YOUNG S MODULUS AND STRAIN DISTRIBUTION IN A WOODEN BAR USING A STRAIN GAUGE

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1 YOUNG S MODULUS AND STRAIN DISTRIBUTION IN A WOODEN BAR USING A STRAIN GAUGE AIM: (a) To determine the Young s Modulus of the given material (half meter wooden scale ) (b) To investigate the strain distribution on a wooden 1 meter scale. Pre-Lab questions: 1. What is a strain gauge? 2. Why measure strain distribution and Young s modulus? 3. What is the advantage of measuring the above quantities using multiple strain gauges? REQUIREMENTS: Part (a) Part (b) Basic Theory: Young s modulus:- A half metre scale with two identical strain gauges fixed to one end of the scale one strain gauge at the top and the other at the bottomlength wise, a clamp, a circuit board with appropriate terminals to constitute a Wheatstone network, hanger and slotted weights, Digital micro-voltmeter (DPM), Vernier calipers, screw gauge, rheostats, R- Box a constant current source and connecting wires. All the above with a full meter scale with 6 or 7 strain gauges stuck on one side. When an external force, F, is applied along a long bar of length L, (and perpendicular to the cross-sectional area A), internal forces in the bar resist distortion and the bar attains an equilibrium when the external force is exactly balanced by the internal forces with a change in length, L. The tensile stress is force per unit area (in N/m 2 ) and the longitudinal stress is the change in length to the original length and it is dimensionless quantity. The ratio of the tensile stress (F/A) to the tensile strain (L/L) is given by, F/A Y= ΔL/L (1) where Y is the Young s modulus of the bar. Strain Gauge: A strain gauge is a transducer whose electrical resistance varies in proportion to the amount of strain in the device. The most widely used gauge is metallic strain gauge

2 which consists of a very fine wire or, more commonly, metallic foil arranged in a grid pattern. The grid pattern maximizes the amount of metallic wire or foil subject to strain in the parallel direction (Figure 1). The cross sectional area of the grid is minimized to reduce the effect of shear strain and Poisson Strain. The grid is bonded to a thin backing, called the carrier, which is attached directly to the test specimen. Therefore, the strain experienced by the test specimen is transferred directly to the strain gauge, which responds with a linear change in electrical resistance. Figure 1 A fundamental parameter of the strain gauge is its sensitivity to strain, expressed quantitatively as the gauge factor (). Gauge factor is defined as the ratio of fractional change in electrical resistance to the fractional change in length (strain): ΔR/R λ= ΔL/L (2) The Gauge Factor for a metallic strain gauge is typically around 2. Wheatstone Bridge: Measuring the strain with a strain gauge requires accurate measurement of very small changes in resistance and such small changes in R can be measured with a Wheatstone Bridge. A general Wheatstone bridge consists of four resistive arms with an excitation voltage, E, that is applied across the bridge (Figure 2). Figure 2 The output voltage of the bridge, e, will be equal to:

3 (3) From this equation, it is apparent that when R 1 /R 2 = R 4 /R 3, the voltage output e will be zero. Under these conditions, the bridge is said to be balanced. Any change in resistance in any arm of the bridge will result in a nonzero output voltage. Therefore, if we replace R 4 in Figure 2 with an active strain gauge, any changes in the strain gauge resistance will unbalance the bridge and produce a nonzero output voltage. If the nominal resistance of the strain gauge is designated as R G, then the strain-induced change in resistance, R, can be expressed as R = R G strain. (4) Assuming that R 1 = R 2 and R 3 = R G, the bridge equation above can be rewritten to express e/e as a function of strain. Ideally, we would like the resistance of the strain gauge to change only in response to applied strain. However, strain gauge material, as well as the specimen material on which the gauge is mounted, will also respond to changes in temperature. Strain gauge manufacturers attempt to minimize sensitivity to temperature by processing the gauge material to compensate for the thermal expansion of the specimen material for which the gauge is intended. While compensated gauges reduce the thermal sensitivity, they do not totally remove it. By using two strain gauges in the bridge, the effect of temperature can be further minimized. For example, in a strain gauge configuration where one gauge is active (R G + R), and a second gauge is placed transverse to the applied strain. Therefore, the strain has little effect on the second gauge, called the dummy gauge. However, any changes in temperature will affect both gauges in the same way. Because the temperature changes are identical in the two gauges, the ratio of their resistance does not change, the voltage e does not change, and the effects of the temperature change are minimized. The sensitivity of the bridge to strain can be doubled by making both gauges active in a half-bridge configuration (how?). Figure 3 illustrates a bending beam application with one bridge mounted in tension (R G + R) and the other mounted in compression (R G - R). This half-bridge configuration, whose circuit diagram is also illustrated in Figure 3, yields an output voltage that is linear and approximately doubles the output of the quarter-bridge circuit. Figure 3a & b

