1.10 Explanatory notes on terminology and notation
1.10.8 Limit state terminology and notation
Because this Guide may be used by some designers who do not have a modern structural engineering training, the following short explanation is given of the limit state terminology and notation found in this Guide. Three subscripts are normally used, the first to indicate the direction of a stress, the second whether it is on the loading or resistance side of the assessment, and the third the significance of the stress in the probability distribution [ENV 1991–1, 1994].
The stresses or stress resultants (forces and moments) in the structure act in different directions, so the first subscript is used to indicate a direction. In general, this is x when in the direction of the axis of the silo, θ when in the hoop or circumferential direction, and ϕ when in the sloping direction of the meridian of the hopper. For shears, two subscripts are used in this role. Sometimes, it is necessary to combine stresses acting in different directions into a von Mises effective stress, and this is given the subscript e.
The structural evaluations undertaken in this Guide are based on the Eurocode limit state format. The second subscript identifies whether it is a load effect or resistance that is being determined. The actions (loads) on the structure induce a set of stress resultants within the structure which are given the subscript S (or for stresses E). The strengths or resistances that the structure can offer to those stress resultants are given the subscript R. The third subscript indicates the probability of this magnitude arising. The actions (loads) and resistances are both evaluated as values that have a notional 5% probability of occurring: these are termed “characteristic” values and given the subscript k. When values for checking the safety of the design are obtained, the actions have been increased by γF from the characteristic value to a “design” value, and the resistances have been decreased by
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γM from the characteristic value to a design value. Both of these design values are given the subscript d. This
subscript is also used for simpler measures of resistance, as in c for the elastic buckling strength of a perfect structure and p for the plastic collapse resistance.
The check for the structural integrity of the design is that the values S should always be less than the values R. 1.10.9 Consistent units
The equations used throughout this Guide are dimensionally consistent. This means that it is not necessary to be sure to adopt special units in any particular equation. However, a consistent set of units for calculations is desirable.
The following two alternative sets of recommended consistent units are taken from Eurocode 3 Part 4.1 [ENV 1993–4–1, 1999].
dimensions and thicknesses : m mm
unit weight : kN/m3 N/mm3
forces and loads : kN N
line forces and line loads : kN/m N/mm
pressures and area distributed actions : kPa MPa
unit mass : kg/m3 kg/mm3
acceleration : km/s2 m/s2
membrane stress resultants : kN/m N/mm
bending stress resultants : kNm/m Nmm/mm
stresses and elastic moduli : kPa MPa
Conversion factors: 1 MPa=1N/mm2 1 kPa=1kN/m2 1 bar ≈ 100 kPa
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Chapter 2
Definitions and notation 2.1 Definitions
The following definitions apply throughout the Guide. Where the term has more than one distinct meaning, the alternatives are separated as (a), (b), (c), etc. Terms which are defined elsewhere in the definitions list are shown in italics.
Aeration Where air is entrained in a fine particulate solid (powder), the solid is said to be aerated. This can arise either from deliberate air injection, from pneumatic conveying or from the filling or
discharge processes.
Angle of repose
The steepest angle that can be measured on a free surface on the top of the granular solid (defined in Section 4.9).
Apex of hopper
A hopper is terminated by the outlet. However, the geometry of the cone is defined in terms of the geometric apex, which is a point in space below the outlet (Figs 1.9, 7.7 and 8.3).
Arching Formation of an arch, bridge or dome in the stored solids over the silo outlet, usually in the hopper (Fig. 1.8) and with a void beneath it. This Guide refers only to stable static arches, which arrest the flow.
Axisymmetric shell
A shell structure whose geometry is defined by the rotation of a meridional line about a central axis (cylinder, cone, sphere, etc.).
Barrel sectionThe upper part of a silo with vertical parallel walls. Bending
moment
The bending moment per unit width of wall is a couple within the shell wall (no net force resultant). A shell carries bending moments in both the vertical and circumferential directions. The direction of the bending
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moment is defined by the direction of the stresses it induces (note the contrast with beam theory where bending is about an axis within the beam).
