Loads on Silo Hopper According to EN 1991-4

Technical Article

001450

12 June 2017

This article describes the actions on silos according to DIN EN 1991-4. Using the example of a free-standing cylindrical silo with a conical funnel for cement, the filling loads on the hopper are calculated.

System and structural components

The system is shown in Figure 1.

[1] , 6.1.1 (2) taking into account the inclination of the funnel walls according to the following classification:

• A plane floor is assumed if the inclination angle of the floor relative to the horizontal α is less than 5 °.
• You can assume a funnel with a slight inclination if the two other cases mentioned above are not applicable.
• A steep funnel is given if the following criterion is fulfilled:
$$\tan\;\mathrm\beta\;<\;\frac{1\;-\;\mathrm K}{2\;\cdot\;{\mathrm\mu}_\mathrm h}\;(6.1)$$

$$\tan\;39,8^\circ\;=\;0,83\;>\;\frac{1\;-\;0,450}{2\;\cdot\;0,458}\;=\;0,60$$

The funnel is considered as flat.

Characteristic depth z o according to the method according to Janssen
$${\mathrm z}_\mathrm o\;=\;\frac1{\mathrm K\;\cdot\;\mathrm\mu}\;\cdot\;\frac{\mathrm A}{\mathrm U}\;(5.75)$$
$${\mathrm z}_\mathrm o\;=\;\frac1{0,450\;\cdot\;0,458}\;\cdot\;\frac{19,63}{15,71}\;=\;6,07\;\mathrm m$$

Vertical distance h o
The vertical distance h o between the equivalent material surface and the highest contact location of the stored bulk material with the wall can be assumed for a symmetrically filled circular silo with:
$${\mathrm h}_\mathrm o\;=\;\frac{{\mathrm d}_\mathrm c}6\;\cdot\;\tan\;{\mathrm\phi}_\mathrm r\;(5.78)$$
$${\mathrm h}_\mathrm o\;=\;\frac{5,00}6\;\cdot\;\tan\;36,00^\circ\;=0,61\;\mathrm m$$

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 = h c = 8.00 m 6.1.2 (2)

$${\mathrm p}_\mathrm{vf}\;=\;\mathrm\gamma\;\cdot\;({\mathrm h}_\mathrm o\;-\;\frac1{\mathrm n\;+\;1}\;\cdot\;({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o\;-\;\frac{(\mathrm z\;+\;{\mathrm z}_\mathrm o\;-\;2\;\cdot\;{\mathrm h}_\mathrm o)^{\mathrm n+1}}{({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o)^\mathrm n})\;(5.79)$$
$${\mathrm p}_\mathrm{vf}\;=\;16,00\;\cdot\;(0,61\;-\;\frac1{-1,55\;+\;1}\;\cdot\;(6,07\;-\;0,61\;-\;\frac{(8,00\;+\;6,07\;-\;2\;\cdot\;0,61)^{-1,55+1}}{(6,07\;-\;0,61)^{-1,55}})\;=\;69,27\;\mathrm{kN}/\mathrm m²$$

Soil load augmentation factor C b
C b = 1.0 (6.3)
The soil load increase factor C b has been applied for silos of requirement class 2 on the condition that the supported bulk material has no tendency to dynamic behavior.

Mean vertical loads at the funnel transition
p vtf = C b · p vf (6.2)
p vtf = 1.0 · 69.27 = 69.27 kN / m²

Mobilized Friction
In a hopper that is inclined at a slight angle, the wall friction is not fully mobilized. The mobilized or effective coefficient of wall friction should be applied 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) · μ heff · cot β (6.28)
S = 2 (6.9)
n = 2 · (1 - 0.2) · 0.33 · cot 39.8 ° = 0.634

Parameter F f
$${\mathrm F}_\mathrm f\;=\;1\;-\;\frac{\mathrm b}{\displaystyle\frac{1\;+\;\tan\;\mathrm\beta}{{\mathrm\mu}_\mathrm{heff}}\;}\;(6.27)$$
$${\mathrm F}_\mathrm f\;=\;1\;-\;\frac{0,2}{\displaystyle\frac{1\;+\;\tan\;39,8^\circ}{0,33}\;}\;=\;0,943$$

Parameter n
n = S · (F f · μeffeff c + β) + 2 (6.8)
n = 2 * (0.943 x 0.33 x cot 39.8 ° + 0.943) - 2 = 0.634

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

This load can be entered as a free variable load in RFEM. The load input can be seen in Figure 3.

 [1] Eurocode 1: Actions on structures - Part 4: Actions on silos and liquid containers; EN 1991-4: 2010-12