11

CHAPTER 4. BEARING CAPACITYIn fig. 4.1 is shown a strip footing, which is a shallow foundation supporting a load-bearing wall. When establishing the area A of contact between the foundation and the soil, two fundamental requirements must be satisfied:

to ensure safety against the risk of shear failure of the supporting soil (fig. 4.1 a),

to limit the settlement s of the foundation to values allowable for the structure and for its normal exploitation (fig. 4.1 b).

a.

b.

Fig. 4.1The problem of bearing capacity, this chapter is dealing with, refers to the first of the two above outlined requirements.

Bearing capacity represents the ability of a soil to carry a load.

The allowable bearing capacity is defined as the maximum pressure which may be applied to the soil such that the two fundamental requirements are satisfied.

The ultimate bearing capacity is defined as the least pressure which would cause shear failure of the supporting soil immediately below and adjacent to a foundation.

As shown in the chapter 7 (ar fi cap1), the problem of ultimate bearing capacity is a special case of limiting or plastic equilibrium in a soil mass.

In the following paragraphs, the particular problem of the ultimate bearing capacity of shallow foundations will be considered.

4.1 Failure modes

Present knowledge concerning the way in which failure of the soil supporting shallow foundations takes place is based on analysis of both causes of accidents in which various structures lost stability and interpretation of experimental data. The experiments were conducted, in general, at small scale in installations allowing to visualize the trajects followed by soil particles during the process of gradual loading until the failure condition was reached.

On that basis, three main modes of failure were recognized, depending, in essence, on the ground conditions.

a. general shear failure

Continuous failure surfaces develop between the edges of the footing and the ground surface (fig. 4.2 a). As the pressure is increased towards the value of the ultimate bearing capacity pf, the state of plastic equilibrium is reached initially in the soil around the edges of the footing then gradually spreads downwards and outwards. Ultimately, the state of plastic equilibrium is fully developed throughout the soil above the failure surfaces. Heave of the ground surface occur on both sides of the footing, although the final slip movement would occur only on one side, accompanied by tilting of the footing, as shown in fig. 4.1 a. The load-settlement diagram, which accompanies this mode of failure, shown in the diagram a in fig. 4.3, puts into evidence clearly the values of the ultimate bearing capacity pf for which deformations increase indefinitely. The transition from the initial, quasi-linear, part of the diagram and the point corresponding to pf is a short one.

Fig. 4.2

The general shear failure (sometimes named complete shear failure) is typical for soils of low compressibility (dense sands, stiff clays) and for rocks.

b. local shear failure

In this mode of failure, there is significant compression of the soil under the footing and only partial development of the state of plastic equilibrium. The failure surfaces do not reach the ground surface and tilting of the foundation is unlikely to occur. The load-settlement diagram (b in the fig. 11.3) shows that the ultimate bearing capacity is not clearly defined and is characterized by the occurrence of relatively large settlements. This mode of failure is associated with soils of medium to high compressibility, (non-cohesive soils of medium relative density, cohesive soils of medium consistency).

c. punching shear failure

This mode of failure occurs when there is compression of the soil under the footing, accompanied by shearing in the vertical direction around the edges of the footing. As the pressure is increased, the foundation penetrates into the soil like a piston. There is no heave of the ground surface away from the edges and no tilting of the footing. The load-settlement diagram (c in fig. 4.3) shows that large settlements are also characteristics to this mode of failure and the ultimate bearing capacity, like in the case b, is not well defined. Punching shear failure, is associated with soils of very high compressibility such as loose sands and soft clays.

Fig. 4.3

In cases of local shear and punching shear failures, the ultimate bearing capacity should be defined based on a deformation criterion. Available experimental data show that settlements of shallow foundations corresponding to a failure load are of the order of (3%...7%) B for clay soils and of (5%...15 %) B for sands where B is the width of the foundation. Hence, a settlement of 10% B could be adopted as a deformation criterion for any soil condition in order to define pf (fig. 4.4). It follows also that plate load tests on compressible soils should be conducted to settlements equal to at least 0.25 B, to be able to define the ultimate load from the load-settlement diagram.

