# Loads on Silo Hopper According to EN 1991-4

### Technical Article

001450 13 June 2017

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.

#### 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.

#### 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.

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.

#### Reference

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

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