Engineered structures are often used
to control the positioning of soil,
water
or granular materials.
These structures can take many forms,
including dams,
retaining walls,
tanks,
silos,
culverts
and tunnels.
To do their job successfully,
they must resist the pressures applied to them
by the materials they support.
In this video, we focus on how these pressures arise
and on the factors that determine their magnitudes.
To understand how granular materials
generate pressures on their containers,
consider a box filled with wooden cylinders.
Collectively, the cylinders produce a downwards force
on the bottom of the container
that is equal to their weight, W.
We could calculate this weight easily
if we knew the weight density
of the material in the container –
a material property we denote using the letter “gamma”.
In engineering, we often instead know
the mass density, rho, of a material, and that quantity
times g, the acceleration of gravity,
gives us the weight density.
The next step in our calculation
is to determine the volume, V, of the material in the container.
This can be found easily by multiplying A,
the base area of the container,
by its height, “h”.
To calculate the total weight,
we simply multiply the weight density and the volume.
When we substitute in the equations we found earlier
for weight density and volume,
we discover that the weight
is equal to the product of the material mass density,
the acceleration of gravity,
the base area, and the height of the material.
To calculate the vertical pressure PV
on the bottom of the container,
we simply divide this weight by the base area.
Substituting in our earlier equation for weight,
and canceling the area term
that appears in both the numerator and denominator,
produces a simple equation.
This equation tells us that the vertical pressure, PV,
is equal to the mass density of the material
times the acceleration of gravity
times the height of the material above the point we are considering.
Notice that if we increase the height of the material,
the pressure increases.
It turns out that the wood cylinders
also push against the sides of their container.
If we stop pushing them together with our hands,
nothing stops them
from just pushing each other out sideways.
To understand why this outwards force arises,
consider three cylinders from the original stack.
As gravity pushes the top cylinder downwards,
it tends to squeeze between the other two cylinders,
pushing them out sideways.
The magnitude of that sideways force
will depend on how hard the upper cylinder is pushed downward,
and that in turn depends on its weight
and on the weight of the material that it supports.
If the lower cylinders are constrained by a structure,
they will exert a horizontal force on that structure.
At a short distance down,
both the vertical and horizontal pressures are small.
At progressively greater depths,
both pressures increase.
Experiments show that for most materials,
the ratio of the horizontal pressure
to the vertical pressure is constant.
We call this ratio the pressure coefficient, K,
and it is usually less than one.
Thus, for granular materials,
the vertical pressure at any depth is typically rho g h,
while the horizontal pressure there is K rho g h.
Clearly, then, both the vertical and horizontal pressures
increase linearly with depth,
as shown in these graphs.
The shapes and packing of the particles in a granular material
affect the contact angles between them.
Those angles and the coefficient of friction between adjacent particles
determine the magnitude of the outwards pressure PH that is generated,
and so they affect the value of K.
If these ellipsoidal candies laid mostly horizontally,
they would produce less horizontal force
than if they were randomly placed or if they were spherical.
Also, if the surfaces of either kind of candy became sticky,
the lateral force would be reduced.
As an extreme but practical example,
sedimentary rock sometimes contains horizontal layers,
like those formed by these wood blocks.
What horizontal pressures would they produce?
What is the associated K value?
Since gravitational forces are transmitted vertically
from one wood block or piece of horizontally-layered rock to the next,
no horizontal pressures result, and K,
the ratio of the horizontal to vertical pressures,
is equal to zero.
Liquids represent the other end of the spectrum.
Experiments show that liquid pressures are the same
in all directions and so K,
the ratio of the horizontal and vertical pressures,
takes on a value of one.
In that case, we simply write that the pressure, P,
acting in any direction at a particular depth, h,
is equal to rho g h.
So, in summary, for granular materials,
the horizontal pressure PH
is usually less than the vertical pressure PV,
as shown by these graphs and formulas.
For water and other liquids, however,
the horizontal and vertical pressures
are equal to a single value P,
as shown in this graph and formula.
We can observe the pressures produced by granular materials and liquids
by using this custom-made apparatus.
It consists of a rectangular chamber
with square windows attached to force gauges
at three different depths along the side,
and one window and force gauge at the bottom.
The gauge readings allow us to determine the
horizontal pressure at three elevations
along the side of the chamber
and the vertical pressure at its bottom.
Using this information,
we can determine the horizontal pressure distribution
along the side of the chamber
and the vertical pressure at its bottom.
When the chamber is filled with marbles,
an example of a granular material,
the indicated horizontal forces
are 7, 14 and 19 Newtons,
and the downward force on the bottom window
is 33 Newtons.
And it is easy to see that the corresponding horizontal pressures
do indeed increase linearly with depth.
Based on a straight-line fit,
the horizontal pressure at the bottom of the chamber would be 2.2 kilopascals.
The measured vertical pressure there is 3.3 kilopascals,
and dividing the horizontal bottom pressure
by the vertical one
gives a K of 0.67.
If we fill the chamber with water,
the gauges again indicate a linear pressure distribution.
To see how the pressures described in this video
affect the design of real-world structures,
we hope you will watch our videos on
"Dams", "Retaining Walls", "Silos and Tanks", and "Tunnels and Culverts".