Loads on Silo Hopper According to EN 1991-4

Technical Article

My previous article describes actions on silos according to EN 1991-4. With an example of a free standing cylindrical silo for cement with a conical hopper, filling loads of the silo hopper were calculated.

Layout and Dimensions

The structural system is shown in Figure 01.

Figure 01 - Layout and Dimensions of Cement Silo

Relevant Parameters for Various Load Applications

The applicable extreme values of particulate solids for the maximum hopper pressures in the full condition are included in the following table.

Figure 02 - Relevant Parameters for Various Load Applications

Physical Properties

The loads on the walls of silo hoppers should be determined according to EN 1991-4 [1] with regard to the steepness of the hopper walls in compliance with the following classes:

  • A flat bottom shall have an inclination to the horizontal α less than 5°.
  • A shallow hopper shall be any hopper not classified as either flat or steep.
  • A steep hopper shall be any hopper that satisfies the following criterion:
    $$\tan\;\mathrm\beta\;<\;\frac{1\;-\;\mathrm K}{2\;\cdot\;{\mathrm\mu}_\mathrm h}\;(6.1)$$

$$\tan\;39.8^\circ\;=\;0.83\;>\;\frac{1\;-\;0.450}{2\;\cdot\;0.458}\;=\;0.60$$

The hopper is classified as a shallow hopper.

Filling Loads

Janssen characteristic depth zo
$${\mathrm z}_\mathrm o\;=\;\frac1{\mathrm K\;\cdot\;\mathrm\mu}\;\cdot\;\frac{\mathrm A}{\mathrm U}\;(5.75)$$
$${\mathrm z}_\mathrm o\;=\;\frac1{0.450\;\cdot\;0.458}\;\cdot\;\frac{19.63}{15.71}\;=\;6.07\;\mathrm m$$

Vertical distance ho
For a symmetrically filled circular silo, the vertical distance between the equivalent surface of the solid and the highest solid-wall contact is calculated as follows:
$${\mathrm h}_\mathrm o\;=\;\frac{{\mathrm d}_\mathrm c}6\;\cdot\;\tan\;{\mathrm\phi}_\mathrm r\;(5.78)$$
$${\mathrm h}_\mathrm o\;=\;\frac{5.00}6\;\cdot\;\tan\;36.00^\circ\;=0.61\;\mathrm m$$

Parameter n
$$\mathrm n\;=\;-(1\;+\;\tan\;{\mathrm\phi}_\mathrm r)\;\cdot\;(\frac{1\;-\;{\mathrm h}_\mathrm o}{{\mathrm z}_\mathrm o})\;(5.76)$$
$$\mathrm n\;=\;-(1\;+\;\tan\;36.00^\circ)\;\cdot\;(\frac{1\;-\;0.61}{6.07})\;=\;-1.55$$

Coordinate z
z = hc = 8.00 m 6.1.2(2)

Vertical pressure pvf
$${\mathrm p}_\mathrm{vf}\;=\;\mathrm\gamma\;\cdot\;({\mathrm h}_\mathrm o\;-\;\frac1{\mathrm n\;+\;1}\;\cdot\;({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o\;-\;\frac{(\mathrm z\;+\;{\mathrm z}_\mathrm o\;-\;2\;\cdot\;{\mathrm h}_\mathrm o)^{\mathrm n+1}}{({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o)^\mathrm n})\;(5.79)$$
$${\mathrm p}_\mathrm{vf}\;=\;16.00\;\cdot\;(0.61\;-\;\frac1{-1.55\;+\;1}\;\cdot\;(6.07\;-\;0.61\;-\;\frac{(8.00\;+\;6.07\;-\;2\;\cdot\;0.61)^{-1.55+1}}{(6.07\;-\;0.61)^{-1.55}})\;=\;69.27\;\mathrm{kN}/\mathrm m²$$

Bottom load magnifier Cb
Cb = 1.0 (6.3)
The bottom load magnifying factor Cb applies to silos of Action Assessment Class 2 under the condition that the stored solids do not tend to dynamic behaviour.

Mean vertical pressure at the hopper transition
pvtf = Cb · pvf (6.2)
pvtf = 1.0 · 69.27 = 69.27 kN/m²

Mobilized friction
In a shallow hopper, the waIl friction is not fully mobilized. The mobilized or effective wall friction coefficient should be determined as:
$${\mathrm\mu}_\mathrm{heff}\;=\;\frac{1\;-\;\mathrm K}{2\;\cdot\;\tan\;\mathrm\beta}\;(6.26)$$
$${\mathrm\mu}_\mathrm{heff}\;=\;\frac{1\;-\;0.450}{2\;\cdot\;\tan\;39.8^\circ}\;=\;0.33$$

Parameter n
n = S · (1 - b) · μheff · cot β (6.28)
S = 2 (6.9)
n = 2 · (1 - 0.2) · 0.33 · cot 39.8° = 0.634

Parameter Ff
$${\mathrm F}_\mathrm f\;=\;1\;-\;\frac{\mathrm b}{\displaystyle\frac{1\;+\;\tan\;\mathrm\beta}{{\mathrm\mu}_\mathrm{heff}}\;}\;(6.27)$$
$${\mathrm F}_\mathrm f\;=\;1\;-\;\frac{0.2}{\displaystyle\frac{1\;+\;\tan\;39.8^\circ}{0.33}\;}\;=\;0.943$$

Parameter n
n = S · (Ff · μheff · cot β + F) - 2 (6.8)
n = 2 · (0.943 · 0.33 · cot 39.8° + 0.943) - 2 = 0.634

Normal pressure
pnf(x) = Ff · pv(x) (6.29)
$${\mathrm p}_\mathrm{nf}(\mathrm x)\;=\;{\mathrm F}_\mathrm f\;\cdot\;(\mathrm\gamma\;\cdot\;\frac{{\mathrm h}_\mathrm h}{\mathrm n\;-\;1}\;\cdot\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h}\;-\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h})^\mathrm n)\;+\;{\mathrm p}_\mathrm{vft}\;\cdot\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h})^\mathrm n$$
pnf(0.00) = 0.00 kN/m²
pnf(1.00) = 52.97 kN/m²
pnf(2.00) = 63.72 kN/m²
pnf(3.00) = 65.33 kN/m²

This load can be entered in RFEM as a free variable load. The load input is displayed in Figure 03.

Figure 03 - Normal Pressure pnf

Hopper frictional traction
ptf(x) = μheff · Ff · pv(x) (6.30)
ptf(0.00) = 0.00 kN/m²
ptf(1.00) = 0.33 · 52.97 = 17.48 kN/m²
ptf(2.00) = 0.33 · 63.72 = 21.03 kN/m²
ptf(3.00) = 0.33 · 65.33 = 21.56 kN/m²

This load can be entered in RFEM as free variable load. The load input is displayed in Figure 04.

Figure 04 - Hopper Frictional Traction ptf

Reference

[1]  Eurocode 1 - Actions on structures - Part 4: Silos and tanks; EN 1991-4:2010-12

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