HYDROSTATIC PRESSURE
Samantha Daniell
Team #4
ABSTRACT.
It is necessary to be able to accurately estimate the magnitude and location of a hydrostatic force
exerted on a submerged surface for the purpose of designing systems that utilize partially or fully
submerged surfaces. A hydrostatics pressure module with weights of varying mass was used to
measure the mass required to equalize the hydrostatic force exerted on surface submerged at
various depths. Theoretical equations were used to calculate the height of the water in the
module, as well as the magnitude and location of the hydrostatic force. The theoretical height
was compared to the observed height and Excel was used to calculate relative error and represent
the data graphically. The theoretical equations were found to be slightly more accurate at greater
depths and to have accuracies of 95%, proving their validity.
INTRODUCTION.
Pressure is an intensive property specific to fluids that is analogous to the normal stress of a solid
(Cengel and Cimbala 2006). The pressure in a fluid is dependent on the density of the fluid, as
well as the depth. The pressure of a fluid at rest increases with depth in order to balance the
“weight” of the fluid above the deeper layer (Cengel and Cimbala 2006). Oftentimes it is
necessary to calculate the magnitude and location of the force caused by the pressure of a fluid
on a submerged object, such as a dam (Fasina 2015). If calculations are done wrong, or the
material fails, underwater repair may be necessary. These types of repairs require underwater
concrete installation, which is a difficult process due to the constant hydrostatic force
(Horszczaruk and Brzozowski 2014). In order to accurately design, install, or repair submerged
systems, the magnitude of the hydrostatic force must be estimated using the density of the fluid
and the depth of the object, as well as the effect of gravity. These factors are incorporated into
the basic equation for calculating hydrostatic force, Equation 1. The location at which this force
acts, or the center of pressure, must also be estimated. The center of pressure is determined based
on the shape of the surface as well as the depth of the fluid as compared to the free surface, as
shown in Equation 2. These equations for estimating the magnitude and location of the
hydrostatic force become more complex with the systems. It is essential that these equations are
valid in order to avoid system failures due to design. The validity of these hydrostatic force
equations can be determined by comparing observed and calculated values and analyzing them
for accuracy.
𝐹 = 𝜌𝑔ℎ𝐴
(1)
𝑦𝑝 = 𝑦𝑐 +
𝐼𝑥𝑥,𝐶
𝑃0
[𝑦𝑐 + ⁄(𝜌𝑔𝑠𝑖𝑛𝜃)]𝐴
(2)
1
OBJECTIVES.
The objectives of this lab were to measure the magnitude and location of a force resulting from
the hydrostatic pressure of water at rest on a partially and fully submerged vertical rectangular
surface. In addition, the validity of theoretical equations used to estimate the force and its
location on the surface was to be evaluated.
MATERIALS AND METHODS.
Equipment.
An Edibon Hydrostatics Pressure Module with weights ranging from 5 g to 100 g.
Procedure.
Water was added to the tank until the meniscus was at a height of 0 mm. The counterweight was
adjusted so that the balance bridge arm was horizontal. To gather data for the partially
submerged surface, a series of 6 different masses were added to the arm on the opposite side of
the counterweight such that the height did not exceed 100 mm. The height reading was obtained
at each new mass. To gather data for the fully submerged surface, a new combination of masses
that totaled 225 g was added to the arm. Water was added to the tank until the meniscus was at a
height of 100 mm. A series of 7 different masses were added to the arm until the water height
reached the maximum of 160 mm. The height reading was obtained at each new mass. The
recorded height values at each mass were then used to calculate a theoretical value of the water
height, center of pressure, and magnitude of the hydrostatic force exerted on the submerged
surface. These values were calculated using Equations 3-5 for masses totaling less than 225 g
and Equations 6-9 for masses totaling above 225 g, with dimensions given by Figure 1. The
relative error was then calculated using Equation 10 to evaluate the accuracy of the theoretical
equations based on the observed values.
𝑚=
𝜌𝑏
2𝐿
𝑦
(𝑎 + 𝑑 − 3) 𝑦 2
ℎ
ℎ𝑝 = 𝑎 + 𝑑 − 3
(4)
1
𝐹 = 2 𝜌𝑔ℎ2 𝑏
𝑚=
𝜌𝑏𝑑
𝐿
(5)
𝑑2
𝑑
(𝑎 + 2 + 12𝑟) 𝑟
𝑟 = 𝑦 − 0.5𝑑
ℎ𝑝 = 𝑎 + 2 +
𝑑
2
12(ℎ− )
𝑑
𝐹 = 𝜌𝑔𝑏 (ℎ − 2 ) 𝑑
𝑅𝐸 = 100% ×
(6)
(7)
𝑑2
𝑑
(3)
|𝑦−ℎ|
𝑦
(8)
(9)
(10)
RESULTS AND DISCUSSION.
The hydrostatic force on a surface can be calculated by multiplying the pressure at the centroid
by the surface area. Because the pressure within a liquid increases with depth due to the
downward force of gravity, the hydrostatic force will increase with the depth of the fluid. Figures
2
1 and 2 are evidence of this fact, since the hydrostatic force increases as the height (depth) of
water increases. For the Edibon module, mass must be added to the balance bridge arm in order
to equalize the hydrostatic force. Since the hydrostatic force increases with depth, the mass
required to equalize the force will increase with depth, as well. This fact is shown in Figures 3
and 4. In each plot, mass increases with the theoretical height (depth) of the water. The point at
which the hydrostatic force acts, the center of pressure, also varies with the depth of the fluid.
