Sheridan Institute of Technology and Advanced Learning

Faculty of Applied Science and Technology

School of Mechanical and Electrical Engineering & Technology

_____________________________________________

ENGI32651: Fluid Mechanics

Lab Report #: 2

Title : Center of Pressure Apparatus

Class #: 42153 ; Group #: ………

Group Members (actively involved in the work): A: Paritosh Chahal (PC); B: Ashish Jain (AJ);

C: Harmeet Singh (HS); D: Richard Kalsi (RK);

Date of Experiment: 25th February 2015

Date of Submission: 13th March; Time: 10:00 am.

Instructor: Lewis Mununga

Declaration: I/we hereby declare that the work presented in this report is mine/ours and not plagiarized or copied except where cited and appropriately referenced:

Signatures: A: .........................; B: …….....................; C: …….…….………; D: …………….………..;

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ContentsExecutive Summary:............................................................................................................................1

Objective:............................................................................................................................................1

Theory:................................................................................................................................................1

Materials:.............................................................................................................................................3

Procedures:.........................................................................................................................................4

Data:....................................................................................................................................................5

Sample Calculations:...........................................................................................................................6

Results:................................................................................................................................................8

Discussion:........................................................................................................................................11

Conclusion:........................................................................................................................................12

Appendix............................................................................................................................................13

References.........................................................................................................................................17

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Executive Summary:The purpose of this lab was to see whether the force applied by the static fluid on the plane of the center of pressure apparatus (fig 1) applies a moment to cancel out another moment applied by force due to the weight plate (some distance from the pivot point) on the rod where the apparatus and the weight hanger is connected. After following the procedure of the lab, collecting data, and doing calculations; it was noticed that the above purpose was achieved since the percentage errors of the results were really low (between ranges of 0.2%-25%). The main results also helped the observer notice two important points. The first point that the “h” (distance between the top surface of the apparatus and the surface of water) increases as the weight on the weight hanger increases; and that as the angle “Θ” (the amount the apparatus rotated clockwise from its reference) increases, the moment by the force applied by the water on the plane (to balance the rod) decreases.

Objective:The objectives of the lab were:

1. Check if moment (created by the force of water on the plane) equals to the moment created by hanging weights (some distance from the pivot point).

2. The relationship between the rotation of the apparatus (“Θ”) and the force by water on the bottom plane of apparatus.

Theory:When a fluid pushes against a wall, the wall experiences a force and pressure based on the area submerged into the fluid. The wall will experience the maximum force at the bottom but the wall will always break where the resultant force is. That is where the center of pressure is located. It is the location at where the resultant force would act. The figure below shall explain the resultant force and where it acts. It also explains how the resultant force and the center of pressure is used to create a moment using the center of pressure apparatus.

Figure 1: force body diagram explaining how the force in a fluid acts and where the resultant force is located.

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When a fluid is in a container, you can assume there are forces applied on the surface of a container based on the forces due to static fluid. In this lab the forces applied by the fluid are observed and compared by the forces applied on the weight arm of the apparatus. Both the weight measured on the arm and the forces due to the static fluid are compared to create moments and create equilibrium as both forces on both sides of the apparatus have to be equal to create balance. These moment forces are created by the rotation of the apparatus.

In order to understand how to calculate these moment forces acting on the quadrant of the apparatus, first one must understand the apparatus.

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Figure 2: Free body diagram of the Centre of Pressure Apparatus

As water fills in the quadrant, the liquid creates a resultant force about the rotational axis. This force causes a moment about the axis. This diagram above shows a free body diagram of the apparatus. This free body diagram will help explain the equation that is used to calculate the moment force of the quadrant. In order to do so the equations used are:

Equation 1:

For h<R1cosθ−Plane Face Fully Submerged

Moment ForceWater=wB cosθ

3(R2

3−R13 )− wB

2(R2

2−R12) h

Equation (2)

For h>R1 cosθ−Plane Face Partially Submerged

Moment ForceWater=wB cosθ R2

3

3−

wB R22 h

2+ wB sec2θ h3

6

2

L arm = 10 inches

Ө

dy

h

Weight

Y

Where:

