Mechanical Properties Rev 5.0

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1 McMaster Faculty of Engineering Hamilton Ontario Canada Materials 2H04 Measurement and Communications Introduction Mechanical Properties Rev 5.0 Tasks 5, 6 Lab tasks 5,6 link closely to the course 2P04 Mechanical Properties which you will also be taking in Term 1. Make good use of this lab to help you understand the terminology, concepts and methodology. The lab is included in the Materials course to help you understand how to make the most efficient use of structural material. The lab also links to the choice of materials using software such as the Cambridge engineering selector which you explored in Engineering 1M03. For this course the lab is a further exercise in measurement in an engineering system, making use of a data logger and the LabVIEW software. Outline Task 5 Measurement of strain in a cantilever beam Location: JHE 247 three different materials OR three different cross sections. Task 6 Simulation of stress and strain in the beam using FEMLAB Location JHE 238 simulation results are compared with measurements from the beam. Expected Learning Outcomes Task 5 o o o o Be able to use strain gauges arranged in a full bridge to measure normal strain. Be able to calculate bending moment and normal stress in the beam. Be able to calculate the elastic (Young s) modulus for the beam materials Be able to recognize, observe, and account for noise in data sets Task 6 o Be able to set up a finite element simulation and interpret the stress and strain profiles in the cantilever beam. Advanced Learning Outcomes o Be able to select a material for a cantilever beam to meet certain design criteria. o Understand the dynamic capability of a beam and data logger and analyze how the frequency is changed by varying the mass. o Understand the concept of section modulus and its application to the efficient use of materials in load bearing applications. Materials_2H04_2005_Lab_Task5_6_v2.doc July

2 Materials 2H04 Measurement and Communications page 2 Reporting for Assessment Individual reports (one report for Task 5 & 6) to be submitted by the due date, as notified on the class timetable on the course web page (usually 1-2 weeks from completion of the tasks) General guidelines for presentation of reports are given on the course web page. Safety There is minimal risk with this lab, but students should still observe and comply with safety signs. Safety glasses should be worn in labs where the signage requires it, and at any times when there is a potential hazard. The Cantilever Beam In this lab you will use a cantilever beam with a support and an applied load to measure the elastic modulus of the material in the beam. Length is a scalar quantity, but engineering requires that we work in three dimensions. It is good practice to always define a frame of reference, as shown in Figure 1. A frame of reference comprises an origin and three (usually orthogonal) directions. Note the X-Y plane, which will be used in the FEMLAB simulation. Y Z X SUPPORT. Figure 1: Frame of Reference for Cantilever Beam APPLIED LOAD This lab assumes that you already understand the concepts of normal stress, shear stress, normal strain and shear strain. See Callister pp to refresh your understanding. The loads applied to the beam used in this lab will be kept within the limit of elastic deformation of the beam. Most engineering design requires the materials to be used within the elastic range of deformation. See Callister pp , p 143. Measurement of Young s Modulus Young s modulus is measured in a three-step sequence using a simple cantilever beam with an applied load. 1. The tensile stress at the top of the beam for a known load is calculated (Equation 1). 2. Strain is measured by strain gauges bonded to the top and bottom surfaces of the beam.

