AERO 214. Lab II. Measurement of elastic moduli using bending of beams and torsion of bars

Transcription

1 AERO 214 Lab II. Measurement of elastic moduli using bending of beams and torsion of bars BENDING EXPERIMENT Introduction Flexural properties of materials are of interest to engineers in many different applications. For example, a bridge structure bends when loaded with traffic. Designers use the modulus of elasticity, which is a material property, to predict the structural deformation associated with the load. We extract material properties from tests performed on small, carefully prepared and handled samples of the material. Solid mechanics lets us design a specimen for extraction of reliable property data. Young's modulus, or the modulus of elasticity, can be measured using different specimen geometries and loads. Tensile extension of cylinders and flexural bending of thin sheets both provide a measure of the Young's modulus. This material property is the same for both tests if the material is isotropic, but for some materials, such as brittle materials, the flexural test is easier to conduct. In this laboratory experiment, you will determine the flexural modulus of elasticity for different materials. Objective To determine the modulus of elasticity for different materials using a flexural bending beam test. To understand how a flexural bending test is performed and how to obtain the relevant material property. Theory A discussion of the mechanical properties of ceramics and flexural testing is provided in Chapter 7 of the Callister text. Figure 1 shows a 4-point bending apparatus used to determine the modulus of elasticity in flexure by applying a load (P) and measuring deflection ( ) with a dial gauge. There is only a negligible upward force from the dial gauge, and this force is NOT shown in Figure 1. For a rectangular beam, experimental values of and P will allow E, the elastic modulus, to be determined from the 4-point bend test. The following equation gives the deflection of the center of the beam: Pa 2 2 deflection (L 4a ) (1) 48EI

2 P h a a a b where P, L and a are as shown in Figure 1; E is the Young s modulus; and I is the moment of inertia. The moment of inertia I for a rectangular cross section is calculated from the equation below: bh I (2) 12 where b and h are given in Figure 1. L Figure 1. Schematic of 4-point Bending Beam Set-up The following expression may be used for a 4-point bending setup (Figure 1) that has equal distances (a) between the loading points and the supports; i.e., when a is L/ as shown (ASTM method D6272): 0.21L m E () bh where m is the slope of the load P (vertical axis)-displacement (horizontal axis) plot from experimental data. The assumptions are that 1. the dial gauge spring exerts no (negligible) upward force 2. the mass of the top sample is negligible. the deformation is elastic (linear relationship between load and deformation) Every experiment has controlled inputs, measured outputs, and parameters that must be known. In this bending experiment, we have Controlled Input: P Measured Output: deflection Parameters: b, h, a 2

3 Materials Your kit contains the following: 4-point bend fixture, dial indicator, weights, micrometer, and test specimens of different materials. The dial gauge is a simple and inexpensive indicator of displacement. The gauge contains a spring that keeps the tip of the gauge in constant contact with a surface so that the relative position of that surface is known. This spring force counteracts the applied load. Hopefully, the force is small and insignificant. Figure 2 is a picture of a similar set-up. Figure 2. The 4-point bending apparatus has a dial gauge that indicates the displacement at the center of a loaded slender beam. Test Procedure 1) Find the four strip specimens for your team and identify the materials. 2) Measure the dimensions necessary for analysis (b, h, and L) ) Measure the deflection as a function of load in a 4-point bend experiment using several different masses a) Create a graph of load vs. displacement. b) Make sure the deflection is linear with respect to load to ensure the equations for the elastic analysis apply (Equation ). c) Repeat the experiment so that you can determine the scatter in the results. See section 7.19 for more details on standard deviation computations. Reporting Requirements 1. Plot load vs. deflection for each material 2. For each material, determine the modulus of elasticity using Equation. Estimate the scatter in your data standard deviation. Also, if you fit a line to the data with Excel or other software, what are the R 2 value and the equation for the line? What does the R 2 value tell us? 4. Discuss the errors in the modulus of elasticity caused by the small force against deflection from the dial indicator (does the error increase or decrease the Modulus?) 5. Compare the calculated values of modulus of elasticity values with those reported in the literature (Appendix in Callister for example)

4 6. Identify and describe another application for any material where the property determined in this experiment is needed (and a flexural test might be easier to conduct) 7. Prepare results as a technical report (Format: see Handout; Maximum Length 5 pages plus tables and graphs). Be sure to address all reporting requirements. 8. Please include a comment about the connection between the modulus of elasticity, which is a measure of material stiffness, and the bond energy curves that we have discussed in class. Note: Those who miss the lab or do not contribute to the lab report will not get credit. 4

5 TORSION EXPERIMENT Introduction Members that transmit torque, such as shafts of motors, drive shafts of automobiles, the shafts that connect the turbine to the compressor in a jet engine, etc. are commonly circular in cross section. However, the torque-carrying components of aircraft are rarely circular in cross section. You will learn how to find the stresses and strains in torqued structures of circular cross section in AERO 214. You will study non-circular cross sections in AERO 04 and 06 and in your technical electives pertaining to materials and structures. Figure 1. A typical torsion test set-up. Objective To determine the shearing modulus of elasticity for different materials using a torsion test. 5

6 To understand how a torsion test is performed and how to obtain the relevant material property. Theory In AERO 214 you will derive the following equation for the relation between the torque T, and the angle of twist,, for a bar T k where the torsional stiffness is given by GJ k L and G is the elastic shearing modulus, L is the length of the bar between the support and the applied torque, and the polar moment of inertia J is given by J r 2 4 for a solid circular cylinder of radius r, and 4 4 ro ri J 2 for a hollow circular cylinder of inner radius r i and outer radius r 0. For a solid square cross section of width b, J xx J yy b 6 4 For hollow shafts of square cross section with inner and outer dimensions a and b, J xx J yy b a 6 The torque is applied by hanging a weight W from a radial distance R from the axis of the cylindrical rod: T WR For this experiment: Controlled input: W Measured output: Parameters: R, r, r, r, L i o 6

7 AERO 214 Technical Laboratory Report Format Each group submits one report. The following sections must be included in each Technical Laboratory Report. 1. TITLE PAGE This is the cover page and should present the report in a professional manner. The following should be included: a. Title of Report b. Authors (Team #, Team Leader(s) for this report, other Team Members) c. Date 2. TABLE OF CONTENTS This page should include all sections of the report, figures, tables, and appendices and their corresponding page numbers.. ABSTRACT This is a brief summary (6-10 sentences) of what was done, what was found, and why it is important. 4. INTRODUCTION This section provides motivation and a general summary of the experiment. definitions, theory, and background literature are included in this section. Any 5. EXPERIMENTAL PROCEDURE This section describes step-by-step the procedures used to conduct the experiment(s). If an ASTM standard test method was used, only the standard test number needs to be cited along with any deviations from this standard method. 6. RESULTS This section includes summary data in tables and figures used in the analysis (next section) and text to describe these tables and figures. Each table and figure must have a number and be cited in the text. This section does not include raw data. 7. DISCUSSION In this section, results presented in the previous section are analyzed and discussed in response to the reporting requirements. Additional figures may also be added. Any problems encountered in the experiment should also be highlighted as well as the effects of these problems on the results and analysis and methods by which to avoid these problems in the future. 8. CONCLUSIONS AND RECOMMENDATIONS This section contains summary conclusions based on the analysis, positive aspects of the experiment in terms of learning, and recommendations for improving and/or expanding the experiment. 9. APPENDICES: This section (if necessary) contains specific calculations and raw data. 7