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

Transcription

1 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 Consortium 1 (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

2 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: Page

3 where ε = axial strain, in/in (mm/mm) L = change in length due to load, in (mm) L = original unloaded length, in (mm) ε = L / L (1) 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, in 2 (mm 2 ) 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: where σ = axial stress, psi (Pa) ε = axial strain, in/in (mm/mm) E = modulus of elasticity, psi (Pa) σ = E ε (3) 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 materials 2. 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 section is defined by the quantity EI. The modulus of elasticity, E, represents the material Page

4 stiffness; the moment of inertia, I, represents the contribution of the cross sectional geometry to bending stiffness. I is defined as follows: I x 2 = y da (4) where I x = moment of inertia with respect to the neutral axis (i.e., x axis in Figure 4), in 4 (mm 4 ) 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 text 2. For the common case of a rectangular beam with a width, b, and height, h, I = bh 3 /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. For the loading and support conditions shown in Figures 1 and 4, the vertical elastic deflection at the midspan of the beam can be determined from the following equation: where y = elastic beam deflection at midspan, in (mm) P = applied load, lbs (N) L = span length of beam, in (mm) E = modulus of elasticity, psi (Pa) I = moment of inertia, in 4 (mm 4 ) The physical meaning of the equation should be emphasized to students: namely, that the deflection is directly proportional to the load, proportional to the cube of the span length, and inversely proportional to the flexural rigidity, EI. IV. Experimental Procedure All steps of the experiment including the test setup, loading, and data reduction can easily be performed by students. Many variations to the test objectives and procedures are possible. A. Preparing the Test Setup 3 PL y = (5) 48EI As shown in Figure 1, a very simple test setup is intentionally used to facilitate hands-on participation and repeatability as well as to accommodate a low budget. Materials required for test specimens, supports, and loading are easily obtained. Local hardware stores or hobby shops normally have a stock of wood, steel, aluminum, brass, and plastic rods of various cross sectional Page

5 shapes that can be used for testing. Figure 6 shows a small wooden test frame that was easily built for testing a balsa wood beam. Simple support conditions such as those shown in Figures 1 and 6 are recommended for initial testing since such supports, by definition, do not restrict rotation at the ends of the beam and are thus most easily and accurately modeled. Loading devices do not need to be elaborate (Figures 1 and 6). Any material of known weight can be used for loading. A bucket with metal weights is shown in Figure 1, whereas small bags of rice weighed out on a food scale are shown in Figure 6. For testing of stiffer beams, heavier weights are more convenient. Teachers should perform sample calculations ahead of time to determine a desirable magnitude of beam deflection for the assumed test specimen and setup. A ruler or tape should be used to accurately place the load on the beam (e.g., at the beam midspan for Equation 5). Because deflection equations are based on beam lengths, the distance between supports (distance, L, in Figure 4) not the total beam length, should be used in calculations. The moment of inertia for the beam cross section can be calculated from measured beam dimensions and appropriate equations from a mechanics of materials text. Instrumentation for measuring deflections can range from a hand-held ruler to a dial gage. Incremental beam deflections less than approximately 0.1 in. (2.5 mm) will likely require the use of a dial gage (Figure 2), which can accurately measure deflections of in. (0.025 mm) or less and total deflections of 1 in. (25 mm) or more. Such dial gages can be purchased for approximately $30-$60. Students who use a dial gage are often impressed that structures do in fact deflect when subjected to even the smallest load and that accurate measurements are easy to make. On the other hand, students also benefit greatly from seeing the effect of load upon a structure and appreciate the significance of the deflection formula more by visualizing structural behavior. With proper planning, tests can be designed to allow students to experience both aspects. For example, deflections as large as 1 in. were measured for the wooden beam shown in Figure 6 and with an accuracy of approximately in. (0.64 mm) using a small ruler with an engineering scale. B. Loading the Specimen After preparing the test setup, students can conduct the test in teams of three to four persons. Prior to testing, teachers should establish the maximum load that will be used to achieve the desired deflection. Some judgment is necessary to ensure the elastic limit of the material is not exceeded so that the beam remains elastic during testing. The maximum applied load may be divided into five or more increments. For example, if the maximum applied load is 10 lbs, then the load stages might be: 1, 2, 4, 5, 8, and 10 lbs. If a team consists of four persons, one member of the team can be responsible for applying the load, another for measuring the deflection, a third for recording data, and a fourth for plotting the data (load vs. deflection) as the test proceeds. At each load stage, all members should observe the beam response to load and the third member should record any observations made. Teammates should also check each other s accuracy in applying the load and measuring the deflection. Because of the simplicity of the test procedures, each experiment can easily be conducted twice to obtain a level of confidence in the accuracy of testing. Teachers may ask Page