3 Materials 2H04 Measurement and Communications page 3 3. The stress-strain graph is plotted from data obtained by applying several different loads and measuring the strain using a strain gauge. The slope of the stress/ strain graph is the modulus of elasticity. (Equation 2) Cantilever Beam Stress Equation The tensile stress in a simple cantilever beam can be calculated analytically. When a load (Fv) is applied at the end of the beam, the tensile stress (σ x ) along the x-axis at the top surface is given as: where: σ x = M I c M = bending moment (units of N-m). M is the product of the effective beam length and the force (Fv) applied at the end of the beam. c = distance from neutral axis of beam (m). For rectangular geometry, c = h/2, where h is the beam thickness. I = moment of inertia for a cross section of the beam (units are m 4 ). The moment of inertia is I = 12 3 bh. (1) Elastic Modulus The quotient I/c is called Sectional Modulus of the beam. Fv = load (N). (Remember 1 kgf = 9.8 N) L = effective beam length = distance between the point where the load is applied and the center of the strain gage (m). b = beam width (m). t = beam thickness (m). The elastic modulus (E) of a material, usually known as Young s modulus is the ratio of stress to elastic strain. E = Δσ Δε Units of elastic modulus are N/m 2 : (or MPa) which is the same as units of stress The modulus of elasticity (E) is an index of the stiffness of the material. When a load (that is, force) is applied to metal such as an aluminum alloy or steel, the strain of the material changes linearly as a function of the stress over a certain, usually small, range. Within this range, the beam exhibits elastic deformation (that is, elastic or reversible strain). When the deformation is not permanent, it is called elastic deformation. Thus, when a force is applied, the beam will bend downward and elongate. When the load is removed, the beam will return to its original shape. (2)

4 Materials 2H04 Measurement and Communications page 4 If in the experiment you load the beam beyond its elastic limit, or yield stress, the beam will not return to its original shape when the load is removed. The elastic deformation will be recovered, but there will be a permanent plastic deformation. Elastic modulus varies widely for the range of materials, as shown in Figure 2. Metals have a relatively high elastic modulus compared to other engineering materials. If you look closely at the shape of the regions for steel, and aluminium alloys you will see metals can be strengthened with little change in elastic modulus. Timber on the other hand gets stiffer as it gets stronger. Figure 2: Elastic Modulus for a Range of Engineering Materials Timber is interesting in that it is highly anisotropic, with properties in the direction of the grain being nearly an order of magnitude better than across the grain. Metals can exhibit a small amount of anisotropy if the microstructure develops a preferred orientation during mechanical deformation, but for most purposes metals are isotropic. Optional: Callister p has a design case study that considers 5 different materials for a torsionally stressed shaft.

5 Materials 2H04 Measurement and Communications page 5 Design of Strain Gauges Strain gauges are used as sensing elements in displacement or load measurement systems. When arranged in a Wheatstone Bridge they can be used to measure extremely small deformation 1E-6, and have a maximum strain limit of about 3%. Figure 3 Strain gauges, used to perform measurements. E.O.Doebelin Measurment Systems Application and Design McGraw Hill New York See also Pressure and Strain Handbook Omega, Strain gauges are made of fine wire with cross-sectional area (A), initial length (L), and resistivity (ρ). The resistance of the wire changes when the wire is stretched or compressed due to strain in the substrate to which it is glued. The resistance (R) of the strain gauge wire is simply R = ρ L A It is common to use a gauge factor (G) to describe the changes in resistance (ΔR) of the strain gauge due to changes in length (ΔL) of the wire. The gauge factor, which is supplied by the manufacturer, is also used to compare various strain-gauge materials: Gauge factor = G = ΔR / R ΔL / L where ε = ΔL L is the strain Strain is dimensionless so it may be measured as (in/in) or (mm/mm). Sometimes the term μstrain (micro-strain) is used, meaning 10-6 in/in. Figure 3 shows some typical strain gauges used to build instruments and carry out measurements. Mounting Strain Gauges The strain gauges are bonded with glue to the surface of the beam. By mounting strain gauges as shown in Figure 4 you can measure the axial strain ε x in the beam when a vertical load (Fv) is applied to the end of the beam.

6 Materials 2H04 Measurement and Communications page 6 Figure 4: Cantilever beam with strain gauges mounted on the top and bottom. Note: The location and orientation of each strain gauge is determines if axial, shear, torsion, or bending strains, or any combination thereof are measured. Bridge Circuits Usually, strain gauges are connected in a Wheatstone bridge circuit as shown in Figure 5. The change in resistance of the strain gauge due to an applied force can be measured as the output voltage of the Wheatstone bridge circuit. A simple quarter-bridge circuit has a single strain gage and three fixed resistors. A better signal is obtained with a full bridge with 4 strain gages which is balanced when R 1 R 3 = R 2 R 4 For a full bridge: ΔV / V bridge = ΔR / R = gauge factor * strain Figure 5: Wheatstone Full Bridge with Strain Gauge The bridge voltage, V bridge, is measured across the V exc ground pathway. In the equipment you will use the voltage is fixed at 5V. Data Logging In this lab you will use a data logger to capture data from a cantilever beam equipped with strain gauges. While the use of a data logger for static measurements is a somewhat trivial application, you can explore the application to dynamic measurement by making the beam oscillate.

