where 9cv,d is the ground's design constant-volume (aka 'critical state') angle of shearing resistance; y9 is defined above; and k = 1 for concrete cast in situ or k = b for precast concrete. Values of 9cv,d are rarely measured, but are usually assessed from simple rules-of-thumb.20

Figure 10.8 illustrates key differences between a soil's characteristic peak angle of shearing resistance 9 k and its constant-volume counterpart 9cvk. The former is inherently more variable than the latter and is only significantly larger in dense (dilatant) soils while strains in the soil are small. By applying a partial factor y9 = 1.25 to the cautious estimate of tan 9pk, Eurocode 7 ensures that the calculation of sliding resistance is sufficiently reliable for design. However, applying the same partial factor y9 = 1.25 to a cautious estimate of tan 9cvk may be overly conservative.


Figure 10.8. Variability of peak and constant volume angles of shearing resistance

According to the theory of critical state soil mechanics,21 ^cv represents the smallest angle of shearing resistance mobilized at large displacement (assuming residual shear surfaces do not form in the soil). Hence the UK National Annex to BS EN 1997-1 notes that:

'it might be more appropriate to select the design value of y'cv directly.'

One way of determining ^cv,d would be to replace the partial factor y9 in the previous equation with a partial factor y9,cv < y9, i.e.:

v Y (p,cv where y9,cv might be as low as 1.0 (depending on how cautious your selection of 9cv is). If this approach is taken, then an additional check that ^cv,d is smaller than should also be made.

It is often the case that a portion of the horizontal load is derived from an inclined load on the footing. In this case, both the horizontal and vertical components of load come from the same source. However, the vertical component is favourable as far as an ultimate limit state of sliding is concerned, whereas the horizontal component is unfavourable. Except in Design Approach 1 Combination 2 (see Chapter 6), a favourable permanent action attracts a partial factor YGfv = 1.0 and an unfavourable permanent action a partial factor yg = 1.35. If the components of the inclined action are treated differently, then the inclination of that action will change. This is a prime example of where the Single Source Principle discussed in Chapter 3 should be invoked and the whole action should be treated either as unfavourable or as favourable (whichever is more onerous).

10.5.3 Undrained sliding resistance

Under undrained conditions, the characteristic shear resistance TRk shown in Figure 10.7 (ignoring passive pressures) is given by:

TRk — cuk where cuk represents the characteristic undrained shear strength of the soil.

The design shear resistance TRd (ignoring passive pressures) is then given by:

YRh Ycu *YRh where YRh is the partial factor on horizontal sliding resistance and Ycu is the partial factor on undrained shear strength.

The table on page 318 summarizes the values of these partial factors for each of Eurocode 7's three Design Approaches (see Chapter 6).

10.5.4 Passive earth pressure - favourable action or resistance?

The characteristic shear resistance TRk shown in Figure 10.7 is given by:

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