## Abstract

A steady state, continuum model for stresses in a circular silo with an off-centre circular core has been developed. The model proposes a geometry for the principal stress cap, resulting in the definition of local and general 3-dimensional co-ordinate systems. Three steady state, orthogonal force balances were conducted which gave partial differential equations in terms of the three principal stresses.

Initial solutions were limited to completely filled systems in order to test the model and compare to existing data and models. Solutions to systems with finite, off-centre cores will be considered in a subsequent paper. Numerical solutions of the three equations were implemented. The solutions were tailored to give outputs consistent with the Janssen equation for symmetrical stress systems in completely filled silos. The solutions required assumptions about the nature of internal stress distributions. There are few data on internal stress distributions in hoppers and silos, and so the assumptions were necessary to facilitate the solutions. Other solutions to the fundamental force balance equations could be implemented as more information becomes available.

The models were then applied to systems with eccentric stress distributions, and a simple model of surcharge was proposed.

Two types of stress eccentricity were identified:

surcharge eccentricity, where an underlying symmetrical stress system is surmounted by an eccentric surcharge.

inherent eccentricity, where the underlying stress distribution is eccentric with an eccentric centre of stress.

The two types of system have different properties and it is not possible to identify the underlying stress distribution by the shape of the solids surface within the hopper.

The model gave a discontinuity in principal stress, σ3, at the centre of stress, leading to the concept of a virtual core. This presents theoretical and practical issues. The model was then compared to an extensive set of DEM data generated for an eccentrically loaded silo. The outputs, in terms of wall shear stresses, compared reasonably well with the DEM data, but it was possible to get reasonable agreement over a range of model parameters, including underlying symmetrical stress.

The inability to identify underlying stress distribution from surface profiles, plus the fact that a given wall stress distribution can be modelled by a range of internal parameters, make the identification of internal stress systems from surface and external measurement extremely challenging.

Initial solutions were limited to completely filled systems in order to test the model and compare to existing data and models. Solutions to systems with finite, off-centre cores will be considered in a subsequent paper. Numerical solutions of the three equations were implemented. The solutions were tailored to give outputs consistent with the Janssen equation for symmetrical stress systems in completely filled silos. The solutions required assumptions about the nature of internal stress distributions. There are few data on internal stress distributions in hoppers and silos, and so the assumptions were necessary to facilitate the solutions. Other solutions to the fundamental force balance equations could be implemented as more information becomes available.

The models were then applied to systems with eccentric stress distributions, and a simple model of surcharge was proposed.

Two types of stress eccentricity were identified:

surcharge eccentricity, where an underlying symmetrical stress system is surmounted by an eccentric surcharge.

inherent eccentricity, where the underlying stress distribution is eccentric with an eccentric centre of stress.

The two types of system have different properties and it is not possible to identify the underlying stress distribution by the shape of the solids surface within the hopper.

The model gave a discontinuity in principal stress, σ3, at the centre of stress, leading to the concept of a virtual core. This presents theoretical and practical issues. The model was then compared to an extensive set of DEM data generated for an eccentrically loaded silo. The outputs, in terms of wall shear stresses, compared reasonably well with the DEM data, but it was possible to get reasonable agreement over a range of model parameters, including underlying symmetrical stress.

The inability to identify underlying stress distribution from surface profiles, plus the fact that a given wall stress distribution can be modelled by a range of internal parameters, make the identification of internal stress systems from surface and external measurement extremely challenging.

Original language | English |
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Pages (from-to) | 330-348 |

Number of pages | 19 |

Journal | Chemical Engineering Research and Design |

Volume | 93 |

Early online date | 18 Jul 2014 |

DOIs | |

Publication status | Published - Jan 2015 |

## Keywords

- bulk solids handling
- wall friction
- circular silo