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where the derived expressions allow Kaq, Kpq, Kac, and Kpc to be derived from values of Kay and K^, which are readily available from charts.

The auxiliary coefficient Kn in these equations is given by: K = 1 ± sin^x sin(2m^ ^^+p_mw

" 1 + sin^x sin(2mt ±q>) and the terms mt and mw are:

Active coefficients are obtained by using the upper sign (+ or - ) where a choice is shown (± or +) in the expressions for Kn, mt, and mw; and passive coefficients by using the lower signs. Note that, for vertical walls, Kaq = Kay and Kpq = KpY, since 0 = 0°.

In all these equations, 9 is the soil's angle of shearing resistance; 5 the angle of friction between the wall and the soil; p the slope angle of the ground surface; and 0 the inclination of the wall to the vertical (see Figure 12.6).

Figure 12.6. Sign convention for determining active and passive earth pressures from numerical procedure in Eurocode 7

We have provided in Appendix 2 a series of charts for vertical walls (i.e. with 0 = 0°) based on these expressions, showing the variation in Kay and K^ with angle of shearing resistance 9 for various values of interface friction 5 and slope angle p. Most charts of this kind (e.g. the ones published in CIRIA C58026) plot values of KaY and KpY against 9 for various ratios of 5/9 and P/9. The format of our charts is therefore unusual but - in our opinion -preferable, since it emphasizes the valid range of values for 5 and p. The following example illustrates the use of these charts.

Consider the cantilever retaining wall shown in Figure 12.7, which is embedded in sand with characteristic angles of shearing resistance 9k = 32° (peak) and 9cvk = 30° (constant volume). The ground behind the wall slopes upwards at a gradient of 1:3 (i.e. 18.4°) and the formation is horizontal. The characteristic angle of wall friction 8k is calculated as: Sk = ,k = 20°.

Figure 12.7. Earth pressures acting on embedded wall supporting sloping ground

Figure 12.8 shows one of the charts from Appendix 2, giving values of Ka versus angle of shearing resistance (9), for an angle of interface friction 5 = 20°. The numbers on the lines represent various slope gradients, tan p = 0 (i.e. flat), 1:10, 1:5, etc. The dashed lines are for slope angles < 0°.

For 9 = 32° and tan p = 1:3, the chart in Figure 12.8 gives Kay = 0.34.

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