Kby

Figure 7.5 illustrates the consequences of the above for a dam retaining various depths of soil, assuming ys = 20kN/m3, yc = 24kN/m3, yw = 10kN/m3, 9 = 25° (hence Ka = 0.406), and 5 = 35°.

The lines labelled 'Traditional' are based on Fs = 1.5 for sliding and Fo = 2.0 for overturning. As Figure 7.5 shows, overturning governs the minimum width of the dam for all heights of retained soil. (Of course, had we assumed a smaller value of interface friction 5 then sliding might have governed the dam's width.)

An additional line for Fo = 3.0, which ensures the resultant force lies within the middle third of the foundation (i.e. e < B/6), is also shown on Figure 7.5. This middle-third rule, which prevents tension occurring between the dam and the founding rock, is more critical than the requirement for Fo = 2.

Eurocode 7 requires us to verify the stability of rigid bodies according to limit state EQU. We can do this by replacing Fs and Fo in the inequalities above by the ratio Yc,dst/ Yg, stb (— 1.1/0.9 — 1.21) and by applying a material factor y9 (— 1.25) to tan 9 and tan 5, reducing 9 to 20.5° (and consequently increasing Ka to 0.482) and 5 to 29.2°. As Figure 7.5 shows, sliding governs the minimum width of the dam for all values of hs/H - a result that is somewhat counter-intuitive, since EQU deals with static equilibrium! hs/H

Figure 7.5. Sizing of dam of Figure 7.4 to avoid sliding and overturning

A fundamental principle of limit state design is that all limit states are of equal importance. Consequently, we should check all design situations for all possible limit states. We can check the situation shown in Figure 7.5 for limit states GEO and STR (see Chapter 6) by replacing Fs and Fo in the inequalities above by the ratio YG/YG,fav (= 1.35 for Design Approach 1, Combination 1, and 1.0 for Combination 2) and by applying a material factor Y9 (— 1.0 or 1.25) to tan 9 and tan 5. As Figure 7.5 shows, overturning governs the minimum width of the dam for values of hs/H < 0.5 and sliding governs for hs/H > 0.5. (This change-over depends on the value of interface friction 5 assumed.) Once again, this result is counter-intuitive, since GEO and STR deal with the strength of materials!

Many engineers would instinctively consider overturning to be controlled by an EQU limit state and sliding by GEO. The simple calculation above demonstrates that Eurocode 7's partial factors lead to the opposite conclusion in some situations. We believe that sliding should be governed by limit state GEO and EQU should only be used to guard against overturning when any actions from the ground are minor.

It is generally unwise to allow tension to occur either within the structure or between its base and the ground. To prevent this from happening, the resultant force must remain within the middle third of the foundation base. Satisfying this condition - which is not required by Eurocode 7 - is likely to govern the design.

7.4 Uplift

Ultimate limit state UPL is defined as:

loss of equilibrium of the structure or the ground due to uplift by water pressure (buoyancy) or other vertical actions [en 1997-1 §2.4.7.1(1)P]

Because uplift involves predominantly vertical actions, verification of stability against uplift is expressed in Eurocode 7 by the inequality: Vd,dst = Gd4st + Qd dst — Gd ,sib + Rd [EN 1997-1 exp (2.8)]

in which Vd = the design vertical action, Gd = design permanent actions, Qd = design variable actions, and Rd = any design resistance that helps to stabilize the structure. The subscripts 'dst' and 'stb' denote destabilizing and stabilizing components, respectively. This inequality is merely a more specific version of the general equation given in Section 7.1:

Eurocode 7 Part 1 allows resistance to uplift to be treated as a stabilizing permanent vertical action, thereby simplifying the expression above to:

However, doing so will lead to a different outcome to that obtained with the previous equation, since in the first case the resistance is obtained by dividing material strengths by their appropriate partial factors (e.g. y9 = 1.25 or Ycu = 1.4), whereas in the second case the resistance is multiplied by the partial factor on stabilizing permanent actions (YG,stb = 0.9). Since it gives a more conservative result, we believe it may be better to treat resistance explicitly as resistance and not as a favourable action.

Destabilizing design vertical actions Vd,dst are obtained from destabilizing characteristic permanent (Gk,dst) and variable (Qkdst) actions by first multiplying by combination factors ^ where appropriate (see Chapter 2) and then multiplying by partial factors yg and yq greater than or equal to 1.0:

Stabilizing design vertical actions Vdstb are obtained from stabilizing characteristic permanent actions (Gkstb) by multiplying by a partial factor less than or equal to 1.0:

There is no term for variable actions in this expression, since it would be unsafe to include it (mathematically, YQ,stb = 0).

The design resistance Rd, if any, that helps to stabilize the structure can be obtained in one of two ways: either directly from design material properties Xd with a resistance factor yr = 1 or from characteristic material properties with yr > 1. See Chapter 6 for further discussion of the way design resistance may be calculated.

Finally, design dimensions ad are are obtained from nominal dimensions anom by adding or subtracting a tolerance Aa. Eurocode 7 does not give any specific recommendations for the value of Aa to use in uplift verifications.

Consider the depressed highway shown in Figure 7.6, which is subject to uplift owing to a naturally high water table outside the constructed section.

Partially relieved section Unrelieved section

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