ENGINEERING TRIPOS PART IIA 3C7: EXPERIMENTAL STRESS ANALYSIS

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1 ENGINEERING TRIPOS PART IIA 3C7: EXPERIMENTAL STRESS ANALYSIS Experiment takes place in BNB-06 (follow downward stairs opposite Baker Building reception). OBJECTIVES To develop an appreciation of two different techniques of experimental stress analysis (i.e. discrete vs. full field measurements). To obtain experimental measurements of the stress concentrations induced by the presence of a circular hole in a thin plate. To note that the stress concentrations are sensitive to the location of the hole in the plate. To compare measured concentration factors with simple analytical predictions. 1. INTRODUCTION Structural failure due to, for example, fracture or fatigue is initiated at the most highly stressed points in a structure, often near a hole, notch or sharp corner. Hence, estimating the stress concentrations induced by different types of geometric discontinuities is of great practical importance. This experiment and its companion, Experiment 3D7, investigate the stress distribution in the vicinity of a circular hole, in a thin plate loaded in uniaxial tension. The plate is in a state of plane stress. This experiment constitutes the first stage of the investigation, where measurements are taken of the stress concentrations induced by the presence of (i) a symmetrically located hole and (ii) an asymmetric hole. For both cases, the stress components at 16 different points of the plate are derived from strain measurements at these points, and an overall view of the stress distribution is obtained by a photoelastic method. These experimental results are compared to the stress distribution near a circular hole in an infinite plate, for which there exists a simple analytical solution. The second stage of the investigation will be carried out next term in Experiment 3D7, where finite element solutions will be obtained. All plates considered in this experiment are made from Aluminium-alloy and have the following dimensions and elastic properties: Length Width Thickness Hole radius Offset from E ν (mm) (mm) (mm) (mm) centre (mm) (N/mm 2 ) Case (i) Case (ii) APPARATUS The tests are done in three Hounsfield Tensometers. Tensometers 1 and 2 are used for tests on strain-gauged plates with symmetric and asymmetric holes, respectively. They are linked to computers that collect, process and display data. Tensometer 3 is used for tests on plates coated with photoelastic material. A polariscope produces isochromatic fringes, i.e. contours of the difference between principal stresses, from which the stress concentration at the most highly stressed points can be obtained. 1

2 2.1 Tensometers 1 and 2 The set-up for both Tensometers is the same, only the position of the hole in the plates and the position and number of the strain gauges changes. The symmetric plate (Tensometer 1) is fitted with 14 strain gauges and the asymmetric one (Tensometer 2) with 29. The strain gauges are connected to a computer through a Microlink 3000 strain measurement system and a National Instruments GPIB board. This system also measures the strain in the loading beam in each Tensometer, from which the total axial force on the plate is obtained. 1 All measurements, data processing and display are controlled using the National Instruments Labview package. Thus, each computer displays the current value of the load and either the strains measured by the strain gauges, at various points on the plate, or the stress components at the same points, obtained by converting the measured strains. The values of the Young s Modulus and Poisson s ratio, used to convert strain to stress, are shown in the bottom-right part of the screen. These values can be changed simply by double-clicking on the current values and typing in new values. RUN STOP Plots of stress/micro-strains load: to change scale, double-click on limits and type in new values. Stop & Set Switch from print zero stress to micro-strain Load (kn) Current readings of stress (N/mm 2 ) or micro-strain: click on buttons to plot Values of E (N/mm 2 ) and ν Figure 1: Schematic layout of virtual instrument corresponding to Tensometer 2. The layout is similar in Tensometer 1. When the display switch is in strain mode, the current readings from the channels are displayed, in micro-strains, on the right-hand-side of the screen. Each value displayed corresponds directly to the linear strain measured, at a particular point of the plate, by a particular strain gauge. The position and numbering system of the gauges are shown in Figures 2 and 3, for the two plates. Strain gauge rosettes, consisting of 3 gauges going through the same point, have been used at many internal points. The first two gauges of each rosette are parallel to the x and y axes, respectively, while the third gauge is parallel to the line y = x. Denoting by ɛ a, ɛ b, ɛ c the linear strains measured by these 3 gauges, it can be shown that ε xx = ε a ε yy = ε b γ xy = 2ε c + ε a + ε b Cross gauges have been used at points that lie on a symmetry line. Because γ xy = 0 at these points, only ε xx and ε yy need to be measured. The two gauges are parallel to the x and y axes, and hence they measure ε xx and ε yy, respectively. Finally, single gauges have been used at points that lie close to a free edge of the plate, where only the strain parallel to the edge needs to be measured. When the display switch is in stress mode, the strains measured by the gauges are converted into stress values and then displayed. The details are as follows. At a general point of the 1 A direct measurement of the load on the plate is provided by the mercury gauge on the Tensometer. 2

