László Mester
The new physical-mechanical theory of
granular materials
2009
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Contents
Introduction ………………………………………………………………………….3
Granular material as a distinct state of matter …….…………………………………4
Physical properties of the granular material in relation to
the different states of matter …………………………………………….…..6
Granular material as a state of matter …….………..…………………………10
Physical-mechanical basic laws of the non-cohesive granular materials …………..14
Law I ………………………………………………………………………….15
Law II ………………………………………………………………………...15
Law III ………………………………………………………………………..17
Law IV ………………………………………………………………………..24
Stresses in the non-cohesive granular materials ……………………………………26
Active stress state …………………………………………………………………..31
Development of the active stress state ………………………………………..31
Pressures acting on the vertical retaining wall ……………………………….36
Arch formation of granular materials …………………………………………...…39
Condition of the arch formation ……………………………………………...39
Character of the discharge ……………………………………………………44
Mechanism of the arch formation …………………………………………….46
Geometric equation of the arch ………………………………………………47
Principle of the hopper design ………………………………………………..50
Experimental results ………………………………………………………….53
Stresses in cohesive granular materials …………………………………………….55
Lateral pressure ………………………………………………………………55
Inclination angle of the free slope ……………………………………………59
Active stress state …………………………………………………………….64
Summary …………………………………………………………………………...69
Bibliography ……………………………………………………………………….73
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Introduction
Following Coulomb’s and later Rankine’s work the physical-mechanical theoretical
research of granular materials has been characterised by the use of stress analysis
deduced for solids since the 18th century. Others seem to detect the characteristic
features of viscous liquids in granular materials, therefore they describe the physical
behaviour of granular materials using the laws pertinent to viscous fluids. In my
opinion most of the theorems, which were put for cotinuums, cannot be applied to
the aggregation of separate, solid granules. Only those natural laws can be
considered as the starting point of examination, which are also valid for the universal
material.
This work – breaking away from the previous tradition – would like to approach the
physical mechanical properties of granular materials from a new point of view. As a
result, the critical analysis of theories formulated earlier in this research area is not
the objective of this paper, since the new principles were laid down irrespective of
those hypotheses. Contrary to the continuum theory, by examining the equilibrium
and kinetic state of individual granular particles this new thesis is based on simple
experiments, on the Newtonian laws, and on an empirical law, the law of friction.
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Granular material as a distinct state of matter
The physical appearance of materials found in nature is quite varied. The most
substantial part of the Earth’s surface is covered by oceans, seas, lakes, that is to say,
covered by water. The dry land is more diverse: one can find rocky mountain ridges,
surfaces covered by gentle slopes and deserts with sand dunes. In places the earth is
covered by snow or ice in winter. Above the surface level the wind is blowing, or we
can experience a period of calm, that is to say we can feel the air. The sun is shining
above us and we know that inside the sun one would find another state of matter.
The outward form of the water that covers substantial part of the Earth is in itself
diverse. At normal temperature and pressure water is liquid, but with the increase of
temperature it evaporates more and more quickly, and it turns into water vapour. Fog
or clouds form. When water vapour freezes, and precipitates in cold, then snow falls,
and it condenses into a granular material. When snow melts, the result is liquid,
which in turn becomes solid when it freezes. That is to say water can exist in liquid,
vapour (gas), snow (granular) and ice (solid) states. In each of its phases water has
different physical properties, and behaves conforming to different laws.
Physics differentiates among the most prominent forms of appearance of material by
classifying them into the different states of matter: plasma, gas, liquid and solid
Some material cannot be put strictly under one category, because they bear the
physical properties of two or more states of matter. These materials, however, can be
described by applying the laws pertaining to materials in a similar state of matter.
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Granular material cannot be put under any one of the above mentioned four
categories. Furthermore the physical-mechanical properties of granular material do
not make it possible to describe its behaviour successfully using the physical laws of
one or more phases.
A granular material is a conglomeration of large number of solid particles related to
one another, where the granules – as the constituent of the aggregate – in spite of the
affecting forces retain their form, and the incidentally arising cohesive force between
the granules is substantially smaller than the inner cohesion of the individual
granules.
The overall, coherent physical system of granular material has not been set up yet,
scientific analysis is available only for a few prominent, primarily soil mechanical
problems. Several theories have been applied for the handling of these problems,
which speculations, however led to contradictory results. Furthermore no connection
resting on firm, uniform physical foundations exists between the theories, or if there
is a relation, it is disputable. The majority of mechanical theories dealing with
granular materials apply the method of stress analysis deduced for solids, which
procedure presupposes, that the granular material is a solid phase continuum.
There are theories, according to which granular materials can be approached with
the laws pertaining to viscous liquids, since granular material exhibits viscoelastic
and viscoplastic properties. Although, the difficult theoretical notions provide an
approximate solution to individual mechanical problems, they cannot be applied to
an overall, reliable description of granular material behaviour.
The opinions concerning the state of matter of granular materials are not unanimous.
This is reflected in the fact that granular material does not have a single uniform
name, for example the following designations: scattering, powderlike, loose,
granulated, grainy, particulate, granular and gritty are all used.
The physical behaviour and properties of granular materials exhibit substantial
qualitative difference from the materials in other states of matter, and should
therefore be considered an additional state of matter in its own right
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The idealised notion of granular material makes the simplified explanation of its
physical behaviour, properties and origin possible, similarly to the hypotheses
applied to perfect gases, ideal liquids and crystalline solids. The ideal granular
material is a conglomeration of large number of solids (granules) where the mass of
the solid particles is small compared to the mass of the material, in this aggregate
attractive force does not operate between the particles, Coulomb friction rule
governs them.
Physical properties of the granular material in relation to the different
states of matter
The basis for classification according to states of matter depend on the question
whether the material can hold its own shape and volume or not. The basic criteria of
classification of the three classical states of matter are the following:
- gases: have no definite shape or volume;
- liquids: have no definite shape, but have definite volume
- solids: have definite shape and volume.
Regarding the question of definite shape and volume it is the characteristic of the
granular material that:
- In part it has definite shape, the granular aggregate holds its shape in the angle
of repose, but under this angle it takes the shape of its container. This attributive
places the material between liquids and solids.
- In part it has definite volume, but it can be compressed to a limited extent. The
compressibility of granular materials stands between the compressibility of gases
and solids.
The researches on substance structure concerning the states of matter found that the
determining factors in the question whether a material holds a definite shape or
volume lie in the physical properties of its constituents and the nature of interaction
between these particles. For this reason modern physics studies the kinetic state of
the smallest particles attributed to the material, their relative position and the particle
interaction when defining the different states of matter. This made it possible that in
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natural science, in addition to the three classical states of matter, a fourth state of
matter, was accepted, the plasma state.
It seems to be necessary to emphasise the expression smallest constituent
characteristic of the material, since it is of primary importance in the definition of
the different states of matter. Consequently
- A material in plasma state consists of the disintegrated parts of molecules or
atoms, the molecule ions or atomic ions. The plasma state is characterized by the
interactions of atomic or molecule ions and electrons, and not by the other parts of
the atom or molecule.
- The physical properties of a material in the gas state are determined by the
interactions of gas molecules – in case of noble gases the interactions of atoms.
- Regarding liquids the determining factors are again the movements of the
atoms or molecules, and the nature of relation between them. In the case of water it
is the interactions of H
2
O molecules and not the hydrogen or oxygen atoms, or the
water drop, which characterize the liquid.
- In the case of crystalline solids the physical properties of a material in the solid
state can be explained with the nature of interaction of the atoms, molecules or ions
positioned in the lattice nodes of a crystal structure, and cannot be described for
example with the interactions of elemental crystals and crystallites, or with the
individual properties of the atoms and its parts, which constitute the molecules
positioned in the lattice points.
- In the case of granular material the smallest constituent characteristic of the
material is the granule. The atomic particles, the atoms and molecules that constitute
the granules, are not direct characteristics – at least in physics they cannot be
regarded as significant physical properties– of a material, just like as in the case of
gases and liquids, where material is not characterised by the atomic particles, which
constitute the atom or molecules, or by the individual physical properties and
interactions of atoms either.
The most significant characteristics of the three classical states of matter can be
summarised in the following way:
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Gaseous state: The molecules of gases – in case of noble gases the gas atoms –
move freely in the space available for them, they collide elastically with one another
in a random motion. The average distance between the constituting particles of gases
is relatively big in proportion to their size, the intermolecular forces between the
particles are very weak. In the case of ideal gases intermolecular force can be
disregarded. Molecules move with a translational, rotational and vibrational motion.
Gases evenly fill the space available for them, that is to say they have no definite
shape or volume.
Liquid state: The intermolecular forces between the smallest constituents
characteristic of liquids, between the atoms or molecules are strong enough to
prevent the particles moving away from each other as a consequence of thermal
motion, but not strong enough to prevent their change of position. Compared to
gases, the translational motion of the molecules are smaller, while they also carry
out rotational and vibrational movement. Due to their motion and proximity the
constituting particles collide elastically with one another all the time, thus touch one
another, therefore liquids have a definite volume. The force of attraction between the
particles is so small compared to the Earth’s gravitational force that it is not enough
for individual shape formation, as a result liquids have no definite shape.
Solid state: The smallest constituting particles, characteristic of solids are the atoms,
molecules or ions. Their position is fixed and geometrically determined in a
crystalline structure, particles carry out only vibrational motion. The intermolecular
forces are strong, which prevent their permanent displacement from their state of
equilibrium. As a result solids have definite shape and volume.