At any point in a shell, there are three bending moments (two direct moments and a twisting moment).
Bending stress The concept of a bending stress is useful under conditions where the silo wall is elastic. The
bending stress then varies linearly through the wall thickness from a tensile value on one face to an equal compressive value on the opposite face. Bending stresses affect conditions of yielding of the wall but do not affect the buckling response (cf. membrane stresses).
The bending stress distribution is characterised by the value of bending stress occurring on the outer surface (tensile positive on the outer surface).
Buckling The ultimate limit state for the structure, where it suddenly loses its stability under compression and/or shear. It leads either to very large displacements or to structural failure.
Bulk density The mass of a quantity of particulate solid divided by the total volume including interstitial voids. The bulk density depends on the packing structure and the stress history.
Bulk solid This term is used to refer to particulate solids when they are handled in large quantities. Whilst the definition is not strict, it is widely used when the conceptual image of the solid does not take account of the individual particles but treats them as a continuum.
Circumferential direction
The horizontal tangent to the silo wall at any point. It varies in direction around the silo, lies in the horizontal plane and is tangential to the silo wall irrespective of whether the silo is circular or rectangular in plan.
Cohesion The component of a particulate solid’s shear strength which is independent of the applied normal stress. A solid is termed one of low cohesion if the unconfined yield stress is less than 14 kPa after the solid has been precompressed to a stress of 100 kPa (Eurocode 1 Part 4).
Critical state The state in a particulate solid where large changes in deformations can occur without change to either the stress state or the total volume of the assembly of particles.
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Conical hopper
A hopper in the form of a cone with its apex slightly below the outlet. In conical hoppers, the solid is forced to converge on the outlet from all directions, making flow stoppages more likely, but the outlet is conveniently small and the structure is very strong. The other common form of hopper is wedge shaped.
ConsolidationThe process by which the particles of a particulate solid are squeezed together into a tighter
packing, principally by the application of stress (both direct and shear stress). In soil mechanics, this term relates to the movement of interstitial water, and is therefore very time-dependent. For dry granular solids in silos, consolidation may involve interstitial air movement, but it is usually taken to depend only on the stress.
Consolidation stress
The consolidation stress is the highest principal stress applied to a particulate solid during a stress history to obtain the flow function (Fig. 4.8). (see also major principal consolidating stress.)
Continuously supported
A continuously supported silo is one in which all positions around the circumference are supported in an identical manner. Minor departures from this condition (e.g. a small opening) need not affect the applicability of the definition.
Core flow Alternative term for funnel flow. Internationally, the term funnel flow is more widely used (e.g. Eurocode 1 Part 4) and is adopted here.
Dead zone A zone of bulk solid within a silo which does not freely discharge under gravity flow, but remains in the silo at the end of discharge.
Design situation
An event, combination of events, or set of physical conditions occurring during the lifetime of the structure for which the design must demonstrate that relevant limit states are not exceeded (see Eurocode 1 Part 1).
Discrete support
A discrete support is a point at which a silo is supported using a local bracket or column, giving a limited number of narrow supports around the silo circumference. Four or six discrete supports are commonly used, but three or more than six are also found.
Eccentric filling
A silo is said to be eccentrically filled when the top of the pile of solids at the top of the silo is not close to the
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central axis of the silo when the silo is in the full condition. The location of the top of the pile depends on the solids trajectory on entry.
Eccentric pipe flow
An eccentric flow involving a pipe flow in contact with the wall of the silo. The flow channel
boundary at any level includes part in contact with the wall and part involving a solid-solid
rupture surface.
Effective angle of internal friction
The friction angle within the solid that governs its flow and pressure behaviour (defined in Section 4.7).
Effective transition
The position on the vertical wall of a mixed flow silo where the flow channel boundary strikes the barrel section vertical wall (Figs 6.3, 6.5 and 6.6).
