Introduction 1 : The purpose of this experiment is to demonstrate the existence of stress and strain concentration in the vicinity of a geometric discontinuity in a cantilever beam, and to obtain an approximate measure of the elastic (theoretical or geometric) stress concentration factor, K t. 2 In this case, the discontinuity is simply a circular hole, drilled through the beam on its centerline. Figure 1: Stress Concentration in a Cantilever Beam The presence of any geometric irregularity in the shape of a loaded mechanical part or structural member impedes the orderly flow of stress trajectories, causing them to crowd together, and locally increasing the stress above the nominal level as calculated by conventional mechanics of materials formulas. Such an irregularity or discontinuity is referred to as a stress-raiser. Figure 1 shows the stress distribution at two sections of a cantilever beam and illustrates the presence of stress concentration. 1 This lab is based on E-104 STRESS AND STRAIN CONCENTRATION, Vishay Measurements Group, Inc., 1982, printed March 2002. Portions of this text were taken verbatim from that document. 2 K t - stress concentration factor. This is not to be confused with K t strain gage transverse sensitivity. Page 1 of 10
At section A, the stress is uniform across the width of the beam, and calculable from the following relationship: (1) where: σ = stress, psi (N/m), M = bending moment, in-lbs (mn), I = moment of inertia on beam cross section, in 4 (m 4 ), c = half-thickness of beam, in (m), P = load, lbs, (N) At section B, the nominal stress, based upon the net area of the section is: If the location of the hole is selected so that (2) (3) the nominal stress at section B is the same as that at section A. The maximum stress at section B, however, is much greater, due to the stress concentration effect. As shown in the sketch, the maximum stress exists at the edge of the hole, on the transverse diameter, and the stress decreases rapidly with distance from the hole. By definition, the stress concentration factor, K t, is the ratio of the maximum stress at the stress-raiser to the nominal stress at the same point. That is, (4) Since the nominal stress at both sections of the beam and the peak stress at the edge of the hole are all uniaxial, the strain and stress are proportional if the proportional limit of the beam material is not exceeded in the experiment. Thus, the stress concentration factor is equal to the ratio of the maximum to nominal strains at section B. Therefore, (5) Equipment and Supplies: - Vishay Flexor cantilever flexure frame - High-strength aluminum alloy beam, 1/4 x 1 x 12 ½ (3.2 x 25 x 318 mm), outfitted with preinstalled strain gages and a ¼ diameter stress concentration hole. - Vishay P3 Strain Indicator Page 2 of 10
Pre-Lab: A) The Flexor loading micrometer will be used to load the beam. Calculate the micrometer motion necessary to induce 2000 µϵ at the location of gage 4: Deflection in a cantilever beam b inches wide by t inches thick is: (6) where: y = deflection (in), P = force, lbs, L = clamp to micrometer distance (10.0 in), E = Young s modulus for aluminum (10.4E6 psi), t = beam thickness (0.25 in). Strain at strain gage 4 (section A) is given by: (7) where: x = micrometer to gage 4 distance (9.0 in). P, b and t are as defined above. Using equations (6) and (7) above, determine the deflection required to induce 2000 µϵ in the beam at gage 4. (Do your work in the space provided below.) y(pre-lab): Micrometer deflection required for 2000 µϵ: (in) B) Engineering texts and handbooks often include tables of stress concentration factors. Use one of these tables to determine the theoretical stress concentration for a ¼ diameter hole drilled through the center of a ¼ x 1 rectangular beam in simple bending. Kt Source Page 3 of 10
Procedure General: In this experiment, the beam will be loaded with the Flexor until a predetermined nominal axial strain level of 2000 µϵ is reached at section B (see Figure 1). As described in the introduction, the nominal strain at section B will be measured, not at that point on the beam, but instead at section A, where the measurement can be made more conveniently and accurately. It is important not to exceed a nominal strain of 2000 µϵ, since the actual strain at the edge of the hole is much higher than the nominal, and excessive strain could produce local yielding. The actual strains in the region of stress concentration will be measured with three very small strain gages placed in section B at varying distances