Granular material exhibits significant differences from the aggregational properties
of the three classical states of matter. The constituents of an ideal granular material,
the granules are at a relative rest. There are no forces of attraction between the
particles, the material is kept in an aggregate by the compressive forces originating
from the gravitational force, by the shear forces arising on the surface of the
granules, and by the static friction force. Due to these forces the ideal granular
material remains stable until the angle of repose is reach, thus it has only partly a
definite shape. The constituting particles are in constant contact, therefore in
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quiescent state granular material has definite volume. Under pressure the material is
compressed, the granules take up a more efficient space filling position. The
compressibility of granular material is small compared to gases, but it is big in
comparison to solids.
The physical properties of ideal granular material exhibit the following significant
differences in characteristics compared to the features of other states of matter:
- In contrast to gaseous state: the constituting particles are in constant contact
with each other, and it has definite volume;
- In contrast to liquid state: granular material has in part a definite shape;
- In contrast to gaseous and liquid state: The constituting particles are in a
relative collision free, quiescent state and static friction force – shear force – arise in
them;
- Solid state: there is no attractive force between the constituting particles,
therefore granular material has only in part a definite shape.
Granular material exhibits such qualitative differences concerning the most
substantive characteristics of the different states of matter that its definition as a
separate state of matter in its own right becomes justified.
The brief, straight to the point definition – with no pretence to completeness – of the
idealised case of the states of matter is the following.
Perfect gas: disordered aggregate of molecules (in case of noble gases atoms) with
no intermolecular forces, where the molecules move far apart from each other,
undergoing random elastic collisions.
Ideal liquid: Aggregate of molecules moving close to each other, undergoing
constant elastic collisions.
Crystalline solid: The ordered aggregate of vibrating atoms, molecules or ions,
which are fixed in their structure with great force.
Ideal granular materials: the aggregate of relatively static particles, which are in
constant contact with each other, in this assembly the force between the constituting
particles is composed of the compressive force arising from the gravitational force
and of the friction force, which is proportional to it, there is no cohesion force
between the particles.
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The notion of granular material, as a separate state of matter is primarily important
from a mechanical viewpoint. (The basis for categorization into different states of
matter has a mechanical origin: the reason for having a definite shape or volume can
be deduced from the intermolecular forces between the constituting particles.) The
mechanical properties of the different states of matter show distinctive, substantial
differences:
- gases respond to an increase in pressure with the significant reduction of their
volume (at constant temperature the multiplication product of volume and pressure
is constant), therefore they can withstand compressive stress only in part, at the
expense of volume change. In perfect gases there is no attractive force between the
constituting particles, therefore no tensile stress can arise in the material. In static –
not flowing – gases no shear stress arises.
- liquids has small compressibility, from a mechanical point of view they can be
regarded incompressible, therefore they can withstand compressive stress. The
intermolecular forces are strong enough to prevent the constantly colliding
molecules from moving far away from one another. The state of equilibrium or
stability can only be attained under a given outside pressure, that is to say, from a
mechanical standpoint a liquid cannot withstand tensile stresses. (When the pressure
is around p=0 the liquid breaks up, its molecules fly apart and turn into gaseous
state) There is no friction in ideal liquids, in real liquids static friction does not arise
either.
- solids can be regarded as incompressible, the constituting particles join
together with great force, therefore they can withstand tensile, compressive and
shear stress.
- ideal granular material has small compressibility, therefore it can withstand
compressive stress. In the non-cohesive granular materials only shear stress arises in
addition to compressive stress, no tensile stress manifests itself.
Granular material as a state of matter
The notion that granular material must be regarded as a separate state of matter can
be justified not only because its distinct physical properties, which differentiate it
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from other states of matter, but also because granular material is one of the existing
outward forms of raw material, that is to say it is one of defined states of matter.
Granular material – as the conglomeration of large number of solids, where the
constituting solids are small in proportion to the total mass of the material –
generally comes into being when large-sized solids are mechanically cut up, or when
the solids themselves break up into smaller pieces. Its formation, that is to say, the
bringing of the material into a granular state can be achieved not only in a
mechanical way, as it is also true for the granular materials in nature, which were
formed not exclusively by mechanical disintegration either. Granular material can be
produced via a thermodynamic method.
It is known, that if the kinetic energy of the molecules of a liquid exceeds a
threshold it changes into gaseous state and if it goes below another threshold the
liquid turns into solid and the process of crystallization begins. The threshold values
of the thermodynamic state parameters characteristic of the different states of matter
can be illustrated in a p – t (pressure-temperature) diagram. In Figure 1 the p – t
diagram of H
2
O can be seen.
Figure 1. p-t diagram of H
2
O
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The curve that joins the triple point and the origin of the p – t diagram is called the
sublimation curve. The material passes from the solid ice phase into the vapour
phase of the gas state by for example pressure reduction and by the crossing of the
sublimation curve. Crossing the sublimation curve backwards, from vapour phase
granular snow is formed not ice. The granular material – in case of water, the snow –
comes into being as a result of crystallization in the local clusters. The process can
be called local crystallization, the physical explanation of which lies in the
phenomenon that the molecules which move with slow translational motion (under
low temperatures) cannot leave the attraction field of the van der Waals type forces –
for example as a result of heat loss – therefore the translational motion of the
molecules ceases.
The molecule pair - bonding the new molecules, which collide into them - form a
crystal lattice, the growing crystals then bring about the granules. The density of
molecules in the gaseous state is very low in comparison to the molecular density of
the solid state, therefore the local crystallization processes, which are relatively far
from one another bring about the multitude of separate granules, which after having
precipitated form a granular conglomeration.
Under constant temperature the process of getting from gas phase to granular phase
is accompanied by heat loss, which is the sum of the melting and the evaporation
heat.
The states of matter change at the phase boundaries of the p – t diagram. If the
matter crosses the sublimation curve from the solid phase toward the gas phase, we
will get a gas, however changing the direction crossing the curve from the gas phase
we will obtain a granular material. Thus the states of matter, in compliance with the
direction of crossing the phase boundaries are the following:
gas → local crystallization → granular → melting → liquid → evaporation →gas;
nevertheless, from the other direction:
gas → condensation → liquid → freezing → solid → sublimation → gas.
From granular phase to solid phase we can get by crossing the same phase boundary
twice:
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granular → melting → liquid → freezing → solid.
Granular material can be produced directly from liquid phase if we place a multitude
of crystal nuclei - which are approximately at an equal distance from one another -
into a supercooled liquid at the same time. The crystal growth is hindered by the
neighbouring crystals, whose geometric crystal position is not symmetrical or
congruent, therefore no or only occasional lattice forces develop between the
crystals, the inner cohesive force of the individual granules are substantially greater
than the incidental cohesive forces acting between the granules.
The substantial physical properties of the granular material differ significantly from
the characteristics of those materials whose chemical properties are identical,
nevertheless belong to the solid, liquid or gaseous state. Its volume weight, its
refractional, thermodynamical, acoustic, electric and mechanical properties and
behaviour, and the fact that most material can be brought into granular state justifies
the classification of the granular material as a distinct state of matter by its own
right.
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Basic physical-mechanical laws of the non-
cohesive granular materials
I. In the non-cohesive granular materials only compressive and shear stresses
can develop.
II. In the non-cohesive granular materials at a quiescent state the stresses
developed by the vertical-direction compressive stresses act downwards in the
0
90
zone measured from the vertical direction. (
is the angle of friction of
the material.)
III. The value of the lateral pressure rising from the self-weight of the non-
cohesive granular material is (
2
h
), the half of the product of the depth (h) and
volume weight (γ), its direction deviates from the horizontal downwards with the
angle of friction developed in the material, if the surface is horizontal and over the
given depth the material fills the space evenly closing an angle
with the
horizontal.
IV. The non-cohesive granular materials conform to the physical-mechanical
laws characteristic of them until their constituting elements, the grains keep their
relative quiescent state. When the grains go into motion – collide with each other -,
the granular materials behave according to the physical-mechanical laws of the
liquids.
The physical-mechanical laws of the non-cohesive granular materials prevail with a
statistical character, because the material itself consists of a multitude of different
grains.
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Law I
Law I is the physical-mechanical definition of the non-cohesive granular materials.
In ideal liquids only compressive stresses develop, the non-cohesive granular
materials are capable of withstanding compressive and shear stresses, while solids
are capable of bearing compressive, shear and tensile stresses. The non-cohesive
granular materials differs from the solids in the respect that they are not capable of
withstanding tensile stress, and they are distinct from the ideal liquids because shear
stresses also develop in them. At the same time the components of the liquids are in
constant relative motion – collide with each other -, while the components of the
granular materials, the grains are in a relative quiescent state.
Law II
Law II formulates the direction of the spreading of the vertical compressive stresses.
The natural stability of the free slope provides its experimental proof (Figure 2).
Figure 2. The boundary equilibrium position of the grain located
on the side of the slope
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In the conglomeration of grains the self-weight of the grains produces the vertical
compressive stress. If a stress vector acted on the grain marked A located on the side
of the slope– in boundary equilibrium position - inclined at an angle of more than
from the vertical then the grain would loose its equilibrium position and
slide down.
Figure 3. The stresses developed by the vertical compressive stresses
incline at an angle bigger than
to the horizontal.
Further experiments prove the correctness of the Law II (Figure 3). If a compressive
stress making a smaller than
angle with the horizontal acted on the grain marked
A, the angle of inclination angle
of the natural slope would be smaller than
. If
the vertical compressive stress induced, for example, a horizontal stress, it would
thrust down the grains located on the side of the slope. The material would spread
and would take a kind of shape that is illustrated in Figure 4. However, it does not
exist.