4 In this experiment we aim to determine the Young s modulus of a half-metre wooden bar by loading it with a mass of m gm. For a beam of rectangular cross-section with breadth b and thickness d, the moment of inertia I, is I = bd 3 /12 (5) The moment of force/ restoring couple is Y.I/R where R is the radius of curvature of the bending beam. The Young s modulus Y is calculated by assuming that at equilibrium, the bending moment is equal to the restoring moment. Source:

5 Wheatstone Bridge Driven By a Constant Voltage We could work with one strain gauge, two strain gauges or four: all four resistances in the bridge could be strain gauges (What advantage does using multiple strain gauges provide?). We will first look at the case when all four arms of the bridge have strain gauges. The other situations can be easily derived from this. Such a configuration is shown below: Figure 4 So if the resistance of a strain gauge R changes by dr, the differential change in output voltage is e e e e de dr dr dr dr R R R R G1 G2 G3 G4 G1 G2 G3 G4 Gi R G2 drg1 dr G2 R G4 drg 3 dr G4 de 2 2 E R R G2 G1 RG 2 R R G4 G3 R G4 RG1 1 RG3 1 RG1 R G3 If the bridge is balance such that R G1 /R G2 = R G4 /R G3, (use fig. 2 as reference with all resistances replaced with strain gauges) we have, R G2 drg 1 drg 2 drg 3 dr G4 de 2 E R G2 RG 1 RG 2 RG 3 RG 4 RG 1 1 RG 1 ΔR G/RG RG2 L1 L2 L3 L4 Since λ=, de 2 E ΔL/L R L G2 1 L2 L3 L4 RG 1 1 RG 1 Most of the time we assume that the two strain gauges have nearly the same resistance. 1 If we have only two strain gauges and if they are mounted on opposite sides of a beam as shown in fig. 3, the strains produced will be opposite to each other. Assuming identical strain gauges, we will assume the strains to be equal and opposite. So we have RG 2 L1 L1 de 2 Which gives 2 E E de straingauge E (6) R L L1 2 G RG 1 1 RG 1 or Gi 1 check and see this if this assumption is valid.

6 Wheatstone Bridge Driven a Constant Current Since strain gauges and the measuring instruments are connected by long wires, if we apply a constant voltage the resistance of the wires will also feature in the bridge balancing and may introduce errors. Therefore instead of using a constant voltage we use a constant current source. This is what you will doing in class. However you must be prepared to use a constant voltage as well. In this case the bridge acts like a current dividing circuit. If the bridge current, I B = I 1 +I 2, where I 1 and I 2 are the current through the arms of the bridge (use fig. 2 as reference again but with voltage sourced replaced by a current source). Analogus to eq. 3 we have, R1 R3 R1 R 4 e I1R1 I2R4 I B R1 R2 R3 R4 So the bridge balance condition is the same as before. The change in bridge ouput due to resistance changes in the bridge arms (assuming all four are strain gauges) is given by R R R R R R R R R R e I B R1 R1 R2 R2 R3 R3 R4 R4 R1 R2 R3 R4 R1 R3 R2 R4 If we have only two strain gauges of nearly equal resistance (R 1 R 2 =R) and if they are mounted on opposite sides of a beam as shown in fig. 3 9also see fig. 6 below), the strains produced will be opposite to each other. Further if the other constant resistances, R 3 = R 4 = P, we have 2 RP R B 2 RP L e I IB 2I RP B straingauge (7) 2R 2P R 2R 2P L 2R 2P The Loaded Beam Consider a beam clamped at one end and loaded at the other end. Let r be the radius of curvature of the cantilever at a point P distance x from the loaded end. AB represents the unstrained neutral layer (Fig.5b & c) at P. CD is a layer at a distance y above the neutral layer. O is the centre of curvature. The thickness of the beam is d = 2y. Fig. 5a Fig. 5b Fig. 5c Then CD AB y d l strain AB r 2r l Beam

7 This represents the strain at P in terms of the radius of curvature at P and the thickness (d) of the beam. The balancing equation for the internal bending moment and the external bending moment is 2 Yak Mgx r Y Young s modulus of the material of the beam ak 2 Geometric moment of inertia of the cross-section of the beam=8by 3 /12=bd 3 /12 for a = bd rectangular cross section. r radius of curvature at a point distant x from the loaded end. Mgx external bending moment. 3 Ybd So we have Mgx 12r 6Mgx and Y 2 bd strain or 6Mgx strainbeam (8) Ybd 2 Beam Assuming that the strain in the beam is the same as the strain developed in the strain gauge, we can find the Youngs modulus 2 and the strain distribution 3 in a loaded beam. These are the two parts of your experiment. We use eq. 6 if a constant voltage source is used and eq. 7 is a constant current source is used. As mentioned earlier we use a constant current source and hence using eq. 7 we have, strain Gauge Const Current e R P I RP and B Y 6gxRI B 2 Re bd 1 PM (9) 2 Use a 0.5 m scale for this 3 Use a 1 m scale for this

