KONA Powder and Particle Journal No. 41 (2024) 108–122/https://doi.org/10.14356/kona.2024013

Review Paper

A Review of Analytical Methods for Calculating Static
Pressures in Bulk Solids Storage Structures †George Dyck 1, Adam Rogers 2 and Jitendra Paliwal 1*
1 Department of Biosystems Engineering, University of Manitoba, Canada
2 AGCO Corporation, Canada

The Janssen equation is a widely used method for calculating pressures in bulk storage
structures. This review explores the historical legacy of Janssen’s equation and its applications
in both planar and three-dimensional structures. Our focus is on the limitations of the original
formulation of Janssen, extensions made to avoid these deficiencies, and alternative models
that have been developed. The motivation behind these modifications is to improve therepresentation of shear stress within a grain bin in both the horizontal and vertical directions.
Modifications to Janssen’s basic assumptions include the vertical-to-horizontal stress ratio(k), the coefficient of friction between the wall and the stored bulk material (μ), internal angleof friction (ϕ), and bulk density (ρ). We also discuss recent developments in pressure theories,
which have provided new insights into pressure fields in bulk storage bins. These modern
approaches include the continuum elastic theory and microscopic theory. Finally, we discuss
recent developments in pressure theories which provide new insights into the storage of bulk
solids. Overall, this review provides a comprehensive overview of the Janssen equation andits historical development, limitations, and extensions, as well as recent advancements inpressure theory that offer a more accurate representation of pressure fields in bulk storage
structures.
Keywords: powder, particle, bulk solids, Janssen effect, stress distribution, granular matter

1. Introduction

A bulk solid is an aggregate material consisting of solid
particles and fluids (gases/air). Considered as a whole, thematerial has properties common to both fluids and solids,i.e., it can change its shape to fill volumes like a liquid but
responds to internal shear stresses like a solid (Walker,
1966). Bulk solids occupy an intermediate position on the
spectrum between liquids and solids. These materials exhibit many novel, counter-intuitive properties, for example
the stress (pressure) distribution within the material when it
is stored in bins. This behavior is associated with “theJanssen effect”, as pressure is commonly described by H.A.
Janssen’s equation. The Janssen effect predicted that thebottom of a bin did not experience the full force of theweight of the commodity stored below some critical height.
This phenomenon resulted from the friction between the
sidewall and the stored material: the walls carried some of
the material weight. Knowledge of internal and wall friction is essential to Janssen’s theory. Coulomb’s work on
friction and failure of soils due to shearing led to the firstunderstanding of the relationship between internal stress†*Received 31 May 2022; Accepted 22 March 2023
J-STAGE Advance published online 29 July 2023
Corresponding author: Jitendra Paliwal;
Add: Winnipeg, Manitoba, MB R3T 2N2, Canada
E-mail: j.paliwal@umanitoba.ca
TEL: +1-204-474-6700

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Depth

and the orientation of failure planes within a material. Rankine synthesized Coulomb’s work and Mohr’s descriptionof failure planes and provided an analysis of material failure. This paper will first discuss the historical developments leading up to pressure models before discussing
Janssen’s work, and then investigate important refinements
and critiques of the model over the past century. There arethree modern works with a similar historical perspective on
the topic of pressures in bulk storage structures. The firstwas the translation of Janssen’s entire original paper into
English, 111 years after its original publication and a century’s worth of citations (Sperl, 2006). This paper contains
original data and figures as well as a short commentary thatspeaks to the legacy of the original publication. The second
and third publications were conference papers by A.W.
Roberts presented during the PARTEC Congress held in
Nuremberg, Germany, 21–23 March 1995. He provided a
detailed description of Janssen’s work, and a review of thepast 100 years of bulk solids research (1895–1995; Roberts,
1995). The work was formally rewritten and later published(Roberts, 1998). The historical section is indebted to Milo
Ketchum’s “The Design of Bins, Walls and Grain Elevators”(Ketchum, 1911). While this paper provided a novel interpretation of Janssen’s equation, the alternative models havenot been as resilient to time as Janssen’s solution. Furthermore, the paper was written in 1998, and advancements incomputational technologies have allowed for new

Copyright © 2024 The Authors. Published by Hosokawa Powder Technology Foundation. This is an open
access article under the CC BY license (https://creativecommons.org/licenses/by/4.0/).

George Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

approaches to modeling bulk solids, which were before“insoluble problems” as noted by Jenike (1961). There will
be a brief overview of the experiments covered in this literature review for historical context. Information about thepressures in a bin can also provide insight into a wide rangeof physical phenomena. The pressure field affects bulk
solid’s packing, flowability, porosity, and tortuosity, toname a few. For example, Janssen along with empirical
packing equations has been used to predict the densitychange within a bin (Thompson et al., 1991; Bhadra et al.,2015, 2018; Turner et al., 2016; Cheng et al., 2017).

2. Historical Pressure Theories

This section focuses on a review of pressure models forgrain storage bins. Historically, the “Janssen Effect” wasknown before Janssen’s mathematical description, noted
anonymously by Gotthilf Heinrich Ludwig Hagen. Hagen’s
account is the oldest published theory of pressure in bulk
materials, 40 years before Janssen in 1852 (Hagen, 1852).
Hagen attempted to predict saturation pressure with respect
to height (sand). The paper was translated into English by
Tighe et al., which provided a detailed view of this seminal
work (Tighe and Sperl, 2007). Hagen suggested that thefriction on the side of the wall acted in the upward direction, opposing to the gravitational force. The pressure felt
by a “disk which is easily movable but seals tightly” wasgiven by the expression,

σ disc  r 2 πρ g y  2rρ g μy 2

(1)where r is the radius of the disc (bin), ρ the bulk density, y
the height of bulk material, and μ the friction coefficient.
(1)This equation reached a maximum at a characteristicheight—related to the diameter of the bin, height, and friction coefficient—and began to decrease afterwards, amathematical prediction but physically incorrect; Hagen
stated that the pressure followed this curve until this maximum was reached. After, the bin pressure remained constant at this maximum, which was a similar behavior to the“Janssen Effect”. The difference from Janssen’s formulation was that Hagen used a quadratic to explain the change
in pressure, whereas Janssen’s approach solved a differential equation whose solution was an exponential function,
which allowed for an asymptotic approach to the maximum
pressure. A noteworthy reference found in Hagen’s work
regarded an observation Huber-Bernand published in 1829,
likely the first published research on the so-called “Janssen
Effect,” which was highlighted in both Hagen and Janssen’s
publications. Huber-Bernand noticed that when he filled acontainer with eggs, added sand in the voids between theeggs extending several inches above the eggs, as well as a
25 kg weight on top, the eggs placed at the bottom of thecontainer remained unbroken, even though all this weightshould crush them. He concluded that only a portion of thepressure was transmitted to the bottom of the container,

