MET 301 EPERIMENT # 2 APPLICATION OF BONDED STRAIN GAGES 1. Objective To understand the working principle of bonded strain gauge and to study the stress and strain in a hollow cylindrical shaft under bending, torsion and combined bending and torsion. 2. Resistance Strain Gage The underlying concept of the strain gage is very simple. In essence, an electrically-conductive wire or foil (i.e. the strain gage) is bonded to the structure of interest and the resistance of the wire or foil is measured before and after the structure is loaded. Since the strain gage is firmly bonded to the structure, any strain induced in the structure by the loading is also induced in the strain gage. This causes a change in the strain gage resistance thus serving as an indirect measure of the strain induced in the structure. A typical strain gage is shown in Figure 1. The orientation of the grid defines the strain sensing axis of the strain gage. Electrical connections are made by soldering lead wires to the strain gage "solder tabs." The entire strain gage is bonded to a thin polymeric backing which helps protect and support the delicate metal foil. Compression causes resistance to decrease Figure 1. A typical bonded strain gage dl The normal strain,, where dl = change in length and l = original length. Since the l electrical resistance (R) is proportional to the length (l) of the conductor R l, then dl dr, where dr is the change in resistance and R is the original resistance of the gage. l R The proportionality constant of the strain gage is called the gage factor (GF): Thus, dr dr GF, or,.(1) R R GF The strain gages used in this experiment is MM Type WA-06-250WT-120 with GF = 2.05 and resistance R = 120 ohms 0.4%. These measurements are supplied by the manufacturer. 1 of 10
Consider the magnitude of the resistance change which must typically be measured. Assume a measurement resolution of 10 x 10-6 m/m = 10 µm/m is required (a typical measurement). The change in resistance which corresponds to a strain of 10 µm/m can be calculated using: Eq. 1: dr = (GF )(R )(ε) = (2.05)(120Ω)(10x10 6 m / m) = 0.00246Ω Thus, a resistance change from 120 Ω to 120.00246 Ω must be measured...a very small change indeed!!! In fact, it is very difficult to measure such small changes in resistance using "normal" ohmmeters. Instead, special "strain gage circuits" are used to measure these small resistance changes accurately and precisely. The most widely used strain gage circuit is the "Wheatstone bridge" which we have studied previously. The digital strain indicator instrument (explained later) will implement the Wheatstone bridge circuitry and provide the necessary controls to determine the change in resistance, and when it is calibrated properly, it will the show the strain digitally. 3. The experimental setup Figure 2. The experimental setup for the measurement The experimental setup consists of a hollow cylindrical shaft ABC (Figure 2) supported by two bearings at A and B, and keyed at B to prevent rotation of the shaft. The BC end of the shaft is a cantilever (no support in the C side), and thus when a weight hung from C, it will cause the shaft to be loaded in pure bending. A torque arm CD is attached to the shaft at C. When a weight is hung from D (at the end of the torque arm) the shaft will be loaded in combined bending and 2 of 10
torsion. The other half of the shaft, the AB section of the shaft is supported at both A and B ends, and it is fixed from rotation at B end, thus a weight hung from E (at the end of the torque arm in AB section) will load the shaft in pure torsion. Twelve strain gages (#1 to #12) are bonded at various locations on the shaft surface to register strain. These strain gages are electrically connected to the four rotary switches (RI, RII, RIII & RIV) (Figure 2) on the connector box. Three additional non-load bearing strain gages (#13, #14 & #15) are placed inside the connector box, and those are also connected to the rotary switches. The non-load bearing strain gages serve to compensate any change in electrical resistance from the change in room temperature. When this connector box is connected to the digital strain indicator, by dialing appropriate numbers on the four rotary switches, any four strain gages can be placed on a Wheatstone bridge circuit within the digital strain indicator. In our experiment, we will dial the number of an active strain gage (any one of #1 to #12) on RI switch on the left, and set #13, #14 & #15 in rest of the three rotary switches (RII, RIII & RIV). Thus, the Wheatstone bridge will always have one active gage and three compensatory gages on its four arms. This configuration of four gages on the four arms is known as full bridge configuration. The shaft is made from 6063-T6 aluminum tube, with the following dimensions: Outside diameter d o = 0.825 in Inside diameter d i = 0.635 in Length of the cantilever from the centerline of the gages 1-6 