3 13 14 Grid: 8 mm 8 mm y x R=14.8 mm 100 mm Figure 2: Plate with symmetric hole (Tensometer 1). plate, 3 strain components are required to obtain the corresponding 3 stress components. For an elastic plate in plane stress the stress components are σ xx = E 1 ν 2 (ε xx + νε yy ), σ yy = E 1 ν 2 (ε yy + νε xx ), σ xy = E 2 (1 + ν) γ xy and they are displayed in the order σ xx, σ yy, σ xy. On a symmetry line σ xy = 0 and hence is not displayed. At a point lying close to a free edge of the plate it can be assumed that only the stress in the direction tangent to the edge is non zero σ tt = E ε tt, σ nn = σ nt = 0 and hence only one stress component is displayed. Therefore, when the virtual instrument of Tensometer 1 is set to stress mode, channel 1, for example, displays σ xx at the point with coordinates (0, 48), while channels 4 and 5 display σ xx and σ yy at the point with coordinates (0, 32), respectively. 3

4 Grid: 8 mm 8 mm y 13 x R=14.8 mm 34 mm 66 mm Figure 3: Plate with asymmetric hole (Tensometer 2). 2.2 Tensometer 3 The third Tensometer is set up to test Al-Alloy plates whose geometry is identical to the plates tested in Tensometers number 1 and 2. These plates have been coated on one side with a photoelastic material (Type PS-1-B, manufactured by Measurements Group Inc., NC, USA). When this material is stressed and exposed to a polarised beam of light, it splits the light beam into two orthogonally polarised components, each parallel to a principal stress direction, whose propagation velocity is proportional to the stress magnitude. The two components of the beam are then passed through another polarizer, in order to produce two sets of fringes, when the two components are 180 out of phase. The first set of fringes are the isochromatics, which join points of the plate with the same principal stress difference. The second set of fringes are the isoclinics, which join points of the plate with the same principal strain directions. In this experiment the isoclinics are removed by using two additional polarizers, called quarter-wave plates. In this experiment a white light source is used, so that a spectrum of colours appears for the isochromatics. The coating has been calibrated for a wavelength of 575 nm, corresponding to the red/blue transition, which is known as the tint of passage. The principal stress difference required to produce one fringe at this wavelength is 63.5 N/mm 2 (this value depends on the 4

5 sensitivity and thickness of the particular type of photoelastic material used). The second fringe will correspond to a principal stress difference of 127 N/mm 2, and so on. 2 A set of colour photographs of the fringe patterns in the two plates, corresponding to a load of 10 kn, will be handed out by the demonstrator. 3. EXPERIMENTAL PROCEDURE Each group of students carries out three tests, one on each Tensometer. Each test should take about 30 minutes. 3.1 Tensometers 1 and 2 The experimental procedure for Tensometers 1 and 2 is similar. 1. Apply a pre-load of about 2 kn to the tensometer. 2. Check that the virtual instrument is set to STRESS. 3. Initialise the stress measurement. - Tensometer 1: switch OFF the plotting of all channels, press RUN ( ), press ZERO readings and then CLEAR GRAPH. - Tensometer 2: switch OFF the plotting of all channels, press RUN ( ) and then RESET. It may take a few moments until all readings go to zero (they may subsequently drift a little). 4. Turn ON the plotting of all channels. 5. Slowly increase the load up to approximately 5 kn. 6. Check that the graphs are linear, that all gauges are responding properly, and that the graphs fit on the screen scales. Turn OFF the plotting of all channels except σ xx at points across the section x = 0, PLUS the far-field stress components (13, 14 for the symmetric hole and 17, 18, 19 for the asymmetric hole). 7. When all other graphs have disappeared press STOP & PRINT. 8. Calculate the far-field uniaxial stress in the plate,. N.B. The values of ALL stress components are available in the printout, not only σ xx. 3.2 Tensometer 3 The procedure for the photoelastic test is as follows. Mount the plate with the symmetric hole on the Tensometer. Switch on the light source and gradually increase the load on the plate. Observe the fringes as they appear at the edges of the hole and travel across the plate as the load is increased. Identify the first and second tints-of-passage as blue just appears at the edge of the hole. Increase the load up to 10 kn, then stop and compare the fringe pattern to that shown in the photograph handed out by the demonstrator. The zero-order fringe, which is the locus of points with zero principal stress difference, will remain black. Unload the plate. Then, gradually increase the load again and record the loads at which the second fringe reaches (i) the 2 Further details on photoelastic methods can be found in Chapter 5 of S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill 1970, or Chapters 4 and 5 of G.S. Holister, Experimental Analysis, CUP 1967, or 5

6 most highly stressed point in the plate, and (ii) the point corresponding to strain gauge 8. This point is marked on the plate. Repeat this procedure for the asymmetric plate, but record the loads at which the second fringe reaches (i) the most highly stressed point in the plate, (ii) the point diametrically opposite to it, (iii) the point corresponding to strain gauge 8, and (iv) the point corresponding to strain gauge 21. These last two points are marked on the plate. Use these recorded loads to estimate the stress concentration factors σ xx / at these points. Recall that two fringes correspond to a principal stress difference of 127 N/mm 2, and σ yy 0 near the edge of the hole. N.B. The load on any photoelastic plate should not exceed 10 kn. 4. ANALYTICAL SOLUTIONS There are several analytical solutions for the stress distributions in thin plates with circular holes. The best known is the solution for a circular hole of radius R in an infinite plate, see Figure 4, which is obtained in the lectures. 3 The stress components on the centre lines are given by x-axis: σ xx = 1 5R2 2x 2 + 3R4 2x 4 (1) σ yy = R2 2x 2 3R4 2x 4 (2) σ xx y - axis: = 1 + R2 2y 2 + 3R4 2y 4 (3) σ yy = 3 ( ) R 2 2 y 2 R4 y 4 (4) both axes: σ xy = 0 (5) x y R Figure 4: Infinite plate with a hole. The stress concentration predicted by Equations (1)-(4) decays rather quickly away from the hole. Therefore, this solution can be used for non-infinite plates, provided that the edges of the plate are sufficiently far from the hole that the stresses on the edges are approximately equal 3 S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, p