8 Procedure: IMPORTANT: The procedure given below describes a Wheatstone s bridge with 2 two strain gauges. You need to set up a Wheatstone s bridge with (a) 1 strain gauge (b) 2 strain gauges (c) 3 strain gauges and (d) 4 strain gauges Calculate the Youngs Modulus in each case and explain your results with error analysis. 1. Clamp the beam to the table in such a way that the strain gauges are close to the clamped end. Load and unload the free end of the beam a number of times. 2. Make the connections as given in the circuit diagram (Fig. 6). P R A Rh Q Rh DPM D R B Figure 6 I B P = 100 resistor; I B = 10 ma current source, DPM = a Voltmeter with digital panel R = strain gauges of resistance ~ 121 ohms with a gauge factor =2.2, Q = 82 + (20 and 10 rheostats) all in series (100 and 82 are connected internally under the board). 3. Switch on the constant current source and the DPM. 4. Balance the bridge using the two rheostats in tandem. At this stage the DPM will read 0 or very nearly zero. Note that this is done with no load. 5. Load the beam with a hanger of mass m gm suspending it as close to the free end as possible. Note the DPM reading. Note that as you are about to take a reading the last digit will be changing about the actual steady value. Take at least 10 readings continuously and take the average of these ten.

9 6. Increase the load in steps of m gm, up to 5m gm and take the readings each time. 7. Unload the beam from 5m down to zero in steps of m gm at a time and note the DPM reading each time. 8. To check reproducibility, repeat all the above processes taking readings while loading and unloading in steps of m gm. 9. Draw a graph between m along X-axis and unbalanced voltage dv along Y-axis. Determine the slope of this graph (dv/m). 10. Note the distance between the center of the strain gauges and the point of application of the load (L). 11. Measure the breadth of the beam using slide calipers (b). 12. Measure the thickness of the beam using a screw gauge (d). 13. Young s Modulus of the material of the beam is given by eq Tabulations Load/gm DPM reading 0 m 2m 3m 4m 5m 1) loading V 1 / mv 2) Unloading V 2 / mv Mean of V 1 +V

10 Part (b) : Study of Distribution of strain in a loaded beam AIM To study the distribution of strain along the length of a loaded cantilever. The strain will be evaluated using strain gauges. Five strain gauges are fixed at various points along the length of the cantilever as shown in Fig.6. The strain at different points can be calculated using eq. 8. and the Youngs modulus of the beam obtained from the first part of the experiment. The strain developed in the strain gauges is measured and compared with the calculated values. Free end Clamped end Figure 6 Measurement of resistance of the strain gauges: Load Pass current from the constant current source through a standard resistor provided. Measure the voltage obtained to determine the value of the constant current. Now pass current through each strain gauge separately, find the voltage drop and hence the resistance of the strain gauges. Now measure the resistance of the gauges directly using a multimeter. Do the values agree? Why bother measuring resistance this way when a multimeter is convenient? Measurement of variation of resistance of the strain gauge when the cantilever is loaded Suspend the hanger(dead weight) at the load point of the cantilever. Form a wheatstone bridge with any one of the strain gauges say SG(5) and two equal 100 Ω resistors and a variable resistor. Use the 10 ma constant current source and the DPM 4 ½ digit 20 mv input digital panel meter (detector) (Digital microvoltmeter). The 10 Ω rheostat, 50 Ω rheostat and R-Box are necessary for fine adjustment for initial balancing of the bridge. You can also use a R-.Box in series with the above two. Initially keep the DPM in 100 mv Range, balance the bridge for a value less than 10 mv. Then set to 10 mv Range, again balance the bridge for a very small value (close to Zero). The minimum voltage at the best balancing condition can be made to be about to mv Now load the cantilever with a mass of 250gms or 100gm(2 slot, each slot is 50gm) and suspend it as close to the free end as possible. The strain gauge gets strained.

11 The unbalanced voltage (due only to the change in resistance of the strain gauge) can be measured from the same DPM. V The change in resistance R of the strain gauge is R ohms I The experiment has to be repeated for all the strain gauges that are stuck on the scale one by one in ascending order from the clamped end. Initial balancing must be done for all strain gauges. Reproducibility of results must be checked by repeating the experiment a few times (say 3 times for each gauge). The results are to be recorded in the (previous) tabular column. Obtain the theoretical strain at the same points, assuming the Young s modulus of the material of the rod. Plot a graph with distances on the X axis and the corresponding strains (both experimental and theoretical) on the Y axis, connect the points by a smooth graph. The graph represents the distribution of the strain at various points along the length of the cantilever. Table II Sl. No Distance of SG From loaded end m V SG volts V SR volts R Ohms R Ohms R R