Crack detection and repair of beams

CRACK DETECTION AND REPAIR OF BEAMS BY THIDA KYAW (B E) DEPARTMENT OF CIVIL ENGINEERING A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2005 I ACKNOWLEDGEMENT It is a great pleasure and to show my respect to Buhhda, Dhamma, Sangha, my parents, my eldest brother and all my teachers, who had brought me to this level I would not be able to guide myself without their teachings In the attempt to complete this thesis, I like to thank to all those who have helped me I could not have done it alone Special thanks go out to the following people: Associate Professor Ang Kok Keng, for guiding me throughout this study His extensive knowledge, serious research attitude and encouragement were extremely valuable to me; My colleagues at the Department of Civil Engineering of NUS In particular,Sithu Htun ,Tun Myint Aung and Cui Zhe, for their encouragement, and friendship assistance when things went difficult; My parents-in-law and my elder sister who bear responsibilities to look after my daughter so that I felt free to emphasize my study; I am indebted to the National University of Singapore for the award of a Research Scholarship during the period of candidature Finally, above all else, my loving husband, Naing Win Tun, for his invaluable support throughout my study Without him, my research study would not be possible I dedicate this thesis to him II TABLE OF CONTENTS ACKNOWLEDGEMENTS…………………………… ……………I TABLE OF CONTENTS…………………………………………… II SUMMARY………………………………………………………… VI NOTATIONS……………………………………………………… VII LIST OF FIGURES……………………………………….…….… XI CHAPTER INTRODUCTION 1.1 Background ……………………………………………………………… 1.2 Literature review ………………………………………………………… 1.2.1 Crack detection in the structure …………………………………… 1.2.2 Repair of cracked beam using piezoelectric material …………… 1.3 Objective and scope …………………………………………………… 1.4 Organization of thesis …………………………………………………… CHAPTER CRACK DETECTION UNDER STATIC LOADING 2.1 Introduction ……………………………………………………………… 2.2 Stress intensity factor and local flexibility …………………………… 10 2.3 Modeling of cracked Euler-Bernoulli beam ………………………… 12 III 2.4 Crack detection by deflection difference ……………………………… 13 2.4.1 2.4.2 2.5 Beams with one crack …………………………………………… 14 2.4.1.1 Simply supported beam 14 2.4.1.2 Cantilever beam 21 2.4.1.3 Simple-clamped beam ………………………………… 23 2.4.1.4 Clamped-clamped beam ………………………………… 25 Beams with two cracks ………………………………………… 27 Results and Discussion ………………………………………………… 2.5.1 31 Beams with one crack …………………………………………… 32 2.5.1.1 Simply supported beam ………………………… 32 2.5.1.2 Cantilever beam 33 2.5.1.3 Simple-clamped beam 34 2.5.1.4 Clamped-clamped beam 35 2.5.2 2.6 Simply supported beam with two cracks 35 Summary ………………………………………………………………… 36 CHAPTER REPAIR OF CRACKED BEAM USING PIEZOELECTRIC ACTUATOR 3.1 Introduction ……………………………………………………………… 51 3.2 Repair of cracked beam using piezoelectric patch …………………… 51 3.3 Repair of cracked Euler-Bernoulli beam ……………………………… 53 3.3.1 Simply supported beam ………………………………………… 54 3.3.2 Cantilever beam ………………………………………………… 58 IV 3.3.3 Clamped-clamped beam ………………………………………… 58 3.4 Example ……………………………………………………………… 59 3.5 Repair of Timoshenko beam …………………………………………… 60 3.5.1 Relationship between Euler-Bernoulli beam and Timoshenko beam ……………………………………………………………… 61 3.5.2 3.6 Example ……………………………………………………… 62 Summary ……………………………………………………………… 63 CHAPTER REPAIR OF CRACKED BEAM USING MOMENTS 4.1 Introduction …………………………………………………………… 72 4.2 Numerical modeling …………………………………………………… 73 4.2.1 Statically determinate beams with one crack …………………… 74 4.2.1.1 Simply supported beam ………………………………… 74 4.2.1.2 Cantilever beam ………………………………………… 76 4.2.2 Statically indeterminate beams with one crack ………………… 77 4.2.2.1 Simple-clamped beam ………………………………… 77 4.2.2.2 Clamped-clamped beam ………………………………… 79 4.3 Finite element model ………………………………………………… 80 4.4 Numerical examples and discussion ………………………………… 81 4.5 4.4.1 Effect of applied