A Simple Beam Test: Motivating High School Teachers to Develop Pre-Engineering Curricula

Session 2326 A Simple Beam Test: Motivating High School Teachers to Develop Pre-Engineering Curricula Eric E. Matsumoto, John R. Johnston, E. Edward Dammel, S.K. Ramesh California State University, Sacramento Abstract The College of Engineering and Computer Science at California State University, Sacramento has developed a daylong workshop for high school teachers interested in developing and teaching pre-engineering curricula. Recent workshop participants from nine high schools performed “hands-on” laboratory experiments that can be implemented at the high school level to introduce basic engineering principles and technology and to inspire students to study engineering. This paper describes one experiment that introduces fundamental structural engineering concepts through a simple beam test. A load is applied at the center of a beam using weights, and the resulting midspan deflection is measured. The elastic stiffness of the beam is determined and compared to published values for various beam materials and cross sectional shapes. Beams can also be tested to failure. This simple and inexpensive experiment provides a useful springboard for discussion of important engineering topics such as elastic and inelastic behavior, influence of materials and structural shapes, stiffness, strength, and failure modes. Background engineering concepts are also introduced to help high school teachers understand and implement the experiment. Participants rated the workshop highly and several teachers have already implemented workshop experiments in pre-engineering curricula. I. Introduction The College of Engineering and Computer Science at California State University, Sacramento has developed an active outreach program to attract students to the College and promote engineering education. In partnership with the Sacramento Engineering and Technology Regional Consortium1 (SETRC), the College has developed a daylong workshop for high school teachers interested in developing and teaching pre-engineering curricula. With the guidance of faculty and students, recent workshop participants from nine local high schools performed “hands-on” laboratory experiments and received a workshop manual containing handouts and ideas for lesson plans, as well as other literature. The highly-interactive workshop format allowed teachers to explore ideas which can be easily implemented at the high school level to motivate students to study engineering. This paper describes one experiment that can be used to introduce fundamental structural engineering concepts to students through a simple beam test. In addition to explaining experimental procedures, the following sections review basic structural engineering concepts necessary for high school teachers to successfully implement the experiment. The approach Page 6.98.1 Page Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright © 2001, American Society for Engineering Education taken in this paper is to present basic engineering concepts and experimental procedures in a manner similar to that used in the actual workshop. II. Overview of Experiment In this experiment, a load is applied at the center of a simply-supported beam and the resulting deflection at midspan is measured. Figure 1 shows the test setup, including the specimen, supports, loading device, and dial gage. The specimen shown in the figure is a steel beam of rectangular cross section that rests on a wide flange stubs. The simple supports at each end prevent vertical displacement but not rotation. Load is applied using a bucket filled with steel weights or any other convenient material. Figure 2 shows a cable loop that is placed around the beam at midspan to apply load. The dial gage is used to measure the deflection corresponding to the applied load. Figure 3 shows workshop participants recording data during a beam test. Load versus deflection response is then plotted using test data. From this plot, the elastic stiffness of the beam is determined and compared to published values for various beam materials and shapes. Beams can also be tested to failure to observe the failure mode and to compare elastic and inelastic response. III. Basic Concepts The beam test provides an effective means to illustrate several important structural engineering concepts. The plot of load vs. deflection provides a direct means to experimentally determine the structural stiffness, K, of the beam as well as the material stiffness, referred to as the modulus of elasticity, E. While the parameters themselves have important practical significance, this experiment provides a springboard for discussion of additional topics of significance in engineering practice, such as elastic and inelastic behavior, influence of materials and structural shapes, stiffness, strength, and failure modes. To help students develop an understanding of these concepts, several underlying principles are briefly introduced. A. Elastic Behavior and Hooke’s Law To understand how a beam responds to load, it is first necessary to understand material behavior. Many engineering materials exhibit what is referred to as elastic behavior—the property by which a material returns to its original dimensions during unloading. For example, if you pull on both ends of a rubber band, it will stretch. When you release the load (or force), the rubber band will return to its original shape. This similar to how a steel bar stretches, although the change in length for the stiffer steel bar would be much smaller, even imperceptible, in many cases. Elastic behavior for many materials is the same whether the material is pulled upon (i.e., subjected to tension) or pushed upon (i.e., subjected to compression), assuming buckling of the specimen is prevented in compression. When a load acts along the length, or axis, of a specimen as described above, the load is referred to as an axial force. This is in contrast to bending, shearing, or twisting of a structural member. When a material acts elastically, the change in length due to an axial force is directly proportional to the force. The change in length per unit length is known as the strain, ε: 6.98.2 Page Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright © 2001, American Society for Engineering Education ε = ∆L / L (1) where ε = axial strain, in/in (mm/mm) ∆L = change in length due to load, in (mm) L = original unloaded length, in (mm) The force per unit cross-sectional area is known as the axial stress, σ: σ = F / A (2) where σ = axial stress, psi (Pa) F = applied axial force, lbs (N) A = cross-sectional area of the specimen, in2 (mm2) When the axial stress is plotted against axial strain, many materials display a linear relationship between stress and strain within a certain range. This linear relationship is shown for steel, aluminum, and wood in Figure 5. The elasticity of the material is evident by the fact that when a specimen is loaded and then unloaded, the stress and strain both return to a zero value. Thus, in the case of linear elastic material behavior, axial stress and strain are directly proportional and are related by a proportionality constant, E, defined as follows: σ = E ε (3) where σ = axial stress, psi (Pa) ε = axial strain, in/in (mm/mm) E = modulus of elasticity, psi (Pa) The equation σ = E ε is known as Hooke’s law. E is known as the modulus of elasticity and is simply the slope of the stress-strain diagram. As shown in Figure 5, E varies for different materials and represents the material stiffness. Based on Figure 5, the wood would be expected stretch about 15 times as much as a steel rod of the same cross sectional area when subjected to the same axial load. Published values for E are available for many common materials2. If the stress exceeds the elastic limit, the slope of the stress-strain plot becomes nonlinear, as discussed in Section V. B. Bending, Moment of Inertia, and Deflection The foregoing discussion provides a basis to understand beam behavior. When a beam is subjected to a transverse load such as that shown in Figure 1, the beam bends and deflects in a curved shape. For a beam with linear elastic material properties, the load produces a linearly varying strain through the depth of the beam, resulting in compression on the top surface and tension on the bottom surface. Based on Hooke’s law, stresses also vary in the same manner. The magnitude of the elastic deflection of a beam depends on how stiff the beam is. The stiffer the beam, the smaller the deflection will be for a given loading. Bending stiffness of a cross Page 6.98.3 Page section is defined by the quantity EI. The modulus of elasticity, E, represents the material Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright © 2001, American Society for Engineering Education stiffness; the moment of inertia, I, represents the contribution of the cross sectional geometry to bending stiffness. I is defined as follows: = ∫ 2 I x y dA (4) where 4 4 Ix = moment of inertia with respect to the neutral axis (i.e., x axis in Figure 4), in (mm ) y = distance from the neutral axis to a differential area dA of the cross section, in (mm) The moment of inertia is a geometric property, based on the second moment of the area about the neutral axis (i.e., the axis where the strain equals zero). This quantity is more fully understood using calculus. However, for many shapes the moment of inertia can be easily calculated from simple formulas found in any mechanics of materials text2. For the common case of a rectangular beam with a width, b, and height, h, I = bh3/12. It is important to note that the definition of b and h depends on which axis of the cross section the beam bends about. It is also shown in mechanics of materials texts that the deflection of a beam due to bending is a function of the magnitude of the load, span length, support conditions, and flexural rigidity.

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