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Specialist article
01.01.2020  |  1076x
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Active and Passive Stress States – Relevance to Bulk Material Behavior

A grasp of the difference between active and passive stresses is necessary to understand how bulk materials behave in storage hoppers, silos and mechanical equipment, such as feeders and mixers. An active stress is one that presses onto a contact surface due to the forces generated within the body of the material. If the surface were to be slightly moved away from the material, such stresses will follow and continue to act with virtually the same pressure. A passive stress is caused by the resistance offered to a bulk material against any force trying to compact the mass of product. If the surface is withdrawn slightly, this pressure ceases. A simple case to illustrate the difference is that of retaining walls for a hopper or stockpile. Product piled against the walls exerts a force caused by the horizontal stresses generated within the bulk material. If the wall is withdrawn slightly, the material will normally collapse and form a new shape pressing with a similar force. An exception may occur if the material were cohesive, in which case the pile may stand as a vertical cliff because the internal strength of the material is sufficient to contain these internal stresses.

A contrasting situation is if an attempt were made to push the walls into the pile of material. This would give rise to a passive resistance with a magnitude much larger than the original active stress. It is easy to see that this basically is required to overcome the original active force pushing out against the wall, but also have the extra work of compressing the bulk and/or pushing up the level of the pile. Contrary to what the words ‘active’ and ‘passive’ may suggest, this example graphically indicates that passive stresses can be substantially greater that active stresses.

What generates an active stress in the first place? Well, a void gas under pressure is a classic form of an active stress and may be one component of this class of stress generated by a bulk material. When a bulk material is delivered into a container it is usually in a dilate condition. The particles move closer together as the bulk settles into a stationary bed. Air is expressed from the voids through the remaining interstitial gaps, the escape path becoming longer as the bed depth increases. With fine particles it can be a long process for the void air pressure to come to equilibrium with the surrounding ambient atmosphere. Until this happens, the void pressure supports part of the superimposed weight of the mass, thereby delaying the particle-to-particle contact pressure from achieving its ultimate value. Initially, the particle contact forces maybe so small that they can move against each other with ease, and the bulk can behave as a fluid. In these circumstances the active pressure on the wall is hydrostatic, dependent on the bed depth and the effective density of the product in this state.

With coarse materials this effect is less dramatic and short lived because the air can readily percolate through the larger void gaps. Nevertheless, there is almost invariably a temporary, declining void pressure in a recently filled bed of loose solids. The other, longer lasting source of active pressure is that arising from the particles being ‘squashed’ by the overburden. Any solid will tend to deform under stress, reducing in dimension in line with a compressive stress and, unless restrained, normally expanding to a less degree at right angles to the axis of the compressive stress. The ratio of this dimensional change is termed the ‘Poisson ratio’, and, for granular types of material, is generally of the order of 0.4. A simple compressive load of this type causes the ‘major principal stress’ in the mass and the transverse stress generated by this is called the ‘minimum principal stress, which is the stress needed to contain the ‘sides’ of the material to its initial dimensions.

During the filling of a silo or hopper the bulk material forms a growing pile that settles against the container walls with an active pressure in both the lower, converging section, and in the parallel body part of the storage facility. Now consider what occurs when discharge commences. As the restriction to flow from the outlet is removed, by opening a valve or starting a feeder, the pressure reduction at the outlet initiates a ‘wave of dilation’ to rise through the stored mass. Withdrawal of support allows the material to move downwards under the influence of gravity. In a mass flow hopper the total contents move, with material held in the body section of the container moving en-mass from the parallel section into the converging section. In the parallel section, the material moves in a ‘bed flow’ manner without changing in cross section, so the walls need only contain the active bulk stresses that remain virtually unchanged during this phase. As the bulk material enters the converging section, passive resisting forces are exerted by the walls to compress the material to a smaller cross-section. This passive resistance to deformation requires additional work input to that needed to contain the original active stresses. Furthermore, as it is considerably more difficult to overcome the initial structural resistance to deformation than it is to continue the failure process, there is a initial high ‘kick stress’ at the transition from the vertical walls, which then reduces to a lower passive stress as the deformation continues. The wall stress then continues to reduce further as the material approaches the final outlet, because the material is dilating and becoming weaker in the approach to the less confined conditions of the material flowing though the outlet.

