## Y

Integration of these equations allows the active effective earth and water thrusts (P'a and Ua on Figure 11.6) to be calculated:

P' = j<adz and Ua = j udz 0 0 where H is the height of the virtual plane above the base of the wall. Both of these actions are unfavourable for bearing, sliding, and toppling of the wall.

### 11.4.1 Bearing

Section 6 of Eurocode 7 Part 1 requires the design vertical action Vd acting on the wall's foundation to be less than or equal to the design bearing resistance Rd of the ground beneath it:

which, as discussed in Chapter 10, may be re-written as:

qEd < qRd where qEd is the design bearing pressure on the ground (an action effect) and qRd is the corresponding design resistance.

The design bearing pressure qEd beneath the base of the wall is given by:

qEd=-A-=yg I ■AI+Z y q ,i ViqQk ,i where WGk is the wall's characteristic permanent self-weight (including backfill); VQk is a characteristic variable vertical action imposed on the wall (to the left of the virtual plane); A' is the effective area of the base; yg and yq are partial factors on permanent and variable actions, respectively; and ^ is the combination factor applicable to the ith variable action (see Chapter 2).

The wall's self-weight is simply the sum of the weights of its stem and base, plus that of the backfill to the left of the virtual plane (Wstem, Wbase, and Wbackfill on Figure 11.7). Since these are unfavourable actions in bearing, the characteristic weight densities should be selected as upper (or 'superior') values.

centre-line of base uuiuunuuumuuiiu

eccentricity

Figure 11.7. Inclined, eccentric resistance to actions on reinforced concrete wall

Bearing resistance eccentricity

Figure 11.7. Inclined, eccentric resistance to actions on reinforced concrete wall

Bearing resistance is calculated using the methods discussed in Chapter 6 for footings subject to vertical actions. However, whereas most footings are subject to simple vertical loading, the footing of a reinforced concrete wall has to resist an eccentric, inclined action owing to the combination of the wall's self-weight and the horizontal thrust on the virtual plane (P'a + Ua on Figure 11.7). These factors have a strong detrimental influence on the bearing resistance that the ground provides. They also make the calculation of that resistance much more complicated than for a simple footing subject to vertical loading, as the worked examples at the end of this chapter show.

### 11.4.2 Sliding

Section 6 of Eurocode 7 Part 1 requires the design horizontal action Hd acting on the virtual plane to be less than or equal to the sum of the design resistance Rd from the ground beneath the footing and any design passive resistance Rpd on the side of the wall (see Figure 11.6):

which may be re-written as:

HEd ^ HRd where HEd is the design horizontal action effect and HRd is the corresponding total design horizontal resistance.

The design horizontal action effect HEd is given by:

HEd = Hd = Pad + Uad where P'ad and Uad are the design values of P'a and Ua shown on Figure 11.6.

When drained conditions apply, the magnitude of the design horizontal resistance Rd is given by:

YRh tan S,

Yh where WGk = the wall's characteristic permanent self-weight, including backfill; UGk = the characteristic permanent water upthrust beneath the base; 5k = the characteristic angle of interface friction between the base and the ground; YGfav and yg = partial factors on favourable and unfavourable actions, respectively; YRh = a partial factor on sliding resistance; and y9 = a partial factor on shearing resistance.

In this equation, the wall's self-weight is regarded as a favourable action since an increase in its value would increase the sliding resistance, Rd; whereas the water upthrust is treated as an unfavourable action since an increase in its value would decrease Rd. Hence a larger factor yg > YGfv is applied to UGk than to WGk.

Another reason why the water upthrust should be treated as an unfavourable action is the fact that the horizontal water thrust Ua shown on Figure 11.7 is clearly an unfavourable action. Since both forces arise from the same source, for consistency they should be factored the same way. This argument could be extended to the weight of the backfill, since it contributes both to the effective earth thrust P'a and the weight of the wall. However, it is not at all certain that the backfill on top of the wall heel will be compacted as much as that outside the virtual plane, and we therefore recommend treating the earth pressure and the self-weight of the backfill as separate actions (this is also a more conservative assumption).

If the previous equation had been written in terms of submerged weight, as: r = wgd x tan St

YRh then only a single partial factor (presumably Yc,fav) could be applied to the term W'Gd (= WGd - UGd) and the design resistance would have been overestimated. Because of this, we recommend that the traditional use of submerged weights in gravity wall calculations should be discontinued.

The weight of the imposed surcharge should be omitted from the calculation of design resistance, since it is (usually) a variable action and therefore a more critical design situation arises when it is absent.

### 11.4.3 Toppling

Verification of resistance to toppling requires the design destabilizing moment MEd,dst acting about the wall's toe (point 'O' on Figure 11.7) to be less than or equal to the design stabilizing moment MEdstb acting about the same point:

MEd,dst < MEd,stb

The forces that contribute to the destabilizing moment are the effective earth thrust (P'a) and the water thrust (Ua) behind the virtual plane, plus the water uplift (U) - as shown on Figure 11.7. The effective earth thrust should include the contribution from the imposed surcharge acting at ground surface behind the virtual plane.

The forces that contribute to the stabilizing moment are the self-weights of the wall stem (Wstem) and base (Wbase), plus that of the backfill on the wall heel (Wbackfill) - again, as shown on Figure 11.7. The weight of the imposed surcharge should be omitted from the stabilizing moment, since it is (usually) a variable action and therefore a more critical design situation arises when it is absent.

### 11.4.4 Reinforced concrete walls with narrow heels

The discussions of bearing, sliding, and toppling in the previous sections have assumed that the heel of the reinforced concrete wall is large enough for a Rankine zone to form within the area bounded by the so-called 'virtual plane' (see Figure 11.8, top). This assumption greatly simplifies the calculations that need to be undertaken to verify the wall's strength and stability (because active forces can be resolved vertically and horizontally along the virtual plane and friction ignored).

The assumption is valid when the heel's width 'b' satisfies the inequality:

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