moment position on deflected shaped ………… 81 4.4.2 Effect of applied load position on moment required …………… 83 4.4.3 Effect of crack position on moment required …………………… 84 Summary ……………………………………………………………… 85 V CHAPTER CONCLUSIONS AND RECOMMENDATIONS 5.1 5.2 Conclusions …………………………………………………………… 94 5.1.1 Crack detection ………………………………………………… 94 5.1.2 Rrepair of crack ………………………………………………… 95 Recommendations for future study …………………………………… 96 REFERENCES ……………………………………………………… 97 VI SUMMARY This study can be divided into three portions Firstly, an analytical method for crack detection based on the difference in the measured static deflection of a damaged beam and the corresponding theoretical deflection of healthy beam of various boundary conditions is primarily presented in this study Equations for the damaged and healthy beams are formulated by analytically How to repair of cracked Euler-Bernoulli and Timoshenko beams with one crack can be made through the application of voltage to piezoelectric patch actuator placed on the surface of the beam over the crack surface is secondly presented Repair is considered to be effected when the discontinuity in the slope at the crack section completely eliminated Finally, a numerical investigation of crack in a beam is presented Repair was carried out by applying a pair of oppositely directed moments to eliminate the discontinuity of slope induced at a crack section The crack is modeled in the finite element model as an equivalent rotational spring The spring is allowed to rotate about the Z directional moment of the beam with transverse load along the Y axis The finite element software COSMOS/M is used for the numerical study Notations VII NOTATIONS A area of beam a depth of crack a H crack depth ratio B width of beam c1 , c , c n integrating constants E modulus of elasticity, Young’s modulus e31 electric constant for piezoelectric voltage F transverse vertical point load Gs modulus of rigidity H depth of beam I moment of inertia of beam J strain energy release rate K rotational spring constant KI stress intensity factor for first mode of fracture K II stress intensity factor for second mode of fracture K III stress intensity factor for third mode of fracture k shear correction factor L length of beam Notations L1 VIII crack location for beam with one crack first crack location for beam with two cracks L2 applied load position for beam with one crack second crack location for beam with two cracks L3 applied load position for beam with two cracks ME moment at a section of Euler-Bernoulli beam MT moment at a section of Timoshenko beam M a1 applied moment at the left side of crack M a2 applied moment at the right side of crack M a1 Mc ratio of required moment Mc moment at crack section of healthy beam M c' moment at crack section of damaged beam ML reaction moment at left end support of healthy beam MR reaction moment at right end support of healthy beam M L' reaction moment at left end support of damaged beam M R' reaction moment at right end support of damaged beam p1 distance from left end of piezoelectric patch to crack for first crack p2 distance from right end of piezoelectric patch to crack for first crack Notations p3 IX distance from left end of piezoelectric patch to crack for second crack p4 distance from right end of piezoelectric patch to crack for second crack RL reaction force at left end support of healthy beam RR reaction force at right end support of healthy beam RL' reaction force at left end support of damaged beam RR' reaction force at right end support of damaged beam β distance between two concentrated moments, M a1 and M a β distance from M a1 to crack section (or) distance from crack section to M a Va Electric voltage applied at piezoelectric patch x position of the beam section of consider y deflection of damaged beam y' rotation of damaged beam y" curvature of damaged beam Κ curvature of beam Θ local flexibility δ thickness of piezoelectric