RESIDUAL STRESS MEASUREMENT IN STEEL BEAMS USING THE INCREMENTAL SLITTING TECHNIQUE

Transcription

1 659 RESIDUAL STRESS MEASUREMENT IN STEEL BEAMS USING THE INCREMENTAL SLITTING TECHNIQUE DZL Hodgson 1, DJ Smith 1, A Shterenlikht 1 1 Department of Mechanical Engineering, University of Bristol University Walk, Bristol BS8 1TR, UK corresponding author, david.hodgson@bristol.ac.uk ABSTRACT This paper examines the application of the incremental slitting technique for residual stress measurement to plastically deformed four-point bent beams and autogenously edge welded steel beams. Initially a study into a number of factors affecting the series expansion approach was conducted. Due to the over-determined nature of the series expansion approach appropriate selection of polynomial series order was necessary to develop the best prediction. The effect of elasto-plastic properties in experimental situations creates local slit tip yielding which could led to errors particularly at shallow depths and in near slit gauges. The specimens used for the plastic four-point bending were manufactured from ferritic steel with dimensions of width 10mm, depth 25mm and length 250mm. These experiments were conducted to create a better understanding of the incremental slitting procedure and its experimental application. The measured stresses were found to be in reasonable agreement with the FE predictions. A set of autogenously welded ferritic beams of width 10mm, depth 50mm and length 180mm were also measured. These beams were edge welded to create a through depth residual stress field. Residual stress measurements for two different weld torch speeds were obtained. We show that a suitably chosen set of polynomials and gauge locations produce residual stress measurements with an uncertainty of about 7MPa. INTRODUCTION The slitting method has been developed over the last 30 years into a well established method for the measurement of residual stresses in both metal and composite structures. The first example of the method was Vaidyanathan[1] however this work used a cumbersome photo-elastic method. During the 80 s several authors [2, 3] began using strain gauges, finite element analysis and powerful desktop computing to increase accessibility of this method. The method has since been applied to various materials and geometries [4, 5, 6]. The method can be broken up into three areas, the forward solution, experimental cutting

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3 660 and the inverse solution. The forward solution develops a compliance matrix for the specimen finding out the effect of stresses on the strain output. The forward solution used in this report applies a series of Legendre polynomials to a finite element (FE) model of the specimen beams. Other methods that have been used are a distributed method similar to the integral method for hole-drilling [3], stress intensity approaches [7], body force method[8] or dislocation density[9]. The current method for slitting is the use of wire electro-discharge machining (WEDM) with strain gauges applied to the surface of the specimen. Jewellers saws and slitting saws were used in the past though the stress-free cutting of WEDM is used in preference now. The inverse solution maps the recorded strains from the experimental slitting into a stress field prediction. This is done using a least squares fit of a series expansion using the compliance matrix developed by the forward solution. Closed form solutions have been developed [1] however these were for specialised geometric configurations that are generally not applicable. The work conducted for this paper investigated the analysis methods of the slitting method, investigating the effect of polynomial order, elasto-plastic properties and stress field mapping. The slitting method was then applied to plastically bent beams, Figure 1, and autogenously edge-welded beams. Autogenous welding is a form of tungsten inert gas (TIG) welding which does not use any filler material to create the weld bead. INITIAL METHOD INVESTIGATION The initial studies were conducted into incremental slitting method using finite element simulation of the whole experimental process; forward solution, slitting and inverse solution. This allowed various factors and stress fields to be investigated. All the stress fields throughout this work were applied using the SIGINI user sub-routine within ABAQUS[10]. The models used in these investigations were 2D models however they were modelled on the specimens used for the plastically bent beam whose dimensions are shown in Figure 1. The beams were thin so plane stress could be assumed hence the use of 2D models. Figure 1: Plastically bent beam dimensions and gauge locations

4 661 POLYNOMIAL ORDER To determine the stresses from slitting strains accurately the orders of polynomials must be selected to contain enough turning points to map the stress field however an infinite number of polynomials cannot be used either. This is due to the over-determined nature of the solution approach leading to instability in the compliance matrix particularly when inverted. This instability can be summarised by the condition number of [C], larger condition numbers indicate greater instability. The condition number is calculated by the equation 1, where κ is the condition number and C is the compliance matrix: κ(c) = C C 1 (1) Figure 2 is an example of the how the condition number increases exponentially with the increasing number of polynomials used. This instability leads to the propagation of errors from experimental measurements into the results. An example of this condition number effect is shown in Figure 3 which compares the stress predictions for a plastically bent beam using 2nd - 8th order polynomials and 2nd to 13th order polynomials. Figure 2: Effects of Legendre polynomial orders on condition number of compliance matrix This creates the need for a balance between suitable number of polynomials to map the stress field and minimisation of the error propagation. The approach suggested by Prime and Hill[11] of uncertainty analysis and strain misfit is a reasonable approach and is shown later in this paper to be successful in selecting appropriate polynomial orders. ELASTO-PLASTIC MODELLING The effect of elasto-plastic material properties on the experimental results were investigated. It could be seen in the FE models that local yielding occurs at the slit tip due to the stress