7 Materials 2H04 Measurement and Communications page 7 Task 5 Measurement of Elastic Modulus Objectives Task 5 To measure the strain in a cantilever beam using a strain gauge bridge and a prewritten LabVIEW program (Strain_Stress.llb). Equipment A computer with National Instruments LabVIEW software National Instruments Data Acquisition (DAQ) board (inside computer), cable, and a connection block. Three cantilever beam of different materials (brass, aluminium and wood) with mounted strain gage connected to Connection Block. System for applying force to the beam. A simple system of a beaker part filled with water will be used. The mass is measured using a top loading balance. Figure 6: Screencap of LabVIEW Environment

8 Materials 2H04 Measurement and Communications page 8 Lab Procedure Task 5 1. Run LabVIEW Open Strain_Stress_dsc_v6.vi, posted in C:\windows\Desktop\ 2. On the front panel of the Strain_Stress_dsc_v6 set: Sample rate = 1000 [samples / s ] Number of Samples = 100 (data logger continuously stores 100 data points in a buffer when the stop button is pressed, the buffer is written to file. Amplifier gain: 1000 (default) Record here the data displayed in the front Strain-Stress VI panel: Value Units Strain Gage Factor Resistance Tolerance Note that the bridge circuit type is a FULL BRIDGE (4 strain gauges) 3. In the LabVIEW window, use the beam selector to select the beam you wish to study; the numbers go in order from left to right on the beams even if the ones you are using do not match the qualitative description on the drop-down menu. 4. Balance the Bridge a) With applied weight = 0 (no additional mass on the beam), press the run icon on the top left of the menu. Record the bridge output in mv. Balance the bridge by entering the zero load output value in the offset window. Check that the strain output displays a value < 0.05mV. NOTE There will always be some level of noise in your bridge output the aim is to minimise it. Unstable or non reproducible readings may result from poor connections in the system or other faults. A good experimentalist will always check for reproducibility in the readings. NOTE If for any reason you change beams, or reset the computer, this step to balance the bridge must be repeated. b) Press the STOP button in VI panel to calculate an average reading from 100 data points. Save your data to a disc. 5. Measure the bridge output with a loaded beam: Choose a mass between 100 and 600 gm and weigh the entire mass and hanger on the balance. Record the mass. Note that you are not weighing the beam should you? NOTE: Choose increments in mass of approximately 100g, but make sure you record the actual mass and use the actual mass when you plot your data. Hook the mass on the end of the cantilever beam. Be sure that the units you use are consistent! Be sure the mass is not swinging. When the mv output is stable, press the STOP button in VI panel. NOTE: If there was noise in your bridge output it may well continue. 6. Repeat step 5 for at least five more weights. Make sure you get values across the range of 0 to 600 g. DO NOT EXCEED 650 g TOTAL!

9 Materials 2H04 Measurement and Communications page 9 7. Take one last measurment with no weight on the beam. The mv output should be within ±0.02mV of the original zero value. Press the STOP button. Note: If you followed the lab instructions carefully, you did not exceed the elastic limit of the beam. Any change in the zero load strain of the beam can therefore be attributed to drift in the bridge circuit. Any difference in the two no-load readings should be noted in your lab report. 8. Repeat steps 3 to 7 for each beam. 9. When finished gathering data, press the Save and Quit button on the front panel. Exit LabVIEW. Retrieve your labelled floppy disk (if used) for data analysis. 10. For each of the beams, using an engineer s steel ruler, measure the width, thickness, the cantilever beam length, the load point and the strain gauge locations to calculate the effective beam length (distance from the point where the mass is applied to the center of the strain gauge). Record the values in the table below 11. Clean up your area. Table 1: Beam 1 Data BEAM IDENTIFICATION Width (m) Thickness (m) Beam length (m) Effective beam length (m) Load point (m) Mass Bridge mv Max Stress at Gauges Strain kg 0 0