Equivalent surface
The level to which the silo would be filled if the same volume of material were arranged with a level surface (Fig. 6.6).
Expanded flow A flow pattern combining mass flow within a lower hopper with funnel flow in the upper part (Fig. 1.4d).
Fatigue The ultimate limit state for the structure where cyclic loading causes cracking or failure of the shell plate.
Feeder A device conveying material away from the outlet of the silo.
Flat-bottomed silo
A silo in which the base is flat. Eurocode 1 Part 4 groups these silos with those where the hopper has a slope of less than 20° to the horizontal.
Flow function The relationship for a particulate solid that describes the variation of the unconfined yield
stress with consolidation stress, as measured using a Jenike shear cell (see Section 4.8). Flow pattern The geometry of the moving solids when flow is well established (Figs 1.3 and 1.4). The silo
is still close to its full condition.
Flow channel, Flow channel boundary
The flow channel is the zone of moving material when a silo displays funnel flow. The flow channel boundary is the boundary between the flowing and static material (Fig. 6.6).
Fluidised solid A state of a stored fine particulate solid when its bulk contains a high proportion of interstitial air, with a pressure gradient which supports the weight of the
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particles. The air may be introduced either by aeration or by the filling process. A solid may be said to be partially fluidised when only part of the weight of particles is supported by the interstitial air pressure gradient.
Free flowing material
A particulate solid that does not develop significant cohesion when subjected to the stress levels typical of those occurring in silos.
Frictional traction
A stored solid in a silo applies both a normal pressure and a frictional shear stress on the silo wall. To give a clear distinction between the applied loading from the solid and shear stresses developing within the wall (of which there are many in different directions), standards for silo loading term this frictional shear the frictional traction on the wall.
Full condition A silo is said to be in the full condition when the top surface of the stored solid is at the highest
position considered possible under operating conditions during the design lifetime of the structure. This is the assumed design condition for the silo.
Funnel flow A flow pattern in which solid adjacent to some part of (core flow) the wall remains stationary whilst other parts move, when the silo is effectively in the full condition. The stationary parts may either be only near the outlet or may extend as high as the surface. This category is subdivided in this Guide into mixed flow and pipe flow (Fig. 1.4b and 1.4c).
Granular solid
See particulate solid.
Homogenising silos
A silo that is used for mixing processes and in which the stored solid is, at least at times, fully
fluidised.
Hopper A hopper is a converging section towards the bottom of a silo. It is used to channel solids towards a gravity discharge outlet. In this Guide the hopper is generally conical.
Hopper flow factor
A measure of the capacity of a hopper to discharge a particulate solid of a given cohesion under gravity (see Section 5.4).
Hopper half angle
Angle between the vertical axis of a conical hopper and the hopper wall.
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Incomplete cleanout
The condition in which a stable body of solid remains in the bottom of the silo after all gravity flow (whether enhanced by flow aids or not) has ceased (Fig. 1.8). It may occur in hoppers, and usually occurs in a flat-bottomed silo. This material may be termed dead. Flow is arrested.
Intermediate aspect ratio silo
A silo in which the overall aspect ratio hb/dc of the stored solid lies between 1.00 and 1.50 (Fig. 1.9).
Internal pipe flow
When pipe flow occurs in a silo with a concentric or only slightly eccentric outlet, the flow may be termed ‘‘internal”. The material in a vertical or nearly vertical channel above the outlet is in motion, but is completely surrounded by stationary solid. For the flow to be termed internal, the flow channel must not intersect the silo walls before it reaches the top surface of the solid (Fig. 1.4c).
Junction A junction is the point at which any two or more shell segments meet. It can include a stiffener or not: the point of attachment of a ring stiffener may be treated as a junction.
Kick
pressure at the
transition
A local increase in pressure at the transition in a mass flow silo during discharge (Fig. 8.3). This term in not used extensively in this Guide, as the local increase is seen as part of a complete pressure distribution.
Limit state A condition in which the structure ceases to fulfil one of its principal design requirements. Limit states may be either a functional limit (serviceability) or a structural failure (ultimate) (see Section 1.6).