5 662 Figure 3: Comparison of different polynomial orders and their effect on stress prediction concentration. This is thought to affect the strain measurements and therefore introduce error into the stress prediction. This local yielding is also evidenced in the experimental slitting results and is thought to be a source of error particularly in highly stressed components. This yielding is likely to cause the most error in the gauge situated near to the slit. MAPPING The mapping of stress fields defined by polynomials was successful due to the perfectly fitting coefficients for Legendre polynomials. The slitting measurement of plastically bent beams was also successful, Figure 4. There is a slight error surrounding the turning point and near surface measurements but that is to be expected. The prediction of a linear, non-equilibrium stress field however was unsuccessful. This stress field was created by simulating a fixed displacement experiment where a beam is placed into a four-point bend rig, elastically bent and fixed. Adding the 1st order Legendre polynomial to the compliance series did not improve the prediction. It is believed that the lack of complete system modelling in the compliance matrix leads to this failure. APPLICATION TO PLASTICALLY BENT BEAM MATERIALS AND PROCEDURE Experimental slitting measurements were conducted on plastically bent ferritic beams using strain gauges and a WEDM. The beams were manufactured from work hardening ferritic steel. They were bent in an Instron servo-hydraulic compression machine to a peak load of 25kN creating a standard saw-tooth residual stress profile with a surface stress of 79.5MPa and an internal peak stress of 78.7MPa, Figure 5. The beam dimensions and gauge locations are shown in Figure 1. 4 specimens were prepared however 2 specimens were lost during slitting due to waterproofing failure. The failed

6 663 Figure 4: Plot of the FE based stress prediction for a plastically bent beam waterproofing was a polyurethane based coating that was too brittle to survive close to the protective water-jet of the WEDM. This failure of the coating caused the gauges to short and produce no useful data. This coating was replaced by a dual layer acrylic/butyl rubber coating. A 0.25mm diameter electrode was used in the WEDM cutting a slit 0.3mm across in 0.25mm increments to a depth of 15mm. This depth was selected as the stress field is similar in the upper and lower half so complete cutting is excessive. The compliance matrix was developed using 2D FE models of the slitted beams containing 40,000 elements and nodes. Legendre polynomial stress fields were applied to the beams whose orders ranged from 2nd to 20th. RESULTS The uses of all four gauges lead to poor stress predictions so different combinations of gauges were used. During this work it was noted that the prediction from the gauge closest to the slit was particularly poor. The best combination of gauges was to use the side gauge; centred on the mid-line, 10mm from the slit, and the back face gauge. The stress prediction was made using the 2nd to 8th order Legendre polynomials, an example of the results for specimen 2 is shown in Figure 5. This number of polynomials was decided with help from the error calculation algorithm developed by Prime to stress uncertainty and strain misfit. For specimen 1 the average uncertainty over the depth profile was 2.50MPa with a maximum of 33.08MPa and a minimum 0.31MPa, the maximum uncertainty is always for the top surface measurement. Specimen 2 had slightly higher figures of 3.53MPa, 58.28MPa and 0.36MPa for average, maximum and minimum uncertainties respectively. Figure 5 shows the blind test result calculated by Prime using a the pulsed integral method compared to the 2nd to 8th order Legendre series expansion fit.