6 students before (of after) loading how much additional deflection is expected when, for example, the load is doubled. This will help students better grasp the significance of the terms in the deflection equation. C. Plotting, Analyzing and Interpreting Test Data Students should plot load-deflection response of the beam as the test proceeds. The interpretation of this plot is readily understood if Equation 5 is rearranged and solved for P as follows: Students should be able to recognize this formula as the equation of a line with a slope of 48EI/L 3 and a y-intercept of zero. The slope represents the stiffness of a structure for the particular case of a simply-supported beam subjected to a concentrated load at midspan. By plotting load-deflection test data, students can: 1) visualize the linearity of the system, 2) determine the beam stiffness, and 3) calculate the actual modulus of elasticity of the material. Test data for the steel beam shown in Figure 1 are plotted in Figure 7. If a computer is available, these results can be conveniently plotted during testing using spreadsheet software. This plot provides a valuable means to quantify beam response and to correlate such response with physical observations. Load-deflection response should be monitored to ensure the beam is not loaded beyond the elastic limit, indicated by a reduction in the slope. After the maximum load is applied, students can also collect data as the beam is unloaded to verify linear elastic behavior and measure any residual displacement. Figure 7 shows a linear regression line plotted through test data, with the y-intercept set to zero. The slope of this line represents the experimental beam stiffness. Tables 1 and 2 show calculations for moment of inertia and experimental modulus of elasticity based on test results. These calculations show an experimental modulus of elasticity within one percent of the commonly published value. The r 2 term is shown to be very nearly 1.0, confirming the linearity of the system. A hand-drawn line will also work. V. Additional Experiments The beam experiment provides students an excellent opportunity to make many interesting comparisons that have practical engineering significance. The following sections highlight several useful experiments beyond that illustrated in the previous section. A. Modulus of Elasticity P 48EI y L = 3 (6) An important aspect of the structural design process is the comparison of alternative structural systems, including systems using different materials. Concrete, steel, masonry, wood, and combinations of these materials as well as others are commonly considered. One simple way for students to appreciate differences in material behavior is to test beams of different materials. Page

7 The effect of different moduli of elasticity on elastic behavior of beams can be demonstrated by conducting beam tests using different materials for the same beam cross section, length, and support conditions. Figure 8 compares load-deflection response for beam specimens of steel, aluminum, and wood. (Actual test data are shown for the steel beam, but hypothetical data, consistent with realistic values of E, are shown for the aluminum and balsa wood beams.) All beams exhibit linear elastic response, but the slopes of the regression lines differ greatly, indicating a significant difference in beam stiffness related different values of E. For a given load, the aluminum beam deflects about 3 times more than the steel beam, and the balsa wood beam deflects an enormous 61 times more. From this experiment, students can: 1) gain a physical feeling for different material stiffnesses, 2) recognize that material stiffness (modulus of elasticity, E) is the sole factor changing the beam stiffness (48EI/L 3 ), and 3) compare experimental moduli of elasticity for the different beams. B. Moment of Inertia Structural engineers often limit deflections and bending stresses by using beams with advantageously-shaped cross sections. A wide flange beam (sometimes called an I-beam) is a good example of this principle. The vertical web of the beam increases the distance between the horizontal flanges and the neutral axis, thereby significantly increasing the moment of inertia. This particular shape has a relatively large moment of inertia for the amount of material used. In addition, the shape resists bending stresses efficiently because the large flange areas are located in the most highly stressed region of the cross section. Equation 6 shows that the beam stiffness, 48EI/L 3, is directly proportional to the moment of inertia, I. Students can easily observe the influence of different moments of inertia by testing a beam with a rectangular cross section. Figure 9 compares the deflection of a ½ in. x 1 in. (13 mm x 25 mm) steel beam for two cases. When the beam is oriented with the long side vertical, the moment of inertia, I x, is calculated as follows: I x = bh 3 /12 I x = (0.5)(1.0) 3 /12 = 1/24 But when the beam is laid flat, the moment of inertia is: I y = bh 3 /12 I y = (1.0)(0.5) 3 /12 = 1/96 It is evident that I x is four times larger than I y. Students can easily conduct both experiments and compare the predicted and experimental deflections based on a change in geometry alone. Students should also try bending the beam specimens about both axes by hand to get a feel for the difference in stiffness related to the moment of inertia. C. Elastic vs. Inelastic Behavior and Failure Modes A particularly important aspect of material and structural behavior occurs when the applied loads are large enough to produce inelastic response and failure. Demonstrating elastic behavior of a Page