This is also due to the downward force of gravity. If gravity were nonexistent, the hydrostatic
force would act at the centroid of the fluid. But because gravity causes the pressure to increase
with depth, the center of pressure is pulled farther away from the centroid. If both the depth of
the fluid and the distance to the center of pressure are measured from the bottom of the tank, the
center of pressure increases with the depth. For the Edibon module, the center of pressure is
measured from the top of the water tank while the height of the water is measured from the
bottom of the tank. Because of this, the distance to the center of pressure decreases as the height
of water increases, as shown in Figures 5 and 6.
Because the validity of the equations used to calculate the magnitude and location of the
hydrostatic force (Equations 4 and 5) vary only by the observed height and the theoretical height,
the relative error of the theoretical height and observed height (Equation 3) can be used to
confirm the validity of Equations 4 and 5. If the equations were 100% accurate, a plot with the
calculated height of water vs. the observed height of water would have a slope of 1. Figures 7
and 8 show slopes of ~1.04 with high coefficients of determination, which shows a high
closeness of fit. Although the slopes are close to 1, there is some error. Figures 9 and 10 show
how the relative error varies with the observed height. In both plots, the relative error decreases
slightly with the water height, implying that the theoretical equations are more accurate with
greater depth. The average relative error for the partially submerged equations and the fully
submerged equations are 4.43 and 4.56, respectively. This shows that the theoretical equations
for both fully submerged and partially submerged surfaces are 95% accurate.
TABLES AND GRAPHS.
Figure 1. Diagram of hydrostatic pressure module with dimensions.
3
35
Hydrostatic Force (N)
30
25
20
15
10
5
0
0
0.02
0.04
0.06
0.08
0.1
Height of Water (m)
Figure 1. Hydrostatic force vs. height of water in the tank for partially submerged surface.
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Hydrostatic Force (N)
70
60
50
40
30
20
10
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Height of Water (m)
Figure 2. Hydrostatic force vs. height of water in the tank for fully submerged surface.
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0.25
Mass (kg)
0.2
0.15
0.1
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Theoretical Height of Water (m)
Figure 3. Mass added to arm vs. theoretical height of water in tank for partially submerged
surface.
0.5
0.45
0.4
Mass (kg)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Theoretical Height of Water (m)
Figure 4. Mass added to arm vs. theoretical height of water in tank for fully submerged surface.
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Distance to Center of Pressure (m)
0.195
0.19
0.185
0.18
0.175
0.17
0.165
0
0.02
0.04
0.06
0.08
0.1
Height of Water (m)
Figure 5. Distance from the top of the tank to the center of pressure on the surface vs. height of
water in the tank for partially submerged surface.
Distance to Center of Pressure (m)
0.168
0.166
0.164
0.162
0.16
0.158
0.156
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Height of Water (m)
Figure 6. Distance from the top of the tank to the center of pressure on the surface vs. height of
water in the tank for fully submerged surface.
6
Theoretical Height of Water (m)
0.12
y = 1.0402x
R² = 0.9998
0.1
0.08
0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
Recorded Height of Water (m)
Figure 7. Theoretical height of water in the tank vs. recorded height of water in the tank for
partially submerged surface.
Theoretical Height of Water (m)
0.18
y = 1.0469x
R² = 0.9991
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Recorded Height of Water (m)
Figure 8. Theoretical height of water in the tank vs. recorded height of water in the tank for fully
submerged surface.
7
8
7
Relative Error (%)
6
5
4
3
2
1
0
0
0.02
0.04
0.06
0.08
0.1
Recorded Height of Water (m)
Figure 9. Percent relative error vs. recorded height of water in the tank for partially submerged
surface.
6
Relative Error (%)
5
4
3
2
1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Recorded Height of Water (m)
Figure 10. Percent relative error vs. recorded height of water in the tank for fully submerged
surface.
CONCLUSIONS.
There are widely used theoretical equations to calculate the location and magnitude of the
resultant hydrostatic force for submerged surfaces. Equations 1-3 are equations specific to the
submerged surface of the Edibon Hydrostatics Pressure Module. By comparing observed results
to calculated results, the accuracy of both equations was found to be higher at greater depths,
with an overall accuracy of 95%. It can be concluded that these equations are valid for estimating
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the magnitude and location of hydrostatic force exerted on a partially and fully submerged
surface.
NOMENCLATURE.
𝐹 = ℎ𝑦𝑑𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐 𝑓𝑜𝑟𝑐𝑒 (𝑁)
𝐴 = 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 (𝑚2 )
𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 9.81
𝑚
𝑠2
ℎ = ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 (𝑚)
𝑦𝑝 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑚)
𝑦𝑐 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 (𝑚)
𝑃0 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑
𝐼𝑥𝑥,𝑐 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑎𝑡 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑 𝑜𝑓 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
𝑚 = 𝑚𝑎𝑠𝑠 (𝑔)
𝑘𝑔
𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 = 1000 𝑚3
𝑦 = 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑖𝑛 𝑡𝑎𝑛𝑘 (𝑚)
ℎ𝑝 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡𝑜𝑝 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑡𝑜 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (𝑚)
𝑅𝐸 = 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑒𝑟𝑟𝑜𝑟 (%)
𝑚
𝑔 = 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 = 9.81 2
𝑠
REFERENCES.
Cengel, Y. A., and J. M. Cimbala. Fluid Mechanics Fundamentals and Applications. 3rd ed.
New York: McGraw Hill, 2014. Print.
Fasina, O. Chapter 3: Pressure and Fluid Statics [PDF document]. Retrieved from Canvas:
< https://auburn.instructure.com/courses/922317/files/55457839/download?wrap=1>.
Horszczaruk, E., and P. Brzozowski. "Bond strength of underwater repair concretes under
hydrostatic pressure." Experimental Thermal and Fluid Science (2014). Science Direct.
Elsevier. Web. 22 Sept. 2015.
<http://www.sciencedirect.com/science/article/pii/S0950061814009313>.
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