Review Paper

indicating some kind of saturation depth (Sperl, 2006),(Tighe and Sperl, 2007). A Welsh engineer Isaac Roberts
published a crucial pre-Janssen study that observed an asymptotic trend in bulk solid bin pressures. He claimed thatthe pressures at the bottom of bins stopped increasing at theheight of two diameters, which was an empirical demonstration of the Janssen effect (Roberts, 1883). He conducted
two experiments on this topic. The first was prone to measurement errors due to the mechanical nature of the scales
used, an issue other researchers (e.g. Prante) would struggle with and overcome; the second experiment produced
reliable results that were widely referenced (Ketchum,1911). Ten years later, H.A. Janssen formulated his seminal
equation analytically modeling Roberts’s work, although
the work was not cited. Both Hagen and Roberts worked onmathematical descriptions of stresses of earth pressure
systems and experimental data from agricultural commodities. H.A. Janssen would be the first to formulate a simple
way to calculate pressures in deep bins, the standard reference for well over a century on the topic. This equation’s
influence was enormous and has been used in almost all
design codes and standards in the world, such as theEuropean bulk storage building codes (European Standard
EN 1991-4, 2006). The expression was also used in thecalculation of apparent weight for bin capacities by the
ASABE (American Society of Agricultural and BiologicalEngineers; ASABE, 2005). Janssen’s work has been usedby a vast community of researchers and industry because it
explained the complicated phenomenon of bin pressures
via a straightforward expression (Eqn. (7)). Janssen was anengineer from Germany; not much is known about the individual and his work outside of this one publication, a seminal paper for the science of bulk solids. Roberts postulates
that it was likely his work and records were destroyed
during the Second World War (Roberts, 1995). The life andwork of Janssen is accessed through his 1895 paper,“Experiments on Corn Pressure in Silo Cells.” One year
after its publication, an English translation of Janssen’s
abstract was published in the Proceedings of the Institutionof Civil Engineers, the work often cited as Janssen (1896).
Soon, the equation was added to the building practices
outline in the “Des Ingenieurs Taschenbuch or The Hütte Das Ingenieurwissen”, a book of references for practicing
engineers, initially written in German and now translated
into many languages (Ketchum, 1911). During this time,
many aspects of the work were validated, critiqued, andmodified, which is discussed later in this review. It should
also be noted that M. Koenen worked on a similar theory ofpressure but published it a year after Janssen. Koenen’s
work is challenging to find because it was never translated
from German to English, and the conclusions were passed
down by secondary literature. His most significant contribution suggested the lateral to vertical pressure ratio could
be calculated using established equations, such as Rankine’s

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active pressure state (Eqn. (14)), while Janssen experimentally determined its value (Reimbert M. and Reimbert A.,
1976; Schulze, 2008; Koenen, 1896). Koenen explicitly
connected Janssen’s work to the Mohr-Coloulmb failure
criterion by relating it to the principal stresses acting on amass. It is an interesting historical point that Janssen referenced two other works incorrectly, which Sperl addressedin the translation of Janssen’s original document (Sperl,
2006). The first was a fellow German researcher, C. Arndt,
and was used to discuss the then-new developments ingrain storage in North America. The paper’s title is “TheSilos of Galati and Braila”, which are Romanian cities that
hold the oldest concrete grain bins. These were built in thelate 1880s, designed by an engineer Anghel Saligny, andwere two of the earliest examples of reinforced concrete
bins. Janssen described these bins as “...iron-strengthened
brickwork, and in six-fold profiles like honeycombs. The
grain was introduced through the upper end of the cellthrough a hatch. For discharge, stock transfer, or embarkation, the bottoms of the cells provided close-able openings”(Sperl, 2006). This sounds similar to modern grain storage
bin design; Janssen saw these as the future of bin design
and wanted to provide mathematical models for predicting
pressures. The second citation was Hagen, who has been
discussed thoroughly above (Sperl, 2006). Ketchum discussed the design of concrete bins in Chapter XVIII of his
text “The Design of Bins, Walls and Grain Elevators”. Unfortunately, he did not mention Saligny and only discussed
the concrete bins of North America.

2.1 Quantitative description of the Janssen effectJanssen began with a force-balance equation of a thin
horizontal layer of bulk material in a deep bin. Note twothings: first, Janssen was investigating deep bins but thetheory can be applied to shallow bins (Xu and Liang,2022). Second, the following is not directly Janssen’s formulation but a modernization of the derivation for claritythat was made popular by Ketchum, but is essentially thesame (Ketchum, 1911). The first difference lies in incorporating a hydraulic radius RH, which generalized the container geometry from Cartesian (rectangles) to arbitrary
shapes. Secondly, in Janssen’s work, he expressed K = μk
and solved for K experimentally. The lateral pressure ratio,k, is defined as the ratio of the horizontal stress to the vertical effective stress. Within a filled container (bin), the force
balance on an infinitesimal thin disk element of granularmaterial in the vertical direction is given as:
Aσ y  g ρA d y  A σ y  d σ y   τ w C d y(2)where A is the cross-sectional area and C the circumferenceof the bin, ρ the bulk density of the material, g the gravita(2)tional acceleration, σx and σy the horizontal and verticalstresses, respectively, and τw the shear stress acting on thematerial at the wall, which opposes the gravitational force.

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Review Paper

Here, x and y are the horizontal and vertical coordinates,
respectively. It should be noted that the depth of the stored
material is given by y, increasing in the downward direction. The shear stress is related to the normal stress in thehorizontal direction through the wall friction coefficient μ
as follows:τ w  σ x μ  σ x tan w(3)with μ = tanϕw given by the wall friction angle ϕw. Finally,
the ratio of horizontal (σx) to vertical stress (σy) is assumed(3)
to be constant,

k

σx
σy

(4)

Defining the hydraulic radius
(5)

A
(4)
 RH
C

substituting Eqns. (3) and (4) into Eqn. (2) yields theJanssen differential equation,
(5)
dσy
μk

σ y  gρ(6)dy
RH
Solving for vertical pressure givesμyk(6) ρ g R 


H
RH
σy 
1
e



μk 


(7)

Janssen’s equation may also be derived from the fundamental(7) (standard) static equilibrium equations for an infinitesimal element of material in 2D Cartesian or cylindricalcoordinates. For a 2D Cartesian coordinate system, theequilibrium equations on an infinitesimal element of material in the vertical and horizontal directions are given as:
σ y
y