- h=height- R1=Radius 1

- R2=Radius 2

- w=is the specific weight of the fluid- y=sla nt depth- dy=width- θ=angle of the quadrant

This equation will give the theoretical moment force from the apparatus. After finding this Moment force applied by the liquid, find the actual moment force applied by the arm using the following equation:

Moment Forc earm=Weight∗LARM

- Weight=the weight placed on the weight arm- Larm=Lengthof arm

Using equation 2, it is possible to calculate the specific weight. This will give a theoretical value for the specific weight of the fluid. Using the actual value of the fluid, it is possible to compare the values.

Materials:1. Centre of pressure apparatus (NOTE: Detailed Diagram is placed in the appendix)2. Weight hanger: Used to insert weights on the weight arm (rod), and create a moment on one end

of the rod.3. Metal tray: For storing the flushed water4. 1/8” steel pin: To hold the rod in equilibrium while setting up the assignment. 5. Weighted plates: Used to create a moment on one end of the arm (10 inches away from the pivot)6. Balancing weight block: Used to balance the arm such that it is not just in contact with the grub

stool.7. Measuring cup: Used for pouring water in the quadrant.

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Figure 3: measuring cup, weight plates, and weight hangers

NOTE: 1 100 ml graduated cylinder was used to transfer small amount of water during the final state of the trails (picture not available). The observers can use any 50-100 ml of graduated cylinders.

Procedures:1. The apparatus was set onto the metal tray, and feet were adjusted at the bottom of the apparatus

until leveled.2. The weight hanger was placed 10 inches away from the pivot point

i. The members of the apparatus and the weight arm were locked with the steel pin. 3. A grub screw was installed on the top of the weight arm over the 1/8” steel pin. When the end of

the screw touched the arm, the arm was completely horizontal.4. Once the weight arm was locked, the angle of the apparatus was adjusted to 0o by loosening the

large screws on the top of the apparatus.5. The balancing weight block on the weight arm was adjusted until the grub screw was just not in

contact with the weight arm and was just floating above it. i. The steel pin was removed

6. A 1/20 lb. weighted plate was placed on the weight hanger.7. Some water was poured into the tank until the system is in equilibrium again from the measuring

cup.i. If there was too much water in the apparatus, the drainage system was opened to flush

the water.ii. A 100 ml graduated cylinder was used to transfer small amounts of water during the final

states of the trail.

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Grub Screw

Drainage

Figure 4: Center of Pressure Apparatus

8. Once system was in equilibrium , the data was collected9. Weights of 1/20 lb. were added one by one.

i. Steps of 7 and 9 were repeated after the addition of each weight for around 9 readings. 10. The water was drained from the apparatus after the 9th reading.11. The angle of the apparatus was adjusted to 20o by loosening the large screws on the top of the

apparatus.12. Steps 5-10 were repeated for the second part of the experiment.13. The apparatus was cleaned and kept away.

Data:

1. Experiment 1 when θ=0o

Table 1: Table showing the data collected for the first experiment

Trail Weight (lbs) Pivot length (inches) “h” (in)1 0.15 10 5.752 0.2 10 5.253 0.25 10 54 0.3 10 4.755 0.35 10 4.56 0.60 10 3.757 0.65 10 3.58 0.70 10 39 0.75 10 Not recorded

NOTE: The reason the observers could not record data for the 9th trail was because the large screws on the top were not tightened properly, this made the apparatus to “fall down”.

2. Experiment 2 when θ=20o

Table 2: Table showing the data collected for the second experiment.

Trail Weight (lbs) Pivot length (inches) “h” (in)1 0.15 10 5.352 0.2 10 53 0.25 10 4.744 0.3 10 4.45 0.35 10 4.156 0.45 10 3.77 0.55 10 3.228 0.65 10 2.759 0.75 10 2.29

NOTE: Pivot length in Table 1 and 2 is the distance between the weight hanger and pivot point. The red shaded results represent when the bottom plane (where the water applied force) of the apparatus is fully submerged.