Fig. 4.4Besides the nature of the soil, the mode of failure depends also on other factors such as:

the depth of the foundation; punching shear failure will occur in a soil of low compressibility, for instance dense sands, if the foundation is located at considerable depth (deep foundation);

the kind of loading; a dense sand subjected to cyclic loading will exhibit punching shear failure;

the rhythm of loading; a saturated, normally consolidated clay, exhibits a general shear failure under a sudden loading, when no volume change takes place, and a punching shear failure when the rhythm of applying the load is slow and after each load stage the time required for the consolidation of the soil is provided.

4.2 General hypothesis adopted for computing the ultimate bearing capacity

For the computation of the ultimate bearing capacity pf the following hypothesis are adopted:

a continuous failure surface characteristic for the general shear failure mode (fig. 4.5);

Fig. 4.5

the failure condition is fulfilled in each point of the failure surface;

the shear strength of the soil between the level of the foundation and the ground surface (part CD of the failure surface) is neglected;

the friction between the soil above the level of the foundation and the lateral face of the foundation (EB) is neglected;

the friction between the soil located above and below the foundation level (on the line BC) is neglected;

the friction between the base of the foundation (AB) and the soil to which it c.. in contact, is neglected.

With these hypothesis, the soil located above the foundation level is replaced by a surcharge q = D, where D is the foundation depth.

4.3 Ultimate bearing capacity in the case of a failure surface made by two planes

The two failure planes (fig. 4.6) have the inclinations in respect to the horizontal of and , corresponding to the development in the mass of soil under the footing of two Rankine zones on both sides of a imaginary, fictitious, perfectly smooth (frictionless) wall BD, namely the active zone on the left of the wall and the passive zone on the right of the wall.

Fig. 4.6

Computing pf is based on expressing the active earth thrust Pa behind a vertical wall BD limited by an horizontal ground surface, on which a surcharge pf is applied, and the passive resistance Pp in front of the same wall, limited by an horizontal ground surface on which a surcharge q = D is applied (fig. 4.7).

Fig. 4.7

(4.1 a)

(4.1 b)

To find pf, the condition Pa = Pp is written, considering that:

(4.2)

The expression (11.2) can be put into the form:

(4.3)

where , named bearing capacity factors, are depending on the angle of internal friction , and have the following expressions:

(4.4)

4.4 Ultimate bearing capacity in the case of a curved failure surface

The problem is solved in three phases, corresponding to the following conditions:

a. cohesionless, weightless soil (

b. frictionless, weightless soil ()

c. soil with weight ()

a. In the case of a soil without cohesion and weight, a suitable failure mechanism for a strip footing is shown in fig. 4.8. The footing, of width B and infinite length, carries a uniform pressure on the surface of a mass of homogeneous, isotropic soil. When the pressure becomes equal to the ultimate bearing capacity pf the footing will be pushed downwards into the soil mass, producing a state of plastic equilibrium, in the form of an active Rankine zone, below the footing, the angles ABC and BAC being (). The downward movement of the wedge ABC forces the adjoining soil sideways, producing outward lateral forces on both sides of the wedge. Passive Rankine zones ADE and BGF develop on both sides of the wedge ABC, the angles DEA and GFB being (). The transition between the downward movement of the wedge ABC and the lateral movement of the wedges ADE and BGF takes place through zones of radial shear ACD and BCG. In his solution, Prandtl admits that the surfaces CD and CG are logarithmic spirals, to which BC and ED, or AC and FG, are tangential. The equation of the spiral is where is the angle between the initial radius ro and the one corresponding to a point on the spiral; is the angle made by the radius with the normal in any point of the spiral. A state of plastic equilibrium exists above the surface EDCGF, the remainder of the soil mass being in a state of elastic equilibrium.

Fig. 4.8

To find pf, first the equilibrium of the wedges ABC and BDE, as equilibrium of forces on vertical direction, will be considered. Then, the equilibrium of the transition zone BCD, as equilibrium of moments toward the point B, will be written.

On the conjugated failure planes AC and CB are acting the reactions RI, making an angle with the normal (fig. 4.9 a).

