QEk A A
where Vrep is a representative vertical action; VGk, VQk, and WGk are as defined above; A' is the footing's effective area (defined in Section 10.4.2); and ^ is the combination factor applicable to the ith variable action (see Chapter 2).
If we assume that only one variable action is applied to the footing, this equation simplifies to:
since ^ = 1.0 for the leading variable action (i = 1).
The design bearing pressure qEd beneath the footing is then:
where yg and yq are partial factors on permanent and variable actions, respectively.
10.4.2 Eccentric loading and effective foundation area
The ability of a spread foundation to carry forces reduces dramatically when those forces are applied eccentrically from the centre of the foundation.
To prevent contact with the ground being lost at the footing's edges, it is customary to keep the total action within the foundation's 'middlethird'. In other words, the eccentricity of the action from the centre of the footing is kept within the following limits:
where B and L are the footing's breadth and length, respectively; and eB and eL are eccentricities in the direction of B and L (see Figure 10.5).
Eurocode 7 Part 1 requires 'special precautions' to be taken where:
... the eccentricity of loading exceeds 1/3 of the width of a rectangular footing or [60%] of the radius of a circular footing. [en 19971 §6.5.4(1)P]
Note that this is not the middlethird rule, but rather a 'middletwothirds' rule. We recommend that foundations continue to be designed using the middlethird rule until the implications of Eurocode 7's more relaxed Principle have been thoroughly tested in practice.
Bearing capacity calculations take account of eccentric loading by assuming that the load acts at the centre of a smaller foundation, as shown in Figure 10.5. The shaded parts of the foundation are therefore ignored. The actual foundation area is therefore reduced to an 'effective area' A', which can be calculated from:3
where B' and L' are the footing's effective breadth and length, respectively; and the other symbols are as defined above.
10.4.3 Drained bearing resistance
The drained ultimate bearing capacity of a spread foundation qult has traditionally been calculated from the socalled 'tripleN' formula, which in its original form4 is given by: y ' BN
where c' is the soil's effective cohesion; q' the effective overburden pressure at the foundation base; y' the effective weight density of the soil below the foundation; and Nc, Nq, and NY are bearing capacity factors.
The overburden and cohesion factors Nq and Nc were established in the 1920s by Reissner5 and Prandtl,6 respectively, in terms of the soil's angle of shearing resistance 9:
and these equations are used almost universally in geotechnical practice. However, there is no consensus regarding the value of the factor NY.
Design practice in many parts of Europe7 has traditionally used BrinchHansen's8 equation for NY: Ny= 1.5 (Nq 1) tan while in America designers typically employ Meyerhof's9 equation: Ny=(Nq 1)tan(1.4)
and offshore structures engineers10 use Vesic's11 equation: Nr= 2 (Nq +1) tani which recent research12 suggests may overpredict NY. Chen's13 equation:
Nr= 2(Nq 1)tani is also popular and appears in Eurocode 7 Annex D. Note that Chen's equation assumes a rough base with interface friction > 0.5 times the soil's angle of shearing resistance.
Values of these bearing capacity factors for different angles of shearing resistance are illustrated in Figure 10.6. The curves for Meyerhof's and BrinchHansen's NY are virtually coincident for 9 < 30° and diverge only marginally as 9 approaches 60°. Chen's formulation for NY is slightly more conservative than Vesic's but significantly more optimistic than BrinchHansen's, particularly at large angles of shearing resistance.
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