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8.2.1 Form of the design pressures

The pressures defined here follow the pattern defined in the draft Eurocode [ENV 1991–4, 1995]. The discharge pressures on vertical walls are again defined in terms of a “fixed load” and a “free load’’, called a patch load. The free load is to be placed at whatever point on the silo wall is deemed to be most damaging to the silo. The fixed load corresponds to Janssen filling pressures amplified by a flow pressure multiplier or load magnifier C. 8.2.2 Symmetrical discharge pressures on vertical walls

The normal pressure acting on the barrel section of the silo during discharge at any level is related to the filling pressure as:

phe=Chphf

(8.1)

where phf is given by Eq. 7.1.

The frictional traction on the wall at any level is given by:

pwe=Cwpwf

(8.2)

where pwf is given by Eq. 7.2.

Where the silo is discharged from the top, the values of Ch and Cw are given by:

Ch=Cw=1.0

(8.3)

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Where the silo is discharged from below, the values of Ch and Cw are given by:

Ch=Co

(8.4)

Cw=1.1

(8.5)

in which the reference value of normal pressure multiplier Co is deemed to be a material parameter, and is given in Table 4.1.

For solids which are not listed in Table 4.1, the value of Co is found as:

Co= 1.35+0.02 (ϕi−30°)

(8.6)

but not less than Co,min= 1.35.

Equation 8.6 indicates that solids with higher angles of effective angle of internal friction (higher strength) are believed to cause higher pressures during discharge than solids with lower angles, but the result is not thought to be very sensitive to the internal friction when the latter is low. The equation is an empirical fitting to the best available data at the present time, and should not be regarded as precise [Rotter, 2000a].

The normal pressure multiplier is high to indicate that the pressure may locally rise considerably at any given level, but a high pressure in one place is commonly balanced by a lower one at another position. The pressures are all increased because it is not known in advance where the high pressure may occur. This pressure definition takes into account knowledge that the structural response to a uniform rise in pressure will be principally

confined to the corresponding level.

By contrast, if the pressure rises at one level and the friction is fully mobilised (and does not change from the filling value), the resulting frictional traction will rise too, but only locally. The stress resultant which arises from the wall friction is the integral of tractions all down the wall, so the resulting vertical force which develops in the wall will not rise by the same proportion as the local rise in normal pressure. Thus, if the multiplier Cw is to be used on tractions at all heights in the silo wall, it is appropriate to use a smaller increase for wall friction than for normal pressures.

The outcome of the above discussion is that the multiplier Ch on normal pressure is larger than that on wall friction Cw. Alternative interpretations of these equations might be that the wall friction has either fallen during discharge (not reliably demonstrated for most solids) or that the wall friction is imperfectly mobilised during flow. Neither of these propositions is an appropriate deduction from the test data.

8.2.3 Symmetrical pressure reductions for squat and intermediate silos

Where the silo has a squat or intermediate aspect ratio (Fig. 8.1), the values of Ch and Cw can be reduced. This reduction attempts to account for the

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Figure 8.1 Aspect ratio of a silo.

effects of flow channel types without a formal prediction of them. The revised multipliers are shown in Fig. 8.2. In squat silos, where hb/dc < 1:

Ch=Cw=1.0

(8.7)

In intermediate aspect ratio silos (1.0 < hb/dc < 1.5), a linear interpolation is used between the squat and slender limiting cases:

(8.8)

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Figure 8.2 Flow pressure multiplier or load magnifier for silos of different aspect ratios.

8.2.4 Additional local patch pressure on barrel section during discharge

The pattern of patch pressure to be applied to the silo barrel is shown in Fig. 7.5. The patch extends over a height s, and consists of a pressure which varies in a sine wave around the circumference from an outward value pp on one side to an inward value pp on the opposite side. The patch pressure consists only of a normal pressure on the silo wall, and the changed wall friction component can be ignored. The magnitude of the patch pressure is increased when the silo is eccentrically filled or discharged.

The value of the discharge pressure ppe to be added to the fixed pressures is given by:

ppe=0.2 β phe

(8.10)

in which phe is found from Eq. 8.1 and:

β=1+emax/dc

(8.11)

where emax is the relevant eccentricity, which is defined as the larger of the eccentricity of filling ei and the eccentricity of the outlet eo.

The patch pressure distribution around the silo follows the pattern defined for filling (Fig. 7.5) and is given by:

ppes=ppe cosθ

(8.12)

The height over which it extends and the location of the patch are given by Eqs 7.16 and 7.17, and the equivalent horizontal force Ppe at the depth zp below the surface is:

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ppe=π s r ppe

(8.13)

In squat silos (hb/dc < 1), the patch pressure may be ignored (i.e. ppe=0).

In intermediate aspect ratio silos (1.0 < hb/dc < 1.5), the patch pressure may be reduced to:

(8.14)

where ppe is taken from Eq. 8.10.

This procedure gives a poor representation of the effects of eccentric discharge, and may not be used where the discharge eccentricity from the silo axis to the centre of the outlet eo exceeds 0.25dc (=r/2) and no part of the outlet may lie at an eccentricity greater than 0.30dc (=0.6r). By contrast with the treatment of filling

eccentricities, this procedure overestimates the seriousness of eccentricity in relatively squat silos whilst underestimating its consequences for tall silos. For the latter, it may represent an unsafe approach when the outlet eccentricity is close to the limit defined here (which is a large eccentricity) and designers are advised to approach this limit to the design eccentricity with caution.

Advice on the assessment of pressure regimes where greater eccentricities occur is given in Chapter 9, though this topic is outside the scope of Eurocode 1 Part 4.