Let It Fly!

Aerospace Engineering Applications of Mathematics 10

Lesson Idea by: Cindy Botnen, Van McPhail, Dave Ward and Leanne Zorn
School District #23 (Central Okanagan)

Aircraft designers always strive to balance weight and strength. "If you save one pound of weight on a plane, that can save the plane's operators $5,000 over 20 years," says Phil Evans, an aerospace engineer at Avcorp Industries, a Richmond, B.C. aerospace company.

The strength of a plane's parts is a critical issue in aircraft design, but not the only one. For example, minimizing weight is also very important. "We use mathematics all the time," says Evans. Math is used to calculate the strength of each of a plane's parts. Math is also used to predict when the material making up a part will fail. If calculations show that one kind of metal isn't strong enough, designers usually have to choose another lightweight, strong material.

Ensuring a part's strength is a two-part process. First, the aircraft designers do preliminary calculations to determine whether the part they've designed can withstand the load that will be placed upon it. Then it goes to the "stress department" to do more precise calculations.

"The designers just have to get (the load calculation) close," Evans says. They don't have to absolutely confirm that a part can withstand a certain load - that's up to the stress department. The stress analysts use a computer program called "finite element analysis" to calculate precisely whether a part is strong enough - or too strong! If the part is too strong it is likely too heavy. Designers have to strike a balance between weight and strength.

With your classmates, brainstorm answers to the following questions. Also discuss the concept of load. Remember that load is not the same as mass. Load is a force that includes mass along with other inputs like acceleration, gravity, and centrifugal forces.

"How will you know if a material can sustain a load?"
"What happens if the material in an airplane is too strong and too heavy?"
"What are the consequences of an inadequate load resistance? (i.e., what if the material is not strong enough?)"
"What does bridge-building or roofing have to do with load?"

In small groups, perform a series of "load" tests using strands of spaghetti and weights. Place a single piece of spaghetti between two desks set 15 to 20 centimetres apart. Tie a small string to the centre of the spaghetti. Hang weights of increasing mass to the string until the spaghetti breaks. Record the mass at which the spaghetti breaks, using the attached record sheet. Repeat the test with two, three, four and five strands of spaghetti, bundled together with elastics.

Compare the results of your tests with other groups in the class. What mass was required to break different diameters of spaghetti? What conclusions can you make regarding the weight-bearing capacities of the spaghetti?

Use the following formula to calculate stress:

stress = load divided by area. ( S =	L	)
	A

Review all the key concepts needed to use this formula. These should include the formulae for area, the concepts of radius versus diameter, and pi. (For your spaghetti calculations, use approximate measurements for the diameter of the spaghetti bundles. Each strand has a diameter about 1 millimetre.)

Also discuss the units of measurement that will be used (i.e., Newtons and Pascals). How do these formal units of measurement compare with the weights used in the experiment?

Apply the data from the spaghetti experiment to the stress formula. Do so for each of the five tests. Record the results in Pascals. Discuss any pattern in the results with your classmates. Is there a relationship between the area of the material and the weight it can bear?

Now, apply what you've learned to a real-life situation:

You work as a designer at an aerospace engineering company. You're designing a support rod for the inside of an aircraft wing. The support rod is a metal bar. It's one of many that make up the internal structure of the wing. This particular support is called a "stringer." The stringer runs the length of the wing through another kind of support beam called the ribs. (See drawing below) The stringer helps add rigidity to the wing in addition to the skin, which is the outside covering of the wing.

The special shape of a wing is created by the ribs. The ribs run in the same direction as the airplane's motion. The stringers go the other way - they run through and between these ribs, but perpendicular to them most of the length of the wing. (In three dimensions, the stringer runs at right angles, or in an "orthogonal" fashion).

Your current design specifications call for the rod to be made out of aluminum and to be a diameter of 20 mm. (Hint: the units must be consistent, so convert millimetres to metres.) The rod will be supporting a load of 50 x 10^3 Newtons (a measurement of force). According to your stress analysis tables, the allowable stress on aluminum is 100 x 10^6 Pa (Pascal N/m^2). This is the maximum load that can be placed on aluminum before it will break. You can't exceed this number.

The formula for stress on a rod is: Stress equals Load divided by Area. The formula for area of a circle is: Area equals pi r ^ 2.

(Please note: This symbol, ^, called a caret, is used to indicate "to the power of.")

Calculate the stress on the aluminum rod.

Can you use an aluminum rod?

Is it within the allowable stress for aluminum?

If the rod is not within the allowable stress for
aluminum what are your alternatives? Note, you cannot modify the design.

The Solution

area = 3.14 (20 mm/2)^2  [radius = ½ of diameter]
         = 314 mm^2

1 mm = 10^-3 m
Area = 314 x 10^-6 m^2

stress = load divided by area
          = 50 x 10^3 / 314 x 10^-3 m^2
          = 159 x 10^6 Pa

The allowable stress on aluminum is 100 x 10^6 Pa. Since your calculations showed the load is 159 x 10^6 Pa, you cannot use the aluminum rod as it exceeds the allowable stress. As the designer, you may choose to increase the diameter of the aluminum rod. This may have other design implications, however. Or you may choose a new metal, something stronger and lighter than aluminum, like titanium.

Course/Grade: Applications of Mathematics 10
Curriculum Organizer(s): Number Operations Patterns and Relations Problem Solving	Curriculum Sub-organizer(s): Solving problems with formulae
Prerequisites: Scientific Notation Shape and Space (Area) Powers Simple Equations Units of Measure	Resources: · spaghetti · a variety of weights · calculators

Student Activity Sheet Let it Fly! Participants 1.____________________ 2.____________________ 3.____________________ 4.____________________ Stands of Spaghetti Weight Stress 1.     2.     3.     4.     5.     stress = load divided by area. ( S = L ) A