τ xy

 gρ

x

(8)
σ x τ xy

0
x
y
(8-1)
For the axisymmetric conditions where a cylindrical coordinate system (r,θ,z) is used, the following equilibrium
equations(8-2) are used:σ z τ rz τ rz


 gρ
r
z
r
(9)
σ r τ rz

0
r
z
(9-1)
These formulae were used by Rahmoun et al. (2008) and
Schillinger & Malla (2008) among others to derive pres(9-2)equations. The equilibrium approach is wellsure
established in the field of continuum mechanics (Negi andOgilvie, 1977), and Janssen’s results can be obtained fromthese expressions, with certain assumptions. For example,
assuming the shear stress is related to the normal stress byGeorge Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

equation τrz = μσr and using the ratio of principal stresses,
σr/σz = k, we obtain(10)

τ rz  z  μkσ z  zCombining Eqns. (9) and (10) with the definition of thehydraulic radius RH = b/2 reproduces the Janssen’s differ(10)Eqn. (6).
ential

2.2 Lateral pressure ratio k

In addition to producing the equation that bears his
name, Janssen also gathered experimental data to validate
his model for calculating pressures in bulk storage structures (Eqn. (7)). With novel instruments and measuring
techniques, Janssen measured μ, the pressure exerted on thebottom of the bin (σy) and had planned to measure the horizontal force on the wall (σx) but failed to succeed due tothe arching effects of the material and the construction of
the apparatus. He suggested an improvement that wasadapted by Jamieson in 1906. To solve Eqn. (6), Janssen
assumed that k was a constant, an assumption that many
would question (Ketchum, 1911). Typically, the verticalto-horizontal stress ratio k is assumed—unrealistically—tobe constant throughout a bin, in the range 0 for rigid solids
and 1 for liquids. For granular materials, the Rankine coefficient is often used which gives the stress ratio in terms of
the internal angle of friction ϕ,
k

(11)

1  sin 
1  sin 

In fact, even with the assumption of constant k there is
still a wide variety of values cited throughout the literature.
(11)
This
lack of consensus exists because k is not a fundamental, innate physical property of bulk solids; it is an emergent result of equilibrium, arising from the “fluid” nature ofbulk materials. The value of k is not realistically expected
to be constant within a storage structure at all. The simplifying assumption is made because Janssen’s equation
works well within a narrow column of material, but thevariation of k through a large volume of bulk material has
not been well-studied experimentally. It has been difficult
to include a varying k analytically. Coulomb, Rankine andJanssen all made the simplifying assumption of constant0 < k < 1. Coulomb’s approach produced a stress ratio
which related lateral and vertical pressure of soil, and wasdetermined by using the Moore-Coulomb failure criterion(Labuz and Zang, 2012) as follows:cos 2   ω
ka 

2


cos 2 ω sin w  β 1 (12)

sin w    sin   β cos w  ω cos ω  β 
(12)Review Papercos 2 (  ω)
kp 

2


cos 2 (ω) sin w  ω 1 sin w    sin   β cos w  ω cos  β  ω 

(13)where ka is the active pressure ratio (the bulk solid exerts(13)
pressure on the wall), kp the passive pressure ratio (the wall
exerts pressure on the bulk solid), ϕw the friction angle ofwall, ω the slope of the wall, ϕ the internal angle of friction,and β the angle of the backfill surface. Rankine provided asimplification of the above equations by assuming a frictionless and non-adhesive interface between the verticalwall and the bulk solid. Under such conditions the following stress ratios can be formulated (Rankine, 1856),π (14)
k a  tan 2   
4 2
π 
(15)
k p  tan 2   
4 2
(14)
Note that ka, shown in Eqn. (14), is mathematically
identical to Eqn. (11). Rankine’s formulation of k was oneof (15)the first theories of pressure for shallow bins, because
the pressure distributions within these structures behave
analogously to retaining walls. The assumptions fail when
the bin height is increased, as the pressure exerted on thebottom and sides of the bin no longer fit the observations
because the bin walls were assumed to be frictionless (Xuand Liang, 2022). This phenomenon is a property of bulk
solids, but not considered until Janssen. Pleissner convincingly established the variation of k at the beginning of the20th century. Many different numerical methods have adopted simplified versions of the Rankine formulation (14).
For example, the Eurocode (European Standard EN 19914, 2006) usedk  1.10  1  sin (16)

A mathematical model to account for vertical and radialchanges in stresses and k is discussed in detail later throughthe(16)
work of Zhang et al. (1998a). They provided a literature
review on the topic in the context of agricultural commodities, up to the publication date, and reported that there wasno consistent agreement on how pressure, and the resultingk, varied radially. Rusinek investigated the variation inpressure of rape seeds using a hydraulic pressure transducer to determine the mean normal stress and mean shear
stress on the wall, as well as the vertical pressure distribution, and found that k was significantly lower in the middleof the bin than the wall, and when wall friction increased,so did k (Rusinek, 2003). Horabik et al. discussed the behavior of k within bins (Horabik and Rusinek, 2002), andreported that the pressure ratio was mostly dependent on
the internal angle of friction but also the shape of the seed(particles), and inversely related to moisture content; theexperimental uniaxial compression test produced similar

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values to theoretical predictions given by Eqn. (16). Qadir
et al. (2010) showed experimentally that k increased withthe ratio of the individual grain and diameter of the bin. Sun
et al. presented four independent approaches to k and applied them to deep and shallow bins. The results werecompared with experimental data and three national standards (Sun et al., 2018). They also compared different failure criteria to the typical Mohr-Coulomb criterion andrecommended the Lade-Duncan (L-D) criterion to be used
to calculate wall pressure. Uniaxial compression tests
showed, for cereal grains, k decreased with increased moisture content (Horabik and Rusinek, 2002), whereas forcalcareous sand, using a similar procedure, increased watercontent initially increased k, which reached a maximum
and eventually declined again (Wang et al., 2020). Numerical studies that used discrete element methods suggested
that k did not change with polydispersity (Wiącek andMolenda, 2014). Zhang et al. applied a two-parameter
failure criterion that was typically used for soils to determine failure points (Zhang et al., 1994). When both parameters were nonzero, the lateral to vertical stress ratio
increased with the vertical stress. At the same time, in otherconfigurations, it was observed to be roughly equal to that
determined from the Mohr-Coulomb failure criterion. Xu& Liang considered the elastic deformation of the bin walls
in both static and dynamic states and demonstrated that the
Rankine formulation of k would underestimate lateralforces (Xu and Liang, 2022). Back et al. studied the effectof friction on the pressure ratio k. Specifically, they focusedon particle-particle friction, which the lateral pressure ratio
was associated with through analytical arguments (Back,2011). They established that k increased with particle–
particle friction and packing.

2.3 After Janssen

The reaction to Janssen’s work was immediate, andmany people rushed to both validate and critique his analysis in the decades after the paper was published.

