Building loads are transmitted by columns, bearing walls, or by other bearing member to foundations. A foundation is the lower part of a structure which transmits loads to the underlying soil without causing a shear failure of soil or excessive settlement. Thus, the word foundation refers to the soil under structure as well as any intervening load carrying member. If the soil near the surface has adequate bearing capacity to support the structural loads it is possible to use spread foundation such as footing or raft. If the soil near the surface is incapable of supporting the structural loads, piles or piers are used to transfer the loads to soil laying at greater depth capable of supporting such loads. The foundations are classified as shallow and deep foundation, according to the depth of construction.
Bearing Capacity and Stability of Foundations The ability of a soil to support a load from a structural foundation without failing in shear is known as its bearing capacity. The stability of foundation depends on:
There are, therefore, two independent stability conditions to be fulfilled since the shearing resistance of the soil provides the bearing capacity and the consolidation properties determine the settlement.
Bearing Capacity The supporting power of soil is referred to as its bearing capacity. It may be defined as the largest intensity of pressure which may be applied by a structure to the soil without causing failure of soil in shear or excessive settlement. Consider a footing placed at depth D below the ground surface, the overburden pressure at the base of the footing is q_{o}=γD . The total pressure at the base of the footing due to the self weight of the footing, weight of the superstructure and due to the weight of earth fill over the footing is known as the gross pressure intensity. The difference in intensities of gross pressure after the construction of the structure and of the original overburden pressure is known as the net pressure _{ .} The ultimate bearing capacity of soil may be determined by analytical methods (i.e., by bearing capacity theories) and field tests, or approximate values may be adopted from Building Codes which are based on experience. Ultimate Bearing Capacity q_{u}_{} The ultimate bearing capacity q_{u}_{} is defined as the least gross pressure intensity which would cause shear failure of the supporting soil immediately below and adjacent to a foundation. Three distinct modes of failure have been identified and these are illustrated in Fig.1, they well be described with reference to a strip footing In the case of general shear failure, continuous failure surfaces developed between the edges of the footing and the ground surface as shown in Fig.2. As the pressure is increased towards the value q_{u}_{} 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. Heaving of the ground surface occurs on both sides of the footing although the final slip movement would occur only on one side, accompanied by tilting of the footing. This mode of failure is typical of soils of low compressibility (i.e. dense or stiff soils) and the pressure settlement curve is of the general form shown in Fig.2, the ultimate bearing capacity being well defined. In the mode of local shear failure there is significant compression of the soil under the footing and only partial development of the state of plastic equilibrium. The failure surfaces, therefore, do not reach the ground surface and only slight heaving occurs. Tilting of the foundation would not be expected. Local shear failure is associated with soils of high compressibility and, as indicated in Fig.2, is characterized by the occurrence of relatively large settlements (which would be unacceptable in practice) and the fact that the ultimate bearing capacity is not clearly defined. Punching shear 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. There is no heaving of the ground surface away from the edges and no tilting of the footing. Relatively large settlements are also a characteristic of this mode and again the ultimate bearing capacity is not well defined. Punching shear failure will also occur in a soil of low compressibility if the foundation is located at considerable depth. In general the mode of failure depends on the compressibility of the soil and the depth of foundation relative to its breadth.
Net Ultimate Bearing Capacity q_{nu} The net ultimate bearing capacity is the minimum net pressure intensity causing shear failure of soil. q_{nu}=q_{u } q_{o} q_{u=}q_{nu+}q_{o} Net Safe Bearing Capacity q_{n}_{s} The net safe bearing capacity is the net ultimate bearing capacity divided by the desired factor of safety F.
Safe Bearing Capacity q_{s} The safe bearing capacity is the maximum pressure which the soil can carry safely without risk of shear failure.
Allowable Bearing Capacity The allowable bearing capacity is maximum pressure which is considered safe both with respect to shear failure and settlement. When the term bearing capacity is used without any prefix it may be understood to refer to the ultimate bearing capacity. BEARING CAPACITY THEORIES Broadly, there are two approaches for the analysis of stability of foundations. The first of these is known as the conventional approach which generates from the work of Coulomb (1977). This is based on the assumption of a certain shape for the rapture surface. The other approach which stems from the work of Rankine (1857) and Kotter (1903) is based on the assumption of simultaneous failure at every point in certain zone of the soil mass. This is referred here as plasticity theory approach. However, there is found to be reasonably good agreement between the two approaches Terzaghi's Bearing Capacity Theory Assumptions: Based on Prandtl's theory (1920) for plastic failure of metal under rigid punches Terzaghi derived a general bearing capacity equation. All soils are covered in this method by two cases which are designated as general shear and local shear failures. General shear is the case wherein the loading test curve for the soil under consideration comes to a perfectly vertical ultimate condition at relatively small settlement as shown by curve 1 in Fig.3. Local shear is the case wherein settlements are relatively large and there is not a definite vertical ultimate to the curve as in curve 2 in Fig.3. (Soil is loose relative to a general shear failure). The following assumptions were made in the analysis.
