BSEN 3310 Hydrostatic Pressure
Joel Ham
Abstract: Hydrostatic pressure is the pressure exerted on a submerged surface by water that is not
moving. This pressure and its relationships with water depth, force exerted, and mass supported can be
studied using an Edibon Hydrostatics pressure system. The relationships can be explored through the
three equations for a partially submerged surface, which are as follows:
𝑚=
𝜌𝑏
𝑦
(𝑎 + 𝑑 − )𝑦 2
2𝐿
3
ℎ𝑝 = 𝑎 + 𝑑 −
ℎ
3
1
𝐹 = 𝜌𝑔ℎ2 𝑏
2
Where m is the mass applied to the system, ℎ𝑝 is the height of pressure applied to the surface, and F is
the hydrostatic force on the surface. These equations are useful in analyzing pressure and how it is
applied. These equations are backed up based on data recorded from observing a hydrostatic pressure
system.
Introduction: The study of hydrostatic pressure is very important in biological systems, particularly in
those involving drainage and irrigation. Knowing the pressure involved in a system is very important in
determining how to build that system and what materials to use. Applications such as center-ofpressure gates in irrigation systems rely heavily on information related to hydrostatic pressure. Such
gates operate under the laws relating to hydrostatic pressure (Humpherys, 1991). Using equations
relating to hydrostatic pressure can aid in future design of systems, and save time from manually
measuring every property of a system.
Objectives: The objectives were to determine the center of pressure and the force on a vertical surface
when the surface is both partially and fully submerged in stationary water in a tank.
Materials and Methods: An Edibon Hydrostatics Pressure System and small weights were used to
measure height of water on the surface and force pushing against the surface. The tests began with
ensuring that the water surface was level with the bottom of the device. Weights were added in varying
increments. Each time a weight was added, water was also added to the tank to make the balance arm
level. The mass and the corresponding water level were recorded for the partially submerged surface.
Masses were added until the water level approached 0.1 meters. The highest reading recorded was
0.094 meters. To measure forces for when the surface was fully submerged, 0.22 kilograms was added
to the end of the balance arm. Weight was again added in varying increments. Water was added each
time to bring the surface back to 90 degrees with the bottom of the tank. The masses and corresponding
heights were recorded. The following are the equations used in the experiment for when the surface
was partially submerged in water:
𝑚=
𝜌𝑏
(𝑎
2𝐿
𝑦
+ 𝑑 − 3 )𝑦 2
(1)
ℎ
ℎ𝑝 = 𝑎 + 𝑑 − 3
(2)
1
2
𝐹 = 𝜌𝑔ℎ2 𝑏
(3)
If: m = mass on the end of the balance arm
ρ = density of water, kg/m3
b = width of the surface, m
L = length of the balance arm, m
a = distance from the top of the surface to the top of the device, m
d = height of the surface, m
y = calculated height of the water from the bottom of the surface, m
h = measured height of the water from the bottom of the surface, m
g = acceleration due to gravity, m/s2
ℎ𝑝 = height of applied pressure from the bottom of the surface, m
Equation (2) was derived first, based on the fact that the pressure applied under water has a triangular
pressure profile. Pressure in a fluid increases with depth because there is more fluid at deeper layers.
The effect of the extra weight on a deeper layer is shown by a pressure increase (Cengel et al, 2014). The
applied pressure is 1/3 of the height of the triangle from the bottom of the triangle. The sum (a+d) is
the distance from the top of the device to the bottom. (h/3) is the distance from the bottom of the
device to the point of pressure. Therefore (a + d – h/3) is the height of pressure given from the top of
the device.
Equation (3) was derived next, using 𝑝𝑐 = pressure at the centroid, 𝐴𝑠 = area of submerged surface, and F
= applied hydrostatic force on the surface. The following relationship is determined:
𝐹 = 𝑝𝑐 ∗ 𝐴𝑠 = 𝜌𝑔ℎ𝑐 𝐴𝑠 , where ℎ𝑐 =
1
ℎ
2
= height of centroid of the submerged surface, and 𝐴𝑠 = ℎ ∗
𝑏.