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Sample Calculations:1. Given Values:

Radius1 ( R1 )=4 inches

Radius2 ( R2 )=8 inches

Breadth of Quadrant (B)=2.5 inches

h=Distance ¿ refernce ¿ surface of water

SpecificVolume of Wat er=62.4lb

ft3=0.0361

lb

¿3

2. Finding the specific weight:

*Specific Weight determined graphically by plotting the Moment against the Height (Graph 1 in results) from Sample Data

Moment mass=Moment Water=wB cosθ

3(R2

3−R13 )−wB

2( R2

2−R12) h

NOTE: Equation 1 is in terms of “y = mx + b”, where m = −wB

2(R2

2−R12 ) and b =

wB cosθ3

( R23−R1

3 )

m=−wB2

( R22−R1

2)

−2.0184=−w 2.52

(82−42 )

−2.0184=−60 w

w=0.03364lb

i n3

Average of the two specific weights = 0.03364+0.0362

2=0.03492

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b=wB cosθ3

( R23−R1

3 )

13.5=w 2.5 cos(0)

3( 83−43 )

13.5= w 2.53

(488 )

13.5=373.33 w

w=0.0362lb

i n3

The average experimental specific weight of water was calculated to be 0.03492lbs

¿3

3. SampleCalculations : For Whenθ=0 , h=3.75 inches

Forcemass∗Moment Arm Length=TheoreticalValue

10∈¿0.60 lbs=6.0∈¿ lb s

Equation (1)

For h<R1cosθ−Plane Face Fully Submerged*

3.75 inche s<4.0 inches cosθ−Plane Face Fully Submerged

*Assuming w = experimentally calculated = 0.03492lbs

¿3

Moment Water=wB cosθ

3( R2

3−R13 )−wB

2(R2

2−R12 )h

0.03492lbs

¿3∗2.5∈¿cos (0 °)

3¿

4. SampleCalculations : For Whenθ=0 , h=5.75 inches

Forcemass∗Moment Arm Length=TheoreticalValue

10 inches∗0.15lbs=1.5∈¿ lb s

Equation (1)

For h>R1cosθ−Plane Face Partially Submerged

5.75 inches>4.0inches cos θ−Plane Face Pa rtially Submerged

*Assuming w = experimentally calculated = 0.03492lbs

¿3

Moment Water=wB cosθ R2

3

3−

wB R22h

2+ wB sec2 θ h3

6

0.03492lbs

¿3∗2.5∈¿ cos(0 °)∗¿¿¿

7

¿1.60∈¿ lbs

5. Percentage Error: When h = 5.75 inches (Plane partially submerged)

Percent Error=|Theoretical Value−Experimental ValueTheoreticalValue |∗100 %=|1.5−1.6

1.5 |∗100%=6.67 %

Results:

1. ResultsTable :Whenθ=0o

Table 3: Table showing the results of all the trails performed in the first experiment.

Trial #

h¿ h3(¿3) Moment(¿∗lbs)

Moment (lb*in) [Actual value]

Moment+wB R2

2 h2

(¿∗lbs)

Percentage Error

Specific weight of

water (lb/in3)

Percentage Error for Specific

weight of water

1 5.75 153.13 1.60 1.5 17.67 6.67 0.03492 5.67 %2 5.25 125 2.33 2 17.05 16.5 0.03492 5.67 %3 5 106.50 2.75 2.5 16.72 12.5 0.03492 5.67 %4 4.75 85.18 3.19 3 16.46 6.33 0.03492 5.67 %5 4.5 71.473 3.65 3.5 16.23 8.75 0.03492 5.67 %6

3.75 50.65 5.17 6- 13.84 0.03492 5.67 %

73.5 33.39

5.76.5

- 12.3 0.03492 5.67 %

83 20.80

6.757

- 3.71 0.03492 5.67 %

2. Graph for relationship between Moment and “h” when θ=0o :

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

14

f(x) = − 2.0184473481937 x + 13.5

Moment VS "h" when the plane is fully submerged

Moment Vs "h"Linear (Moment Vs "h")

