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MOMENT OF INERTIA Prerequisites: Understanding of Mass vs. Weight, "Beam Teams" Lesson LEARNING OBJECTIVES Students will: Concepts: - Define and be able to explain "inertia" - Understand and explain "moment" - Understand the meaning, symbolic representation (with the letter "I"), and units of "moment of inertia" - Understand how moment of inertia relates to the stiffness of structural elements - Understand that the moment of inertia of a beam or any structural element is based on the shape and dimensions of its cross section - Understand the concept of "area of a cross section" - Be able to calculate moment of inertia of a rectangular and I-shaped beam Scientific Method and Tools - Analyze a geometric figure - See a relationship between the graphic representation of a beam's cross section (its cross-sectional area) and a measurable physical property (its moment of inertia) - Express a physical property in symbolic form (the formula for moment of inertia ­ "I")) and to be able to calculate "I" given various constraints - Correctly express the units of moment of inertia Collaboration Skills - Share results of individual work with other students and lead a discussion. - Be able to deepen and contribute to the whole-class discussion.

ASSESSMENT The student: - Understands how the stiffness of a beam is related to different characteristics of its cross section, and understands how to maximize the stiffness of a beam given the area of its cross section. - Can manipulate the formula for moment of inertia to design beams with the desired stiffness - Can represent the results in graphic form - Carefully completes all calculations and can predict what will happen if certain parameters are changed. - Raises questions, e.g. How do you calculate "I" for other shapes? How can we prove that this formula is accurate? - Communicates results and reasoning clearly to others

KEY CONCEPTS 1. Inertia is the property of an object at rest that causes it to remain at rest unless acted

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MOMENT OF INERTIA upon by an external force. The amount of mass an object has is a measure of its inertia; the more mass a body has, the greater its inertia. 2. A moment is the product of a force acting on a lever and its perpendicular distance or lever arm from the pivot (expressed in foot-pounds, inch-pounds, kilogrammeters, etc.) When you turn a nut with a wrench, you exert a moment equal to the force of your arm movement multiplied by the distance from your hand to the nut (the length of the handle on the wrench = the lever arm). 3. Structures have their own inertia that varies according to the mass of the structure. For example, a beam bends when a load acts it on. The load is the external force that moves the beam by overcoming its inertia. The larger the load, the more the beam bends. To limit the bending of the beam, you can manipulate its inertia by rearranging the mass in the beam. 4. Moment of inertia is a quantity that gauges the stiffness of a beam, depending on how much of the beam's mass has been moved away from the neutral axis. It varies with the cross-sectional shape of the beam. The moment of inertia tells you how stiff an element is in its resistance to bending (It is also referred to as the moment of resistance of a structural element). The further the mass is from the neutral axis, the larger the moment of inertia. 5. Center of gravity is the point at which the mass of an object is thought to be concentrated. Gravity acts on an abject as if its weight were all concentrated at the center of gravity. For a solid square beam, the center of gravity is at the center of the area of the cross-section. 6. The formula for the moment of inertia (I) of a rectangular beam is as follows:

B H

I

= (BH) H2 12

= BH3 12

(units = cm4)

7. The formula for the moment of inertia of an I-beam is as follows:

B A H

2

I =

2(AB)

H+A 2 2

+ CH3 12

(units = cm4)

C

The formula above has an additional component, compared to the rectangular beam formula. This component takes into consideration the flanges, and is proportional to the square of the distance from the neutral axis of the beam to the center of the

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MOMENT OF INERTIA flange. The further the flange is from the neutral axis, the larger the moment of inertia and the stiffer the beam. This is why I-beams are so popular. 8. If the scale of a beam changes - the size of the beam increases by a factor of 2 ­ the moment of inertia goes up by a factor of 16 or 24. VOCABULARY Center of Gravity, Cross-section, Efficiency, Inertia, Lever Arm, Mass, Moment, Moment of Inertia, Neutral Axis, Structural Elements RESOURCES Thompson, Marilyn, Charles W. McLaughlin and Richard G. Smith. Merrill Physical Science. Glencoe/McGraw Hill, 1995. www.efunda.com/math/areas/ features a Moment of Inertia calculator for checking work Beams in the built environment