to the centerline of the weight pan (see Figure 3): Moment arm L = 9.875 in Length of the torque arms L T = 12.764 in Nominal load P = 3 lb (use the stamped load value for your calculations) Modulus of Elasticity E = 10x10 6 psi Poisson s ratio = 0.33 4. Strain on various points and direction (i) Pure Bending: For pure bending, we will apply the load directly on the shaft at point C (shown in Figure 2), and we will measure strains in gage #1, #2, #3, #4, #5 and #6 (see Figure 3). Strain gage #1, measures longitudinal stress at the top layer of the beam, #2 crosswise at the top layer, #3 and #4 are similar positions but at the bottom layer of the shaft, #5 and #6 are at 45 o crosswise from the neutral axis. Remember that the strain gage can only measure the normal stress in the direction it is attached. From the applied load P, the normal bending stress ( x )at the top (tensile) and the bottom (compressive) layer of the shaft will be 32P L x... (2) 3 4 d o (1 ) where, L = Moment arm, d o = Outer diameter, d i = Inner diameter & d i / d o The strain gages #1 and #2 attached lengthwise () and crosswise(y) directions (See Figure 3), respectively. The strains ( ) registered by these two gages should be x 1, and E 3 of 10
2 x (poison s effect), respectively. [E = Modulus of elasticity, and = Poisson s ratio of the shaft material] Since the bending stress in the bottom layer of the shaft is equal in magnitude but opposite in sign compared to the stress in top layer, the strain gages #3 and #4 should register strains, 3 1, and 4 2. At the neutral axis there will be only transverse shear stress ( xy ) in & Y direction. However, the transverse shear strain at the neutral axis will be very small (of the order of 3 to 4 in/in) and will be ignored, and we will expect close to zero strain in these two gages 0). ( 5 6 Figure 3. Bending stress and strain 4 of 10
(ii) Pure Torsion load: For pure torsion, we will apply the load on the left side torque arm at point E (as shown in Figure 2) and we will measure strains in gage #7, #8, #11 & #12 (Figure 4). When pure torque T is applied on the cylindrical beam, only shear stress xy will be developed on the outer surface. This stress is constant along the length of the beam. The shear stress 16T xy... (3) 3 4 do (1 ) Where, T = Torque= P*L T and L T = length of the torque arm. The strain gage #7 and #8 affixed along the and Y axes, will not register any strain, since there is no normal stress in and Y directions. However, the gage #11 and #12, affixed at 45 o to the axis, will register strain. xy xy 11 1 and 12 1 E E d o d i 7 8 12 11 T 45 O Y xy xy xy xy xy xy 12 11 45 O 45 O xy xy 12 Stress 11 Figure 4. Bar under pure torsion Strain 5 of 10
(iii) Combined bending and torsion load: We will apply the load at torque arm on the right side (point D, as shown in Figure 2), and measure strains registered in gage #1 and #5 (Figure 5). The bending load will produce normal bending stress ( x ) and very small transverse shear stress, which will be neglected. However, torque load will produce shear stress ( xy ). Strain gage #1 only will register the normal stress ( x ), and as a result, strains should remain unchanged from the pure bending load condition. Transverse shear and pure shear stress will be in the same direction, and the total shear stress 16T xy 3 4 d o (1 ) The strain registered by gage #5 will be xy 5 1 E 2 1 A 1 Strain 2 x x 1 5 6 2 45 O L Stress d o d i Y xy xy 5 6 T P 45 O xy 45 O xy 5 Stress 6 Strain Figure 5. Bar under combined bending and torsion 6 of 10
5. Measurement of Strain by Digital Strain Indicator The Wheatstone bridge circuit is established by connecting the binding posts (P+, P-, S+, S-) on the connector box of the beam assembly to the corresponding binding posts (P+, P-, S+, S- ) on the digital strain indicator (Vishay Digital Strain Indicator, model Vishay/Ellis-20A) (Figure 6). Once the binding posts are connected you can select any four strain gages in the four sides of the Wheatstone bridge by turning the 4 switches on the biding post. The digital strain indicator is powered up by moving the Function Selector switch from off to zero position. The digital display should read 000. If there is a drift from instrument zero, then the Instrument Zero switch is adjusted until the digital display reads 000. This instrument zero drift should be checked periodically during the experiment. This is not balancing the bridge. The Range Selector switch on the digital strain indicator should be turned to full bridge configuration (denoted by 1), since we are using four strain gages (one active R a, and three non-active R 13, R 14 & R 15 ) in the four arms of the bridge. When the Function Selector switch is turned to read position and the strain indicator digitally displays the bridge unbalance voltage (V o in figure 6). Although the four strain gages have same resistance values in unloaded condition, different lengths of