Factors affecting earth pressure
3.3 EartH PrESSUrE PrINCIPLES
3.3.8 Effect of wall flexibility on bending moments applied to embedded walls
Classical solutions generally implicitly assume rotation of a rigid wall about its base. In practice, many reinforced concrete walls behave in an approximately rigid manner, but steel sheet-pile walls can be much more flexible. Experience in the first half of the twentieth century showed that, for anchored steel sheet-pile walls, designs based on the classical earth pressures were over-conservative in terms of the thickness of the steel section required to support the bending moments applied by the soil.
Early explanations considered vertical arching, in much the same way as Terzaghi’s General Wedge Theory. Rowe (1952), however, pointed out that for anchored sheet-pile walls’ vertical arching is unlikely because the anchor will yield sufficiently to restore hydrostatic pressures in the soil.
It is now accepted that bending moments in the sheets are affected by the deflected shape of the wall below dredge level and that this is a function of the flexibility of the wall relative to the soil.
Figure 3.24a shows the conventional assumed earth pressures on an anchored sheet-pile wall. The pressure from the supported soil causes wall movement, and it is assumed that full active and passive pressures can be mobilised on the wall. Thus, the centre of passive pressure is at one third of the embedded depth from the base of the wall, and the wall can be simpli-fied into the beam shown in Figure 3.24b, where the maximum bending moment will be a function of (approximately) the square of the span, L.
Figure 3.24c shows the simplified ‘assumed-rigid’ passive pressure dis-tribution for a rigid wall rotating about its base. This is the assumption made to obtain the pressure distribution in Figure 3.24a. Figure 3.24c also shows the type of pressure distribution observed by Rowe (1952) on model
(a) (b)
Average yield of wall Average yield of wall ∆t
∆t/H
Figure 3.23 Idealised relationships between the average yield of a wall and the coefficient of earth pressure. (a) Wall rotates about its base. (b) Wall translates away from the soil. Shaded areas represent the resulting distribution of earth pres-sure immediately after the total lateral prespres-sure has become equal to the Coulomb value. (From Terzaghi, K., J. Boston Soc. Civil Eng. 23, 71–88, 1936.)
flexible walls on sand. Generally, there is a point of contraflexure in the wall some distance below dredge level. For very dense sands, the point of contraflexure may be at, or slightly above, dredge level. For loose sands, it will be lower. Because the deflections at the bottom of the wall are small, passive pressures are not obtained. Therefore, the pressure distribution for a medium-dense sand might be parabolic, as shown. For this condition, the resultant force acts near to D/2 from the base of the wall, and therefore the equivalent span, L, and hence the maximum bending moment applied to the steel sheets, are both reduced.
The deflected shape of the wall is a function of the stiffness of the sheets relative to the stiffness of the soil. As the wall becomes more flexible relative
(a) dense sand on flexible wall Factor of safety of 2
on passive pressure
Figure 3.24 Mechanism of moment reduction due to wall flexibility. (a) Assumed (design) earth pressures. (b) Equivalent simply supported beam. (c) Wall deflection and passive pressure distributions below dredge level for rigid and flexible walls.
to the soil, the position of the resultant passive force, Qp, moves up, pro-gressively reducing the applied maximum bending moments.
Rowe (1952) carried out tests on model walls, 500–900 mm high, of differing metal thickness, supporting various soils in loose and dense con-ditions. For similitude between the model and the prototype (Figure 3.25), there must be geometrical similitude. If slopes of the deflected shape of the wall are to be equal at corresponding points,
dy dz
dy
model dz prototype
=
(3.18)
but if slope = dy/dz then bending moment
M d y
dz since M EI
d y
∝ 22 =dz22 (3.19)
and shear force
S d y
∝dz33 (3.20)
and load
P d y
∝dz44 (3.21)
H
Deflected shape of wall αH y
βH
τ z
Figure 3.25 Terms used in Rowe’s analysis.
Therefore, from the above
i.e. at any depth, for similitude Mz
If z = H, and as, for a triangular pressure distribution, horizontal stress α H, then
so that (H4/EI) must be the same for both model and prototype. Rowe termed this ‘ρ’ and thus was able to use the results of his model tests to provide design curves for moment reduction in full-scale structures (see Section 10.3.4).
A recent contribution to the effects of wall flexibility has been made by Diakoumi and Powrie (2009), using simplified kinematically admis-sible strain fields in the Mobilised Strength Design approach (Osman and Bolton 2004) to analyse embedded retaining walls propped at the crest.
For the assumptions and simplifications made by Diakoumi and Powrie, the method shows that as the wall flexibility or soil stiffness increase, the bending moments may fall by about 20% of the values calculated using the Free Earth Support method (described in Chapter 10), whilst the prop force may reduce by about 15%. In contrast, Rowe’s moment reduction factors suggest reduction of maximum bending moment by as much as 70% when wall flexibility is high.
At this stage, caution should be exercised in applying large moment reductions. As long ago as 1953, Skempton, mindful of the fact that Rowe’s moment reduction factors were derived from model tests, suggested that when used in design the amount of reduction should be as follows:
Sands: use 1/2 of the moment reduction suggested by Rowe.
Silts: use 1/4 of the moment reduction suggested by Rowe.
Clays: use no moment reduction.
3.3.9 Stress relief during in situ