Figure 4. If the compressive stresses induced horizontal stresses, this
would be the position of the granular material.
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Law III
In the cases described in the Law III the lateral pressure is
2
h
, and its direction
inclines from the horizontal downward with the angle of friction rising in the inside
of the material. Its proof is as follows:
Figure 5. Infinite Quadrant of the Horizontal Terrain
Figure 5 shows that part of the non-cohesive granular material aggregate of infinite-
expansion and horizontal-terrain, which is cut out theoretically by two vertical
planes perpendicular to each other, (consequently, it shows an infinite quadrant of
the horizontal terrain,) which makes the planar execution of the mechanical tests
possible. According to the Law II, from the material part under the section AB only
reaction stresses produced by the material part over the plane AB can act on the
plane OA. If we took off the granular material located in the triangle OAB, then the
material would remain stable in the natural angle of repose AB inclining at an angle
to the horizontal. On the plane OA, stresses can only rise from the self-weight of
the granular material located in the space part OAB. On the plane AB an equilibrium
boundary position exists; the material having a friction coefficient of
tg
does
not slide as yet on the slope with an inclination of angle
. If we increased the slope
angle with
of a very low value, then the material above it could slide down with
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constant acceleration on the slope with the inclination angle
, i.e. it would
exert a force in direct proportion to its mass and acceleration on the plane OA. Due
to the physical properties of the granular materials there are infinitely many slopes
with an inclination angle
, above which angle, on the slopes increased by the
angle
, the weight of the materials exerts slope-direction stresses on the plane
AO. According to Figure 5, on the slope with an inclination angle of
,
produced as a result of the depth increased by Δh, the granular material ADC weighs
heavily on the slope with its self-weight (ΔG) and with the weight of the material
located above it (G). The material amount ADC is supported on section Δh. The
projection of the surface section Δh, perpendicular to the slope is
)cos(
hF
. Considering that the Δh is very little, therefore, the stress
distribution can be considered even, so it can be stated for the slope-direction stress
developed there:
,
)cos)(sin(
F
GG
In the unit-length space part
,
2
2
ctg
h
G
and
,cos
sin2
1
h
h
G
that is,
,
2
ctgh
h
G
furthermore,
).cos(
hF
The (
cossin
) in the equation can also be expressed:
)cos()sin(
tg
,
),sinsincos(cos
cos
sin
sincoscossin
,sin
cos
sin
sincos
2
- 19 -
)sin(cos
cos
sin
22
,
.
cos
sin
Substituting the values of the G, ΔG, ΔF and
cossin
into the relation
written for the
:
)cos(
cos
s in
s in
cos
22
2
h
hh
ctg
h
,
,
)cos(
cos
sin
)(
2
h
hhctg
h
,
)sinsincos(cos
cos
sin
)(
sin
cos
2
h
hh
h
,
s in
s in
cos
coss in
2
h
hhh
,
sin
cossin
2
2
h
tg
h
hhh
but the
tg
can be expressed from the triangle ACE of Figure 5:
,
sin
sin
cos
h
h
h
tg
,
sin
cossin
2
hh
h
tg
therefore,
,
sin
sin
cossin
cossin
2
2
2
h
hh
h
h
hhh
,
sinsin
2
22
hhh
hhh
- 20 -
,
2 h
hhh
,
2
hh
if the Δh is very little, that is Δh → 0, then α →
, consequently
→
, i.e. the
direction of the stress inclines at an angle
to the horizontal. Consequently,
,
2
lim
0
h
h
that is,
2
h
.
As a result of the deduction it can be established that the stress distribution is linear
against the depth.
However, in case of granular materials one cannot speak of a stress in the classical
sense, since the force effects are transmitted at the contact points of the granules, i.e.
from one point to another, not on a surface perpendicular to the given direction. Not
on a surface, because the material is a discontinuum and the grains touch each other
only at points. Therefore, the meaning of the stress can be interpreted as the average
force imparted to one surface and these average forces are transferred from one
granule to another. The direction and size of these forces manifest themselves as a
statistical average on a given surface.
Figure 6. Division of the average force acting on the grain located
next to the wall into a horizontal and a vertical component
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Let us examine what the magnitude of the force is, which the above deduced stress
, inclining at an angle
to the horizontal, exerts on a vertical wall. Through the
contact points force arrives from the neighbouring granules at the grain –which is
supported by a frictionless vertical wall – shown in Figure 6. The resultant of this
force effect should be equal with the product of the stress
calculated for the
surface
1
F
, i.e. considering its magnitude and direction the compression force
acting on this grain corresponds to the statistical average of the force exerted at this
depth in the given granular material. This granule presses the vertical frictionless
wall with a horizontal
x
F
2
force on the wall section
2
F
at the contact point. The
vertical-direction force
3
F
y
is received by the grain or grains located under it.
The surface of sections
1
F
,
2
F
and
3
F
are equal surfaces on statistical average,
because, considering their shape and position, the grains are spheres on statistical
average; the projections of the spheres from any direction are of equal surface area.
(If a granular material – for example, rice – consists of oblong grains; considering
the random arrangement of the grains the average of their projection taken in any
direction is a circle, i.e. the grains must be considered as spheres on statistical
average.)
Consequently, it can be written for the vector triangle of the Figure 6
cos
12
FF
x
,
but since
21
FF
,
therefore
cos
x
.
It comes from the result of the above consideration that in granular material the
stresses – the average forces calculated for a given surface – can be decomposed or
added as vectors. From the
cos
x
equation the factor of static pressure is
received after the
substitutions, that is
h
y
cos
2
h
x
,
so
2
cos
.
- 22 -
Up to this point, only the shear stresses generated by the stresses acting
perpendicularly to the direction inclining at an angle
to the horizontal and rising
from the self-weight were taken into account at the deduction of the static pressure.
The shear stresses
y
, produced by the horizontal stress components -
cos
2
h
x
- reduce the vertical stresses hγ with
tg
h
y
cos
2
,
that is
,
cos
sin
cos
2
h
y
so
sin
2
h
y
.
Considering that the stresses
act in pairs on the theoretic plane OA assumed
inside the granular material (Figure 5), therefore, the vertical-direction stresses hγ
are reduced by 2
y
, consequently
yy
h
2
,
that is
,sin
2
2
h
h
y
and
)sin1(
h
y
.
Consequently, the figure of the stresses acting inside the non-cohesive granular
materials at quiescent state can be constructed (Figure 7).
Figure 7. Stresses acting at a quiescent state
- 23 -
If the plane OA according to Figure 5. is a theoretic plane assumed inside the
granular material, then the angle of friction is
there. In this case a compressive
stress of
, inclined at an angle
to the horizontal acts on both sides of the plane
OA (Figure 7.). On the plane OA the
’s can be divided into horizontal and
vertical components (Figure 8.). The horizontal components have the value
cos
2
h
x
,
they are perpendicular to lane OA and satisfy the action-reaction law. The vertical
stress components of
y
complement the vertical stresses to hγ symmetrically to
plane OA in a reciprocal way.
Figure 8. The horizontal and vertical stress component at a quiescent state
If the plane OA according to Figure 5 is a frictionless wall, then that is capable of
taking only horizontal stress, i.e. the horizontal component of the
, which is
cos
2
h
x
.
At the same time the vertical component of the
complements the vertical stress
component of the material OAB to hγ.
If the plane OA is an actual rough, rigid wall, which serves to support the OAB
material amount, then the development of the static pressure can be interpreted as
follows.
- 24 -
After filling up the OAB material amount behind the OA wall the value of the
2
h
is not reached immediately, since due to the inclined stress effect and
because of the friction rising on the wall a weight force intake is realised on the
plane OA. As a result of the weight-force intake the force acting on the plane AB is
reduced; so it is not capable of producing stresses with the value of
2
h
and in the
direction AB. The freshly filled-in material comes to a standstill by finishing its
consolidation motion. As a result of the quiescent state the friction rising on the wall
is reduced to zero, consequently the stresses
are decomposed into their
horizontal and vertical components. In the OAB material the vertical components
complement the vertical stress components to hγ, and supplement the weight of the
OAB material to
ctg
h
2
2
. At this time the horizontal component of stress
acts
on the OA wall. Consequently the horizontal component of the static pressure is:
,cos
2
h
x
and, therefore the factor of static pressure is (λ):
2
cos
.
Law IV
Law IV can be proved experimentally.
As a result of the experiment demonstrated in Figure 9, due to the collision of the
grains, the granular material behaves according to the laws of the communicating
vessels.
- 25 -
Figure 9. a) state of rest b) state formed as a result of vibration
In Figure 10 as a result of the vibration, and due to the collision of the grains the
body with bigger volume weight γ
1
and the body with smaller volume weight γ
2
, -
which were place into the granular material with volume weight γ - sinks to the
bottom of the vessel, or floats on the surface of the granular material respectively;
consequently the law of Archimedes prevails.
Figure 10. a) state of rest b) state developed as a result of the vibration
As a result of the vibration the components of the granular material, the grains
collides into each other, and due to this effect the pressure changes to hγ in every
direction.
- 26 -
Stresses in the non-cohesive granular materials
We performed the examination of the horizontal plane quadrant cut out by the
vertical plane in the proving procedure of the Law III. If this plane inclines towards
the material compared to the vertical, – closing an angle β with the horizontal, - and
the terrain is horizontal, then, generalizing the former deduction the magnitude of
lateral pressure of this granular assembly can be determined in the plane of angle β.