from the hole, with one of the gages directly adjacent to the edge. The strains indicated by the three gages will be plotted on the graph sheet (included in the Appendix of this experiment) at the locations of the respective gage centerlines. A smooth curve can be drawn through the resulting three data points to show the strain distribution in the vicinity of the hole. Since the centerline of the closest gage to the hole cannot physically coincide with the edge of the hole, it is necessary to extrapolate the data to the edge to obtain an approximate value for the technique given in the Analysis and Presentation of Data section. The ratio of the maximum to the nominal strain at section B is the strain concentration due to the disruptive presence of the hole. If the proportional limit of the beam material has not been exceeded during the experiment, the stresses are proportional to the strains, and the same ratio represents the stress concentration factor, K t. In measuring stress concentration with a strain gage, it must be kept in mind that the gage tends to indicate the average strain in the area covered by the grid. Since the strain in the immediate vicinity of a stress-raiser decreases very steeply with distance, it is obviously necessary to select the smallest practicable strain gage and bond it in place as close as possible to the edge of the stress-raiser in order to minimize the error in sensing the peak strain. Even with this technique, the strain indicated by the gage may be significantly lower than the peak strain. The three gages selected for measurements in the steeply varying strain field near the hole should have grids no larger than 0.031 x 0.031 in. Ideally, gage grid dimensions should be 10% or less of the radius represented by the stress concentration configuration. The 4 th gage, which is located at section A, remote from the hole, and out of the region of steep strain gradient, can be considerably larger. At a distance along the beam from the stress raiser, the axial strain is essentially uniformly distributed across the width of the beam, and varies linearly along the length. The average strain indicated by the gage at section A is thus equal to the strain at the center of the grid. Because of the relationship expressed by Eq. (3) in the introduction, this is also equal to the nominal axial strain at section B. Micro-Measurements foil strain gages employed on the pregaged beam are intrinsically temperature compensated for use on the material from which the beam is made. Because of this, the four gages can be connected to the P3 Strain Indicator individually in quarterbridge arrangements, completing the bridge circuit each time with the precision resistors built into the instrument. For quarter bridge operation, a three-wire circuit is ordinarily used with each gage in order to obtain compensation for temperature-induced resistance changes in the leadwires by placing equal lengths of leadwire in adjacent arms of the bridge Page 4 of 10
circuit. It is often convenient in minimizing the lead and connection requirements to combine the leads from one solder tab of each strain gage into a common lead. When threewire circuitry is used, this becomes a pair of leads which is common to every gage in the system (see Wiring Diagram included in the Appendix). Acquisition of Data: The gages will be connected (via the flexor cable) to the strain indicator one at a time, first with the beam undeflected, and again with the beam deflected. An initial reference reading of the strain indicator readout will be obtained for each gage with the beam undeflected, and a final reading with the beam deflected. The differences in these two sets of readings will give the strains at the respective gage locations. 1) Record gage factor G and resistance Ω for all four gages: G Gages 1 thru 3 Gage 4 Ω 2) Install the beam in the flexor with gages facing up on the clamp end. Make sure the micrometer is clear of the beam and that the beam is as far in as it will go. 3) Connect the leads from the beam to the flexor terminals and gage 1 to Channel 1 on the P3 indicator as shown in the Appendix. 4) Turn on the P3. Set it for Channel 1 active, Channels 2-4 inactive, quarter bridge. 5) Set the gage factor to that for gage 1. 6) Balance the gage and save settings. 7) Record the initial reading for gage 1 (0 µϵ) in the Data Table. 8) Connect gage 2 in place of gage 1. (Swap leads 3 and 4, leaving 1 and 2 alone.) 9) Record the gage 2 initial reading in the Data Table. 10) Connect gage 3 in place of gage 2. (Swap leads 4 and 5.) 