Introduction: Engineering Problems: Force of Water on a Dam

Dams are used around the world to stop rivers and force them to become much larger bodies of water. These reservoirs serve two main purposes:

1. irrigation: for farmlands, cities, other rural areas

2. generation of hydropower (electricity)

Several factors play into how engineers design dams. These variables dictate the materials engineers use, the thickness of the dam, and where dams are built.

1. Environmental factors: Ice, silt pressure (soil at the bottom of the lake and under the dam), and earthquakes (if those are common in the area)

2. Material factors: the weight of the materials used to build the dam on the earth surrounding it and how they can withstand the other factors.

3. Water factors: the pressure of the water being held back by the dam, and the pressure from waves on the surface of the water.

When these factors are ignored or not accounted for correctly, dams can break leading to flooding and destruction of people’s homes and livelihoods. A recent example of engineers not correctly anticipating and correcting for these factors is the Oroville Dam in California which lead to the evacuation of 200,000 residents when excess rain in California caused the dam to fill to capacity.

Read more at:

http://www.sfchronicle.com/bayarea/article/Releasi...

This Instructable explains how to account for the water factor, or the water pressure exerted on the dam.

materials needed:

1. Basic understanding of algebra (multiplication and division)

2. Pencil/pen

3. Calculator

Step 1: Understanding Up the Problem:

First, you need to understand conceptually what water pressure is, and how it acts on the dam before you can begin. Imagine you are swimming in a pool. Do your ears pop near the bottom of the deep end? This occurs because the weight of the water above you causes an increase of pressure around you causing your ears to pop. The same principle can be applied on a dam. Although, the dam is experiencing a much larger pressure because the depth of the water is even greater.

Step 2: Read and Sketch the Problem:

Now that you have a basic understanding of how water pressure effects a dam, you can approach the problem.

1. Read your problem so you know exactly what you are solving for.

2. Sketch the dam and force acting on the dam

The sketch of the force can be thought of conceptually first, keeping in mind what you learned about water pressure. Since the water pressure increases proportionally to the depth, force, which is drawn as an arrow pointing at the object the force is acting on, is increasing as well. The force can thus be drawn as a triangle of arrows increasing in size.

For the purpose of solving a problem, however, the force is treated as an average of all the arrows of the triangle which acts at the centroid (geometric center) of the triangle which is at (h(height)/3). (label the force F)


Note: we will use this problem as a example.

What is the force acting on a 25 meter width dam and water depth of 15 meters?

More information about centroids:

https://en.wikipedia.org/wiki/Centroid

Step 3: Identifiy the Variables:

Now that you have read and sketched the problem, you need to understand and label the variables.

For our problem, we are given several values:

What is the force acting on a 25 meter width dam and water depth of 15 meters?

1. width of the dam (w)

2. depth (or height) of the water (h)

Label these two variables on your sketch.

There are two other constants (variables which do not change) you will need to know going forward:

1. Density of water (whos symbol is either ρ (the lower case Greek letter rho) or ∝ (the lower case Greek letter alpha) which has a magnitude (amount) of 1000 kg/m^3.

2. Gravity which has a magnitude of 9.81 m/s^2.

(both of these constants are also drawn on the sketch)

Step 4: Derive the Nessacary Formula:

To understand the formula you must think back to when were imagining swimming in the pool. The water above you caused an increase in pressure. Only, in this problem, instead of the water being above you, it is acting on the surface area of the dam. With that in mind, we can understand the force exerted on the dam by the water is:

Force= Pressure (x) Area

We also know that:

P= rho (x) g (x) h

A (referring to the surface area in contact with the water)= (h (x) w) /2

Finally after multiplying area and pressure, you can write down the formula:

F= (rho)(g)(h/2)(w (x) h)

Step 5: Plug in Variables and Solve:

Now that you have the problem, drawn a sketch, labeled all the variables, and developed your formula, you can begin actually solving the problem.

1. Plug all the variables into in the formula, making sure to include units.

2. Algebraically solve by multiplication and division of both the numbers and the variables. Notice how the units cancel out to leave (kg (x) m)/s^2).

To also leave you answer in a more manageable form use unit conversions:

1 (kg (x) m)/s^2)= 1 N = 1000 kN

Step 6: Going Further:

Now you know how to solve for the force exerted on a dam by water pressure. Next time you drive past or over a dam, water you plants, or turn on your lights remember how amazing these man-made structures are and just one of the factors which effect them.

For this example you examined a simple rectangular dam, but dams come in many different shapes and sizes. To learn more about dams visit:

https://en.wikipedia.org/wiki/Dam

To learn more about how to solve for the force acting on differently shaped dams visit:

http://www.codecogs.com/library/engineering/fluid_...