Figure 11. Infinite quadrant of the horizontal terrain confined
with an inclined plane
Using the markings of Figure 1, it can be said that the ADC granular material weighs
on the slope with an inclination angle
, produced as the result of the Δh
- 27 -
depth increase(as it is marked on Figure 11) with its self-weight (ΔG) and with the
weight of the material above it (G). Line segment Δh supports the ADC material
amount in order to prevent its sliding down. Since Δh is very little, therefore, the
stress distribution can be considered even in the area, thus the following can be
formulated for the slope-direction stress rising there:
,
)cos)(sin(
F
GG
where the ΔF is the projection of the surface segment Δh perpendicular to the slope.
Figure 12.
h
part of Figure 11. Figure 13.
F
part of figure 11.
The G given in the equation can be expressed for the unit-long space parts ΔG and
ΔF from Figures 11, 12 and13:
),(
2
2
ctgctg
h
G
and
sin2
hm
G
,
where m can be expressed with the help of Figure 11:
)sin(
zm
, and
,
)90cos(
0
h
z
thus
),sin(
)90cos(
0
h
m
- 28 -
that is
),sin(
sin
h
m
Replacing the value of the m into the relation written for the ΔG:
.
sinsin
)sin(
2
hh
G
The value of ΔF can be expressed with the help of Figure 13:
,)(sin
zF
.)(sin
sin
h
F
The equality
cos
sin
cossin
was deduced during the proof of the Law II.
Replacing the value of G, ΔG, ΔF and
cossin
into the relation written for
the σ
α
:
,
cos
sin
)(sin
sin
sins in
)sin(
2
)(
2
2
h
hh
ctgctg
h
,
)(sin
sinsin
)sin(
)(
cos
sinsin
2
hctgctgh
h
h
,
sin)cos(cos)sin(
sin
sincoscossin
)cossin(
cos
sin
2
hctgh
h
h
,
)cos()sin(
)cossin()cossin(
cos
1
2
ctg
ctghctgh
h
h
,
)cos()sin(
cossin
cos2
ctg
ctg
h
hhh
but
ctg
can be expressed from the triangle ACE of Figure 11 by employing the
triangle ADF of Figure 12:
m
AFCD
m
AE
ctg
,
where:
sin
h
CD
and
)cos(
zAF
,
- 29 -
but
sin
)90cos(
0
hh
z
,
therefore
)cos(
sin
h
F
,
so
,
)cos(
sinsin
m
hh
ctg
since
),sin(
sin
h
m
,
)sin(
sin
)cos(
sinsin
h
hh
ctg
)sin(sin
)cos(sinsin
h
hh
ctg
,
);(
)sin(sin
sin
ctg
h
h
ctg
replacing the value of
ctg
into the relation of the
:
,
)cos(
)sin(
)sin()cos(
)sin(sin
)sin(sin
cossin
cos2
h
h
ctg
h
hhh
,
)cos()cos(
sin
sin
cossin
cos2
h
h
ctg
h
hhh
cossin
sin)cossin)((
2
h
ctghhh
,
),1(
2
tgctg
hh
if Δh is very little, that is Δh → 0, then α →
, consequently
→
, i.e. the
direction of the stress closes an angle
with the horizontal.
Consequently
),1(
2
lim
0
tgctg
h
h
- 30 -
that is
tg
tgh
1
2
.
The horizontal component of the
is:
)
sin
(cos
2
tg
h
h
.
Examining the three special values of the slope angle β of the plane, (which is the
angle at which the plane inclines to the horizontal), it can be established that if
, then
0
, i.e. the non-cohesive granular material will stop in the free
slope without support.
If
0
90
, then the static pressure acting on this plane is:
2
h
, and
cos
2
h
h
.
If
2
45
0
, then the static pressure acting on this plane is:
sin1
1
2
h
, and
2
45
2
0
tg
h
h
.
- 31 -
Active stress state
Development of the active stress state
The small-size horizontal-direction displacement – tilt – of the vertical wall
supporting the non-cohesive granular material being in a state of rest causes
expansion in the material. The motion of the material follows the displacement of
the retaining wall into the horizontal direction loosened up, which appears as a
relatively two-direction displacement from a given point of the interior of the
material. As a result of the displacement following the expansion, the effect of the
shear stresses mobilised by the horizontal stress components of the static pressure
ceases (breaks up). The relatively two-direction displacement inside the material
terminates the vertical-direction shear stresses in pairs, therefore the vertical stress
increases to hγ. At the same time the material begins to carry out a consolidation
motion.
Figure 14. Stress model in quiescent a position
- 32 -
The vertical stress of Figure 7 and 8 (state of rest) can be divided into two stresses of
the same size and direction (Fig. 14). The development of the active stress state can
be explained in the horizontal-surface non-cohesive granular materials by the change
of the so obtained stress model. The shear stresses, which were terminated due to the
effect of the expansion, change the stress model of Figure 14 to that of Figure 15.
Figure 15. Change of the stress model in an active state
The consolidation motion occurs in the direction of the biggest stresses, i.e. in the
direction of the stress resultants. The directions of the resultants of the stress pairs –
which can be read from Figure 15 – incline at an angle
2
45
0
to the horizontal
and at an angle
0
90
to each other. Considering the acting (resultant) stress
directions this stress condition consequently, corresponds to the Rankine active
stress state. Stress
1
R
starts the consolidation motion. This motion is reduced by the
multiplication product of
tg
and stress
2
R
- a stress perpendicular to the stress
1
R
-
,
1
R
mobilises shear stress. The magnitude of stress K inclining at an angle
2
45
0
to the horizontal, consequently is
.
21
tgRRK
Stress
1
R
consists of two stresses.
1
R
and
2
R
can be expressed from the illustrations
of the stress vectors in Figure 16.
Stress
1
R
presents itself as the sum of two stresses; the sum of its stress components
taken for this direction (
2
R
) develops the shear stress.
- 33 -
Figure 16. The motion started by the resultant stress mobilises shear tress
,
2
45sin
2
2
0
1
h
R
and
,
2
45cos
2
2
0
2
h
R
thus
.
2
45cos
2
45sin
00
htghK
The horizontal component of the resultant stress K:
,
2
45cos
0
KK
h
,
2
45cos
2
45cos
2
45sin
0200
tghhK
h
,sin1
2
1
2
452sin
2
1
0
tghK
h
because
,sin1
2
1
2
45cos
02
that is
,
2
sin
2
cos
2
2
2
45cos
2
02
- 34 -
,
2
cos1
2
cos1
2
1
2
45cos
2
02
2
cos1
2
cos1
2
cos1
2
2
cos1
2
1
,
,
4
cos1
21
2
1
2
.sin1
2
1
,s in1
cos
sin
cos
2
h
K
h
,
cos
sin1sincos
2
2
h
K
h
,
cos
sinsincos
2
22
h
K
h
,
cos
sin1
2
h
K
h
2
45
2
0
tg
h
K
h
.
The vertical component of the resultant stress K is
v
K
.
,
2
45sin
0
KK
v
2
45cos
2
45sin
2
45sin
0002
htghK
v
,
since
sin1
2
1
2
45sin
02
and
cos
2
1
2
45cos
2
45sin
00
,
thus
,cossin1
2
1
tghK
v
,cos
cos
sin
sin1
2
1
hK
v
.
2
h
K
v
- 35 -
The resultant stress K is
22
vh
KKK
,
1
2
45
2
02
tg
h
K
,
2
45cos
2
45cos
2
45cos
2
45sin
2
02
02
02
02
h
K
,
,
2
45cos
2
45cos
2
45sin
2
02
0202
h
K
,
2
45cos
1
2
02
h
K
.
2
45cos
1
2
0
h
K
Comparing the value of the K with the value of the
– which is
2
h
- it is
conceivable that the K is bigger. Therefore, in case of expansion, or in case of a
more significant displacement of the wall supporting the granular material the
motion direction of the material inclines at an angle of
2
45
0
to the horizontal.
Due to the expansion the stress starting the motion can be illustrated according to
Figure 17.
Figure 17. Motion starting stress in an active state
- 36 -
The motion is realised towards the direction of the resultants of the stresses,
consequently, in the direction inclining at an angle of
2
45
0
to the horizontal. The
material moves with its whole material amount, i.e. infinitely many slip planes
inclining at an angle of
2
45
0
to the horizontal are developed.
The horizontal component of the resultant stress K is
x
, i.e.
2
45sin
2
45cos
00
KK
x
,
2
45cos
2
45sin
2
0
0
h
x
,
2
45
2
0
tg
h
x
.
Pressure acting on the vertical retaining wall
If the horizontal-terrain non-cohesive granular material is supported by a real,
frictional, vertical wall, and an expansion occurs in the material due to its
displacement, then the stresses acting on the wall can be determined with the
knowledge of the angle (δ) of the friction rising on the wall:
A stress with a magnitude of
, inclining at an angle δ to the horizontal acts on the
wall. The
sin
-fold amount of this stress reduces the vertical stress
2
h
to
sin
2
h
. The horizontal stress component:
cos
is in direct proportion to
- 37 -
the vertical stress, just as the ratio of the vertical
2
h
and the horizontal
cos
2
h
stress components is constant in the stress model acting inside the material.
Consequently the proportion can be written:
cos
sin
2
cos
2
2
h
h
h
.
The
can be expressed:
cossincos
22
cos
2
hhh
,
cos
2
cossincos
h
,
cos
2
)cossin(cos
h
,
cossincos
cos
2
h
.
The horizontal component of the
is
h
:
cos
h
,
cossincos
coscos
2
h
h
,
cos1
cos
2
tg
h
h
.