2.3.1 Wilfrid Airy: 1898

Airy noted at the beginning of his paper both the worksof Isaac Roberts and Janssen, although he admitted he was
not able to find any, “Tables of coefficients of friction ofgrain, wither on grain, or wood, iron, or brickwork...”(Airy, 1898). Moreover, the reference to Janssen was to theEnglish translation of the abstract (Janssen, 1896), which
was the reason why Airy did not have access to the friction
coefficients. The translation only included the dimensionsof the test cell, the equation describing the vertical pressureof grain, empirical data on corn depth and the resulting side
pressure. Regardless, he noted these were reasonable investigations on the pressure of wheat (Janssen investigatedcorn) on the bottom of small bins. In the conclusion, Airycommented on the geometry of Roberts and Janssen’s work

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Review Paperand the limitation of “small bins”, precisely how it would
adjust the coefficient of friction on the rupture plane. Unfortunately, this topic was addressed by Janssen only in theoriginal German publication. Moreover, Airy cited a textbook by Weisbach to describe “semi-fluids” for which, “thepressure on the side of a bin was the maximum pressure
due to a wedge-shaped mass of the grain which might besupposed to separate from the general mass, and the angleof slope of the wedge-shaped mass which exerted the maximum pressure had to be determined” (Airy, 1898;
Weisbach, 1849). The Mohr-Coulomb failure criterion wasused as the basis for the pressure ratio. A slightly different
approach was the specific application to “fluid–
solids”, which are known in the modern literature as “bulksolids”. Airy was a contemporary of Janssen and took adifferent approach to modeling pressure, applicable to bothshallow and deep bins. Airy defined the critical height hL interms of the coefficient of friction on the rupture planeμ′ = tanϕ as well as the coefficient of friction between thegrain and the bin wall μ such that2

1   μ 
hL  D μ  μμ  μ 



(17)

Airy then defined two regimes for shallow and deep bins.
Case 1: Shallow Bin, h < hL
(17)
2


1
σ x  ρy (18)
 μ  μ  μ  1   μ 2 Case 2: Deep Bin, h ≥ hL


ρD 
σx 
1 
μμ 


(18)




2h
 μ  μ  1  μμ D
1   μ2(19)where h was the height of grain bulk solid pile, D the diameter of the bin and y the depth of grain measured from thesurface(19) of the pile. Airy started with the Mohr-Coulomb
failure criterion, which was used to calculate the static
condition of the wedge shape above the failure plane. The
proof and resulting equations were complicated, and although the deep bin equation produces similar results to
Janssen’s. Airy’s approach was not widely used due to thedifficult calculation (Ketchum, 1911).

2.3.2 Milo Ketchum: 1907 and others

Milo Ketchum wrote a book “The Design of Bins, Walls
and Grain Elevators”, which exhaustively covered how tobuild all aspects of a modern elevator for the early part ofthe 20th century. It is an interesting text for a historical
picture of the turn of the century in bin design and still
provides a relevant pedagogical explanation of practices
and equations. In chapter XVI, Ketchum discussed both

George Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

Janssen and Airy’s proposed solutions to grain bin pressure(Ketchum, 1911). It is from this text that we find the formulation of Eqn. (7). In Janssen’s original text, the formulation was different, but only superficially. However,
Ketchum’s form is dominant because its derivation was
clearly articulated in his text. Ketchum’s work was important because it formalized Janssen’s theory in a text discussing proper engineering practices and organized data thatengaged with Janssen’s theory. Ultimately, he concluded
that Airy’s equations were too complicated for practical use
and recommended a graphical method for obtaining solutions. Ketchum provided summaries of many researchers
who responded directly to Janssen’s work, including
Roberts, Prante, Toltz, Jamieson, Luft, Pleissner, Bovey,
Lufft, and Pleissner. A.W. Roberts provided citations to all
the original works discussed by Ketchum (Roberts, 1995;
Roberts, 1998). This was the primary channel for understanding the development following Janssen’s efforts.
As for the development after Janssen's efforts, a few are
mentioned below to establish issues with Janssen’s conclusions. A.W. Roberts provided a good summary of the work,
which extended much beyond Ketchum’s analysis (Roberts,
1995; 1998). Prante (1896) observed that the flow of bulk
solids increased pressure as much as four times static values. Still, the results might have been unreliable due to theexperimental apparatus, although the general trend wascorrect. These experiments established that grain removal
should be symmetrical, otherwise this pressure increase on
one side of the bin could bring about bin failure. Toltz(1903) utilized Prante’s work while designing a grain elevator to account for fluid pressure within the facility. Toltz
repeated Prante’s experiment but changed the techniqueused to measure the deflection of steel plates and calculated
the resulting pressure during filling and emptying. These
results were considered reliable and confirmed that staticand dynamic pressures were different, with dynamic pressures being significantly higher. Jamieson (1903) validated
Janssen’s model and calculations of lateral pressure ratio
and also showed pressure changes in static and dynamic
states with symmetric and asymmetric outlets. Furthermore, Jamieson’s measurement device was complex, andallowed for both vertical and lateral pressures to be measured; his work added robustness to Janssen’s theory.
Pleissner (1906) conducted many experiments throughout
1902–1905 that showed k was not a constant in a bin andchanged for different commodities. This was one of
Janssen’s assumptions, which most researchers adopted.
Other experiments conducted by Bovey (1904) and Lufft(1902–1904) showed variability in k but less comprehensively than Pleissner. Finally, it should also be noted thatoutside of the enormous project of writing “The Design ofBins, Walls and Grain Elevators,” Ketchum conducted experiments, which showed that the flow of bulk solids was
independent of the grain height (the height of a bulk solid

Review Paperpile). This phenomenon is commonly known throughBeverloo’s model, published in 1961 but is also found in
the second half of Hagen’s work and is commonly called
the Hagen-Beverloo Equation (Beverloo et al., 1961).

2.3.3 Andrew W. Jenike: 1961

Jenike was the first to mathematically describe the flowof bulk materials, which culminated in his book “Gravity
Flow of Bulk Solids” (Jenike, 1961). While the topic of thispaper is static pressure, Jenike developed essential descriptions of flow that determined what future state a system
would take and is thus relevant to this review. Jenike’s
work drew heavily on his contemporaries as well as investigations into mathematical descriptions of plasticity. In
addition to new analytical theories and methods, the adventof computers would make it possible to carry out otherwise“insoluble” mathematical problems. This comment hinted
at the future of bulk solid research and where one finds theforefront of research today, introduced through the work of
P.A. Cundall and O.D.L. Strack, who developed the discrete element method (Cundall and Strack, 1979). The
work “Gravity Flow of Bulk Solids” was outlined into six
parts: 1) the nature of the yield function; 2) differential
equations that described steady-state flow and producepressure, density, and velocity fields; 3) the initiation offlow, which was called incipient failure; 4) necessary criteria for the bulk solid to flow; 5) apparatuses to test flow
properties of bulk solids; and 6) experiments and applications of this flow theory. Jenike categorized different typesof bulk solid flows and how the bin/hopper geometry andother properties could influence flow types. Jenike notedthat there existed no perfect bin; each bin had its own
unique advantages and disadvantages. In part VI, Jenikementioned both Janssen and Ketchum’s work as being themost important contributions to the knowledge of bulk
solids at the turn of the century. Jenike also produced several other important publications. One was written with
J.R. Johanson and J.W. Carson that described bin design,
mass flow, and funnel flow in a four-part series (Jenike etal., 1973a; 1973b; 1973c). Moreover, Jenike summarized
bin loads under different flow states: static, during flow,
and steady flow (Jenike and Johanson, 1969). These later
publications were often cited in both research and engineering practices because they included design formulasand examples of calculations.