10 Materials 2H04 Measurement and Communications page 10 Table 2: Beam 2 Data BEAM IDENTIFICATION Width (m) Thickness (m) Beam length (m) Effective beam length (m) Load point (m) Mass Bridge mv Max Stress at Gauges Strain kg 0 0

11 Materials 2H04 Measurement and Communications page 11 Table 3: Beam 3 Data BEAM IDENTIFICATION Width (m) Thickness (m) Beam length (m) Effective beam length (m) Load point (m) Mass Bridge mv Max Stress at Gauges Strain kg 0 0 OPTION: Explore the dynamic capability of the data logger. With a 400g mass on the beam, make the beam oscillate vertically. Observe the response and capture a data set with the data logger. Now repeat the experiment with a 200g mass and capture a second set of data. You will need to change the sample rate and number of samples to record the time decay curve note the settings you use. Table 4: Sampling Data Sample Rate Number of Samples

12 Materials 2H04 Measurement and Communications page 12 In-Lab Questions: 1. Is there noise in your voltage data? Is it periodic? What are some causes of noise in this experiment? Discuss. Report Task 5 Tasks: Determine the beam stress and strain at the location of the strain gauges, and calculate the elastic modulus for the material. Explore cross section and dynamics. 1. Prepare a dimensioned sketch of the beam noting the frame of reference details in 2. Figure 1. Use (x,y,z) notation to locate the strain gauges and the load. 3. Calculate the stress for each load applied using equation 1. You may choose to do this using Excel, by using setting up the formula, or simply calculate it by hand and enter it into the Excel table. 4. If you have noise in your measurements, create a table from your data, and calculate the mean ΔV for each load. 5. Calculate the strain for each load applied using the gauge factor (see the output screen). You may choose to do this using Excel, by using setting up the formula, or simply calculate it by hand and enter it into the Excel table. 6. Prepare a chart of stress vs. strain and use the Excel functions Add Trendline to perform a linear regression on the data. Look carefully at the trendline. Hooke s law gives a straight line plot through (0,0). Does the trendline go through (0,0)? If not, why not? Look at the data points are they close to the trendline? If not, why is there scatter? 7. Determine the modulus of elasticity from your graph. Use an appropriate number of significant figures in your calculation. Compare the value of the modulus of elasticity with published data. (See Callister Appendix B2 or Femlab data bank) 8. OPTION Process the dynamic data, presenting a static result, and the two dynamic results on the one chart. Calculate the frequency of the dynamic oscillations, and estimate the time to decay to half the peak value. Clearly label your chart. 9. Prepare dimensioned sketches of the different beam cross sections. Note mass per unit length. Rank the beams for material use efficiency, from worst to best. 10. Include your charts, properly formatted and labelled in your report. You TA will show you an example chart. Note: Make sure you label units for each variable. If the units are not consistent, you will not get the right modulus of elasticity. Show all your calculations, including conversion of units. Work in SI units (that is, kg, m, sec, N). Note: It is important to understand the concept of CALIBRATION. In this lab we have a simple cantilever beam (with strain gauges) which can be used as a weigh scale once it is calibrated. We calibrate the beam with several known weights and produce a calibration chart plotting strain gauge reading against known weight. When we have an unknown weight, we can use the beam and the calibration chart to work out the unknown weight.

