Lab for Deflection and Moment of Inertia

Deflection and Moment of Inertia

Subject Area(s) Physics Associated Unit Wind Effects on Model Building Lesson Title Lab for Deflection and Moment of Inertia

Grade Level (11-12) Part # 2 of 3 Lesson # 1 of 1

Time Required 2-3 classes 911research.wtc7.net/.../WTC/I-beams.htm

Summary Students are to discover the relationship with deflection of balsa wood beams and how this affects the moment of inertia of an object. Students will then use this information to help in the design and construction of the model storage facility in Part III of Wind Effects on Model Building.

Engineering Connection Shape and moment of inertia Engineering Category Civil Engineering

Keywords Force, deflection, stress, strain, moment of inertia, torque

Educational Standards AP Physics C Standards http://www.collegeboard.com/student/testing/ap/physics_c/topics.html#newtMech

Pre-Requisite Knowledge Students must have prior knowledge of forces, torque and center of mass. An understanding of integration will be beneficial, but is not required.

Learning Objectives  Students should be able to identify the relationship between stress and strain.  Students should be able to identify the relationship between shape and moment of inertia.  Students should be able to apply the information to the construction and design of the model building. Introduction / Motivation Are you stressed? What is stress? Have the students brainstorm about stress.

Lesson Background & Concepts for Teachers For symmetrical buildings: The APPLIED FORCE is through the CENTER OF MASS and The RESISTING FORCE is through the CENTER OF STIFFNESS in the opposing direction.

Applied force

Top View

For non-symmetrical buildings, the forces balance, but the center of mass and the center of stiffness are not the same point so torque results ( τ = r x F where r is the distance between CM and CS).

Higher Moment of Inertia gives better opportunities for load. From the corner: b I = ∫ y2 ∂A where ∂A = b ∂y with limits of 0 to h

∂A I = ∫ y2 (b ∂y) = b ∫ y2 ∂y = b (y3 / 3)h 0 h

I = bh3 / 3

b For a centroid, the limits change from 0 to h to –h/2 to h/2 so that,

3 Ix = bh /12 h

3 Iy = hb /12

If not at the center, then I = ΣI + Ar2

Here are some examples: 3 I = bh /12 4m I = 4*63 /12

2 I = 72 kg*m

2m 3 I = bh /12

3 3 6m 4m I = 4*6 /12 - 2*4 /12

void 2 I=61 kg*m

Note: Rectangular structure is best for torsion control. 4m

6m 3 I = bh /12

3 3 I = 4*6 /12 - (1.5*3 /12)*2 Vocabulary / Definitions Word Definition Force A push or a pull, it causes objects to change their motion. Deflection The deviation of the indicator of an instrument from the position taken as zero. Torque The magnitude of a torque acting on a body is the product of the magnitude of a force and its force arm (perpendicular distance from the axis of rotation of the body to the line of action of the force). This product is called the moment of the torque about the axis or the torque. Moment of The tendency of an object to resist being accelerated when it is rotating. Inertia Stress The force, or combination of forces, which produces a strain; force exerted in any direction or manner between contiguous bodies, or parts of bodies, and taking specific names according to its direction, or mode of action, as thrust or pressure, pull or tension, shear or tangential stress. Stress is the mutual action between portions of matter. Strain To act upon, in any way, so as to cause change of form or volume, as forces on a beam to bend it.

Student Lesson

Part I: The students must first understand the idea of stress and strain. Watch the video Designing Backpacks at http://www.thefutureschannel.com/dockets/realworld/designing_backpacks/ See Strap-Stress.pdf from The Futures Channel before watching the video. Discuss What qualities or characteristics would you look for in a material that is used to make a backpack?

Part II: The students are to use balsa wood to test deflection and show the relationship between shape and moment of inertia. Student worksheet is shown on next few pages. Block ______Date ______Name ______

Deflection Lab Objective: To determine the relationship between deflection, shape of the object, and load.