MOTIVATION A suggested explanation for the class: "Now that we have explored the different shapes of beams in Beam Team, we are ready to prove those observed results mathematically. The stiffness of a beam can be expressed in a quantity called the Moment of Inertia, represented by the symbol "I". What is inertia? Inertia is the tendency of an object at rest to stay at rest as well as that of an object in motion to stay in motion, unless acted on by an external force. We see examples of inertia all around us. A desk in the classroom has its own inertia that has to be overcome by a student in order to push the desk across the classroom. A skateboard rolling on a smooth flat surface will continue to roll (in a perfect, frictionless environment) until it hits something or someone stops it. These are examples of inertia. Beams have an inertial force that has to be overcome in order for them to bend. The more mass the beam has, the larger its inertial force, and the more resistant it is to bending (stiff). What is a moment? Think of a seesaw on the playground. A person on the end of a seesaw is inducing a moment or rotation. The moment can be calculated by multiplying the weight of the person by the distance from the person to the fulcrum/axis of the seesaw. When the force of the moment exerted by the person on one side of the seesaw is larger than that of the person on the other side, the seesaw rotates. We use the term moment in reference to beams because the bending of a beam is a form of rotation. Putting the terms together, we can say (taking some liberties with the actual

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MOMENT OF INERTIA mathematical explanation of moment of inertia) that the moment of inertia is a measure of how much inertia a beam has to resist bending. Every beam has this resistance or stiffness. What factors affect the stiffness of the beam? How can you increase a beam's stiffness? Why are I beams used so often? Can you prove this numerically? THE CHALLENGE You are a structural engineer on a big skyscraper project. Most of the beams you are using are I-beams. Your junior engineer is skeptical. Why not use rectangular beams which are cheaper, she asks? You will have to explain how to analyze the stiffness of different types of beams. If your junior engineer can prove that another shape is better than your I-beams ­ let her try... PROCEDURE Time frame of lesson: two forty-five minute periods maximum. Materials Worksheet, hand-held calculator, graph paper Download the worksheet and modify it to fit the needs of the class. Begin with discussion of key concepts and Motivation portion of the lesson. Follow with Motivational Questions and a discussion of the Challenge. Distribute worksheets to the students and instruct them to work the problems individually. Students should work the examples and pause to discuss their results, before completing problems 1 through 4. The teacher should prompt students to compare the moments of inertia for the rectangular and I-shaped beam of the same area (examples 1 & 2) and see that the Ibeam has a larger moment of inertia. After students solve all the problems, a class discussion helps to deepen their understanding and identify patterns in the calculations and the results. Discuss why the moments of inertia differ and how that relates to choosing the most efficient beam shape for a give cross-sectional area. The last problem in the handout asks students to design their own I-beam for a given area that has a larger moment of inertia than the example provided. Have the students draw their final designs on the board and label the sizes of flanges and webs. Also have them indicate the corresponding moment of inertia. Compare students' results and discuss with the class the different design options. Students should come away with a deeper understanding of beam behavior. They should also be able to calculate the moment of inertia of rectangular and I-beams and understand how to distribute material in the flanges and web of an I-beam to maximize the moment of inertia. Students should be able to explain the benefit of higher moment of inertia, i.e. greater stiffness and strength, which results from using an I-beam instead of a rectangular beam (for a given area).

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MOMENT OF INERTIA

Have students give each other a pat on the back ­ they have done the same calculations performed by structural engineers! VARIATIONS & FOLLOW-UP Have students explore the website www.efunda.com/math/areas/ and investigate the moments of inertia for different shapes of beams. Add the following problem, for a beam that structural engineers commonly use in buildings - a W12x26 beam. The students can check their calculations on the web at www.efunda.com/math/area/IbeamIndex.cfm by selecting Steel W Type IBeams and then selecting W12x26. A=.38in, B=6.49in, C=.23in, H=11.46in There may be a small difference in the students calculations compared to the actual property, due to rounding. MIDDLE SCHOOL STANDARDS National Standards: New Standards - National Center on Education and the Economy Physical Science - S1a Properties of matter - S1b Motions and forces Scientific Thinking - S5b Uses concepts from Science Standards 1 to 4 to explain observations - S5d Proposes, recognizes, analyzes, considers, and critiques alternative explanations - S5f Works individually and in teams to collect and share information and ideas Geometry and Measurement Concepts - M2a Is familiar with assorted two-and three- dimensional objects - M2d Determines and understands length, area and volume - M2e Recognizes similarity and rotational and bilateral symmetry in two- and threedimensional figures - M2I Reasons proportionally in situations with similar figures - M2k Models situations geometrically to formulate and solve problems Function and Algebra Concepts - M3d Finds solutions for unknown quantities in linear (and exponential) equations Mathematical Skills and Tools - M6a Computes accurately with arithmetic operations on rational numbers - M6b Knows and uses the correct order of operations for arithmetic computations - M6d Measures length, area and volume correctly - M6e Refers to geometric shapes and terms correctly - M6f Uses equations, formulas, and simple algebraic notation appropriately - M6h Uses pencil and paper and calculators to achieve solutions

