WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.1 Acoustic Emissions Monitoring of Aluminum Beams D. Aliaga, M. Madsen, and J. Soller Department of Phsics, Wabash College, Crawfordsville, IN 47933 (Dated: Ma 5, 2011) The ultimate load strength of a beam is dependent upon the elasticit of the material, its moment of inertia of the cross sectional area, and the longitudinal length of the beam. B appling a load, the combination of compressive and tensile forces on opposite sides of beam cause fractures in the material, eventuall leading to full failure of the beam itself. B appling calculated loads to aluminum beams, we hope to predict the failure point of the material beams through the emploment of acoustic emissions monitoring.
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.2 Testing the ultimate load strength of beams is a frequentl encountered situation in the world of professional engineering. Having standard, experimental measurements of a materials strength allows architects and engineers to build and design in confidence with given materials. A common practice used to determine the loading capacit of a structure is the criteria that defines the maximum load that can be withstood when the material reaches its maximum stress ield [2]. The most common wa to monitor the maximum stress ield of a given material in industr toda is b using acoustic emission sensors to detect small tremors in the material as the molecular structure of the material begins to weaken[1]. As stress is placed on a loaded beam, the capacit of the beams strength weakens as the molecular bonds begin to stretch. When the force on the beam begins to grow, the molecular bonds are stretched too far and begin to break, causing high-frequenc tremors to move transversel through the material. Within the acoustic emission sensors, a piezo-electric accelerometer will detect an small acceleration of the beam, due to molecular bond failure, and rela the signal as an electric pulse we can then detect [1]. B using acoustic emission monitoring, we can detect the small tremors that result from molecular bond failure, and relate that failure point to the amount of pressure applied to the beam. The branch of applied mechanics that deals with our subject is called Mechanics of Materials. The following definitions and derivations are from Ref. [4]. Our principal objective is to determine the stresses, strains, and displacements in the beam in the hope that this will give us enough information to predict the beam s breaking point. The stress and strain are the most fundamental concepts that need to be understood in this field. The stress σ is basicall the force per unit of area that our beam undergoes when a load is present. The strain ɛ is a dimensionless quantit that measures the ratio between the length l and the elongation d of the bar due to the load. ɛ = d l (1) The segment depicted in Fig. 1 helps to understand both concepts. It is useful to understand the deformation of the material to obtain the Stress-strain diagram which is sketched in Fig. 2. The linear region obes Hook s law. For the other regions, the model for deformations gets more complicated. Hook s law for materials is expressed b the equation
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.3 a) x b) l l-d l l+d FIG. 1. A segment of a circular bar is presented in a). The same segment under a longitudinal load P with its respective strain d is showed in b). FIG. 2. Initiall, the linear relationship between stress and strain is maintained until the elastic deformation of the material is maxed out, and the material can take a given amount of stress, until it begins to fracture and eventuall fail completel.
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.4 σ = Eɛ (2) where E is the modulus of elasticit. There are two different kinds of stresses: normal stresses (σ) which are perpendicular to the surface of the material and shear stresses (τ) which are tangential. This concept applies even if ou consider onl an infinitesimal volume as shown in Fig. 3. x FIG. 3. Plane-stress element of an infinitesimal x cross section of the beam in Fig. 4. In our case we consider the case where we have a beam that is simpl supported with a concentrated load P at the center as shown in Fig. 4. This setup simplifies our calculations b providing horizontal smmetr. One problem with our model is the fact the we have non-uniform bending rather than pure bending. Pure bending happens when we have a constant bending moment M throughout the bar as shown in Fig. 6. The bending moment is similar to the torque experienced b the structural element. A more rigorous definition states: a bending moment exists in a structural element when a moment is applied to the element so that the element bends. In our model, there is a concentrated load so the bending moment is zero at the ends and increases constantl towards the center where the load P is applied, as seen in Fig. 5. Therefore, we don t have pure bending, nevertheless, experiments show that differences are small and make up a good approximation of the bending. Therefore, we assume constant bending when considering the shape of the beam s curvature (circular). In order to derive the model for the stress as a function of the load we start b considering the shear forces and bending moments.