13 Materials 2H04 Measurement and Communications page 13 Discussion point: How many known weights do you need as a minimum? Discussion point: Could you use the beam to measure the weight of an object greater than the wieght of the largest calibration weight? Discuss. Questions Task 5 1. If the strain gauges were mounted half way along the beam, would you calculate a different elastic modulus? 2. If the strain gauges were mounted at 90º to their present orientation in the X-Y plane could you reasonably use that configuration to determine the elastic modulus? 3. If the load was increased to the point of exceeding the yield stress by 30%, where would the first yielding occur, and what would the strain gauges tell you? 4. OPTIONAL: What was effect of increasing the mass in the dynamic test? Could you estimate the maximum dynamic stress from the strain gauges and compare it to the static value? Why did the oscillation decay? How might you increase the damping? NOTE: I made up the following questions because we discussed similar things. I m not sure of all the answers, though I ve a vague idea. If they re good questions, that s great, otherwise I shall just insert some from Callister somewhere. 5. What factors can cause noise in the data for this experiment? How can they be detected and compensated for? 6. If you had a faulty resistor in the bridge circuit and only two replacement resistors, how could you most efficiently determine which resistor was faulty? 7. List three good applications for strain gauges. Note: when strain gauges are used for load cells to measure the weight of industrial storage bins, dynamic factors can seriously degrade performance. Manufacturers, on the other hand, build electronic damping in to compensate for higher frequency oscillations; however, if the bins are large, oscillation frequency can be very low, and measurements may be compromised. Callister Questions: 6.19, 6.51

14 Materials 2H04 Measurement and Communications page 14 Task 6 FEMLAB Simulation - Cantilever Beam Rev 6.0 Objectives Task 6 To demonstrate the use of finite element simulation. To examine the stress and strain in a cantilever beam Option: Compare simulation results with experiment. Option: Select a material for a given beam to meet a design specification. Finite Element Modeling The response of physical objects to applied loads can be studied using Finite Element Modeling. In many cases a static 2-dimensional simulation of the stress in an appropriate plane of the object is sufficient for the study, but more complex 3D and dynamic simulations are possible. A model is built by first preparing a mesh with a series of nodes and interconnecting elements that represent the dimension and mechanical properties of the material. The intersection points of the mesh act to discretize the beam, and represent it as a finite number of elements. When a load is applied to the object the displacement of the nodes is calculated by using numerical solvers for the differential equations that describe the system. When a calculation is performed on the beam, such as force, the force is calculated at each node. In this way, the calculation becomes a form of integral over the area of the two-dimensional beam, or, if a three-dimensional model was used, a form of integral over the vollume. The forces at each node and the deflection are calculated in an iterative manner until a stable solution is found. Mesh Beam Figure 7 Example of the Finite Element Mesh Used to Solve the Cantilever Beam In this lab we will use FEMLAB to simulate the cantilever beam and study the stress and strain profile in the beam. FEM simulations typically generate results in the form of 2D and 3D plots of a particular property such as stress or strain. A colour scale is frequently used to indicate the magnitude of a given variable or property.

15 Materials 2H04 Measurement and Communications page 15 Experimental Procedure Task 6 FEMLAB software (Version 3.1i) will be used to demonstrate modeling stress and strain in a cantilever beam. The base case example is shown in Figure 8. It is the same as one of the beams used in Lab Task 5. The beam is 0.250m long and has a vertical load ( kg) applied m from the free end. The material is aluminium. Z Y X Figure 8: Cantilever Beam You will work through a sequence of seven steps to build the base case model. Select the class of problem to model (2D Plane Stress, Static) Define grid and axis settings (choose an appropriate grid and scale) In draw mode specify the subdomain (beam geometry and dimensions) Define conditions at the boundaries. (free, constrained or loaded) Create of the finite element mesh. Solve model. Specify what results to display and how to display them. (post-processing).

16 Materials 2H04 Measurement and Communications page 16 Task Run Femlab v3.1i For the cantilever beam we will solve it using structural mechanics, assuming plane stress and using a 2D static analysis. 1. Start FEMLAB and open the Model Navigator screen shown in Figure 9, 2. Expand directory to locate Structural Mechanics, Plane Stress, Static Analysis. 3. Check that the Space Dimension frame is set to 2D. 4. Press OK to confirm the selection and exit the Model Navigator. Figure 9 FEMLAB Model Navigator 5. FEMLAB will now display the main menu shown in Figure 10. Locate the following menu items to note are EDIT, OPTIONS, DRAW, PHYSICS, MESH, SOLVE, POSTPROCESSING icons which select the display mode are from left to right DRAW, POINT, BOUNDARY, SUBDOMAIN, MESH, POSTPROCESSING Figure 10 FEMLAB - Main Menu In the following tasks you will now define the beam and load.