In this converging section the stress pattern has also radically changed from that pertaining before flow commenced. Originally, the greatest stress was due to the vertical pressure of material weight. This is still the case in the parallel section of the bed but in the converging section the maximum stress direction is across the bed, causing the cross section to reduce in width during flow. A shear force also acts on the wall boundary due to the frictional drag on the contact surface. The combination of compressive and shear stress forms a maximum principal stress direction inclined upwards from the wall surface, the stress line bending over in a catenary arc within the bulk, to meet the opposing wall at a similar angle. The magnitude of this stress diminishes as the material nears the outlet, because the material is eventually not confined at the outlet location as the material falls or is taken away. The most likely location for a stable arch to form is near the outlet, where the span is small. If a stable arch forms the underside will not be confined and will be left as an exposed surface. This will happen if the unconfined failure strength of the material in this region exceeds the compressive stress across the arch surface available to cause the material to deform. The arch shape will follow the line of maximum principle stress inclined from the walls according to the angle of wall friction and take the general shape of a catenary. This stressed arch field in the converging section has significant bearing on the pressures acting through the outlet onto feeders. See note on ‘loads on feeders’.

Such a situation is more probable from first-fill condition because, as previously described, the stress required to ‘switch’ the state of wall stress from active to passive is much greater that the stress needed to sustain the passive stress. This situation will apply to material in the whole converging section, including the critical outlet region of smallest span, unless flow has taken place. For this reason, the Jenike method requires that a small amount of material is withdrawn from the outlet to initiate flow stresses in the region local to the outlet, before the vertical stresses raised by the weight of the material being loaded develops a stronger bulk material.
It is not usually important to initiate a flow field through the total converging hopper section because the span of the flow channel at the upper levels is normally greater than that of a potential arch. Flow opposing stresses within the material in this upper region are readily are overcome by the large work content available from the prevailing forces when flow eventually takes place. The stress pattern developed in the bulk during flow remains in place as flow stops, although time consolidation through settlement and other mechanisms according to the nature of the bulk material may increase the bulk strength of the material during prolonged storage.

The Jenike method of hopper design is founded upon measuring this state of stress, on the basis that once this flow state has been achieved it will reliably restart again when similar circumstances re-occur, i.e. when the outlet once again allows material to escape. This is provided the material is not allowed to stand for some time and allow the bulk to gain strength as the dilated flow condition settles to a firmer bed. Measurement of any increase in bulk strength with time is measured by ‘time compaction’ tests, where test samples are loaded in a static condition with a similar compacting load to that prevailing in outlet region the for an equivalent period to the time of standing without discharge taking place. However, this design method does not apply to first-fill conditions, because a ‘stressed arch field’ above the outlet has not been developed until flow has taken place. See notes on ‘the vertical shear cell’. Unless separate provision is made to initiate flow from a freshly filled hopper it is necessary to extract a small amount of material at an early stage of the filling process to affect the transition from an active to a passive stress state in material immediately above the outlet. In practice, the amount of material that needs to be taken out to cause this change, up to a span that is not capable of forming a stable arch, is very small but the effect is crucial to reliable performance.

The position is different with a hopper design that is not mass flow. This is generally called ‘funnel flow’ because of the characteristic shape of the flow channel that develops within the mass. A similar ‘stressed arch field’ develops in the flow channel but the boundary is of static product that invariably has a higher frictional resistance than smooth hopper walls. Such a flow regime can sustain a ‘rathole’, whereas a mass flow system cannot because the support boundary is constantly being taken away. The Jenike method allows an outlet size to be predicted that will generate reliable flow and also determine the maximum size that a stable rathole can hold firm. Not only is a larger size of opening needed to avoid arching than in a mass flow hopper but the design has to stimulate flow at a larger dimension that the largest rathole can exist. This feature illustrates to value of adopting an ‘expanded flow’ design for materials that do not require total mass flow, but exhibit poor flow characteristics and a tendency to form ratholes.

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