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.5 a) O L/2 P L/2 x h z b b) FIG. 4. a) A simplified version of the setup. It is a basicall a beam of rectangular cross section simpl supported under a concentrated load (force) P. On the left side the support A is fixed. On the other side, support B is allowed to move. b)as a load is applied to the beam, the bottom of the beam is exposed to tension forces, while the top of the beam feels a compression force from the pinching of the material. This combination of forces will result in fractures in the beam, until at a maximum displacement, δ, the beam will fail. M PL/4 O L/2 L x FIG. 5. The magnitude of the bending moment as a function of x from the origin. B smmetr the shear forces experienced at each point to the left or right of P is equal to the reaction force of the opposite side, R A and R B in Fig. 6. B sign convention, the left side is taken to be positive and the right is negative. It is useful to start with a static free bod diagram as shown in Fig. 6. In this case the reaction forces R A and R B are
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.6 M Ra x P Rb M L/2 L/2 FIG. 6. The beam under load P is in static equilibrium due to the reaction forces R a and R b.the Bending Moment M is also portraed. In the same wa, the bending moment is equal to R a = R b = P 2. (3) going from left to right and from right to left it is M = R a x = P 2 x (4) M = P (x L). (5) 2 Each of the equations expressed above is onl valid for its respective half. The maximum bending moment is then M max = PL 4. (6) Now we need to find the curvature of the beam. The curvature is defined as the inverse of the radius of curvature of ρ of the center of the beam, κ 1. We derive it, b starting with ρ the relationship ρdθ =ds 1 ρ = dθ ds = κ, (7)
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.7 where s is the arc longitude (see Fig. 7). If we use a small angle approximation (which a) x M A s e m dx f B M n q b) O x M s A e m dx f p Bt s M n q FIG. 7. Deformations of the beam in pure bending: a) undeformed beam; b) deformed beam. is a good approximation since the radius is usuall big compared to s) thens x. Now consider the neutral surface (dashed line) which is defined as the surface along the x axis where the distance between an two planes nm and pq does not change under bending as showninfig.7. Thus, for the neutral surface
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.8 ρdθ = dx. (8) However for an region above or below, the distance either increases or decreases. For that case we derive the strain-curvature equation. Start b considering the distance from e to f in Fig. 7 L ef =dx ds = ρdθ (9) without the applied load (See Fig. 7). Then, once the load is applied, the stress compacts the length so the curved length becomes ds =(ρ )dθ (10) and ΔL =dx ds = ρdθ (ρ )dθ = dθ. (11) Now, we know that the strain on the element legth dx is ɛ x = ΔL dx Note that we use ΔL since >0 in this case. If we consider and object that obes Hooke s Law Eq. (2) then = dθ dx = ρ. (12) σ x = Eɛ = Eκ (13) where σ x denotes the normal strain. Now we derive the moment curvature relationship, again following Ref. (torque) is Now, using Eq. (13), we get M =??, starting with the fact that then the elemental bending moment A dm = (σ x da). (14) M = σ x da. (15) A Eκ 2 da = Eκ 2 da = κei (16) A where A 2 da = I is defined as the cross-sectional moment of inertia. Now combining Eqs. 16. and 13. we get the Flexure Formula σ x = M I. (17)
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.9 Force on da h z b FIG. 8. This is a cross-section parallel to the plane z where we see that the force on da acting perpendicular to cross-section is σ x da. This equation relates the strain as a function of the distance from the neutral xz surface. In our case we have a rectangular prismatic beam so I = A 2 da = bh3 12. (18) Now, we find σ max b replacing M = M max = PL/4and = max = h/2 intoeq.17which gives σ max = M I = 3PL bh. (19) 3 The expression above is the one that we currentl use to calculate the maximum stress produced b the load. It is not exact since it assumes pure bending and elastic behavior of the material which is not true near the breaking point. We compared this to results of our experiment with a aluminum bar and we found that the prediction of the breaking point is not accurate due to our assumption of a linear stress-strain relationship. However, the