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MOMENT OF INERTIA Conventions, Grammar and Usage of the English Language - E4a Demonstrates an understanding of the rules of the English Language in written and oral work New York State Standards: The University of the State of New York ­ The State Education Department Math, Science and Technology 1. Students will use mathematical analysis, scientific inquiry, and engineering design to pose questions, seek answers, and develop solutions. 2. Students will access, generate, process, and transfer information using appropriate technologies. 3. Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in realworld settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry. 4. Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science. 7. Students will apply the knowledge and thinking skills of mathematics, science and technology to address real-life problems and make informed decisions. English Language Arts 1. Students will read, write, listen and speak for information and understanding. HANDOUTS See attached

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Date

Name(s)

Moment of Inertia

How Can You Predict How Stiff a Beam or a Column Will Be Based on its Cross Section?

There is a quantity ­ called the moment of inertia ­ that allows you to tell how stiff a structural element is. When you calculate the moment of inertia you actually measure how far the material in a cross-section of an element is from its neutral axis. First, let's look at a rectangular beam. Its cross section will be a rectangle, of course. You know that the area of a rectangle is equal to the length times the width (or the base times the height). So, to find the moment of inertia (I) you: Rectangular Beam:

Base =B

Take the area of the cross section, multiply it by the height squared and divide by twelve (the constant 12 comes from the derivation of the equation using calculus). I = Area x (Height)2/12 = (BH) H2

12

Height =H

I Area = BH

= BH3

12

I-shaped beam:

B A Flanges H Center of flange Center of beam Web C

For an I-shaped beam, there are many more dimensions to consider. The area of the flange is AB. There are two flanges so multiply by 2. Then multiply the result by the square of the distance from the center of the beam to the center of the flange (that is half of H plus half of A). This gives you the moment of inertia of the two flanges. The height of the web is H. Add the moment of inertia of the web (which is like a rectangular beam so the equation is the same) and your done. I = 2 x Area of Flange x (Web height/2 + Flange height/2) 2 + Web width x (Web height) 3 /12

Area = 2AB + CH

I =

2(AB)

H+A 2 2

2

+ CH3 12

Date

Name(s)

Moment of Inertia

Draw and label your beams and show your work on graph paper for all problems. Example 1 Imagine a rectangular beam with a base (B) that is 7 cm and a height (H) that is 8 cm. What is its cross-sectional area? What is its moment of inertia? Draw and label the beam. Example 2 Imagine an I-beam with a flange that is 2-cm thick (A) and 10-cm wide (B), and with a web that is 2-cm wide (C) and 8-cm deep (H). Draw a picture of the cross section of this beam and label its parts. Calculate its total area, and then calculate its moment of inertia (I) of the beam. How do its area and moment of inertia compare to those of the beam above? Example 3 Double the size of the base and height in Example 1 and recalculate the area and moment of inertia of the resulting beam. By what factor does each change? Why?

Solve the following problems: 1. Another beam has the same flanges as the I-beam in Example 2, but its web is 10 cm deep instead of 8 cm deep. What is its moment of inertia? How much stiffer is this beam than the one above? 2. Calculate the moment of inertia of a beam with flanges that are 2 cm thick and 20 cm wide, and with a web that is 2 cm wide and 20 cm deep (A = 2, B = 20, C = 2, and H = 20). 3. In the problem above, if you could change one of the 2's to a 3 in either A, B or H, which would you change to make the beam as stiff as possible? Prove it by drawing all three possibilities and calculating the moment of inertia of each. 4. A beam's flanges and its web are 2 cm thick (A and C) and its moment of inertia is 299 cm4. The area of its cross-section is 40 cm2. Design a beam with the same cross-sectional area but with a greater moment of inertia. Show your work, indicating the width of the flanges and the depth of the web of your new beam, and make a labeled drawing of the cross-section. Be prepared to discuss your results for problem 4 with the rest of the class!

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