"h" (inches)

Mom

ent (

lb*i

n)

Graph 1: Moment Vs “h” (height from top of apparatus to the surface of water)

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3. Graph for relationship between “M+[γBR22h]/2" and "h3 when θ=0o:

80 100 120 140 160 180 20016

16.2

16.4

16.6

16.8

17

17.2

17.4

17.6

17.8

f(x) = 0.0146555877677834 x + 14.8970040587773

"M+[γBR22h]/2" VS "h3" when the apparatus is still par-tially submerged

"h3" (in^3)

M+[

γBR2

2h]/

2 (lb

*in)

Graph 2: "M + [γBR22h]/2" VS "h3"

4. ResultsTable :Whenθ=20O

Table 4: Table showing the results of all the trails performed in the second experiment.

Trial #

h(¿) h3(¿3) Moment(¿∗lbs)

Actual Moment (in*lb)

Moment+wB R2

2 h2

(¿∗lbs)

Percentage Error (%)

Specific weight

of water (lb/in3)

Percentage Error for Specific

weight of water

1 5.35 153.1304 1.51 1.5 14.94576 0.867 0.03705 2.64 %2 5 125 2.09 2 13.968 4.5 0.03705 2.64 %3 4.74 106.4964 2.55 2.5 13.24166 2 0.03705 2.64 %4 4.4 85.184 3.21 3 12.29184 7 0.03705 2.64 %5 4.15 71.47338 3.73 3.5 11.59344 6.57 0.03705 2.64 %6 3.7 50.653 4.44 4.5 - 1.33 0.03705 2.64 %7 3.22 33.38625 5.51 5.5 - 0.720 0.03705 2.64 %8 2.75 20.79688 6.55 6.5 - 0.769 0.03705 2.64 %9 2.29 12.00899 7.57 7.5 - 0.933 0.03705 2.64 %

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5. Graph for relationship between Moment and “h” when θ=2 0o :

1 1.5 2 2.5 3 3.5 40123456789

10f(x) = − 2.22308065557089 x + 12.6651505372798

Relationship of "M" Vs "h" when the angle is 20 degrees

Moment vs "h"Linear (Moment vs "h")Linear (Moment vs "h")

"h" (inches)

Mom

ent (

lb*i

n)

Graph 3: Moment Vs “h” (height from top of apparatus to the surface of water)

6. Graph for relationship between “M+[γBR22h]/2" and "h3 when θ=20o:

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Graph 4: "M+[γBR22h]/2" VS "h3"

Discussion:The lab consisted of two different experiments where the apparatus was held at two different angles (which where 0o and 20o). There were two objectives to be achieved for this lab; the first one was to check whether the moment created (fig. 2) by the weight hangers equals the moment created by the force applied on the bottom plane of the apparatus (which was “y” distance away from the pivot) when the plane is fully (equation 1) submerged or partially submerged (equation 2). The second objective of the lab was to see whether the relationship between the rotation of the apparatus (“θ”) and the force by water on the bottom plane of apparatus. After following the procedures of the lab, collecting the data, and conducting the calculations; it was observed that the two equations from the theory does bring a theoretical value (moment created by the force that the water applies against the plane) close enough to the experimental value of the moment. This can be seen by the small percentage error (between the ranges of 0%-17% for both when the plane is fully or partially submerged) in tables 3 and 4. The largest error was found to be at experiment 1 (0O elevation) trail 2 which was 16.5%, and the smallest error was found to be at experiment 2 trail 7 at 0.720%. Many of the trails (for both experiments) fall under 0% - 9% range; this shows that the apparatus was almost in equilibrium at all times.

