have to rely on the weight of the soil above the pipeline for its stability, floatation might occur if the soil above the pipeline is taken away by erosion, which is not unlikely. The pipeline then might float to the sea surface, and that should be avoided. In case of a tunnel buried under a river there seems to be more certainty that the soil above the tunnel remains in place. Then the weight of the soil above the tunnel may prevent floatation even if the tunnel is lighter than water ( Yp < Yw). The weight W of the soil above the tunnel is