2.3.4 Walker: 1966

Walker investigated pressure and arching in hoppers,
using Janssen’s model and calculating a k value in both
static and flow states. His work also was an attempt to determine whether a specific hopper would experience blockage through arching of powder across the outlet. In
addition, he investigated the role cohesion played in thegravity flow in a hopper (Walker, 1966).

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2.3.5 Marcel Reimbert and André Reimbert: 1976

Marcel and André Reimbert wrote a book titled“Silos-Theory and Practice”, which presented a new model
for bin pressures (Reimbert M. and Reimbert A., 1976).
The book was written with the same spirit as Milo
Ketchum’s “The Design of Bins, Walls and Grain
Elevators”. It was ultimately interested in the constructionof grain elevators but diverged by presenting a new theoryof bulk solid phenomenon to inform calculations instead of
aggregating established theories. The book also offeredcritiques on the traditional Koenen-Janssen approach andnoted that the Janssen method needed to assume a constantk. Additionally, Ketchum suggested k as a function of bin
geometry and material depth. The model proposed in thebook produced similar results to Janssen’s. However, asignificant difference between the two was that Reimbert etal. used an empirical equation (hyperbola) to describe thechange in pressure as a function of depth. The equation wasdeveloped for cylindrical silos and can also be used forn-sided polygons. Furthermore, while the authors argued
that k changed throughout a bin, they still assumed it to beconstant in calculation, and the value they took (“the characteristic abscissae”) was calculated at the maximum vertical pressure, which resulted in
D
A


π 
4 tan   tan 2   
4 2

h
3

with the horizontal stress,(22)

where y was the depth of the material.

3.(22)
Comparison

Many models have been proposed in the past 100 years
and the immediate differences between these models are
not intuitive. A quantitative comparison was conducted tocompare the results between Hagen, Janssen, Airy, andReimbert. Each model uses the same properties as shown in
Table 1.
Hagen’s work attempted to recreate the “Janssen effect”
using a parabolic increase in pressure until a maximum wasreached, after which it remained constant. This piece-wise
approach used a quadratic equation to describe the increasein pressure, and constant once the maximum was reached,represented by the dashed line in Fig. 1. Hagen’s model isthe crudest of the historical methods but provided an essential first step; subsequent models predicted very similar

114

Table 1 Model variables and values used to generate Fig. 1, with the bin
diameter d, grain height h or y, mass density ρ, stress ratio k, coefficient offriction μ and the internal angle of friction ϕ.

Parameter

Value

Unit

d

6

m

h or y

30

m

ρ

770

kg/m3

k

0.4

N/A

μ

0.7

N/A

ϕ

30

deg

(20)

where D the diameter of the bin. The maximum stress wasgiven by(20)
ρ gD
σ max (21)
4 tan  

2


y
(21)
σ x  σ max 1    1  A  

Review Paper

Fig. 1 Comparing historical bin pressure models. (The data are available publicly at https://doi.org/10.50931/data.kona.23741598)

behavior giving a more gradual increase than Hagen.
Janssen’s model predicted an exponential change in pressure with the material depth that approached an asymptote.
Airy’s model used a complicated geometric approach andapplied the knowledge developed by Coulomb and othersin earth pressure theory to bulk solids in bins. Both Airyand Hagen’s models are separated by a transition point inwhich the pressure changes discontinuously. Reimbert’s
equation found the maximum pressure at the bottom of thebin and modeled the change in pressure using a hyperbolic
approach. Reimbert attempted to create a new variable fork that could account for more factors than the other methods. For example, Janssen assumes a lateral pressure ratio
to remain constant within a particular bin, but we would
expect k to vary from bin to bin as a function of geometry.
Our analysis of these models at different depth/diameter
ratios revealed that the differences in pressure predictions
between the models can be divided into three main regions:low (0–1.5), intermediate (1.5–3), and high (3–5). At thelow h/d ratio region, the equations exhibited the most variation between each other. The following percentage differences were observed: Hagen vs Reimbert, 16.0 %; Hagen
vs Janssen, 31.3 %; Hagen vs Airy, 64.6 %; Reimbert vsGeorge Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

Janssen, 15.6 %; Reimbert vs Airy, 50.0 %; and Janssen vsAiry, 35.1 %. In the intermediate region, Reimbert andJanssen closely approached the Hagen asymptote, with
Airy’s equation being 10.0 % lower during this intersection. At the extreme end of the high h/d ratio region, Reimbert and Janssen were very close, within 2.0 % of eachother, while Airy and Hagen’s asymptote were 5.1 % above
and 7.6 % below them, respectively. It is important to notethat there is a 9.9 % difference between the Hagen and
Janssen asymptote, with Janssen predicting greater pressure.

4. New developments in static pressure
theories

While the Janssen equation produces satisfactory results
for static conditions if the “right” parameters are selected,
it does not represent the full pressure field present within abin. Many new models and approaches to the pressure fieldwithin a bin have provided new insights. Such modern approaches include elastic theory (Bräuer, 2006; Ovarlez and
Clément, 2005; Schillinger and Malla, 2008), ordinarystress linearity (OSL) and incipient failure everywhere(IFE) models (Vanel et al., 2000), microscopic theory (Xuet al., 1996), the principal stress cap approach (Matchett,2006), as well as a plethora of numerical methods (Chen, etal., 1999).