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.10 calculation should be accurate when calculating the load needed to bend the beam. We were not able to prove this since we don t have a wa to measure the applied force on the beam. We tested Images Co. Piezoelectric film transducers, PZ-03, to collect data using the acoustic emissions of the material to which we are appling the load. We calibrated the transducers b measuring the speed of sound within the beams. To do this, we suspended an Aluminum beam (Allo 6061) of length L =1.833 ± 0.002 m and thickness d =0.0127m and attached the piezoelectric transducer to the center of the beam, as shown in Fig. 9. lectric sensor To data collection L 6061 Aluminum Beam FIG. 9. We suspended a bar of length, L =1.833 ± 0.002 m, from two stands, and attached the piezo-electric sensor in the center of the beam. When we strike the end of the beam, the sound pulse propagates down the length of the beam, and registers with the transducer as a voltage difference that was sent to an oscilloscope. A second pulse also appears as the wave reflects off the far end of the beam and returns past the sensor. Using this knowledge, the model for the expected value of the speed of sound in the aluminum was simple because we knew the length of the beam, L, and we find the time between pulses, Δt, using the oscilloscope data. The speed of sound in the beam should agree with the calculation of the speed b relating the Young s modulus, Y Al =10 10 6 psi, and the densit of the aluminum, ρ Al. v s = 2L Δt = Y ρ (20) where v s is the speed of sound in the material, which was aluminum in our case. For the propagation of the waves through the beam, we need to differentiate between what tpe of wave we expect to detect using the piezoelectric transducer. Shear and bulk waves are both tpes of waves that ma travel through media, with shear waves moving translationall across the surface of the material, and bulk waves moving longitudinall through the material
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.11 b compression. B tapping the beam on the end, we expect longitudinal compression waves to travel through the beam, and register with the piezoelectric sensor. In order to properl record the time difference between pulses in the beam, we amplified the 3 mv signal from the piezo-electric sensors using the circuit shown in Fig. 12, with a high-pass filter, so that we could receive a distinct signal from the sensor without background noise from the apparatus. Collecting the data using an Tektronix Digital Storage Oscilloscope, we saw a distinct separation in pulses with a time difference, Δt = 725μs, as showninfig.11. V 0.01 F V s Piezo- Electric Sensor 15 k 47 AD 620 V out V V s FIG. 10. We needed to amplif the signal of the piezo-electric sensors, so we used an amplifing circuit with a high-pass filter. The signal of the sensor was brought through a capacitor that would onl complete the circuit if the signal was of high frequenc. The AD620 amplification chip amplified the signal b a factor of 1000, so that it could be read b a Tektronix TDS 2012B oscilloscope. Using the separation time between the pulses, we calculated that the speed of sound in the aluminum beams is v s =5, 048 ± 35 m/s (95% CI, Gaussian pdf). This value for the speed of sound in aluminum is in agreement with the expected value of v s =5, 014 ± 45 m/s as calculated b using the Young s modulus approach [5], so we were confident in our abilit to properl detect sound pulses through the aluminum beam. This was critical in verifing our use of acoustic emissions sensors to track the maximum load capacit of the aluminum beams we tested. After verifing our detecting technolog, and understanding the process of experimentall breaking beams, we join the two ideas together b appling a load to an aluminum beam and measuring the stresses on the beam using the piezo-electric sensors. We built a hdraulic press apparatus to appl large forces to the beams. The components of the hdraulic press
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.12 1.5 1 0.5 Voltage (V) 0-0.5 t ~750 s -1-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 Time(s) FIG. 11. The repetition of the peaks in the data are the initial and reflected sound waves as the travel through the beam. Their separation was approximatel 750 μs, which we used to calculate the speed of sound in the aluminum beam. are purchased from OTC Tool manufacturers. These components include the 4004 Two- Speed Hdraulic Hand Pump, 10 Ton Hdraulic Shop Press, and the 4105 Single Action, Spring Return Ram, along with the included connections for each element of the sstem. In addition to these three major components, we utilize the OTC 9658 Hdraulic Pressure gauge and a Grainger 4YDL9 Wedge head. Assembling these components of the press correctl was critical to the success of the experiment because hdraulic sstem failure can result in injur. We began b securing the ram in the frame of the press and assembling the quick-couple connections between the input channel of the ram and the hdraulic hose, sealing all threaded connections with teflon tape. After securing the quick-couple assembl, we removed the hose from the ram. Next, we removed the pressure plug from the front end of the hand pump, and connected the