Using this information, equation (3) can be found:
1
𝐹 = 2 𝜌𝑔ℎ2 𝑏 , where b is the width of the surface
Finally, equation (1) was developed by using the relationship between mass on the end of the balance
arm and hydrostatic force (F). The moment created by these two forces about the balance point (above
the partially submerged surface on the balance arm) is equal to zero. The moment equation is as
follows:
𝛴𝑀 = 0
−𝐹(ℎ𝑝 ) + 𝑚𝑔𝐿 = 0
𝑚𝑔 =
𝐹ℎ𝑝
𝐿
Using y = h and the information from equation (3), this progression can be followed:
𝑚𝑔 =
𝐹ℎ𝑝
1
𝑦
=
𝜌𝑔𝑦 2 𝑏 (𝑎 + 𝑑 − )
𝐿
2𝐿
3
Cancelling g from both sides results in equation (1)
𝑚=
𝜌𝑏
𝑦
(𝑎 + 𝑑 − )𝑦 2
2𝐿
3
A similar method was used to develop equations (4), (5), and (6) for the fully submerged surface:
(1) 𝑚 =
𝜌𝑏𝑑
(𝑎
𝐿
+𝑑+
𝑑2
)𝑟
12𝑟
Where 𝑟 = 𝑦 − 0.5𝑑
𝑑
2
(2) ℎ𝑝 = 𝑎 + +
𝑑2
𝑑
2
12(ℎ− )
𝑑
(3) 𝐹 = 𝜌𝑔𝑏 (ℎ − 2 ) 𝑑
Figure 1 gives a visual representation of variables and dimensions and where they come from.
Figure 1: Diagram of an Edibon Hydrostatics pressure system
Results:
Figure 2 contains the relationship between the mass added to the end of the balance arm and the
calculated height of the water. The relationship follows a polynomial trend line due to the (𝑦 + 𝑧)𝑦 2
relationship in the equation relating mass and calculated height. A greater change in mass is required to
raise the height the same amount as mass increases.
0.25
Mass (kg)
0.2
y = 18.94x2 + 0.1734x
R² = 0.9997
0.15
0.1
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Calculated height (m)
Figure 2: Mass on balance arm versus calculated height for a partially submerged surface
Figure 3 shows the relationship between the calculated height of the water and the measured height of
the water based on mass added to the end of the balance arm. Since calculated height (y) and measured
height (h) should be the same for each measurement, the graph follows a linear trend line whose slope
is close to 1. The slope of 1.033 shows a possible error in exactness of measured readings or the balance
arm not being perfectly level.
0.12
Calculated height (m)
0.1
0.08
y = 1.033x
R² = 0.9962
0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
Measured height (m)
Figure 3: Calculated height versus measured height for the partially submerged surface
Figure 4 shows the relationship between the height of the pressure point on the surface and the
observed height of the water level. The slope is negative because h and ℎ𝑝 are measured in opposite
directions. The slope of the trend line (-0.333) comes from the relationship given in the equation ℎ𝑝 =
−1
ℎ.
3
The relationship is linear because there are no exponents in the equation.
0.2
Height of Pressure (m)
0.195
0.19
y = -0.3333x + 0.2
R² = 1
0.185
0.18
0.175
0.17
0.165
0
0.02
0.04
0.06
0.08
0.1
Measured height (m)
Figure 4: Height of pressure versus measured height of water for the partially submerged surface
Figure 5 shows the relationship between the hydrostatic force exerted on the surface and the height of
the water on the surface. The trend line is exponential due to the ℎ2 value in the equation relating force
and height. The measured height of water puts more force at a higher rate on the surface as the height
of the water increases.