4.1 Investigations and modifications of Janssen’s
model

Rigorous validation of Janssen’s model occurred within
the first decade of its publication (Ketchum, 1911; Roberts,
1995). Many modern studies have attempted to validate or
modify Janssen’s model. Di Felice et al. (Di Felice andScapinello, 2010) highlighted several works in the past
decade and identified a literature gap that questioned important parameters in the Janssen model: μ and k. Theirwork showed that while many studies determined the numerical values of these parameters, their physical interpretation was not as clear. Moreover, empirically their study
showed some discrepancies when measuring these parameters. While De Felice et al. highlighted the need to investigate Janssen’s assumptions, others have taken up this task.
Janssen assumed bins to be a static and closed environment, where no forces are transferred from outside thesystem besides the self weight of stored materials. This is
not the case for most bins, which experience external forcesfrom their environments, such as vibrations caused by machinery (Körzendörfer, 2022), earthquakes (Silvestri et al.,2012), intentional flow-inducing inertia forces (Pascot etal., 2020) and bin honking (Buick et al., 2004). Bertho et al.
looked at the applicability of Janssen’s model in dynamic
environments, provided an overview of the current research, and considered the effect of wall movement on
packing (Bertho et al., 2003). They found the classicalReview Paper

Janssen model was valid for bin walls that experienced
several centimeters per second movement. Moreover,
Windows-Yule et al. investigated horizontal wall movement and found that the Janssen model was also valid for awide range of dynamic movement (Windows-Yule et al.,2019). It is worth noting that this topic was often investigated by numerical methods, an important tool for understanding bulk solids but a topic outside the scope of thispaper.
Janssen’s model is a macroscopic model, which treats
the material as a continuum instead of a collection of particles. Bratberg et al. examined the threshold where microscopic behaviors turn into macroscopic behaviors throughnarrow granular columns (Bratberg et al., 2005). They
found that Janssen’s model could not account for small
container diameters. Similarly, for shallow granular pileheights, an interesting result was discovered by Mahajan etal. (Mahajan et al., 2020)—a “reverse Janssen effect”.
Specifically, the wall frictional forces could become compressive, effectively reversing the usual Janssen effect.
Under this condition, the walls increase the effective massof the bulk material at the bottom, which requires modification to Janssen’s equation to predict the changes in effective mass as a function of height. This reverse effect was
only observable when the height of a bulk solid pile wasrelatively small, i.e., h ≤ 30dG with dG the diameter of asingle particle. Thus, we do not generally expect to see thiseffect on large scales such as within bins. The continuum
approach is typically used in analytical models, whereas
numerical models can be applied to both continuum andparticulate materials (Chen et al., 1999; Schulze, 2008).
Many works focused on addressing the inadequacies ofJanssen’s assumptions. Some examples include variablebulk density due to particle packing (Vanel and Clément,
1999; Haque, 2013; Landry et al., 2004), variable angle ofrepose (Pont et al., 2003), horizontal bins (Tang et al.,2021), pressures in the hopper section (Walker, 1966), obstacles in bins (Endo et al., 2017), elastic deformation ofbin walls (Xu and Liang, 2022), moisture content (Zhang etal., 1998b; Chen, et al., 2020), temperature (Lapko et al.,2003), shallow bins (Ooi and Rotter, 1990; Xu and Liang,2022), as well as loading and unloading stresses (Walters,1973), among others (Ayuga, 2008). These modifications
are relevant because they point toward higher fidelity models that can better describe physical systems. Such modifications are potentially important for future industrial
applications as well as advancing development of state-ofthe-art models. While it is not realistic to review the studies
that address all the factors listed above, this paper has selected one significant topic to discuss, namely variable
stress fields.

4.2 Variable stress fields

The most fundamental and controversial assumption in115

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Janssen’s theory is the constant lateral to vertical pressure
ratio, k (discussed at length in Section 2.2). This ratio isfundamentally dictated by the stress field in the bulk solidmass. Many experiments have shown that this ratio is afunction of material properties and location in a bin (Chen,et al., 2020; Liu et al., 2021). In this section, the stress field
is investigated analytically in a 2D Cartesian coordinate
system (x,y), where the y direction points “downward,”
with y = 0 representing the top of the bulk solid surface.
Deriving Janssen’s equation in the Cartesian case requires
the use of a “slip-state” condition throughout the bin suchthat τxy = μσx(y) at the wall. In terms of vertical stress, wehaveτ xy  y  μσ x  y  μkσ y  y(23)

The equilibrium in the vertical direction can be expressedas:(23)
σ y τ xy

 gρ
(24)
y
x
If the stresses are assumed not to vary in the horizontal direction, as in Janssen’s theory, or τxy = τxy(y), the partial(24)
derivativeof shear stress with respect to x vanishes,
(25)

τ xy
0
x

This means the equilibrium equation in the vertical direction is simplified as
(25)
σ y
(26)
 gρ
y
which has the solution of(27)

σ y  g ρy(26)This indicates that the bulk solids behave like a fluid if theshear stress varies only in the vertical direction. Here, the(27)stress changes with depth, just as a fluid does. In this casethe wall friction has no effect on the vertical stress in thematerial. Mathematically, this vanishing of the partial derivative of the shear stress can be observed due to the relationships presented in Eqn. (25). Let’s now consider avariable stress field in the horizontal direction. Specifically,
we postulate a shear stress which varies linearly in thehorizontal direction,

τ xy  x, y x
μσ x  yb(28)

where b is the radius of the bin. Now the partial derivative
term is expressed as
(28)
τ xy
μ
 σ x  y
(29)
b
x
and the equilibrium equation has the form of

(29)

116

dσy

Review Paperμk
σ y  gρ(30)
dybThis equation form is similar to that of Janssen’s differential equation. This means that in order to account for the
Janssen(30) effect, the shear stress in 2D must include radial
dependence. It should be noted that exchanging b to RH in
Eqn. (28) would identically result in the Janssen equation.
The difference between the granular and liquid-like behaviors occurs because the effect of wall friction is not included in the problem when the derivative term drops out.
Deriving the Janssen equation requires a specific choice of
shear stress that links the friction coefficient and the vertical stress through the horizontal to vertical stress ratio
σx(y) = kσy(y). Geometrically, this term naturally arises incylindrical coordinates (r, θ, z) due to the radially dependent volume of infinitesimal cylindrical elements
dV = rdrdθdz. However, this geometrical effect does notoccur in Cartesian coordinates (x, y, z), in which the volume of all infinitesimal elements are independent of theradial coordinate.
Radial variation of the shear stress can be traced back to1948 (Jaky, 1948). In relation to flow problems this termshows up in the work of McInnes (1968) and Savage(1967). Later, it was used by Zhang et al. (1998a) andMilleteet al. (2006) to develop a theory of radial stress behavior in a 2D Cartesian coordinate system and later toexpand to 3D cylindrical coordinates (Rahmoun et al.,2008).
We compare the results of Zhang et al. (1998a) with theusual Janssen approach for a hypothetical bin of 5 m in diameter (Fig. 2). Zhang et al.’s model accounted for the
non-uniformity of stresses in the horizontal and verticaldirections, whereas Janssen’s work only considered change
in the vertical direction. Additionally, in the Zhang model kvaries throughout the bin, whereas variation in k is neglected by Janssen. Zhang’s model suggests that higher
stresses occur at the center of a bin. This may have design
implications for airflow in aeration and drying.
To quantify the differences between the Zhang andJanssen models, we calculated the percentage differences
between their stress values at various depths and distancesfrom the bin center. The analysis revealed a range of percentage differences between the two models in both horizontal and vertical stress distributions. The vertical stress
distribution exhibited more variation than the horizontal, so
we focused on this set of data. At the most distant pointfrom the center of the bin, Zhang’s model predicted lesspressure than Janssen, ranging from 40.0 % to 24.0 %, withthe maximum discrepancy occurring at the 1 m depth anddecreasing afterward without a clear trend. In contrast,
Zhang’s model predicted higher pressure at the center ofthe bin, with differences ranging from 22.5 % to 7.6 %; themaximum discrepancy occurred at the 1 m depth and decreased with depth. At all depths, Zhang’s pressure crossedGeorge Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