8 beam is clearly a simple and appropriate first step in conveying basic structural engineering principles to students. However, inelastic behavior and structural failure should eventually be presented, as these concepts are often more meaningful in the design of structures. In addition, students are usually more excited to test structures to failure. Inelastic behavior can be introduced to students in a fairly simple manner by using a paper clip, as illustrated in the following sample student exercise. Open up a paper clip so that it is a straight wire. Hold one end of the wire down at the edge of a table so that the other end extends out into the air. This represents a cantilever beam. With your finger, gently push down on the free end just a little and then let it go. Notice that the wire quickly returns to its original shape. This is an example of elastic behavior. Now, push the wire down harder so that it bends down permanently. This illustrates inelastic behavior. The beam was stretched so much that a permanent deflection resulted after the load was removed. Finally, bend the wire back and forth until it breaks. This is an example of beam failure following inelastic response. In principle, this is what would happen to a structure that does not have enough capacity for deformability in the event of an overload or earthquake. This approach is preferable to the presentation of complicated formulas. Structures are typically designed with sufficient reserve to avoid inelastic behavior under day-today loads. However, many structures are designed to undergo a certain level of permanent deformation and damage under severe overloading or unusual events such as earthquakes. This deformability of structures in the inelastic range is often referred to as ductility. Some materials, such an unreinforced masonry or plain concrete, exhibit very little ductility. A large factor of safety is thus required to prevent a sudden, brittle failure in the event of overload. A certain degree of ductile behavior, however, is usually more desirable because a greater warning of impending failure is provided and more energy can be absorbed by the system prior to failure. Although permanent damage may result, structures should be designed to prevent collapse and ensure life safety. Figure 10 compares load-deflection response for a beam with a brittle vs. ductile failure mode. The ductile beam initially exhibits elastic behavior, but is followed by a significant amount of inelastic behavior and deflection. This response is achieved because of a ductile material stressstrain response. In contrast, the brittle beam exhibits only linear elastic behavior prior to a sudden failure. After students complete beam testing in the elastic range, specimens can be loaded to failure to observe the full range of elastic and inelastic response as well as the failure mode. Students usually enjoy this part of testing the most! Tests can be conducted in a manner similar to the elastic beam testing, with the following important differences: 1) smaller load increments should be used, especially beyond the expected elastic range, to ensure nonlinearities in beam response are adequately captured prior to failure; 2) load-deflection response should be plotted using a spreadsheet and carefully monitored at each load increment to determine the onset of inelastic behavior, as inelasticities will be noticed from the load-deflection plot before visible signs on the Page

9 specimen, 3) the specimen should be inspected for stretching, cracking, or other signs of damage during loading, and 4) instrumentation such as dial gages should be monitored and removed if sudden failure or excessively large deflections threaten to damage instrumentation. Students should also beware of the possible development of unexpected failure modes during testing, such as lateral buckling of the beam due to compressive stresses on the compression side of the beam. Students should attempt to identify the failure mode and major differences between elastic and inelastic response. An approach to predict the failure load is not presented herein, but can be found in Reference 2. D. Variations of the Beam Experiment Only a few of the many variations of the beam experiment have been mentioned. Students and teachers alike can propose and conduct alternatives that investigate other useful aspects of structural behavior. For example, by inspecting the parameters of the deflection equation (Equation 5) it is evident that beams with different span lengths can easily be investigated, as can deflections at other points along the beam. Support conditions can also be modified. Clamps or vise grips can be used to model fixed-end conditions (i.e., supports which completely prevent rotation). In this way, a cantilever beam or beams with one or both ends fixed can be tested. In addition, multiple concentrated loads can be investigated. Reference 2 provides deflection equations for these and other conditions. A variety of beam cross sectional shapes can also be tested. Round, square, and rectangular shapes are readily available, although other possibilities may be investigated with a little creativity. For example, two channels can be placed back-to-back to represent strong axis (xaxis) bending of a wide flange beam. Simple approaches to calculate the moment of inertia for weak axis (y-axis) bending are available in the event that channels are not welded or bolted together 2. Supplementary exercises may also be developed to illustrate important principles. For example, students can also be challenged to develop the most economical cross sectional shape for a beam, based on a limited amount of material and subjected to certain constraints such as height and width. VI. Summary and Conclusions A simple beam test that can be used to introduce fundamental structural engineering concepts to high school students is described. Background engineering concepts are also introduced to help high school teachers understand and implement the experiment. Based on a simple and inexpensive test setup, an experiment can be performed that not only provides students an opportunity to discover structural behavior in a hands-on format, but also provides a useful springboard for discussion of many important engineering topics such as elastic and inelastic behavior, influence of materials and structural shapes, stiffness, strength, and failure modes. Many variations of the experiment are possible. Page