The results in the tables also show the relationship between the height “h” and the moment created by the force from the weights. As seen in table 3 and 4 (when the plane is fully submerged); as the moment by the weight hangers increases, the distance “h” decreases. When one plots the two against each other, the graph will have a linear relationship which is very similar to equation 1 (see sample calculations). The slope (or the x-intercept) of the graph will gives us the experimental value for the specific weight of water. The experimental value of specific weight of water is very close to the actual value (since the percentage error between the two is only 5.67% for experiment 1; and 2.64% for experiment 2). This

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result also helped the observers understand that the relation stated above can be used to find specific weight of any fluid used in further experiments.

The results from tables 3 and 4 also helped the observers know that as the angle “θ” (the amount the apparatus rotated counter clockwise from its reference) increases, the height “h” (distance between the top of apparatus and the surface of water) decreases at a height “h”. This can be seen by comparing the results table 3 and 4 (at both fully and partially submerged cases); as one can see that as the weights were placed to create the moment, the height “h” was more in experiment 1 (Table 3) than experiment 2 (Table 4). This means that less water was required in experiment 2 (when compared to experiment 1) to create a moment equal to the moment created by the weights.

To find the theoretical value of the moments created by the force of water on the plane of the apparatus,

the value of “h” and “∅” were plugged in the respective equations 1 (when the plane was fully submerged) and 2 (when the plane was partially submerged). Other values like “R1”, “R2”, and “B” were taken from the lab work book; while for specific weight we used the value of 62.2 lb/ft3 (the value was taken from the textbook). In addition, to find the experimental value of the moments (by the force of weight), the observers multiplied the weight (in pounds) times the distance from the pivot point.

While conducting the lab, the observers might have experienced some errors which explained why the theoretical values and experiments values are not the same (especially for the second experiment). Some of those errors were:

1. Measurement Error: i. Increment reading of “h”: The observers might have not read the increment values of

distance “h” properly. This might have affected the theoretical values of moment since the reading of “h” is used in equation 1 and 2.

i. NOTE: This error might have taken placed a lot during the first few trails (when the plane was partially submerged) since due to little amount of water being stayed at the “curved parts” of the apparatus (making it tough to read).

2. Equipment Error: The screws that connects the apparatus with “balance rod: might have been fixed tightly, this might have rotated the apparatus a bit (while water was being filled); thus, changing the angle of the apparatus (“θ”), resulting in affecting the theoretical value of the moment.

3. Other than above errors, there were some “minor errors” which might have changed the experimental or theoretical values by a very small amount. For instance, the temperature of the water was not taken; this might have taken affected the actual value since the specific weight the observers used for their calculation (62.2 lb/ft3) is at temperature 800F (and different and different temperatures). Another minor error is the weights inserted on the weight hanger might not been the value the plate stated. Some of the plates had minor chips off from them which might have affected the observer’s experimental value by a very small value.

Conclusion:The lab was performed to analyze and see if the by the static fluid (water in our case) on the bottom plane of a center of pressure apparatus (fig 1) applies a moment that will cancel out another moment applied by

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force due to the weight plate (some distance from the pivot point) to bring the system (rod where the weight hanger and apparatus is connected) in equilibrium. After following the procedures, and looking at the results; the observers noticed that the theoretical value (moment created by the static fluid force) almost equals to the moment due to the hanging weights (as seen in the result tables 3 and 4). This here shows that the whole system is in equilibrium. The lab helped observers understand about the concepts of the forces by static fluids on any plane, and how a “moment” (created by the static fluid force) can be found using this force. Two different experiments were conducted where the apparatus was rotated at two different angles, this helped the observers understand that rotating the apparatus helps reducing the moment created by the static fluid force since the distance “y” (distance between static fluid force and pivot point) is decreased. Throughout the labs, errors were experienced by the observers; to reduce these errors for next time, one should:

1. To make incremental errors to a minimum (cannot be fully avoided) by telling them to gather 3 different reading of the same reading, and use the mean of the three reading in their sample calculations.

2. A screwdriver should be given to the observers, so that they can tighten the top screws connecting the apparatus with the rod such that it does not move slightly while rotated.

3. Measure the temperature of water before conducting each experiment, that way one will know the exact theoretical value of specific weight of water.