11) Record the gage 3 initial reading in the Data Table. 12) Connect gage 4 in place of gage 3. (Swap leads 5 and 6.) 13) Record the gage 4 initial reading in the Data Table. 14) Turn the loading micrometer clockwise until the strain indication just begins to change. Record the micrometer reading as y(initial) in the table below. 15) Turn the loading micrometer clockwise until gage 4 reads 2000 µϵ plus the initial value recorded in step 13. 16) Record the micrometer reading as y(final) in the table below. Calculate the error as follows: 100% y initial y final y pre-lab Error (%) in in in 17) Record the gage 4 final reading in the Data Table and backtrack steps 12 through 7 to get final readings for gages 3, 2 and 1. Page 5 of 10
18) Back the loading micrometer clear of the beam. The gage 1 strain indication should return within a few µϵ of the initial reading. Record the reading here: The pregaged beams used in this experiment have been tested for gage stability at the time of manufacture, and should perform in a highly repeatable manner unless one or more of the gages has been damaged. If the zero-beam-deflection readings of the strain indicator fail to repeat well, the binding post connections may not have been snug enough to avoid small contact resistance changes. Binding post connections should be snug enough to allow a wiggle test of the leadwires without a zero balance shift. Analysis and Presentation of Data The strain indicated by gage 4 can be corrected for gage factor (if different from the gage factor of gages 1, 2 and 3 and the instrument gage factor setting) by the following relationship: (8) The result of this calculation is the nominal strain at Sections A and B of the beam, and should be entered in the Worksheet. The strains sensed by gages 1, 2 and 3 can be calculated by subtracting the initial reading from the final reading in each case. These numbers should be entered in the Data Table. Even though the centerline of gage 1 is only 0.020 inch from the edge of the hole, the average strain sensed by this gage is considerably lower than the peak strain. A satisfactory estimate of the peak strain can be obtained by extrapolating curvilinearly to the edge of the hole. Assume that the strain distribution can be approximated by an expression of the following form: (9) Where R is the radius of the hole, X is the distance from the center of the hole to any point on the transverse centerline, and A, B and C are coefficients to be determined from the measured strains at three different points along the transverse centerline. Thus, Page 6 of 10
( ) (10) Noting that: R = 0.125 in, X 1 = 0.145 in, X 2 = 0.185 in, and X 3 = 0.325 in and solving Eqs. (10) simultaneously for C, B, and A,.. (11).. (12).. (13) Substituting the measured strains from the worksheet table into Eq. (11) gives C, which can then be used in Eq. (12) to solve for B, etc. Since 1 at the edge of the hole, the peak strain is: (14) The strain (and stress) concentration factor is then: (15) where = the strain indication at gage 4, corrected for gage factor. Plot the strains ϵ 0, ϵ 1, ϵ 2, and ϵ 3, versus the corresponding dimensionless distance X/R on the accompanying graph sheet to visualize the stress distribution in the vicinity of the hole. Write-up: (Worksheet) - The executive summary should include a brief description of the experiment and your value for. - The Results consist of the completed handout. - In the discussion, compare your value for to that found in the pre-lab. Your value is probably considerably smaller than the published value. Discuss possible causes. How did your pre-lab value for the 2000 µϵ deflection compare to the actual value? Mention anything else you deem worthwhile. - For the conclusion, recap everything in 50 words or less. Page 7 of 10
Appendix: Wiring Diagram Page 8 of 10
Worksheet: Date: Lab Partners: Data Table Gage Initial Reading Final Reading Strain 1 0 µϵ 2 3 4 2000 µϵ 1) Correct strain ϵ 4 for gage factor per Eq. (8) and enter the value here: 2) Compute extrapolation coefficients A, B and C per Eqs. (13), (12) and (11) and enter the values here: Extrapolation Coefficients A B C 3) Compute the maximum strain at the edge of the hole,, per Eq. (14), and the stress concentration factor,, per Eq. (15), and enter the values here:, and, 4) Plot the strains ϵ 0, ϵ 1, ϵ 2, and ϵ 3, versus the corresponding dimensionless distance X/R on the accompanying graph sheet. Draw a smooth curve through the points. Page 9 of 10
Figure 2: Strain Distribution Plot Page 10 of 10