17 Materials 2H04 Measurement and Communications page 17 Task Options and Settings FEMLAB can be used to study systems as large as a bridge, or as small as a micromechanical (MEMS) system. Your next task is to set the scale of the view window and the grid. Note that FEMLAB uses SI units. 1. Under Options, Axis / Grid settings set the values according to the following table. AXIS m GRID m X min X spacing 0.01 X max 0.30 Extra X Y min Y spacing 0.01 Y max Extra Y Table 5: Settings Task Draw Mode You will now dimension, locate and name the beam you wish to study. 1. Choose draw mode, select Rectangle. Click and drag to draw any rectangle. 2. Double click inside the rectangle to show the dialog box. 3. Set the exact dimensions and position of the beam. (3.18mm thick, by 250mm long). Note: take care here with the terms width, height and length. You are building a 2D model. The size of the beam in the third dimension is set later as the constant B. 4. Enter the name BEAM - dsc where dsc is replaced by your initials. 5. Press Apply and OK. The beam should appear as shown in Error! Reference source not found.. Figure 11 Rectangle Dimensions Dialog

18 Task Materials 2H04 Measurement and Communications page 18 Force Application Now you must create the element where the vertical load will be applied. Whilst we think of the load as a point load, FEMLAB must analyse it as a distributed load over a small element. We will use 1 mm length to avoid FEMLAB calculating an infinite local stress for a point load. 1. Select point, from point icon drawing menu. 2. Place a new point anywhere near the base line of the rectangle. 3. Double click on the point and then define the precise location by entering the data for point 1 from Table Add the second point by repeating steps 1,2&3. 5. Zoom in to check location of the points. The applied load acts on an area defined by the distance between the points x B. (Note that instead of four boundaries, the model now has 6 boundaries. The Load is applied over boundary 4.) 6. Now is a good time to save your model. Please use a filename format 2H04_surname. Remember to save again at different stages as you progress. Task Edit Constant : AXIS POINT 1 POINT 2 X Y Table 6: Location of Points To make it easier to carry out several simulations for different materials without changing the model, a series of constants will be created using the parametric mode. 1. From the Options menu, select Constants. 2. Define the following names for the constants and enter the values or expressions given in Table Press Set and Apply to view your entries NAME OF CONSTANT EXPRESSION PHYSICAL MEANING g 9.8 gravitational constant LOAD applied load (down is negative) LF 1E-3 length of element supporting the load ( ) B width of beam in Z direction H 3.18E-3 thickness of beam in Y direction AREA LF * B area of element supporting the load Table 7: Constants and Expressions

19 Materials 2H04 Measurement and Communications page 19 Task Boundary Mode: FEMLAB models the forces and strain within the beam. It is now necessary to define the conditions at each boundary of the beam. Most boundaries are free to move in response to the load applied near one end of the beam, but one end of the beam is supported by a rigid mount. The boundary condition for that face must be set such that the elements in that end will not be allowed to move. 1. Select Physics then Boundary Setting. 2. Select boundary 1, choose constraint, check Rx, Ry and enter the Rx, Ry = 0 3. Select boundary 4, choose load and enter the expression for Fy given in Table Check edge load defined as force per unit area. RESTRICTION/ELEMENT Load in X direction (FX) Constraint in X direction (RX) 0 Load in Y direction (FY) LOAD*g/AREA Constraint in Y direction (RY) 0 Table 8: Boundary Settings Task Subdomain Mode In the following steps you will select specific materials and then you will assign the physical properties to your geometry. The material properties are loaded from the material library. From the Physics menu, select Subdomain Settings. Select Load then choose the material you are modeling (Aluminium or Steel or American Red Oak) from Library 1. Press OK.