The equation of projection of forces on the vertical direction:

(4.5)

On the conjugated failure planes BD and DE are acting the reactions RIII, making an angle with the normal (fig. 11.9 b).The equation of projection of forces on the vertical direction:

(4.6)

The equilibrium of the transition zone II (fig. 4.9 c) is expressed in terms of the moment around the point B.

Fig. 4.9

The arc of the spiral CD belongs to the failure surface, therefore the reaction RII makes an angle with the normal to the arc. Hence, the direction of RII coincides with the direction of the radius and RII produces no moment in respect to B. The moment equation becomes:

But r1 = ro

(4.7)

By writing:

equation (4.7) becomes:

pf = q Nq

(4.8)

From (11.8) follows that, in the case of a cohesionless and weightless material, there is a bearing capacity only if there is a surcharge q.

To consider the effect of the cohesion, a normal stress equal to c cot is added to the normal stresses p and q. The equation (11.8) becomes:

(4.9)

By writing

equation (11.9) becomes:

pf = q Nq + c Nc

Nq and Nc are bearing capacity factors depending on .

An additional term should be added to equation (4.10) to take into account the self-weight of the soil. Experimental observations showed that a wedge of soil remaining in elastic state, with faces making an angle with the horizontal, is developed below the foundation and moves downwards together the foundation, tending to produce the lateral movement of the soil along the failure surfaces CDE and CFG (fig. 4.10). The passive resistance of the soil mass above the failure surfaces is mobilized. The problem consists on computing the passive resistance force Pp of a mass of soil (), limited by a horizontal ground surface, behind a wall BC with inclination and height H = .

Fig. 4.10

The failure surface CDE is made of the line DE, corresponding to the passive Rankine zone BDE, and by the arc of logarithmic spiral CD.

The passive resistance force Pp can be expressed:

(4.11)

The equilibrium of the elastic wedge ABC:

(4.12)

The ultimate bearing capacity is:

(4.13)

The following notation was used:

Terzaghi assumed that and obtained the value of the passive resistance force in the hypothesis of a curved failure surface. Adding the additional term bringing the effect of the self-weight of the soil, the expression of the ultimate bearing capacity pf becomes:

(4.14)

Relations of the kind of (4.14) were established by Terzaghi and other authors. Most of them differ only with respect of the third component, introducing the influence of the self-weight of the soil. These relations are theoretically incorrect for a plastic material since they are superposing terms corresponding to different failure figures such as those represented in fig. 4.8 and 4.10. However, the error implied is considered to be on the safe side and is accepted in engineering practice.

4.4 Ultimate bearing capacity in the case of a purely cohesive soil

This is a particular case of the problem previously considered. The failure mechanism shown in fig. 11.8 is transformed, when , in the one shown in fig. 4.11.

Equation (11.10) becomes:

(4.15)

(For , Nq = 1)

Fig. 4.11

One defines as netto ultimate bearing capacity the difference between the critical pressure in the geological pressure at the level of the foundation base:

(4.16)

The problem is to find the bearing capacity factor Nc for this case (.

An approach similar to the one used for the case ( is adopted:

Equilibrium of forces acting on the prism I (fig. 4.12 a)

(4.17)

The normal stress acting on the faces AC and BC:

(4.18)

Equilibrium of forces acting on the prism III (fig. 4.12 b)

(4.19)

Fig. 4.12

The normal stress acting on the faces BD and DE:

(4.20)

pf is obtained by writing the condition that the moment of all forces acting on the failure prism, in respect to the point B, is zero. Normal pressures acting on the circular are CD having the direction of the radius, do not give moment toward B.

(4.21)

But AC = BC = BD = DE = r

Relation (4.21) becomes:

;

(4.22)

(4.23)

(4.24)

Skempton has shown that, in fact, the netto ultimate bearing capacity increases with the depth D of the foundation until a depth D = 5B (fig. 4.13), reaching a limit value 9 for Nc.

Fig. 4.13

For rectangular foundations B x L, for which , Skempton proposed the relation:

(4.25)

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