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.13 pressure gauge, being mindful not to spill the hdraulic fluid out of the pump. Lastl, we connected the hdraulic line to the pressure gauge with the end opposite the quick-couple connection. The completed assembl in Fig.12, was then gravit filled with hdraulic fluid b raising the pump, in the open position, above the hdraulic line, and filling the pump completel with OTC Hdraulic Oil, No. 9637. Pressure Line Ram To Hand Pump Loaded Beam Wedge Head x L Steel Spacer FIG. 12. The applied load of the Ram in the -direction resulted in a displacement, δ, in the beam. We recorded the pressure using the gauge and related that reading to the displacement of the beam. We applied the load to the beam slowl, counting the number of full pumps b the hdraulic pump and checking the output of the piezoelectric transducer with ever press. We recorded the entire process with audio and video so that we could analze the data with realtime audio and track the displacement in the video. A progression of the beam s displacement and change in acoustic emissions is shown below in Fig.13, relating the displacement to the differing wave-forms detected b the sensor. B analzing the wave forms, we can see a qualitative shift in the wave form as the beam undergoes increasing amounts of stress,
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.14 which agrees with the expectation. However, we do not see a repetition of wave forms that we expected. There is no evidence of reflected pulses that pass through the beam. Further experimentation with the piezoelectric transducers will hopefull shed light on this discrepanc in quantitative data. Although we did not break the beam, we were able to significantl stress the beam to the point of elastic deformation. We made this conclusion because after a significant number of presses b the hand pump, the pressure of the sstem did not increase, although the beam was continuing to displace. As shown in Fig.14, the linear relationship between pressure and displacement is onl maintained until a pressure around 900 PSI. After this pressure, the continued increase in displacement was not accompanied b an increase in pressure, which was good proof of elastic deformation. Now that we have successfull deformed an aluminum beam, and setup the acoustic emissions sensing technolog, the direction for future products is to test the load strength of more brittle material, such as reinforced concrete. The aluminum beams we used were ver malleable, which caused problems in tring to fracture the beams. We did however find the point of elastic deformation, which for structural purposes is the failure of the beam. Possible improvements to the process include using a digital pressure gauge for more precise measurements, or using piezoelectric force sensors to relate the displacement and load. We saw some slight movement in the supporting steel spacers during our experiment, so another improvement would be to solidif the frame and structure more. These improvements to the experiment are somewhat minor in the overall scope of the project, and we look forward to continuing this experiment in the future. [1] Dr. D. W. Cullington, D. MacNeil, et al. Transport Research Laborator. 70, 4 (2001).www.puretechnologiesltd.com/pdf/technicalpapers/Bridge11.pdf [2] D. Jia, J. Hamilton, L. M. Zaman, and A. Goonewardene. Am J. P. 75, 111 (2007).www.ntu.edu.sg/mae/Research/programmes/Sensors/projects/Lab-Projects/M312- Report1.pdf [3] S.P.Smithet al. AmJ.P.67, 26 (1999). [4] James M. Gere Mechanics of Materials. Brooks/Cole. Pacific Grove, CA. 2001. p. 14-15, 315-
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.15 0.16 0.14 0.12 Pressure: 0 PSI Voltage (mv) 0.1 0.08 0.06 0.04 0.02 0-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 Time (s) 0.2 0.15 Voltage (mv) 0.1 0.05 0 Pressure: 500 PSI -0.05-0.0015-0.001-0.0005 0 0.0005 Time (s) 0.2 0.001 0.0015-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0.15 Voltage (mv) 0.1 0.05 Pressure: 750 PSI 0-0.0015-0.001-0.0005 0 0.0005 Time (s) 0.25 0.001 0.0015-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0.2 Voltage (mv) 0.15 0.1 0.05 0 Pressure: 900 PSI -0.05-0.0015-0.001-0.0005 0 0.0005 Time (s) 0.25 0.001 0.0015-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0.2 Voltage (mv) 0.15 0.1 0.05 0 Pressure: 1000 PSI -0.05-0.0015-0.001-0.0005 0 0.0005 Time (s) 0.001 0.0015-0.003-0.002-0.001 0 0.001 0.002 0.003 FIG. 13. A progression of the beam s displacement in accordance with the wave-forms detected b the piezoelectric transducer.
WJP, PHY382 (2011) Wabash Journal of Phsics v4.3, p.16 1100 1000 900 Gauge Pressure (PSI) 800 700 600 500 400 1 2 3 4 5 6 7 Displacement (cm) FIG. 14. The continued application of load had a linear relationship between the measured pressure and the displacement until the beam began to deform elasticall around 1000 PSI. 322, 578-580. [5] M. Bobrowsk. Speed of Sound in Aluminum. Universit of Marland, Department of Phsics. http://www.phsics.umd.edu/lecdem/services/demos/demosh1/h1-23.htm [6] U. Esendemir. An Elastoplastic Stress Analsis of Aluminum Allo Metal-Matrix Composite Beams under a Transverse Linearl Distributed Load b Use of Anisotropic Elasticit Theor. Mech. of Adv. Materials and Structures. 12, 7 (2005). [7] F. Abali et al. Journal of Comp. Materials. 37, 5 (2003). [8] B. Schechinger, T. Vogel. Acoustic emission for monitoring a reinforced concrete beam subject to four-point-bending. Construction and Building Materials. 21, 3 (2007). 483-490.