10 Bibliography 1. URL: Sacramento Engineering and Technology Regional Consortium home page. 2. Gere, J.M., and Timoshenko, S.P., Mechanics of Materials, 2 nd Ed., PWS-Kent Publishing Company, Boston, ERIC E. MATSUMOTO Eric Matsumoto is an Assistant Professor of Civil Engineering at California State University, Sacramento. His areas of interest include precast/ prestressed concrete systems and structural dynamics. He previously worked as a structural engineer in the government and private sectors, and is a registered civil engineer in California. He received a Ph.D. in structural engineering from the University of Texas at Austin, and B.S. and M.Eng. degrees in civil engineering from Cornell University. JOHN R. JOHNSTON John Johnston is an Associate Professor of Civil Engineering at California State University, Sacramento. Prior to coming to CSUS, he held civil and environmental engineering positions in both the public and private sectors, and was a member of the CE faculty at CSU Fresno. He is a registered civil engineer in California, with a Ph.D. from the University of California, Davis, and B.S. and M.S. degrees in civil engineering from Stanford University. Currently, Dr. Johnston splits his time between teaching and researching storm water quality issues for the California Department of Transportation. E. EDWARD DAMMEL Ed Dammel is an Assistant Professor of Civil Engineering at California State University, Sacramento. Prior to coming to CSUS, he worked with the University of California, Davis helping develop a research program in storm water quality management for the California Department of Transportation. He holds a Ph.D. and M.S. from the University of California, Davis and a B.S. from the California State University, Chico. S. K. RAMESH Dr. S. K. Ramesh is a Professor of Electrical and Electronic Engineering at California State University, Sacramento and has been the Department Chair since His research interests are in the area of Optical Communication Systems. He teaches courses in Optical Engineering, Fiber Optic Communications and Analog IC Design. He is a Senior Member of the IEEE and is presently the Central Area Chair of IEEE Region 6. Dr. Ramesh graduated with a B.E. (Honors) degree ( 81) in Electronics and Communication Engineering from the University of Madras, India, and received his MSEE ( 83) and PhD ( 86) degrees from Southern Illinois University, Carbondale. Page

11 FIGURES A. B. Figure 1: A) Test Setup; B) Support Conditions for Simply-Supported Beam Figure 2: Cable Loop Used for Loading Beam and Dial Gage Page

12 Figure 3: SETRC Workshop Participants Recording Beam Test Data vertical deflection, y y x applied load, P span length, L Elevation Cross Section Figure 4: Schematic of Beam Elevation and Cross Section Page

13 Steel Aluminum Douglas Fir Wood E steel =29x10 6 psi E alum =10x10 6 psi 500 E wood =2x10 6 psi Strain (in/in x 10-6 ) Figure 5: Idealized Stress-Strain Behavior for Steel, Aluminum, and Wood in the Linear Elastic Range Figure 6: Wooden Test Frame for Testing of Balsa Wood Beam Page

14 Steel P = (1.342)y R 2 = E EXP =28.8 x10 6 psi Steel Beam Data Linear Regression Deflection, y (in x 0.01) Figure 7: Load-Deflection Response for Steel Beam Tested during SETRC Workshop Steel (actual data) Aluminum (hypothetical data) Balsa Wood (hypothetical data) Deflection, y (in x 0.01) Figure 8: Comparison of Load-Deflection Response for Steel, Aluminum, and Wood Beams Page

15 Ix (strong axis) P = (1.342)y R 2 = Iy (weak axis) P = (0.336)y R 2 = Deflection, y (in x 0.01) Figure 9: Comparison of Load-Deflection Response for Strong and Weak Axis Bending Ductile Response Brittle Response Failure Deflection, y Figure 10: Brittle vs. Ductile Load-Deflection Response Page

16 TABLES Section shape: Rectangular Width, b 0.5 in Height, h 1.0 in Moment of Inertia, I = bh 3 / in 4 Table 1: Calculation of Moment of Inertia for Rectangular Beam Span Length, L 35.0 in Moment of Inertia, I in 4 Beam Stiffness, K (Experimental Slope) lbs/in Experimental Modulus of Elasticity, E = K/(48I/L 3 ) x 10 6 psi Published Value for Modulus of Elasticity, E 29 x 10 6 psi Table 2: Calculation of Experimental Modulus of Elasticity Page