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Angle of shearing resistance, cp (degrees) Figure 10.6. Bearing capacity factors Nq, Nc, and Nr

10.4.4 Non-dimensional factors applied to the triple-N formula

Various modifications to the triple-N equation have been proposed, usually involving the introduction of factors to cater for the effects of the foundation's shape, depth, and, base inclination; inclination of the applied load; and inclination of the ground surface.

The 'complete' expression for qult is:

quit = cN cScdcicgcbc + q'Nfydqiqgqbq + ^

where sc, sq, and sY are shape factors; dc, dq, and d are depth factors; ic, iq, and iy are load inclination factors; gc, gq, and gY are ground inclination factors; and bc, bq, and by are base inclination factors.

In Europe, this equation is credited to Brinch-Hansen,14 but in America more usually to Meyerhof15 (who uses just the shape, depth, and load inclination factors).

Annex D of EN 1997-1 gives an equation for the drained bearing resistance of a spread foundation that omits the depth and ground inclination factors, which are commonly found in bearing capacity formulations. The omission of the former is uneconomic, but neglecting the latter is unsafe. The UK's National Annex emphasizes this point and suggests that an alternative method, including depth and ground inclination factors, may be used.

The equations given in Eurocode 7 Annex D for the shape factors sc, sq, and sY are summarized in the table below. The equations for sc and sq are the ones recommended by Brinch-Hansen and Vesic.

Factor Cohesion Overburden Body-weight c q y

Factor Cohesion Overburden Body-weight c q y

Shape

Sx

NcL L L

Depth

dx

No equations given in Eurocode 7

Load inclination

ix

Refer to EN 1997-1 Annex D for details

Ground inclination

gx

No equations given in Eurocode 7

Base inclination

bx

Refer to EN 1997-1 Annex D for details

tk = 0.3 in Eurocode 7 (applies for B/H < 1); Brinch-Hansen and Vesic recommend k = 0.4; a survey16 of European practice showed five countries using k = 0.4, three using k = 0.3; and one using k = 0.2

10.4.5 Undrained bearing resistance

Annex D of EN 1997-1 gives the following equation for the undrained bearing resistance R of a spread foundation:

— = (n+2)cubcscic + q where cu is the soil's undrained shear strength; q the total overburden pressure at the foundation base; and the other symbols are as defined for the drained equation (see Section 10.4.3). The value (n + 2) is obtained by setting

9 = 0 in Prandtl's17 expression for Nc (see Section 10.4.3). The shape factor sc is given by:18

As with the drained equation discussed in Section 10.4.3, R/A' omits the depth factor dc and ground inclination factor gc. The omission of the former is uneconomic, but neglecting the latter is unsafe. The full equation is:

Recent finite element limit analysis19 of the bearing capacity of foundations in clay has found that the following expression provides a good approximation for the depth factor dc for clays:

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