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Fig. 2 Differences between the linear stress variation used by Zhang, Bu and Britton (Zhang et al., 1998a) and the planar Janssen result using the
Rankine coefficient (Eqs. 30 and 11) assuming a bin radius of b = 2.5 m, uncompacted mass density ρ = 834 kg m –3, coefficient of static friction
μ = 0.36 and internal angle of friction ϕ = 22° using h = 1 m. Bin depth is given in the legend. Change in the radial dependence is introduced by thelinear modification to Janssen’s equation. (The data are available publicly at https://doi.org/10.50931/data.kona.23741598)

over Janssen’s between 1.5 and 1.6 meters from the bin
center, and generally, as depth increased, this crossover
occurred closer to the center of the bin. This indicated that
the Zhang model’s predictions could significantly deviate
from those of the Janssen model, depending on the depth
and distance from the bin center. A consistent trend observed across various depths and distances was that
Zhang’s model predicted lower pressure and greater discrepancy on the outside, while the center experienced
higher pressure and relatively less discrepancy comparedto Janssen’s model. This finding further emphasized theimportance of considering non-uniform stress distributions
in grain bin design for optimal aeration and drying performance.
The work of Rahmoun, Millet and de Saxcé (Millet etal., 2006; Rahmoun et al., 2008) included significant expansions over the original Janssen model, specifically applicable to cohesive powder materials. The analysis inMillet et al. (2006) for a planar bin with the 2D Cartesianequilibrium equations (Eqn. (8)) resulted in a differential
equation for the angle between the horizontal axis and theprincipal stress direction. The differential equation was
solved analytically. The model was numerically comparedto Janssen’s equation for low cohesion ∼ 10–6 Pa and theresults were found to agree for the passive and active states.
A similar approach was taken in Rahmoun et al. (2008) inwhich the cylindrical equilibrium equations (Eqn. (9))
were used to produce a more general 3D result. The resulted model could be solved analytically for cohesionlessmaterials, but a numerical approach was required when finite cohesion was included.
An important model used to complement the MohrCoulomb criterion is known as the incipient failure every-

where (IFE), or the rigid-plastic model (Nedderman, 1992;
Wittmer et al., 1996, 1997). The IFE model assumes thatthe material is on the edge of failure throughout the entire
bin. This means that through each point there is some planeover which the shear force is linearly related to the normal
force. The variable vertical to horizontal stress ratio givenby the IFE model has the form of Wittmer et al. (1996)
cot  τ xy  σx 

1  sin 2    2sin  1  α  β 2(31)

cos 2 where α is the vertical to horizontal stress ratio (k–1) and βis (31)the shear stress to horizontal stress ratio,

σ y  ασ xτ xy  βσ x

(32)
(33)

Substituting Eqns. (32) and (33) into Eqn. (31) yields
(32)
(33)
β2
1  sin 2    2sin  1  2
tan 
(34)
α  β cos 2 The α(β) has real solutions if β ≤ tanϕ. This condition is(34) because the stress ratio β is a maximum at the bin
upheldwall β|r=b = μ, so we have generally β ≤ μ ≤ tanϕ. Moreover, under static conditions the vertical stress should exceed the horizontal stress, so we expect α > 1, which selects
the + sign in the quadratic solution. In this case, the variation in α accounts for the change in stress across the bin
radius. Similar to the approach outlined above, Millet et al.
(2006) considered the variation in radial stresses for cohesive materials.

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After Jenike outlined proper designs to avoid arching
and funnel flow, hoppers became a common addition tobins. The stress field in a hopper is variable and Janssen’s
equation is not applicable to hoppers. Walker (1966) andTakami (1975) both used force-balance conditions on aconical slice to describe the stress distribution within aconical hopper and showed vertical and horizontal variations in stresses using Rankine’s formulation of k. Thiswork, similar to Janssen’s theory, considered forces on aplanar differential slice, which was assumed axisymmetric,
which is a reasonable assumption for most cases, but notall. Some bins have eccentric hoppers, which produce a
non-axisymmetric distribution of material and stresses.
While there have been many studies into eccentric discharge of bins in terms of improper flow, there have been
far fewer on structural loads (stresses), but this has changedin the past two decades (Wojcik et al., 2003; Vidal et al.,2006, 2008). A detailed study was reported by Ramírez etal. (2010), which provided a thorough discussion of thebenefits of eccentric hoppers, experimental work, and several numerical models. This work inspired Matchett’s analytical model based on the “principal stress cap” approach,which was capable of accounting for eccentricities in binloads, as well as symmetrical loads (Matchett, 2006;
Matchett, 2020; Matchett and Close, 2021). This research
provided a fundamental start to designing safe eccentric
hoppers.

4.3 Variable density

Janssen’s equation assumed the density of a bulk solid tobe constant throughout the entirety of a bin. There havebeen substantial attempts at numerically incorporatingpacking into pressure models and considering the effect ofparticle impact forces on spatial arrangements of particles(Volfson et al., 2003; Landry et al., 2004; Umbanhowar andGoldman, 2010; Petingco et al., 2022). There are some direct modifications to Janssen’s model to account for density
variations, with varying complexity. Vanel reported thatpacking could affect k and these effects could be quite significant when the porosity changed from 0.43 to 0.39
(Vanel and Clément, 1999). They proposed a modification
to Janssen’s model by introducing a fit parameter to account for inhomogeneous density. Cheng et al. directly
modified Janssen’s equation to account for changes indensity and investigated bins with hopper sections (Chenget al., 2017). They related the degree of packing to theprincipal stress through a quadratic equation and addressed
gaps and ambiguous understandings of pressures in the literature. They concluded that bulk density increased rapidly
with the material depth in the upper cylindrical part of thebin but decreased slowly in the hopper. Haque utilized
hoop stress and hoop pressure, which represent the circumferential stress and force exerted by stored material in acylindrical structure, to analytically describe packing

118

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within a bin (Haque, 2013). Developing a modified
Janssen’s equation, Haque’s model accounted for variations in bulk density when calculating vertical and horizontal pressures within a bin, offering a more comprehensive
understanding of the forces involved in grain storage.