20 Materials 2H04 Measurement and Communications page 20 Task Mesh Mode 1. Select mesh mode and initialize the mesh. The program will automatically create a subdomain. By default the mesh will be created with triangle elements. Another type of elements with a different number of nodes can be selected in Properties Physics. Advanced parameters can be set up in Mesh pull down menu. Task Solve Problem 1. Select Solve the problem from Solve menu. Typically this will take 10 seconds. Task Post Mode Chart 1: This chart will show the vertical deflection of the beam. The data will be presented as both a deformation plot and a color coded surface plot. 1. Select Plot Parameters from the Post Processing menu. 2. Select the General tab, check the Deformed shape plot and Surface plot boxes. 3. Enter the title for your chart which includes you name and the material. 4. Select the Deform tab and choose Scale Factor equal 1.0 in Scaling options to get a realistic deformation graph. 5. Select the Surface tab, then select y displacement and expression (v).

21 Materials 2H04 Measurement and Communications page Press OK. The output should appear as shown in 7. Figure 12. Ask each team member to show 8. Figure 12 for a different material. Compare the colors and deflections. Are they the same? Is this what you would expect? 9. Decide if you are to change any scaling factors, then print Chart 1 for your report.

22 Materials 2H04 Measurement and Communications page 22 Figure 12: FEMLAB Plot of vertical (Y) displacement of the beam. Figure 13: FEMLAB Plot of Normal Stress in X Direction Task Check your Results The following equations can be used to review your simulation using FEMLAB working with Finite Element Model (FEM). The symbols relate to the following picture:

23 Materials 2H04 Measurement and Communications page 23 Figure 14: Strain Gauge Location and Symbols The strain at any position is given by Hooke s Law: σ = E ε where E is the young modulus of the beam material. σ and ε are the stress and strain respectively. The maximum axial stress is given by the bending moment (M) at the section beam divided by the section modulus, (I/c) where I is the moment of inertia of the beam and c is the distance from the neutral fiber at the point measured. σ = M I / c I = 1 12 B H 3 B and H are the width and high of the section beam respectively. For a rectangular geometry c = H/2 The bending moment at a distance x from the free end for a beam fixed at one end only (cantilever) with a concentrated load F at any point, applied at a distance a measured from the free end, is given by: M = F ( x a) Finally the maximum deflection at the free end is given by, 2 F b Δ MAX = 3 6 E I ( L b) Beam deflection

24 Materials 2H04 Measurement and Communications page 24 Figure 15: Beam Deflection The following Table presents the analytical solution, for F = 0.6 kg P E B H l L a X2 I I/c M σ strain Δ max kg Pa mm mm mm mm mm mm mm4 mm3 N mm N/mm2 mm E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 9: Analytical Solution for F= 0.6 kg

25 Task Materials 2H04 Measurement and Communications page 25 Explore FEMLAB Take time to explore the options available and to get the plots and answers to the questions below. FEMLAB is very powerful and lots of different information is accessible if you seek it. Notes: 1. Many plotting packages use auto scaling which suppresses much good comparative information between different plots. This is accessible under post mode, plot parameters, range. You can switch it off if you wish, but you will have to set the max and min values manually. 2. You can hold the mouse button down as you point to a part of the beam and get the local values of stress, strain etc. Click near the top of the beam, then near the bottom of the beam note the change in sign of the stress or the strain. 3. Point with the mouse to get the y displacement where the load is applied 10mm from the end of the beam. Note which material you are using. Repeat for all three materials and compare. Report Task 6 1. Look at the first principal stress (stress x) and the first principal strain (strain x), both of which are in the X-direction. What is the difference between the top and bottom gauge locations? 2. Look at the second principal stress (stress y) and the second principal strain (strain y), both of which are in the Y-direction. Would you align a strain gauge in the Y-direction? 3. Prepare a single graphical output of first principal strain. Label it (by hand if necessary) with material, load, dimensions, your name and date. 4. Compare the analytical and experimental values for maximum deflection, and the stress and strain in the beam at the position where the strain gauges are mounted. Calculate the percentage error.

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