7 664 Figure 5: Plot of stress predictions for plastically bent beams The slitting stress prediction closely follows the finite element prediction. There is some slight error near the surface and surrounding the turning point. The slight turning point error is caused by slight errors in the work hardening curve of the material causing a minor shift in the FE simulation. This FE error probably accounts for the error of the prediction in the first 2mm of the slit depth. AUTOGENOUSLY WELDED BEAM TEST PROCEDURE Experimental measurements of the stress present in autogenously edge welded beams were conducted. These beams were manufactured from SA508, a class 3 reactor pressure vessel steel. Two beams were measured with different welds present, a nominal weld (NW) and a fast pass, high power weld (FHW). They had nominal dimensions of mm. 5 gauges were attached to each of the specimens, 1 top, 3 equidistant on the side and 1 back face gauge. These were waterproofed as for the previous measurements. A 0.3mm slit was introduced in 0.5mm increments to a depth of 49.5mm and 48.5mm for the NW and FHW respectively. The difference in slit length is due the different shaped beads on the surface of the beams. The NW beam had a 1mm higher weld bead than the FHW beam. 2D FE models of the specimens were used to create the compliance matrix using 2nd to 15th order Legendre polynomials. RESULTS The predictions for both beams are shown in Figure 6. The predictions were made using the 2nd to 10th order polynomials for NW and 2nd to 14th order polynomials for FHW. This selection of orders was based completely on the uncertainty and strain misfit results due to the lack of measurements from another source. For the NW beam; the average uncertainty

8 665 is 10.59MPa with and max/min of MPa and 1.38MPa. For the FHW the average uncertainty, max and min are 15.42, and 0.56 respectively. Figure 6: Comparison of nominal and fast pass autogenously welded beams Due to the large and rapidly changing stress field present in these beams the use of polynomial fitting to map the stress field and it has been shown that the pulsed method with Tikhonov regularisation[12] would be a better approach, Figure 6. The large stress field present in these beams means plasticity could be a source of error in the stress prediction. Both the weld beads present on the beams had pronounced radii above the stated design heights of the beams. This was accounted for in the selection of slitting depth however it is believed that the compliances will not have accounted for these radii due to the 2D FE models being used for the forward solution. For welds of this nature it would be suitable to use 3D models to account the effective change of beam width in relation to slit depth. This radii is believed to account for a large portion of the error that can be seen in the first 2-3mm of the measurement depth. CONCLUSION It has been shown that the incremental slitting method can develop accurate stress measurements however the method does need a certain amount of subtlety to achieve the best prediction for the recorded strains. This involves the number of gauges and locations used, the numbers of polynomials used for series expansion fitting and how the appropriate number is selected. The use of the series-expansion method for rapidly changing stress fields is probably not the best due to fitting errors however a reasonable prediction can still be achieved. The use of 2D FE models is not always appropriate for the development of compliances due

9 666 to 3D shape factors of the measurement specimen. The use of 3D models should not be an issue with availability of powerful desktop computing. ACKNOWLEDGEMENTS The authors would like to acknowledge Rolls-Royce Marine for supplying the autogenously welded beams for measurements. We would also like to acknowledge Dr MB Prime at Los Alamos National Lab for conducting blind analyses of the experimental data. REFERENCES [1] S Vaidyanathan and I Finnie. Determination of residual stresses from stress intensity factor measurements. Journal of Basic Engineering, 93: , [2] W Cheng and I Finnie. A method for measurement of axisymmetric axial residual stresses in circumferentially welded thin-walled cylinders. Journal of Engineering Materials and Technology, 107: , [3] D Ritchie and RH Leggatt. The measurement of the distribution of residual stresses through the thickness of a welded joint. Strain, 23(2):61 70, [4] W Cheng and I Finnie. Measurement of residual stress distributions near the toe of an attachment welded on a plate using the crack compliance method. Engineering Fracture Mechanics, 46(1):79 91, [5] MB Prime, P Rangaswamy, and MAM Bourke. Measuring spatial variation of residual stresses in a mmc using crack compliance. Proceedings of the Seventh International Conference on Composites Engineering, pages , [6] M Ersoy and O Vardar. Measurement of residual stresses in layered composites by compliance method. Journal of Composite Materials, 34(07/2000): , [7] T Fett. Determination of residual stresses in components using the fracture mechanics weight function. Engineering Fracture Mechanics, 55(4): , [8] W Cheng and I Finnie. A comparison of the strains due to edge cracks and cuts of finite width with applications to residual stress measurement. Journal of Engineering Materials and Technology, 115: , [9] D Nowell, S Tochilin, and DA Hills. Measurement of residual stresses in beams and plates using the crack compliance technique. Journal of Strain Analysis, 35(4): , [10] Abaqus cae v , [11] MB Prime and MR Hill. Uncertainty, model error, and order selection for seriesexpanded, residual-stress inverse solutions. Journal of Engineering Materials and Technology, 128: , [12] GS Schajer and MB Prime. Use of inverse solutions for residual stress measurements. Journal of Engineering Materials and Technology, 128: , 2006.