4.4 Non-continuum approaches
4.4.1 Microscopic theory

Bulk solids are discontinuous in nature and microscopic
theory may provide a better alternative to the continuum
approach such as Janssen’s, but with greater complexity.
Bratberg et al. investigated the failure of Janssen’s theory
as the bin and material scale transitioned from macroscopic
to microscopic. A few researchers have investigated the binpressure from the microscopic angle. Xu et al. developed abin pressure model accounting for the discontinuous natureof bulk solids (Xu et al., 1996). Their model was based ona more general microscopic approach by Granik & Ferrarithat used principles of thermodynamics to investigate asystem of doublets (pairs of adjacent particles; Granik andFerrari, 1993). The particles modeled by Xu et al. wereidealized spheres in hexagonal packing and held within 2D
frictionless walls. Under a pair of contact forces within adoublet, the two particles experienced microstrains due todeformation, rotation, and slipping. Microstress (elongation, compression, torsion, and shearing) were calculated
from the contact forces, but torsion and shear microstresses
were found to be negligible. Their model results showedthat Janssen’s results were a special case of their microscopic theory. Moreover, k was only a constant value whenthe model used frictionless walls and rigid particles. Xu
had a deeper examination of the application of microscopic
theories in his Ph.D. dissertation (Xu, 1966). Xu et al.
demonstrated how this microscopic theory could be used to
predict moisture induced stresses within the mass of hygroscopic bulk solids. Zhang et al. further developed this theory to explain discharge loads in bins (Zhang and Britton,
2003). Ferrari et al. compiled a review of doublet microscopic theory (Ferrari et al., 1997). The strength of thistheory was the ability to analytically describe the discontinuous nature of particle assemblies, something that was beyond the scope of the continuum approach.

4.4.2 Granular elasticity

Jiang and Liu gave a brief overview of granular elastic
theory (Jiang and Liu, 2007a), which covered the original
work done by Jiang and Liu (2003). Granular elasticity was
distinct due to its definition of strain, which has two parts.
The first accounted for particle deformation (elastic). The
second accounted for irreversible rolling and slippage (aprocess that also causes heat) of particles and was considered plastic. The irreversible plastic phenomenon was themain contributor to the overall deformations. The energy
within the material under loading could be articulated as aGeorge Dyck et al. / KONA Powder and Particle Journal No. 41 (2024) 108–122

function of strains and static stresses. Elastic strains wereused because particles are compressed and sheared, which
stores energy reversibly. These models are based on a relationship between elastic energy, static stress and the deformation of particles. The stress distributions in a granularmaterial could be determined the same way as in elastic
media. Granular elasticity could account for a bulk solid’s
yield, volume dilatancy, and shear-induced anisotropy.
Jiang & Liu discussed the applicability of this theory tobins and compared it to Janssen’s method. They found thatbins were a good fit for granular elastic theory because allsix stress tensor elements could be calculated. Bräuer et al.
validated granular elasticity for its application to bin stress(Bräuer, 2006). Their model showed a saturation pressurethat matched Janssen’s. The remaining components of thestress tensor were also computed, which Janssen’s theory
could not perform. They assumed a constant k and foundlittle variation in vertical stress radially. Other interesting
works on granular elasticity included a slow movement of
deforming particles (Jiang and Liu, 2007b), which accounted for the infamous stress dip in granular piles(Krimer et al., 2006), showing good agreement with experimental data (Jiang and Liu, 2008). Sun et al. applied
granular elasticity to the mesoscopic scale using numericalmethods, which examined packing, pressure, and force
network structures (Sun et al., 2015). While the study was
focused on mesoscopic objects, there was a possibility that
this approach could be applied to many discontinuous systems.

5. Concluding remarks

New perspectives and viewpoints are often required tosolve stubborn, longstanding problems. This was certainly
the state of the field when Janssen’s approach produced arevolution in describing and predicting the behavior of
granular materials in bins. Even in the modern literature,
Janssen’s model forms the cornerstone of our understanding of pressures in bins. Modifications to Janssen’s model
have been sought to generalize it for describing the observed phenomena to higher accuracy, but the starting point
is usually Janssen. Otherwise, it is the first validation caseto test a new theory against. This ubiquity is a monument toJanssen’s success. Just as a new perspective spurred therevolution that led to Janssen’s insights, a survey of theliterature reveals many new promising approaches.
Despite these successes, there are several limitations ofJanssen’s model. First, the vertical stress is assumed to beconstant across a bin. This is not generally true as the pressure is expected to vary from the center to wall of a bin.
Second, the horizontal stress is assumed to be proportional
to the vertical stress through the factor k, which is constantthroughout a bin. Finally, the density is considered constantthroughout a bin. The first of these assumptions is incorrect. The second statement is ad hoc, and generally it

Review Paper

should require some deeper theoretical justification. The
third assumption is generally incorrect.
In recent years the computational power available toperform DEM simulations has significantly increased, andnumerical methods used to simulate granular properties
have been extensively studied. The discussion of material
properties acquisition and standardization has made DEM
a more viable approach for solving bulk solid problems(Xu et al., 2002; González-Montellano et al., 2011;
González-Montellano et al., 2012; Horabik and Parafiniuk,
2016). Generally, DEM simulations need to limit the number of particles for computational purposes, and the particle
shape sometimes needs to be crudely approximated. However, strategies have been developed to mitigate these limitations (Ramírez-Gómez, 2020). DEM provides thenumerical revolution analogous to the analytical revolution
provided by Janssen.

Data Availability Statement

The data from the pressure models are available publicly
in J-STAGE Data (https://doi.org/10.50931/data.kona.
23741598).

Acknowledgments

We acknowledge the effort of the anonymous referee,whose careful reading of our work helped to focus and refine our manuscript.

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Authors’ Short Biographies
George Dyck
George Dyck is a Biosystems Engineering PhD student from the University of Manitoba, Winnipeg, Canada. His research focuses on the application of digital twin technology in bulk solids storage
systems, with an emphasis on the development of comprehensive and efficient models for thesesystems. George’s work is aimed at advancing sustainable and efficient bulk solids storage practices,making a significant contribution to the field of agricultural engineering.

Adam Rogers
Adam Rogers is a Research Scientist with a doctorate degree in Physics and Astronomy from the
University of Manitoba in Winnipeg, Canada. He has worked with AGCO since 2020, where his focus has been on model development and mathematical analysis. His research interests broadly encompass the physics of granular materials.

Jitendra Paliwal
Dr. Jitendra Paliwal is a Professor of Biosystems Engineering whose expertise lies in the storage,
handling, and quality monitoring of cereal grains, oilseeds, leguminous crops, and their derivatives.
His research on developing hardware and software solutions is widely referred to by the designers ofgrain quality monitoring and assessment instruments. In 2018, he was part of a distinguished team ofindustry and academic researchers, who won the American Society of Agricultural and BiologicalEngineering’s AE50 award for adapting a cancer detection imaging technique for remote monitoringgrain bins.

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