Major principal consolidating stress
The major principal stress in a particulate solid at the end of filling and storing (pvf). It is reproduced in a shear test on a solid sample as the stress σm. The value is estimated from the Mohr circle for steady state shearing (i.e. during shear of a particulate solid without change in its bulk density or stress, corresponding to the attainment of the critical state). See Section 4.8 and Fig. 4.8.
Mass flow The condition under which all particles of the particulate solid are in motion when the silo is discharging (Fig. 1.4a) from the full condition.
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Material flow function
See flow function.
Membrane stress
A stress component σm in the shell wall which is invariant through the wall thickness. Membrane stresses are mostly governed by equilibrium considerations, and control the buckling response (cf. bending stresses).
At any point in a shell, there are three membrane stress resultants (two direct resultants and a shear).
Membrane stress resultant
The membrane stress resultant n is the force per unit width of plate in the wall. It is equal to the membrane stress multiplied by the local shell wall thickness (n=σmt).
Meridional direction
The tangent to the silo wall in a vertical plane at any point. It varies according to the structural element being considered. Alternatively, it is the direction that a rain drop would take in sliding down the surface of the structure.
Mixed flowA flow pattern in which the lower part, near the outlet, has stationary particulate solid against the
wall, but the upper part has solid moving across the entire cross-section. The upper limit of the stationary solids is termed the “effective transition”. The silo bottom may be a hopper or a flat
bottom (Fig. 1.4b).
This term (taken from the French) is a preferred term in place of semi-mass flow, which is slightly ambiguous.
Mohr’s circle
The graphical representation of the state of stress in either a particulate solid or in a silo structure, using normal and shear stress as parameters (i.e. in the σ-τ plane: see Fig. 4.8), that permits the stresses acting on planes at different orientations to be found.
Outlet The opening at the base of a silo through which particulate solid discharges under gravity. Flow control devices (rotary valves, feeders, etc.) are normally used at or near the outlet. The outlet opening dimension refers to the opening at the base of the hopper. The outlet can be square or circular, and a transition may be required between the hopper and outlet.
Parallel pipe flow
A pipe flow in which the sides of the flow channel are close to vertical throughout most of the height of the silo.
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Partial factor
Either a multiplier on applied loads (γF) or a divisor on the strength of the structure (γM) that
provides a statistically defined safety margin between the consequences of design situations and the resistance of the structure at a limit state.
Particulate solid
A material formed from many discrete particles which interact with each other through contacts but may move independently. The assembly is assumed to contain enough particles for the statistical mean of any macroscopic property to be independent of the number of particles. This term is used in this Guide to include the terms bulk solid, granular solid, particulate material and powder (the latter has a special meaning within this definition).
Pipe flow A flow pattern in which the particulate solid in a vertical or nearly vertical channel above the outlet
is in motion, but is surrounded by stationary solid. The flow may strike the silo wall if the outlet is eccentric or if specific factors cause the channel location to move from above the outlet.
This term is used to provide a clear distinction between two types of funnel flow. If an internal flow channel does not reach the silo wall within the height of the stored solid, the flow is here termed “internal pipe flow”.
Pipe flows are divided into two categories: internal pipe flow and eccentric pipe flow.
Plastic limit state
The ultimate limit state for the structure in which it develops zones of yielding in a pattern such that its ability to resist increased loading is deemed to be exhausted. It can be related to a small
deflection theory limit load or plastic collapse mechanism.
Powder A particulate solid in which more than 50% of the mass is made up of particles smaller than 50 microns (0.05 mm) in diameter. This limit is given for guidance only: the change in behaviour from coarse solids to powders is gradual as the particle size distribution changes.
Pressure The force per unit area exerted on the silo wall by the particulate solid. The term “pressure’’ is reserved for the interface between the solid and the structure to match accepted structural engineering terminology.
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This restriction ensures that the term “stress” can be reserved for structural stresses within the silo