3.5
3
y = 343.35x2 - 6E-14x
R² = 1
Force (N)
2.5
2
1.5
1
0.5
0
0
0.02
0.04
0.06
Measured Height (m)
0.08
0.1
Figure 5: Hydrostatic Force versus measured height of water for partially submerged surface
Figure 6 shows how the standard deviation between measured height and calculated height vary as
measured height of water increases. Standard deviation gets larger as measured height increases, most
likely due to the larger margin for error that comes with higher readings. The outlying high value is due
to a possible error in reading or an uneven level arm.
Standard Deviation (y vs. h)
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.02
0.04
0.06
0.08
0.1
Measured Height (m)
Figure 6: Standard deviation between calculated height and measured height of water versus
measured height of water for partially submerged surface
Figure 7 shows the relationship between the mass added to the system and the calculated height of
water from equation (4). As opposed to partially submerged, the relationship is linear due to the lack of
an exponential value associated with the calculated height in equation (4). The calculated height
increases at a constant rate in relation to mass added to the end of the level arm.
0.5
0.45
0.4
y = 3.6842x - 0.1637
R² = 1
Mass (kg)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
Calculated height (m)
Figure 7: Mass on balance arm versus calculated height of water for fully submerged surface
Figure 8 shows the relationship between calculated and observed height for a fully submerged surface.
The trend line is linear and should have a slope of 1 because the observed height and the calculated
height are theoretically the same. The slope of 1.0406 is accurate yet not perfect, most likely due to
errors in height readings or a slightly unbalanced level arm at time of measurement.
Calculated height of water (m)
0.18
0.16
y = 1.0406x
R² = 1
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
Observed Height of water (m)
Figure 8: Calculated height of water versus observed height of water for a fully submerged surface
Figure 9 shows the relationship between the height of the pressure point from the top of the device on
the surface and the observed height of water in the device. The relationship is exponential due to the h
value in the denominator of the equation relating ℎ𝑝 and h. The downward trend is due to the fact that
ℎ𝑝 and h are measured from opposite directions and as one increases, the other decreases.
0.168
Height of Pressure (m)
0.166
0.164
y = 0.1269x-0.117
R² = 0.9814
0.162
0.16
0.158
0.156
0
0.05
0.1
0.15
0.2
Observed Height (m)
Figure 9: Height of pressure versus observed height of water for a fully submerged surface
Figure 10 shows the relationship between the hydrostatic force exerted on fully submerged surface and
the observed height of water in the system. The relationship is linear because in equation (6) there are
no exponents connected to the h value. The hydrostatic force increases at a constant rate as measured
height of water in the system increases.
8
Hydrostatic Force (N)
7
y = 68.67x - 3.4335
R² = 1
6
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
Measured Height (m)
Figure 10: Hydrostatic force versus measured height of water on a fully submerged surface
Figure 11 relates standard deviation between calculated and measured height and the measured height
of water in the system. The standard deviation gradually grows as the measured height in the system
increases, due to the larger margin for error with higher readings.
Standard Deviation (calculated vs
measured height)
0.005
0.0045
0.004
0.0035
0.003
0.0025
0.002
0.0015
0.001
0.0005
0
0
0.05
0.1
0.15
0.2
Measured Height (m)
Figure 11: Standard deviation between calculated and measured height versus measured height
Conclusions: The equations relating mass, height of pressure, and hydrostatic force are supported by
information collected from the Edibon Hydrostatics pressure system. For partially and fully submerged
surfaces, the measured height of the water and the calculated height of water are very closely to a 1:1
ratio, which supports equation (1). The height of water and the height of pressure are also linearly
related, as height of pressure decreases as height of water increases.
References:
Cengel, Y.A., Cimbala, J.M. (2014). Fluid Mechanics, Fundamentals and Applications, Third Edition. New
York, NY: McGraw Hill
Humpherys, A.S. (1991). Center-of-Pressure Gates for Irrigation. 7(2): 185-192. (doi: 10.13031/2013.26209)