# Loads on Silo Hopper According to EN 1991-4

### Technical Article

001450 06/13/2017

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.

Janssen characteristic depth zo

$$\begin{array}{l}{\mathrm z}_\mathrm o\;=\;\frac1{\mathrm K\;\cdot\;\mathrm\mu}\;\cdot\;\frac{\mathrm A}{\mathrm U}\;\;\;\;\;\;\;\;\;\;(5.75)\\{\mathrm z}_\mathrm o\;=\;\frac1{0.450\;\cdot\;0.458}\;\cdot\;\frac{19.63}{15.71}\;=\;6.07\;\mathrm m\end{array}$$

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:

$$\begin{array}{l}{\mathrm h}_\mathrm o\;=\;\frac{{\mathrm d}_\mathrm c}6\;\cdot\;\tan\;{\mathrm\phi}_\mathrm r\;\;\;\;\;\;\;\;\;\;(5.78)\\{\mathrm h}_\mathrm o\;=\;\frac{5.00}6\;\cdot\;\tan\;36^\circ\;=\;0.61\;\mathrm m\end{array}$$

Parameter n

$$\begin{array}{l}\mathrm n\;=\;-(1\;+\;\tan\;{\mathrm\phi}_\mathrm r)\;\cdot\;(\frac{1\;-\;{\mathrm h}_\mathrm o}{{\mathrm z}_\mathrm o})\;\;\;\;\;\;\;\;\;\;(5.76)\\\mathrm n\;=\;-(1\;+\;\tan\;36^\circ)\;\cdot\;(\frac{1\;-\;0.61}{6.07})\;=\;-1.55\;\end{array}$$

Coordinate z

$$\mathrm z\;=\;{\mathrm h}_\mathrm c\;=\;8.00\;\mathrm m\;\;\;\;\;\mathrm{according}\;\mathrm{to}\;6.1.2(2)$$

Vertical pressure pvf

$$\begin{array}{l}{\mathrm p}_\mathrm{vf}\;=\;\mathrm\gamma\;\cdot\;({\mathrm h}_\mathrm o\;-\;\frac1{\mathrm n\;+\;1}\;\cdot\;({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o\;-\;\frac{(\mathrm z\;+\;{\mathrm z}_\mathrm o\;-\;2\;\cdot\;{\mathrm h}_\mathrm o)^{\mathrm n+1}}{({\mathrm z}_\mathrm o\;-\;{\mathrm h}_\mathrm o)^\mathrm n})\;\;\;\;\;\;\;\;\;\;(5.79)\\{\mathrm p}_\mathrm{vf}\;=\;16.00\;\cdot\;(0.61\;-\;\frac1{-1.55\;+\;1}\;\cdot\;(6.07\;-\;0.61\;-\;\frac{(8.00\;+\;6.07\;-\;2\;\cdot\;0.61)^{-1.55+1}}{(6.07\;-\;0.61)^{-1.55}})\;=\;69.27\;\mathrm{kN}/\mathrm m^2\end{array}$$

$${\mathrm C}_\mathrm b\;=\;1.0\;\;\;\;\;\;\;\;\;\;(6.3)$$

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 behavior.

Mean vertical pressure at the hopper transition

$$\begin{array}{l}{\mathrm p}_\mathrm{vtf}\;=\;{\mathrm C}_\mathrm b\;\cdot\;{\mathrm p}_\mathrm{vf}\;\;\;\;\;\;\;\;\;\;(6.2)\\{\mathrm p}_\mathrm{vtf}\;=\;1.0\;\cdot\;69.27\;=\;69.27\;\mathrm{kN}/\mathrm m^2\end{array}$$

Mobilized friction

In a shallow hopper, the waIl friction is not fully mobilized. The mobilized or effective wall friction coefficient should be determined as:

$$\begin{array}{l}{\mathrm\mu}_\mathrm{heff}\;=\;\frac{1\;-\;\mathrm K}{2\;\cdot\;\tan\;\mathrm\beta}\;\;\;\;\;\;\;\;\;\;(6.26)\\{\mathrm\mu}_\mathrm{heff}\;=\;\frac{1\;-\;0.450}{2\;\cdot\;\tan\;39.8^\circ}\;=\;0.33\end{array}$$

Parameter n

$$\begin{array}{l}\mathrm n\;=\;\mathrm S\;\cdot\;(1\;-\;\mathrm b)\;\cdot\;{\mathrm\mu}_\mathrm{heff}\;\cdot\;\cot\;\mathrm\beta\;\;\;\;\;\;\;\;\;(6.28)\\\mathrm S\;=\;2\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6.9)\\\mathrm n\;=\;2\;\cdot\;(1\;-\;0.2)\;\cdot\;0.33\;\cdot\;\cot\;39.8^\circ\;=\;0.634\end{array}$$

Parameter Ff

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

Parameter n

$$\begin{array}{l}\mathrm n\;=\;\mathrm S\;\cdot\;({\mathrm F}_\mathrm f\;\cdot\;{\mathrm\mu}_\mathrm{heff}\;\cdot\;\cot\;\mathrm\beta\;+\;\mathrm F)\;-\;2\;\;\;\;\;\;\;\;\;\;(6.8)\\\mathrm n\;=\;2\;\cdot\;(0.943\;\cdot\;0.33\;\cdot\;\cot\;39.8^\circ\;+\;0.943)\;-\;2\;=\;0.634\end{array}$$

Normal pressure

$$\begin{array}{l}\begin{array}{l}{\mathrm p}_\mathrm{nf}(\mathrm x)\;=\;{\mathrm F}_\mathrm f\;\cdot\;{\mathrm p}_\mathrm v(\mathrm x)\;\;\;\;\;\;\;\;\;\;(6.9)\\{\mathrm p}_\mathrm{nf}(\mathrm x)\;=\;{\mathrm F}_\mathrm f\;\cdot\;(\mathrm\gamma\;\cdot\;\frac{{\mathrm h}_\mathrm h}{\mathrm n\;-\;1}\;\cdot\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h}\;-\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h})^\mathrm n)\;+\;{\mathrm p}_\mathrm{vft}\;\cdot\;(\frac{\mathrm x}{{\mathrm h}_\mathrm h})^\mathrm n\end{array}\\{\mathrm p}_\mathrm{nf}(0.00)\;=\;0.00\;\mathrm{kN}/\mathrm m^2\\{\mathrm p}_\mathrm{nf}(1.00)\;=\;52.97\;\mathrm{kN}/\mathrm m^2\\{\mathrm p}_\mathrm{nf}(2.00)\;=\;63.72\;\mathrm{kN}/\mathrm m^2\\{\mathrm p}_\mathrm{nf}(3.00)\;=\;65.33\;\mathrm{kN}/\mathrm m^2\end{array}$$

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

Hopper frictional traction

$$\begin{array}{l}\begin{array}{l}{\mathrm p}_\mathrm{tf}(\mathrm x)\;=\;{\mathrm\mu}_\mathrm{heff}\;\cdot\;{\mathrm F}_\mathrm f\;\cdot\;{\mathrm p}_\mathrm v(\mathrm x)\;\;\;\;\;\;\;\;\;\;(6.30)\\\end{array}\\\;{\mathrm p}_\mathrm{tf}(0.00)\;=\;0.00\;\mathrm{kN}/\mathrm m^2\\\;{\mathrm p}_\mathrm{tf}(1.00)\;=\;0.33\;\cdot\;52.97\;=\;17.48\;\mathrm{kN}/\mathrm m^2\\\;{\mathrm p}_\mathrm{tf}(2.00)\;=\;0.33\;\cdot\;63.72\;=\;21.03\;\mathrm{kN}/\mathrm m^2\\\;{\mathrm p}_\mathrm{tf}(3.00)\;=\;0.33\;\cdot\;65.33\;=\;21.56\;\mathrm{kN}/\mathrm m^2\end{array}$$

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