The obtained result shows, that if no friction were developed on the retaining wall,
then the static pressure would act on it; or looking at it from the other way round: no
friction rises on the retaining wall when static pressure develops. This is proved by
the evidence of the model experiments.
If the friction rising on the retaining wall were equal to the friction rising inside the
granular material, i.e.
, then active pressure would act on the retaining wall,
which pressure also prevails inside the material in the active stress state. If
does
not reach the value of the
, then an intermediate stage – between the static pressure
and active pressure – emerges near the retaining wall.
- 38 -
In the past several people performed experimental measurements with dry sand – a
non-cohesive granular material –. for the determination of the lateral pressure.
Taking accuracy and model size into consideration the experiments started by
Terzaghi in 1929 rose above the other researches. In his experiment the retaining
wall was a 2.1-metre-high and 4.2-metre-long rigid reinforced concrete structure.
The volume of the sandbox was 37
3
m
and the displacement of the wall was
measured with an accuracy of 0.0025 mm. The results of the experiment can be
summed up as follows:
While the retaining wall was motionless, a horizontal, static pressure with the
magnitude of
2
42.0
2
0
h
E
acted on it. At the slight displacement of the wall the
lateral pressure decreased, then due to further displacement, tilt of the retaining wall,
the horizontal component of the lateral pressure became constant near the value of
2
29.0
2
h
, while the tangent of the friction developed on the retaining wall moved
near the value
54.0
tg
. Due to the expansion that occurred in the sand, and as a
result of the loosening the surface sank near the displaced wall.
The measured values correspond well to the result obtained by means of the
previously deduced theoretic formulas:
- the static pressure coefficient was
42.0
,
2
cos
,
42.02cos
,
0
85.32
, which is a value characteristic of the dry sand.
- the horizontal stress component of the pressure acting on the frictional retaining
wall is:
cos1
cos
2
tg
h
h
,
the measured value
84.0cos
and
54.0
tg
,
84.054.01
84.0
2
h
h
,
2889.0
h
h
,
i.e. it is remarkably consistent with the expected measured value of ca. 0.2889≈0.29.
- 39 -
Arch formation in granular materials
The phenomenon of the arch formation is one of the basic questions of the
mechanics of the granular materials. The theoretical clarification of arch formation
provides solution to such direct practical problems, as the bulk storage of granular
materials in silos or their safe discharge. In the hoppers of the silos the material often
coagulates, or an arch is formed, which impedes the gravitational discharge.
Condition of arch formation
In each case it is always the displacement of a part of the material, which generates
arch formation. This motion can originate from consolidation, compaction or, for
example, from the material motion that follows the opening of the gate located on
the bottom of the hopper. Due to the displacement, the stresses in the material are
rearranged in a way that the retaining part of the material that remains in place takes
over also part of the stresses of the moving material part. If the stresses, which rose
this way are big enough and their direction is adequate, an arch will be formed in the
material, which will prevent any further displacement. The arch-forming effect of
the displacement prevails, when the material must undergo specific deformation
during the displacement, i.e. it must pass through for example a narrowing cross-
section.
- 40 -
On the basis of the aforementioned considerations, if we want to follow the process
of arch formation, an infinitely long symmetric trough with narrowing cross-section,
filled with non-cohesive granular material (Fig. 18 ) can be chosen as the starting
point of the examination.
Figure 18. Dimensions of the trough
Let us assume, that the volume weight of the material (
) does not change as a
function of the depth and the material does not compress after the filling. A movable
bottom-plate closes the
b
-wide lower opening of the trough, which has a flat and
rigid side wall inclining outwards with an angle β to the vertical. The assumption of
an infinite length makes the planar examination of the case possible. After the
removal of the bottom-plate of the trough the material moves off – it wants to flow
out – and undergoes specific deformation as a result of the narrowing cross-section;
consequently the model ensures the conditions of arch formation as described
before.
If the granular material is at a quiescent state and the side walls are rigid, then static
pressure develops inside the material; consequently the pressure is hγ in the vertical
and λhγ in the horizontal direction, where λ is the quotient of the vertical and
horizontal pressure i.e. it is the coefficient of the static pressure.
- 41 -
The stresses acting on the side wall inclining outwards at an angle β to the vertical,
and respectively the resultant force of the stresses (E) can be determined according
to magnitude and direction on the basis of Figure 19:
Figure 19. Force equilibrium of the trough with a closed bottom plate,
filled with granular material
0
G
is the weight of the material part between the vertical plane and the side wall,
which is inclining outwards at an angle β:
,
2
2
0
tg
h
G
0
E
is the resultant force of the horizontal static pressure:
.
2
2
0
h
E
The resultant force from the vector triangle is:
.
2
22
2
tg
h
E
For the inclination angle of the resultant force, inclined to the horizontal plane it can
be written:
,
0
0
E
G
tg
that is
.
tg
tg
If the side wall can take up the α-direction force and the stresses due to the lateral
wall friction, then only vertical stresses with a magnitude of hγ act on the bottom
- 42 -
plate. This is only true, if
,
where the δ is the angle of the friction on the
side wall, rising between the wall and the material.
After removing the horizontal plate closing the discharge opening of the trough,
shear planes form inside the material, due to the fact that the material can pass
through the narrowing cross-section only by shear. If
, i.e. only vertical
stresses acted previously on the horizontal plate, then the plane of the shear will be
vertical. Since the material is sheared on that surface, to which the smallest force is
needed. The force necessary for shearing the non-cohesive granular material is
expressed by the relation,
tgAF
n
, using the Coulomb’s equation, where
A ─is the sheared surface,
n
─ is the perpendicular stress acting on the sheared surface,
─ is the friction angle of the material.
The smallest shear force is necessary for the shearing of the vertical plane, since the
sheared surface is the smallest here and it is also the plane where the horizontal
stresses are the smallest. (The horizontal stresses are always smaller than the vertical
or intermediate-direction stresses) Consequently, the plane of the shear is vertical.
From both of the points B and D of the trough a vertical shear plane is formed, if the
vertical-direction force rising from the weight of the material part located above the
opening b is equal with the shear force demand of the two planes:
,
2
2
2
tg
h
hb
that is
tghb
.
If the size of b is bigger than this value, then the material with a
width is
torn off in one part and takes with itself – under the effect of the acting shear stresses
- the other material parts as far as the total opening b of the trough.
If
tghb
, then only two vertical-direction shear surfaces develops after the
removal of the plate closing the discharge opening of the trough. At this time shear
stresses arise inside the material on the shear plane, which are produced by the
vertical weight force
hb
. The material part, which is located between the shear
- 43 -
plane and the side wall of the trough takes up the vertical shear forces F, and
transfers them to the side walls. The side wall can only take up this vertical-direction
plus force entirely if the resultant force (
B
E
) does not exceeds the angle of the
friction arising on the side wall.
Figure 20. Force equilibrium after the removal of the bottom plate of the trough
It is well discernible on the vector polygon of Figure 20 that the resultant force
B
E
does not exceed the friction angle δ, if the inclination angle of the side wall and the
friction arising there is big enough, i.e.
. From the vector diagram the
following can be formulated for the limit case
.)(
0
0
E
GF
tg
The shear force F is the half of the material weight above the opening, since it is
divided into two shear planes, consequently
,
2
bh
F
furthermore
tg
h
G
2
2
0
an
.
2
2
0
h
E
Substituting the values of the F,
0
G
an
0
E
into the relation written for the tg (β+δ)
we get:
2
22
)(
2
2
h
tg
hbh
tg
,
- 44 -
simplified:
h
tghb
tg
)(
consequently, if
tghb
, then starting from the lower opening of the trough,
individual shear planes are formed and the side wall can take up the weight of the
material part located between them by the rearrangement of the shear stresses, if
.)(
h
tghb
tg
In this case the material cannot flow out and an arch is formed above the opening.
From the relation
tghb
and the relation obtained for the
)(
tg
the
relationship
h
b
can be expressed:
tg
h
b
,
tgtg
h
b
)(
.
These two equalities and inequalities formulate the condition of the formation of the
arch in the non-cohesive granular material.
According to the solution of the arch formation, calculated with the help of the
resultant forces, the shear force F diverts the fulcrum of the resultant force E of the
quiescent state towards the lower opening of the trough. The intersection point of the
line of action of the forces E and F determines the position of
B
E
. The deviation is
in direct ratio with the increase of the angle β. This deviation can be left out of
consideration in case of the practical calculations. Since, in the event of a significant
increase of the angle β the arch already rests directly on the material and not on the
side wall (see later).
Character of the discharge
The character of the discharge of the granular material flowing out of the trough can
be interpreted with the previously deduced arch conditions in cases, when any one of
the conditions is not fulfilled.
- 45 -
a) If
, i.e.
tgtg
h
b
)(
, then the side wall cannot take entirely
up the resultant force, which changed due to the rearrangement of the shear stresses.
The free component of the resultant force E, parallel with the side wall, sets off the
slide of the material along the side wall. The material part located between the shear
plane and the side wall also moves off and mass flow occurs (Figure 21/a).
Figure 21. Typical discharges: a) mass flow; b) tunnel flow
b) If
, i.e.
tgtg
h
b
)(
, but
tghb
and
tg
h
b
, then
even though the side wall can take up the resultant force, but the weight force of the
material located above the opening is bigger than the shear force preventing the
break away; therefore, the material with a width of b flows out vertically, while the
material part next to it – the part between the vertical shear plane and the side wall -
remains in its place, and flows from above to the vertically moving material part
afterwards. Tunnel flow develops (Figure 21/b).