7 to the remote stress field. If this is not the case, alternative solutions by Howland and Mindlin may be used. Howland 4 derived semi-analytical expressions for the stress concentration factors at the edge of a symmetric hole of radius R in a plate of finite width 2W. These factors have been calculated for a range of plate geometries and are shown in the table below. σ 0 B x y A R σ 0 W W Figure 5: Plate of finite width. concentration factors R/W Point A Point B σ xx / σ yy / Mindlin 5 derived an analytical solution for the stress concentration factors in the vicinity of a hole near the edge of a semi-infinite plate, Figure 6. For the asymmetric plate used in this experiment Mindlin s theory predicts the following stress concentrations. σ 0 x B y A σ xx = 3.94 at point A σ xx = 0.59 at point B 5. WRITE-UP Figure 6: Semi-infinite plate. Your report should include: 1. A summary of about 50 words. 2. A short introduction and background (do not repeat everything in the handout, summarize and refer to the handout as necessary). 3. The two tables of results attached to this handout, complete with the non-dimensional stress components at the 16 strain-gauged points and at the most highly stressed points in the plates. 4 Howland, R.C.J. (1930). On the stresses in the neighbourhood of a circular hole in a strip under tension. Philosophical Transactions of the Royal Society of London A, 229, Mindlin, R.D. (1948). distribution around a hole near the edge of a plate under tension. Proceedings of the Society for Experimental Analysis, 5,

8 4. Sketches of isochromatics, one for each plate, on sheets of tracing paper. Make sure to label each line traced with the correct stress difference. 5. Plots of measured σ xx / and σ yy / across the sections x = 0 and y = 0, for both plates. To save time, the predictions for a hole in an infinite plate have been plotted on the sheets attached to this handout: plot on these sheets your data and any other analytical predictions that you wish to consider. 6. A short discussion of any discrepancies between the experimental results and the various theoretical solutions. Where they don t agree, carefully examine the theoretical assumptions; don t assume that the theory is always right! Comment on the effect of strain gauge size and position. Notes: Clear presentation of the results is important. The discussion should compare results to guide conclusions, and is meant to show critical thinking and insight. General marking guidelines can be found on the back side of the experiment report coversheet. Submit your report to Miss C Whitaker (cw535@eng), room BE3-39, 3rd floor Baker Building within 15 days of your lab session. Dr F Cirak September

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11 PLATE WITH SYMMETRIC HOLE Load = = Strain Gauge Results Photoelastic Results Finite (stress at 2nd fringe = 127 N/mm 2 ) Element Results Coordinates Component Channel Concentration Load at 2nd fringe at 2nd fringe Concentration Concentration (mm) (N/mm 2 ) (N/mm 2 ) stress/ (kn) (N/mm 2 ) stress/ stress/ (0,48) σ xx 1 (0,40) σ xx 2 σ yy 3 (0,32) σ xx 4 σ yy 5 (0,24) σ xx 6 σ yy 7 (0,14.8) σ xx (0,16) σ xx 8 (12,12) σ tt 9 (16,0) σ yy 10 (24,0) σ xx 11 σ yy 12 (80,0) σ xx 13 σ yy 14

12 PLATE WITH ASYMMETRIC HOLE Load = = Strain Gauge Results Photoelastic Results Finite (stress at 2nd fringe = 127 N/mm 2 ) Element Results Coordinates Component Channel Concentration Load at 2nd fringe at 2nd fringe Concentration Concentration (mm) (N/mm 2 ) (N/mm 2 ) stress/ (kn) (N/mm 2 ) stress/ stress/ (0,32) σ xx 1 (16,32) σ xx 2 (0,24) σ xx 3 σ yy 4 σ xx 5 (8,24) σ yy 6 σ xy 7 (0,14.8) σ xx (0,16) σ xx 8 σ xx 9 (16,16) σ yy 10 σ xy 11 (12,12) σ tt 12 (16,0) σ yy 13 σ xx 14 (32,0) σ yy 15 σ xy 16 σ xx 17 (48,0) σ yy 18 σ xy 19 (12,-12) σ tt 20 (0,-14.8) σ xx (0,-16) σ xx 21 σ xx 22 (16,-16) σ yy 23 σ xy 24 (0,-32) σ xx 25 σ yy 26 (0,-48) σ xx 27 σ yy 28 (0,-64) σ xx 29