lead wires and minute difference in resistances of the gages will cause an initial imbalance in the bridge. When the function selector in the read position, we will balance the bridge by using the balance coarse and balance fine switches until the display reads 000, or close. This should be done in no load condition. Turning the balancing switches adds or subtracts resistance in one of the branches of the Wheatstone bridge circuit, and thereby Balance coarse and fine controls achieves a balanced bridge, that is V o =0, and the digital output is 000, when the function switch is in read more From the balanced bridge (display reads 000) in unloaded condition, when a mechanical load is placed on to the machine element, the strain in the machine element causes a change in resistance (dr a ) of the active strain gage (R a ). This change in resistance causes a bridge unbalance voltage V o, which is displayed via the digital display. You can lightly push the shaft and see how the digital display is changing. The next part is to calibrate the digital readout of V o to display the strain in a given measurement unit. The Instrument zero Digital display R13 Function selector Ra V i Vo R15 R14 Span coarse and fine controls Figure 6. Digital strain indicator and the full bridge Wheatstone bridge configuration. Rs R13 Ra Range selector V i Vo R15 R14 Figure 7. Internal calibration by a shunt resistance 7 of 10
gain of the digital display can be adjusted by turning the Span Coarse and Span Fine switches. If we apply a known level of strain in the active strain gage and adjust gain of the bridge output to display the strain, then the display will be calibrated to read any unknown strain, within the vicinity of the calibrated strain. However, instead of applying a known external strain, we will implement a predetermined small change in the resistance in the active gage (R a ) by placing a comparatively large known shunt resistance R s in parallel (Figure 7). In our case, R s = 200,000 and R a =120 y turning the Function selector switch to internal calibration the shunt resistance R s is connected parallel to the active strain gage R a. Change in resistance in the active leg (R a ) due to shunt dr a R a R eq 120 200,000 120 0.071956826 120 200,000 According to the equation (1) in section 2, this change in resistance in the active arm is equivalent to a mechanical strain, 6 dra 0.07195682610 c 293 in / in R GF 120 2.05 a Thus, after the bridge is balanced (V o =0), we will apply the shunt resistance by turning the Function selector switch to internal calibration position. At this position the digital output of the bridge is set to 293 by adjusting the span coarse and fine switches. Once this calibration is achieved, we will turn back the Function Selector switch to read mode again, which will disconnect the shunt resistance from the active gage. Since the bridge was previously balanced, the display should then again read 000. Now, any change in the resistance in the active gage from an external mechanical loading will be read by the strain indicator in in/in. In summary, to measure the strain from each strain gage, we need to connect one active gage and three dummy gages to form the Wheatstone bridge. When the bridge is in reading mode, balance the bridge using balance coarse and balance fine switches with no load applied on the shaft. Once bridge is balanced, calibrate it output by selecting internal calibration, and bring the number 293 using span coarse and span fine controls. The turn back the switch to read mode, it should still be zero, if not adjust it back to zero using balance fine. Once it is both balanced and calibrated, you are ready for strain measurement. Apply the load corresponding to the strain gage you want to measure, and note the digital output which gives you strain in inch/inch. Repeat this procedure for each strain measurement, 6. Analysis (a) For a pure bending load of approximately 3 lb, calculate the theoretical strains on the beam that should be recorded by strain gages #1, #2, #3, #4, #5 and #6. Compare these values with those measured in the experiment. (b) For a pure torsion load of approximately 3 lb, calculate the theoretical strains on the beam that should be recorded by strain gages #7, #8, #11 and #12. Compare these values with those measured in the experiment. (c) For a combined bending and torsion load of approximately 3 lb, calculate the theoretical strains on the beam that should be recorded by strain gages #1 and #5. Compare these values with those measured in the experiment. 8 of 10
9 of 10
DATA SHEET Date Names of the group members:,,,, Weight No. gram Weight pound 1 2 3 Test # Loading Type 1 Bending 1 2 Bending 2 3 Bending 3 4 Bending 4 5 Bending 5 6 Bending 6 7 Torsion 7 8 Torsion 8 9 Torsion 11 10 Torsion 12 11 12 Bending & Torsion Bending & Torsion Strain Gage # Strain () in in/in R(I) R(II) R(III) R(IV) Reading Calculated 1 5 10 of 10