It is easy to see that the character of the discharge is determined not only by the
inclination angle and wall friction of the trough, but it is also influenced by the shear
resistance, friction angle and the pressure conditions of the material. It can be
observed in silos, that certain granular materials are discharged firstly by mass flow,
then later by tunnel flow, proving that the pressure conditions dominating in the
hopper also influence – though to a smaller extent than the inclination angle and
friction of the hopper – the character of the discharge. In the course of our
experiments performed with wheat and sand with the application of hoppers with
- 46 -
different inclination angles , it could be observed (when the experiment was carried
out in the same hopper and with the same material) that at first mass flow occurred,
but when the discharge was continued, the phenomenon of tunnel flow appeared.
The change of the character of the material discharge occurred at the pre-calculated
height value.
Mechanism of the arch formation
Consequently, the conditions of the arch formation are, that the weight force of the
material above the lower opening of the trough should be smaller than then the sum
of the shear forces arising on the shear planes, and that due to their inclination angle
and wall friction, the side walls should be able to take up the forces acting on the
them.
The shear stresses developed after the removal of the plate closing the lower opening
of the trough are transferred onto the side wall and summed up with the stresses
acting there. These resultant stresses form the arch. The arch surface is formed as a
result of a second stress rearrangement.
Figure 22. The arch is formed as a result of stress rearrangement
Between the edges B and D of the trough the distribution of the horizontal
components of the resultant stresses is as shown in Figure 22/a. At the places where
the horizontal components of the compressive stresses prove to be too small to
- 47 -
support the material against the gravitation – its shear resistance is smaller than its
weight force arising from the gravitation – there the dropping of granules. Due to the
bleed the stresses rearrange, presumably according to Figure 22/b, in order to
provide enough compressive stress for the support of the material. The dropping of
granules ceases when identical horizontal stress components – of critical value in
terms of the dropping of granules on each point of the arch surface. The fact, that on
each point of the arch, the same-value horizontal stress components must act, makes
the determination of the equation describing the geometric shape of the arch
possible.
Geometric equation of the arch
It is known, if for the support of an evenly distributed load such quadratic parabola
is used, in whose end-point only tangential stress develops, then in each point of the
parabola only a stress with a parabola-direction, and equal-size horizontal
component arises. Such a parabola is loaded by no bending moment, which
condition is of vital importance. The bending moments would produce tensile and
compressive stresses, which a solid body can withstand, but the non-cohesive
granular material is not capable of bearing tensile stresses.
In the points B and D of the trough an equal-size stress acts in the β+δ –direction,
while above the arch there is an almost evenly distributed load. The direction β+δ
and the opening width b definitely determine the parabola. The maximum rise of the
parabola that has the aforementioned characteristics is:
)(
4
tg
b
f
.
With the coordinate axis y placed in the symmetry plane of the opening of the
trough, and the axis x leading through the points B and D (Fig. 23) the cuspidal point
of the parabola intersects the axis y at the height of C.
- 48 -
Figure 23. Parabola of the arch
The general form of the equation of the quadratic parabola symmetrical to the axis y
and running downwards is:
CAxy
2
,
since f=C, so
)(
4
tg
b
C
.
The first differential coefficient of the equation of the parabola at the place of
2
b
x
is
)('
tgy
, consequently
CAxy
2
,
Axy 2'
and
2
2)(
b
Atg
.
A can be expressed:
)(
1
tg
b
A
Substituting the values of A and C into the general equation of the parabola:
)(
4
)(
2
tg
b
tg
b
x
y
,
)(
4
2
tg
b
xb
y
we get the geometric equation of the arch. From the equation formulating the arch
condition the
)(
tg
can be substituted:
h
tghb
b
xb
y
2
4
.
- 49 -
In some cases the arch is lower than the form given in the above equation. It occurs
when the inclination angle of the trough is big. In this case the arch is supported by
that plane of the material, which inclines at an angle ε to the vertical, and ε<β. In the
material the angle of the friction is
, therefore,
substitutes the values of β+δ
in the equation that formulates the arch condition and describes the geometric form:
,)(
tgtg
h
b
)(
4
2
tg
b
xb
y
.
The arch leans directly on the material in the case, when
is smaller than β+δ.
If from the two arch conditions the equation
tgtg )(
restricts the ratio
h
b
,
the limit case can be achieved at the identical ratio of
h
b
.)()(
tgtgtgtg
The equation formulated for the limit case gives a solution for the ε only in case of
certain
tg
tg
values, because it is a quadratic equation and its discriminant can be
negative depending on the ratio of δ and
. If the discriminant is negative, then the
arch will continue to lean on the side wall of the trough. If the discriminant is
positive, then the equation
)(
4
2
tg
b
xb
y
determines the geometric form of the arch.
If from the arch conditions
tg
restricts the ratio
h
b
, then the angle ε can be
calculated from the following relation:
tgtgtg )(
.
- 50 -
Principle of the hopper design
The relations formulating the condition of arch formation in granular media makes
the design of such hoppers possible, from which the gravitational discharge of the
material can be ensured and which takes up the smallest possible space.
If none of the arch condition is fulfilled, then the gravitational flow is ensured. If
tg
h
b
,
but
tgtg
h
b
)(
,
then the gravitation discharge takes place in the form of a tunnel flow. If, however,
tgtg
h
b
)(
,
then the discharge is of mass flow type.
At the design of the hoppers, the goal is, in general, to ensure the mass flow with the
smallest discharge opening, in a way that the vertical dimension of the hopper
should be the smallest one possible. The principal procedure of the design for a
hopper with circular cross-section is as follows.
In the case of hoppers with circular cross-section, instead of a width b opening a
radius r discharge opening must be used, so instead of the equation
tg
h
hb
2
2
2
evidently
tg
h
rhr
2
2
2
2
,
is written, from which
tg
h
r
.
- 51 -
Before designing the hopper the internal friction angle of the granular material, the
friction angle developing on the surface of the hopper and the volume weight must
be determined. The friction angles can be determined by shear experiments. The
normal loads applied to the material, which was filled in the shear box, must
correspond to the expected pressure values in the hopper. For the measurement of
the friction angle δ it is advisable to make a packing plate of the material of the
hopper for the shear box. The granular material is filled onto the packing plate
placed in the shear plane of the shear box, and the angle of the friction is determined
by shear experiments. If the material stays in the hopper for a longer period, then the
required rheological measurements must also be performed: the shearing is carried
out by changing the time of the normal loads acting on the sample. From the
tendency of the curves of the material properties drawn in the logarithm of the time
the expectable values of the material characteristics during a longer storage period
can be concluded. If the granular material is also cohesive, or becomes cohesive in
the course of a longer storage, then for the given normal stress value of the shear, the
inclination angle to the horizontal axis of the straight-line drawn from the origin,
can be taken into account (see later), as the angle of the internal shear resistance
(effective friction angle).
Figure 24. Hopper-design construction
- 52 -
The principal procedure of the hopper design is as follows:
1. Let us determine the value of the
h
r
k
1
with the help of the
tg
, which
with the substitution of
2
cos
is:
;
2
sin
1
h
r
k
2. Let us set up the symmetry axis of the hopper according to Figure 24 and
construct straight lines with different
h
r
ratios;
3. Let us calculate the hopper inclination angle
1
belonging to the straight line
1
k
from the relation
tgtgk )(
:
tg
tgktgtgktgk
tgarc
2
44)1(1
22
;
4. Let us draw angle
1
to the given point of the straight line
1
k
. (On the upper
part of the hopper the mass flow can be ensured by using the hopper inclination
angle
1
, by using an angle that is steeper than that.);
5. Let us calculate the value of angle
2
for a ratio
2
k
, which is smaller than the
critical value
1
k
;
6. Let us construct angle
2
on the intersection point obtained on the straight line
1
k
and draw its leg as far as the ratio line
2
k
, then draw the angle
3
calculated on
the basis of the
3
k
on the straight line
2
k
, then continuing the construction and
calculation we get a hopper with a curve-constituent approached with individual line
segments;
7. By determining the proportion for the given discharge-opening dimension, or
for the upper diameter of the hopper, we obtain a dimension-correct hopper shape is
obtained, from which the chosen dimensions can be read.
- 53 -
Consequently, the hopper profile ensuring a favourable discharge is a curve.
If technological difficulties justify the construction of a hopper with a straight
constituent, then the inclination angle calculated in the dimension of the discharge
opening is the decisive factor. If we disregard the mass flow, the only requirement
is, that an angle steeper than the natural angle of repose must be chosen instead of
the ratio
1
k
, the inclination angle of the hopper is indifferent. The great advantage of
the hopper with curve constituents is, that its build-in space demand is the smallest
possible, and it can be inserted as a packing into the existing hoppers, by which the
discharge difficulties can be efficiently improved.
The flow-improving advantages of the hyperbolic hoppers with curve constituents
are known, which are also proved experimentally by the hoppers designed on the
basis of the present theory.
Experimental results
After the elaboration of the theory we carried out measurements with experimental
tanks and hopper designed for the given material to be stored. There was a 2 metres
high and 1 metre diameter material column above the hopper designed for the
material characteristics and friction parameters of the corn-grits. At the straight
conical hopper with an inclination angle β=30º an arch was formed as far as the
hopper opening with a diameter of 150 mm, which impeded the gravitational
discharge. At the same tank but with a hopper with curve-constituents, and with an
opening diameter of 100 mm and with the construction of a shorter hopper obtained
a safe discharge (Figure 25).
- 54 -
Figure 25. Curve-component hopper
The experimental measurements proved our theoretical calculations not only for
corn grits, but also for wetted sand – as a model material – the calculations were
furthermore verified with experiments for fertilisers and mixed feeds, for extracted
soya grits, feed lime and alfalfa flour.
- 55 -
Stresses in cohesive granular materials
Granular materials generally exhibit, smaller or bigger cohesion depending the on
their moisture content. Concerning their mechanical behaviour, the cohesive
granular materials show a considerable difference from the non-cohesive granular
materials, which justifies their separate treatment.
A cohesive granular material is the conglomeration of large number of solid bodies,
which are in constant contact with each other, where the cohesion force between the
grains – as the constituting elements of the assembly – is smaller than the cohesive
force of the individual grains. Coulomb’s friction law pertains to the material and the
individual grains keep their shape in spite of forces acting on them.
Lateral Pressure
According to the Coulomb’s friction law the shear resistance of the cohesive
granular materials can be described by means of the following relation
ctg
n
where:
τ = shear resistance of the material;
n
= normal stress acting on the sheared surface;
- 56 -
= friction angle of the material;
c = cohesional coefficient.
The shear resistance consists of two parts, the friction (which depends on the
pressure acting perpendicularly to the sheared surface) and the cohesion (which is
independent of the normal stress).
Figure 26. Direct relation of the slip limit angle
and the Coulomb’s
straight-line
The relation τ –
n
is linear (Figure 26); consequently, it can be represented a
straight line, the so called Coulomb’s straight-line. The points lying on the straight-
line represent the slip limit state, i.e. boundary-equilibrium state. Inside the cohesive
granular material, which is in a state of rest, a limit angle belongs to each depth, i.e.
to each vertical-direction, hγ-size stress value, rising from the self-weight, where a
slip boundary state can be found. This inclination angle to the horizontal is: Φ,
which is the inclinational angle of the straight line linking a given point of the
Coulomb’s straight-line and the origin, to the horizontal axis (Figure 26). The angle
Φ changes depending on the normal stress, i.e. normal stress produced by the self-
weight inside the cohesive granular material in a state of rest. Consequently, angle Φ
depends also on the depth. On the basis of Figure 26 the following can be formulated
for angle Φ:
n
tg
,
since
ctg
n
,
- 57 -
and
cos
h
n
,
so
cos
h
c
tgtg
.
Consequently, the angle Φ depends on the friction angle, the cohesional coefficient,
as well as on the product of the depth and volume weight, where the relation is no
longer linear. Inside the cohesive material, consequently the surface belonging to the
slip boundary state is a curve, the angular coefficient of whose tangent is tg Φ. As a
matter of fact angle Φ is nothing else but the angle of the shear resistance, similarly
to the friction angle used for the characterisation of the non-cohesive granular
material. The physical content of the angle Φ is identical to that of the angle
of
the non-cohesive granular materials. (Angle Φ, which changes as a function of the
depth, declines into constant
in the case of c=0.)
Considering the angle Φ into consideration, the laws II and III concerning the non-
cohesive granular materials can be applied to the cohesive granular materials, as
well:
II. In the cohesive granular materials at a quiescent state the stresses developed by
the vertical-direction compressive stresses act downwards in the zone
0
90
measured from the vertical direction (where the Φ is the angle of the internal shear
resistance of the material).
III. The static pressure value of the cohesive granular material, rising from the self-
weight, is the half of the product of the depth and the volume weight
2
h
, its
direction deviates downwards from the horizontal with the angle of the internal shear
resistance of the material. The horizontal component of the static pressure:
.cos
2
h
x
The application of the static pressure of the non-cohesive granular material for the
cohesive granular material becomes feasible due to the fact, that in granular
materials the ratio between the vertical stress rising from the self-weight and the
horizontal stress components depends only on the physical parameters of the
- 58 -
material, to be more precise it depends only on the angle of the shear resistance of
the material.
Since that angle Φ changes as a function of the depth; the cos Φ can be expressed
from the relation
cos
h
c
tgtg
:
,cossin
h
c
tg
h
c
tg coscos1
2
,
01cos
2
coscos
22
2
222
h
c
tg
h
c
tg
1
1
cos
2
22
2
2
tg
h
c
tgtg
h
c
,
since
2
2
cos
1
1 tg
,
so
2
22
2
cos1coscossincos
h
c
h
c
,
respectively
sincos
cos
cos
2222
cch
h
.
Substituting the relation obtained for the cos Φ into the formula
x
the horizontal
component of the static pressure is:
sincos
2
cos
2222
cch
x
.
The horizontal stress components
x
have a negative sign till a certain depth
0
h
. In
the depth
0
h
the value of the
x
is 0. The depth of
x
=0 can be expressed:
0sincos
2
cos
222
2
0
cch
,
,sincos
22222
2
0
cch
22222
0
cossin ch
,
c
h
0
.
- 59 -
Up to the depth
0
h
the horizontal-surface cohesive granular material stands also
without support in the vertical wall.
The value of the resultant force
0
E
acting on the h high vertical retaining wall can
be obtained by means of the definite integral of the horizontal stress components
x
, taken from the depth
0
h
to h (if on the retaining wall no friction develops,
which according to the model experiments occurs as a result of rest):
h
h
x
dhE
0
0
.
The result of the integration:
h
h
ch
ch
h
E
0
cossin
2
coscos
4
2222
0
h
h
c
h
c
hc
0
1
cos
cos
lncos
4
22
22
3
2
The term containing the ln can be left out because it is the third power of the cos
,
which is smaller than 1, and due to the fact that product of ln has a relatively small
value , thus
c
h
c
ch
h
E 2cossin
4
cos
4
cos
2222
0
.
Consequently, the resultant force of the static pressure of the horizontal-surface
cohesive granular material, acting on the vertical retaining wall is :
sin2cos
4
cos
2
2222
0
h
c
cch
h
E
.
Inclination angle of the free slope
The angle Φ of the straight line linking a given point of the Coulomb’s straight-line
characteristic of the cohesive granular material and the origin, to the axis
n
characterises the shear strength of the material in the depth belonging to the given
- 60 -
normal stress
n
as the friction angle
characterizes the non-cohesive granular
material. The only difference can be discerned in the fact, that in the cohesive
material this angle changes depending on the normal stress acting on the sheared
surface. The tangent of the angle Φ can be expressed as a function of the depth in the
following way:
Starting from the previously written relation
cos
h
c
tgtg
and
substituting
2
1
1
cos
tg
h
tgc
tgtg
2
1
,
from which relation the tg Φ can be expressed:
02
2222222222
ctghtgtghtghc
,
222
22222222
ch
chtghctgh
tg
,
)(cos
cossin
222
222222
ch
chch
tg
.
Taking into consideration, that in the cohesive granular materials, in a state of rest,
the direction of the stresses rising from the self-weight inclines at an angle bigger
than Φ to the horizontal; the biggest inclination angle of the free slope is determined
by those stresses, which are produced by the vertical stresses rising from the self-
weight, inclining at an angle Φ to the horizontal, which just do not exceed yet the
side of the slope.
The cohesive slope can be constructed from the shear straight-line as follows (Figure
27):
After taking up the Coulomb’s straight-line on the basis of the shear experiments, we
can draw angle Φ, which changes depending on the depth: the angle of the internal
shear resistance corresponding to point A of the Coulomb’s straight-line is
A
. In
the triangle OAD the side length OD is
A
h cos
, i.e. it is the component of the
vertical stress rising from the self-weight, perpendicular to the direction
A
.
Consequently, the hypotenuse of the triangle OA is hγ, so taking the hypotenuse into
the span of the compass, we get point A’ by turning down the hypotenuse to the axis
- 61 -
hγ starting vertically from point O, to which point the angle
A
is copied.
Constructing the angles Φ belonging to the points taken up on the Coulomb’s
straight-line onto the straight line hγ in this way, the envelope curve of the stress
directions can be drawn. If the scale is divided by γ on the axis hγ, then we can
obtain the geometric shape of the steepest slope of the horizontal-surface cohesive
granular material.
Figure 27. Constructing the cohesion slope
As far as the height
c
the cohesive granular material stands without support in a
vertical wall, while the angle Φ is reduced by the increase of the depth, and goes
towards the friction angle
, as a limit. The shape of the steepest slope is of
hyperbolic character. If the determination of the inclination angle of the steepest, but
flat-surface slope, in proportion to the horizontal is the task, then the inclination
angle β of the straight line connecting the given point of the envelop curve of the
stress directions - which curve inclines to the horizontal at an angle Φ – to the O
origin, as a flat slope side to the horizontal, determines inclination angle of the
- 62 -
chosen slope. Constructing the envelope curve of the stress directions inclining to
the horizontal at an angle Φ by the help of the axis h - just directed over it (Figure
28) - , and starting from the origin O, the height belonging to the point cut on the
envelope curve by the leg of a requested angle β provides the slope height, where
under a slope angle β, and in case of a flat slope side the horizontal-surface cohesive
granular material is still capable of standing without support.
Figure 28. Relation between the slope height and slope angle
The envelope curve of the Φ stress directions has a concave character (Figure 27).
The side of the slope can be flat (Figure 28), but convex as well. The extent of the
convexity depends on the requirement that the stress directions examined on a
vertical plane laid through any point of the slope, changing as a function of the
depth, inclining at an angle Φ to the horizontal should not intersect the side of the
slope. The convex envelope curve of the slope with a given height
x
h
can be
constructed according to Figure 29:
The concave envelope curve is constructed by the method shown in Figure 27. The
geometric shape of the
x
h
high concave slope is described by point B cut from the
envelope curve by the horizontal line drawn from the point
x
h
and by the curve
between
0
h
O. The convex slope belonging to the height
x
h
can be obtained by
- 63 -
turning of the curve
BOh
0
in such way that the positions of the points O and B are
exchanged.