The application of the load (Fig.4) tends to push the wedge of soil abc into the ground with a lateral displacement of zones II (radial shear zones) and zones III (plane shear zones). The downward movement of this soil wedge is resisted by the resultant of the passive pressure of the soil and the cohesion , acting along the surface of the wedges ac, bc as it moves. Considering the equilibrium of the wedge abc, Terzaghi presented the following bearing capacity expression for general shear failure:
where
= relates the passive pressure of the soil in zones II and III to the size of the footing, and angle of failure zone I (Fig.4). The values are determined by means of the φ circle or logarithmic spiral. It is proposed that ultimate bearing capacity for local shear failure condition may be computed based on the following soil parameters
Table 1 Bearing Capacity Factors for General Shear Conditions and Local Shear Conditions
Shape Factors Equation 1 is the bearing capacity equation for a long strip footing. It can also be used for rectangular footing of length L equal to or greater than 5 times the width B i.e. . Terzaghi has recommended that Eq 1 could be used for circular and square footings with the following modifications. For circular footing
For saturated clay may be assumed to be equal to zero, and hence:
For cohesionless soils (c = 0.0)
Limitations: (i) The shear strength of soil above the base level of footing is neglected. (ii) This theory gives conservative values for footings whose depths are greater than zero. (iii) Subdivision of the bearing capacity problems in two types of shear is an arbitrary one, since two cases cannot cover the wide range of conditions.
Meyerhof's Bearing Capacity Theory . Assumptions: The bearing capacity of shallow foundations has been derived by Meyerhof (1951) taking into account the shear strength of the soil above the base level of the footing. He assumed a failure mechanism similar to Terzaghi's but extending up to ground surface as shown in Fig. 6.
The following assumptions are made in the analysis: 1. The footing is continuous 2. The failure surface is composed of a straight line and a logarithmic spiral. 3. The soil wedge ABC beneath the base of footing is in elastic state. 4. The principle of superposition is valid. Meyerhof extended the previous analysis of the plastic equilibrium for the surface strip foundation to shallow and deep foundation. In the mechanism of failure shown in Fig.6. there are two main zones on each side of the central zone, ABC, radial shear zone BCD and mixed shear zone BDEF. The shearing resistance of the soil above the foundation level is considered in this analysis. The bearing capacity of shallow foundations with rough bases is expressed as:
where N_{c,}_{q} and Nγ are the general bearing capacity factors which depend on foundation depth, shape and roughness and the angle of internal friction. To calculate the bearing capacity factors, the inclination of the equivalent free surface and the stresses and acting on this surface must be determined. Meyerhof computed the values of N_{c,}_{q } and Nγ for various angles of and . These values for shallow strip footing are shown in Fig.7. The general solution given by Eq. 5 is too tedious for routine application. To simplify the solution and to avoid estimation of the equivalent free surface stresses the bearing capacity factors are combined to give:
For cohesionless soil the bearing capacity of strip foundation is given by
Where N_{γq}_{ } depends on both _{γ}and N_{q} , the former is more important at greater depths, the latter is more important at shallow depths. The values of N_{γq } depends on the coefficient of earth pressure K_{S}_{}. The values of N_{γq } for two values of (30^{o} and 40^{o} ) are shown in Fig.8 and Fig. 9.
For rectangular, square and circular foundations, Meyerhof modified the strip bearing capacity factors N_{C} ,N_{q }and Nγ by multiplying them by an empirical shape factor λ . Values of λ for various values of depth, width ratio and are shown in Fig.10.
Limitations: Bearing capacities computed from Meyerhof's theory are found to be higher than the observed bearing capacities in sands at greater depths.
Skemptnn's (1951) Bearing Capacity for Clays Skempton (1951) recommended the following shape and depth factors, and values of N, for surface footing on clays. (i) Surface footings (D = 0) N_{C}_{ } ≈ 5 for strip footing N_{C}_{ } ≈ 6 for square or circular footing (ii) At depth D
(iii) At any depth, for rectangular footings,
Brinch Hansen's Bearing Capacity Theory A theory, somewhat similar to the Terzaghi's, has been proposed by Hansen (1961). The ultimate bearing capacity according to this theory is given by
The values of bearing capacity factors as well as approximations for the shape, depth and inclination factors are given in Tables 2. and 3. Table 3 provides equations for depth, shape, and inclination factors for use in Eq.9. for more precise computations TABLE 2 Bearing Capacity Factors N_{C} ,N_{q }and Nγ for Use in Eq. 9
Table 3 Shape, inclination, and depth factors for use in Hansen equation Eq. 9
Egyptian Code of Practice for Soil Mechanics and Foundation Engineering (six edition 2001) Based on the above analyses, the Egyptian Code of Practice for Soil Mechanics and Foundation Engineering has proposed a general bearing capacity equation. This equation include the most affecting factors on the calculation of bearing capacity.
For Vertical Centric Load. The ultimate bearing capacity is given by the following formula:

