Engineering task and calculation objective
The EC3Cone module designs conical silo hoppers to DIN EN 1993-4-1, Section 6. The hopper is the most heavily loaded component of a silo: it carries the entire weight of the stored bulk solid via meridional membrane forces into the transition junction with the cylindrical shell, where a ring stiffener additionally takes up the radial deviation forces. The governing checks are the verification of the meridional membrane force resistance at the top of the hopper and the assessment of the plastic mechanism at the transition junction or at a change of plate thickness.
The calculation covers the hopper half angle, the local wall thicknesses of hopper, cylinder and skirt at the transition junction, the wall friction coefficient of the hopper wall, and the bulk solids pressures immediately above and below the transition. Unsymmetrical filling effects are accounted for through the unsymmetrical stress augmentation factor for the meridional force. As results, the module delivers the design value of the meridional membrane force resistance and checks condition 6.3.2.4 for the design membrane forces; in addition, it checks whether buckling becomes relevant for the hopper.
Together with the modules EC3Cyl (cylindrical shell) and EC3Roof (conical roof), this makes it possible to calculate a complete silo to the Eurocode.
Standard and calculation basis: DIN EN1993-4-1:2017-09 Abschnitt 6
Calculation workflow
- Define geometry and material: The inputs are the hopper half angle, the local wall thicknesses of hopper, cylinder and skirt at the transition junction, the radius at the transition, and the yield strength, tensile strength and modulus of elasticity of the material with the partial safety factors for resistance. The hopper apex angle is checked against the limits of applicability.
- Determine bulk solids loads and meridional forces: From the bulk solids pressures (per EN 1991-4) and the wall friction coefficient of the hopper wall, the design values of the meridional membrane force per unit circumference at the top of the hopper and of the local meridional force are obtained. Unsymmetrical filling and discharge are covered by the unsymmetrical stress augmentation factor for the meridional force.
- Verify the meridional membrane force resistance: The design value of the meridional membrane force resistance at the transition junction is determined via the plastic mechanism at the top of the hopper or at the change of plate thickness and compared with the acting design meridional membrane force (condition 6.3.2.4 (3)).
- Verify the transition junction and transition ring: At the nodal line of the transition junction, the effective radial force and the effective moment are calculated from the forces of hopper and cylinder. The ring stiffener at the junction is verified with its cross-sectional area (without any effective contributions from the shell) for the circumferential compressive force; the local bending stress at the top of the hopper is evaluated.
- Check buckling relevance: Finally, it is checked whether buckling can become governing for the hopper – for example for thin-walled, steep hoppers under unsymmetrical loading – and, if necessary, the supplementary stability verification is triggered.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Check hopper apex half angle | Winkel | – |
| Increased partial factor for resistance 0 | γMg0 | - |
| Design value of the local meridional force per unit circumference | nϕh,Ed | N/mm |
| Unsymmetrical stress augmentation factor | gasym | - |
| Design value of the meridional membrane force per unit circumference at the top of the hopper | nϕh,Ed,s | N/mm |
| Factor for design resistance of the hopper at the transition joint | kr | - |
| Local wall thickness | t | mm |
| Material | Wknr | – |
| Tensile strength | fu | N/mm² |
| Partial factor for resistance 2 | γM2 | - |
| Design resistance of the hopper at the transition joint | nϕh,Rd | N/mm |
| Hopper half angle | β | ° |
| Rradius at the top of the plastic mechanism (hopper top or change of plate thickness) | r | mm |
| Wall friction coefficient for the hopper | μ | - |
| Yield strength | fy | N/mm² |
| Design resistance | nϕ,Rd | N/mm |
| Design value of the meridional membrane stress resultant at the top of the hopper | nϕ,Ed | N/mm |
| 6.3.2.4 - (3) - Condition Design stresses | (3) | – |
| Effective radial force acting on the transition ring | Fe,Ed | N |
| Effective radial moment acting on the transition ring | Me,Ed | N·mm |
| Force hopper | Fh | N |
| Force cylinder | Fc | N |
| Local meridian coordinate cylinder | xc | mm |
| Local meridian coordinate hopper | xh | mm |
Frequently asked questions
Where do the bulk solids pressures for the hopper calculation come from?
The actions are provided by DIN EN 1991-4 (actions on silos): filling and discharge pressures depending on the bulk solids properties (unit weight, angle of internal friction, wall friction coefficient) and on the flow pattern (mass flow or funnel flow). The design to EN 1993-4-1 assumes these pressures as given; the decisive point is to apply the filling and discharge states consistently with the same bulk solids properties (upper/lower characteristic values).
Why is the transition from the cylinder to the hopper so critical?
At the transition junction the meridional force abruptly changes direction: the inclined hopper force produces an inward radial component that must be resisted as a circumferential compressive force in the transition ring. At the same time, local bending stresses occur at the top of the hopper. Failure modes are plastic failure of the ring region or buckling of the ring – which is why the standard requires dedicated checks here using the local wall thicknesses of all adjoining shell segments.
What does the unsymmetrical stress augmentation factor for the meridional force do?
Real silos are never loaded perfectly axisymmetrically: eccentric filling, funnel flow eccentricities and patch loads increase the meridional force locally above the mean value. The augmentation factor covers this asymmetry in a lump-sum manner without requiring an elaborate unsymmetrical shell analysis. It depends on the silo slenderness and on the eccentricity of filling or discharge.
When does buckling become relevant in the hopper?
The hopper is predominantly in meridional tension and is therefore less prone to buckling than the cylinder. It can become critical for very thin-walled hoppers when circumferential compressive stresses occur – for example near the transition junction, under unsymmetrical discharge, or due to local bending at a change of plate thickness. The module checks this relevance and reports whether a supplementary buckling verification is required.