It is easy to prove by means of construction that in case of a convex slope the base
point B is a stress-collecting place; therefore, the stability of the convex slope is less
certain than the concave slope. Furthermore it is well discernible from the
construction that the inclination angle of the straight line segment OB, measured to
the horizontal, is at the same time, the biggest inclination angle β of the flat-surface
slope belonging to the height
x
h
.
Figure 29. Constructing the convex slope
In Figure 29 in the part stripped between the geometrical slopes of the free slope
with convex and concave boundary states, of height
x
h
, the geometric shape of the
side of the slope can be so chosen that - assuming a continuous curve –that above the
section OB it should show a convex character, below it a concave one. In case of a
broken-line slope the examination must be performed for the Φ direction stresses
depending on the depth, in order to prevent the stress directions from intersecting the
side of the slope.
The formation of a slope with a convex boundary position can be expected, if the,
for example, vertical wall supporting the material is removed carefully moving
- 64 -
downwards, while the slope with a concave boundary position can be obtained by
removing the wall vertically upwards.
Active stress state
Inside the non-cohesive granular material the horizontal component of the active
pressure is expressed by means of the following relation
2
45
2
0
tg
h
x
In the cohesive granular material the following equation can be written for the
horizontal component of the active stress – on the basis of the analogy of the Φ and
the
pertaining to the non-cohesive material, which occurs in a granular material
with small friction angle and low cohesion already in relatively small depth –:
2
45
2
0
tg
h
x
,
that is
cos
sin1
2
h
x
.
The values of the sinΦ and cosΦ can be expressed by means of the relation obtained
from the Coulomb’s straight-line:
cos
h
c
tgtg
,
h
c
tg cossin
,
tgh
c
tg
sin
cos
.
Substituting the values of the sinΦ and cosΦ into the relation written for
x
:
ch
ctghh
tg
h
x
sin
cos
2
,
but
ctghh cossin
,
- 65 -
so
tg
h
chh
x
cos2
.
For the cosΦ we were able to deduce the following from the Coulomb straight-line:
sincos
cos
cos
2222
cch
h
.
Substituting the value of the cosΦ into the relation
x
, the horizontal stress
component of the active pressure developing inside the material in a given depth is:
sin
sincos
cos2
2222
cch
chh
x
.
If the active stress condition is caused by the displacement – tilting – of the vertical
and friction retaining wall, then the friction developed between the retaining wall
and the material modifies the direction and magnitude of the stresses acting on the
retaining wall.
The friction developed between the cohesive granular material and the wall are
generally composed of the friction factor
tg
depending on the normal stress acting
on the surface and the adhesion
a
independent of that, which is – in accordance with
Coulomb’s friction law – can be illustrated as it is shown in Figure 30.
Figure 30. Friction developed between the cohesive granular material and the wall
- 66 -
At a given normal stress
n
its friction angle is
. On the basis of Figure 30 the
tg
can be expressed:
n
tg
,
n
a
tgtg
.
In the cohesive granular material the vertical stresses rising from the self-weight
generate the Φ direction stresses.
Figure 31. Cohesion stress model
Applying the stress model set up for non-cohesive granular materials to the cohesive
material, and using Φ instead of the angle
(Figure 31) – the relation of
proportionality between the vertical and horizontal stress components can be written:
cos
2
2
h
h
x
y
.
Next to a friction retaining wall, part of the vertical stress components of the
cohesive granular material are transferred onto the retaining wall; on the retaining
wall a weight-force intake realizes. If a stress
acts on the retaining wall inclining
at an angle
to the horizontal, then the retaining wall takes up a vertical-direction
stress
sin
from the material, i.e. reduces the vertical stress of the material part
next to the retaining wall by
sin
. Therefore the horizontal stress component is
also reduced proportionally. At the same time a
cos
size horizontal stress
component acts on the retaining wall the. On the basis of the proportion between the
vertical and horizontal stress components it can be written:
- 67 -
cos
sin
2
cos
2
2
h
h
h
.
The
can be expressed:
cossincos
cos
2
h
.
The horizontal component of the
is
h
:
cos
h
,
cos1
cos
2
tg
h
h
.
In the relation written for the
tg
in case of a vertical retaining wall
hn
, so
h
h
a
tg
h
cos
cos1
cos
2
.
The
h
can be expressed:
cos1
2
2
cos
tg
ah
h
,
where
sincos
cos
cos
2222
cch
h
.
The active pressure of the cohesive granular material, acting on the friction retaining
wall is, consequently, lower than its static pressure. If adhesion develops on the
retaining wall the horizontal stress components are reduced by the following value
cos1
cos
tg
a
as compared to the retaining wall without adhesion. The horizontal
h
E
component
of the active compressive force of the cohesive granular material, acting on the
retaining wall can be calculated by means of the definite integral of the
h
, where
the lower limit of the integration is given from the condition
0
h
:
- 68 -
0
cos1
2
2
cos
tg
ah
.
The equality exists, if
0
90
, which happens at
c
h
0
, respectively if hγ=2a
and so
a
h
2
, that is
a
h
2
0
.
Consequently, the lower limit of the integration is
0
h
, but for
0
h
the higher value
must be taken into account from
c
and
a2
h
h
dh
tg
ah
E
0
cos1
2
2
cos
0
where the
cos
is also the function of the h.
- 69 -
Summary
The basic physical properties of the granular material differ significantly from those
of the chemically identical materials, which are, however in the solid, liquid or
gaseous state, therefore, the definition of granular material as an additional state of
matter in its own right is justified.
The ideal granular material is – similarly to the concept of perfect gas, ideal liquid
and crystalline solid – a non-cohesive granular material.
The basic physical-mechanical laws of the non-cohesive granular materials are as
follows:
I. In the non-cohesive granular materials only compressive and shear
stresses can arise.
II. In the non-cohesive granular materials in a quiescent state, the stresses
developed by the vertical-direction compressive stresses act downwards in the
0
90
zone measured from the vertical direction. (
is the angle of
friction of the material.)
III. The value of the lateral pressure rising from the self-weight of the non-
cohesive granular material is (
2
h
), i.e. the half of the product of the depth (h)
and volume weight (γ), its direction deviates from the horizontal downwards
with the angle of friction rising in the material, if the surface is horizontal and
- 70 -
over the given depth the material fills the space evenly inclining at an angle
to the horizontal.
IV. The non-cohesive granular materials conforms to the physical-
mechanical laws characteristic of them until their constituting elements, the
grains keep their relative quiescent state. When the grains get into relative
motion – collide with each other -, the granular materials behave according to
the physical-mechanical laws of the liquids.
The physical-mechanical laws of the non-cohesive granular materials prevail with a
statistical character, because the material itself consists of a multitude of different
grains.
The stresses – the average forces calculated for a given surface – can be divided or
compounded as vectors.
The factor of the static pressure is:
2
cos
.
In the non-cohesive granular material, the lateral pressure — in a plane inclining at
an angle
to the horizontal and tilting towards the assembly — is
tg
tgh
1
2
and its direction inclines at an angle
to the horizontal.
In the active stress condition arising due to the expansion, the motion is realised in
the direction of
2
45
0
to the horizontal.
The horizontal component of the pressure acting on the vertical friction retaining
wall is
cos1
cos
2
tg
h
h
, where
is the angle of friction developed between
the retaining wall and the material.
The simultaneous existence of two equalities in a trough formulates the condition of
the arch formation:
- 71 -
tg
h
b
,
tgtg
h
b
,
where:
b is the size of the discharge orifice of the trough,
h is the height of the trough,
is the factor of the static pressure,
is the inclination angle of the trough, measured to the vertical,
is the angle of the friction developed between the trough and the
material.
The discharge from the trough can occur by mass flow, if
, that is
tgtg
h
b
. The tunnel flow occurs, if
, that is
tgtg
h
b
, but
tghb
.
The geometric equation of the arch is
h
tghb
b
xb
y
2
4
, and
tg
b
xb
y
2
4
, if the arch is supported by the material in the plane inclining
at an angle
to the vertical. The angle
can be calculated.
Flow-proof hoppers can be designed in the knowledge of the conditions of the
formation of the arch. The present work shows the procedure of the design. The
experiment carried out with use of the curve-component hopper, received with this
design procedure, proved the correctness of the theoretical calculations.
In the cohesive granular material of quiescent state the stresses produced by the
vertical-direction compressive stresses act downwards in the zone
0
90
measured from the vertical direction (where
is the angle of the internal shear
resistance of the material).
- 72 -
The value of the static pressure of the cohesive granular material, rising from the
self-weight, is half of the product of the depth and volume weight
2
h
, its
direction deviates from the horizontal downwards by the angle of the internal shear
resistance of the material. The horizontal stress component of the static pressure is:
cos
2
h
x
,
and
sincos
2
cos
2222
cch
x
,
where the c is the cohesional coefficient.
The resultant force of the static pressure of the horizontal-terrain cohesive granular
material, acting on the vertical retaining wall is:
sin2cos
4
cos
2
2222
0
h
c
cch
h
E
.
The cohesive granular material in a vertical wall without support is stable until the
height
c
h
0
The basic principle of the construction of the steepest slope is that the
-direction
stresses developed by the self-weight should touch the side of the slope. The
procedure of the construction of the cohesive slope can be found in the present work.
The horizontal stress component of the active pressure of the cohesive granular
material, acting on the vertical retaining wall having a friction angle δ and and
adhesion coefficient a is:
cos1
2
2
cos
tg
ah
h
,
where
sincos
cos
cos
2222
cch
h
.
- 73 -
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