Materials: Balsa Wood 3/16” x 3/16” x 36” (2) 3/16” x 3/8” x 36” (4) 3/8” x 3/8” x 36” (1) Two ring stands with clamps Large bungee cord or strap Large amount of mass 15 – 20 kg per group Meter stick Glue (Wood glue if prepared ahead of time) / (Hot glue if prepared during lab)

Procedure: 1. Set ring stands 80 cm apart from one another; will need to move to 40 cm for part of the testing. 2. Construct beams to be tested from the data table. 3. Place beam to be tested between two clamps on meter stick. 4. Place strap over the center of the beam and mark initial zero location with strap in place. 5. Add mass in small measurable increments until the system is no longer static. Each time mass is added, take a deflection reading. Record this value in the data table. 6. Repeat until the data table is full or until you run out of time. Check with your teacher on which beams must be completed.

Sketch of experimental setup: Data: Beam A Beam B Beam C Cross Length Cross Length Cross Length Section 80 cm Section 40 cm Section 80 cm 3/16” x 3/16” 3/16” x 3/16” 3/16” x 3/8” Mass Deflection Mass Deflection Mass Deflection

Beam D Beam E Beam F Cross Length Cross Length Cross Section Length Section 40 cm Section 80 cm (2) 3/8” x 3/16” 80 cm 3/16” x 3/8” 3/8” x 3/8” In T shape Mass Deflection Mass Deflection Mass Deflection

Analysis: Determine the cross sectional areas in cm2. Cross Sectional Beam Area cm2 A B C D E F

Using ExCel, complete the following graphs and attach to laboratory report. Graph 1: Graph the deflection (dependent) versus the load (independent).

Explain why the Beam F has the smallest slope?

What does this show about the relationship between deflection and load?

Graph 2: Using only Beams A, C, E and F, graph the cross sectional areas with the maximum load.

What do these beams all have in common?

Explain the results of this graph.

What does this show about the relationship between area and load?

Deflection Lab Student Reflection Notes http://www.cmiengineer.com/sheetpiling/product-design-theory.php

Moment of Inertia The bending performance of a particular beam is largely controlled by a cross section property known as the second moment of area or more commonly known as the moment of inertia (I).

The moment of inertia is based solely on the shape of a cross-section, or area, and not controlled whatsoever by material properties.

Moment of inertia is calculated as follows:

where the moment of inertia of area A is calculated about axis x.

Deflection The deflection of a beam is principally a function of the moment of inertia of the beam cross section, and the modulus of elasticity of the beam material. Generally speaking, the higher the moment of inertia and modulus of elasticity of a particular beam, the lower the deflection and therefore stiffer the beam will be in bending.

For the situation noted in Figure 1 in the bending moment section, the maximum deflection will occur at the center of the span and can be calculated as follows:

Stress, Strain and Modulus The structural performance of all materials is primarily controlled by two main factors:

Stress (σ) – Applied force over a given area. Usually given in pounds per square inch (psi), or pascals (Pa) Strain (ε) – The amount of deformation or stretch of a material. Usually given in inch per inch (in/in) or percentage (%)

Stress and strain for isotropic materials are related by Hooke’s law:

where E is the modulus of elasticity.

E is a constant for our laboratory experiment since it is a constant for each material, such as Balsa wood. Lesson Closure Go over Building Project Structural Concepts Power Point and do several moment of inertia calculations with students.

Discuss the objectives and the conclusions of the two activities, and how their findings will affect the design of their model building.

Assessment Evaluation of Deflection and Moment of Inertia Lab

Lesson Extension Visit the local Civil Engineering Department to perform actual beam testing. Have students write a reflection paper on how it relates to the small scale testing they did in their laboratory exercise. Wood Timber Beam Test video

Multimedia Support None

Creator Danielle Reynolds / Duncanville High School

Sponsor University of Texas at Arlington / Civil Engineering National Science Foundation under Grant No. EEC-0808687