Appendix

1. Real life application:In many different Engineering applications, the center of pressure of a system is a valuable piece of data to be used in design and analysis since it can help one to determine the amount of force a static fluid applies against a certain “plane”. One fairly common application where center of pressure is used to find force applied by static fluid is in the building of dams in waterways (fig 5). Dams undergo high amounts of stress due to the pressure of the water reservoir pushing against the dam walls. Due to this it is important to understand what designs can help one to reduce the force applied by static water, so that this way the wall can be reinforced to prevent damage, have longer life, and prevent failure in the dam system.

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Fig 5: Dam

Many dams in real life are designed such that they are in angle (less than 90 o) with the horizontal (as seen in fig 5) rather than 900 to the horizontal. Dams are created this way since at an angle (less than 90o) to the horizontal; the force applied by the static fluid will be less towards the center of the pressure. This statement here is very similar to our lab where we did two experiments where the angles were 0 o to the vertical for the first one and 20o to the vertical for the second one. As the data for each experiment were collected, and sample calculated were done; the observers could see that second experiment (20 o), the moment (created by force applied by the static water on plane) is less when compared to the first experiment. Below equations shows the proof for the above statement:

1. Dams with vertical wall (0o to the vertical reference):

Fig 6: FBD for Dams with vertical walls

F=Pressureaverage X Area

Pressureaverage=ρg( h2)

F=ρg( h2 )X B X h

F=ρg( h2

2 )X B

2. Dams with walls at an angle θto the horizontal:

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Fig 7: FBD for Dams with vertical walls

F=Pressureaverage X Area

Pressureaverage=ρg ¿

F=ρg ¿

F=ρg ¿

NOTE: L❑¿

As seen above the two equations are very different from other, where h (in vertical wall dam) is greater than “L❑¿”; since L❑¿is lower, less force will be applied by the static water as seen above.

2. Sample calculation when θ = 20o:NOTE: specific weight of water calculated is 0.03492

1. Equation (1)

For h<R1cosθ−Plane Face Fully Submerged*

3.70 inches<4.0inches cos θ−Plane Face Fully Submerged

Moment Water=wB cosθ

3( R2

3−R13 )−wB

2(R2

2−R12 )h

0.03492lbs

¿3∗2.5∈¿cos (20 °)

3¿

Moment (Theoretical value) = 10 inches X 0.45 = 4.5 lb. * inches

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TheoreticalValue−ExperimentalValueTheoreticalValue

∗100 %=Percent Error

4.5∈¿ lbs−4.44∈¿ lbs4.5∈¿lbs

∗100 %=1.33%

2. Equation (2)

For h>R1cosθ−Plane Face Partially Submerged

5.3 5 inches>4.0 inches cosθ−Plane Face Partially Submerged

Moment Water=wB cosθ R2

3

3−

wB R22h

2+ wB sec2 θ h3

6

0.03492lbs

¿3∗2.5∈¿ cos(20 ° )∗¿¿¿

¿1.513∈¿ lbs

Moment (Theoretical value) = 10 inches X 1.5 = 1.5 lb. * inches

TheoreticalValue−ExperimentalValueTheoreticalValue

∗100 %=Percent Error

1.50∈¿lbs−1.513∈¿ lbs1.5∈¿lbs

∗100 %=0.867 %

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3. Raw Data

NOTE: In experiment, there was an error made by the observers while counting the weight plates which was later resolved; Experiment 1 height is 3 inches when the weight on the mass hanger was 0.65. While for the final trial (weight 0.75 lb.), no height was recorded since the top screws were not tightened properly (causing the apparatus to fall down).

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ReferencesButler, C. Technical Bulletin (pp. 1-5). Nottingham: Lechquipment Limited.

Mott, R. (2000). Applied fluid mechanics (5th ed., pp. 78-82). Upper Saddle River, N.J.: Prentice Hall.

http://www.fotolibra.com/gallery/693210/walking-on-a-concrete-dam/

http://www.madehow.com/Volume-5/Concrete-Dam.html

http://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html

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