patch ω deflection of healthy beam φ rotation of healthy beam Chapter Repair of cracked beam using moments 87 Beam length (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.1 0.9 nocrack a/H=0.3 Deflection (mm) β=0.00001 -0.2 β=0.01 β=0.02 -0.3 -0.4 -0.5 -0.6 Figure 4.3 Deflection of healthy, cracked and the repaired cantilever beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) Beam length (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -0.001 Deflection (mm) -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 -0.008 nocrack a/H=0.3 β=0.00001 β=0.01 β=0.02 β=0.03 β=0.04 -0.009 Figure 4.4 Deflection of healthy, cracked and the repaired SC beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) Chapter Repair of cracked beam using moments 88 Beam length (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Deflection (mm) -0.001 -0.002 nocrack -0.003 a/H=0.3 β=0.00001 -0.004 β=0.01 β=0.02 -0.005 β=0.03 β=0.04 -0.006 Figure 4.5 Deflection of healthy, cracked and the repaired CC beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) 0.0001 0.00008 0.00006 Rotation (rad) 0.00004 0.00002 -0.00002 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 nocrack -0.00004 a/H=0.3 -0.00006 β=0.00001 -0.00008 β=0.01 -0.0001 Figure 4.6 β=0.02 Beam length (m) Rotation of healthy, cracked and the repaired SS beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) Chapter Repair of cracked beam using moments 89 Beam length (m) -0.0001 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 nocrack -0.0002 a/H=0.3 -0.0003 Rotation (rad) 0.9 β=0.00001 β=0.01 -0.0004 β=0.02 -0.0005 -0.0006 -0.0007 -0.0008 -0.0009 Figure 4.7 Rotation of healthy, cracked and the repaired cantilever beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) 0.00003 0.00002 Rotation (rad) 0.00001 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 nocrack a/H=0.3 -0.00001 β=0.00001 β=0.01 -0.00002 β=0.02 β=0.03 -0.00003 Figure 4.8 β=0.04 Beam length (m) Rotation of healthy, cracked and the repaired SC beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) Chapter Repair of cracked beam using moments 90 0.000025 nocrack 0.00002 a/H=0.3 β=0.00001 Rotation (rad) 0.000015 β=0.01 β=0.02 0.00001 β=0.03 0.000005 β=0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -0.000005 -0.00001 -0.000015 Beam length (m) Figure 4.9 Rotation of healthy, cracked and the repaired CC beam (crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) CF 1.8 SS SC Ma1/Mc 1.6 CS CC 1.4 1.2 0.8 0.1 0.2 0.3 0.4 0.5 Distance between two cracks (beta value) (m) Figure 4.10 Effect of applied moment positions to required magnitude of applied moments on various boundary conditions of repaired beam(crack position at L1 =0.4m, concentrated point load 100N at L2 =0.7m, crack depth ratio (a/H) = 0.3) Chapter Repair of cracked beam using moments 91 Ma1/Mc 1.2 L2=0.5m L2=0.6m L2=0.7m 0.8 0.1 0.2 0.3 0.4 Distance between two moments (beta value)(m) Figure 4.11 Simply supported beam (SS); Effect of applied load position to the required magnitude of applied moments (crack position at L1 =0.4m, crack depth ratio (a/H) = 0.3) 1.4 Ma1/Mc 1.2 L2=0.5m L2=0.6m L2=0.7m 0.8 0.1 0.2 0.3 0.4 Distance between two moments (beta value) (m) Figure 4.12 Simple-clamped beam (SC); Effect of applied load position to the required magnitude of applied moments (crack position at L1 =0.4m, crack depth ratio (a/H) = 0.3) Chapter Repair of cracked beam using moments 92 1.8 Ma1/Mc 1.6 1.4 1.2 L2=0.5m L2=0.6m L2=0.7m 0.8 0.1 0.2 0.3 0.4 Distance between two moments (beta value) (m) Figure 4.13 Clamped-clamped beam (CC); Effect of applied load position to the required magnitude of applied moments (crack position at L1 =0.4m, crack depth ratio (a/H) = 0.3) 15000 Mc (M-mm) 10000 5000 0.1 0.2 0.3 0.4 0.5 0.6 SC -5000 CS CC -10000 Beam length (m) Figure 4.14 Moment of healthy beam; (applied load position at L =0.7m) Chapter Repair of cracked beam using moments 93 1.8 Ma1/Mc 1.6 1.4 1.2 L1=0.6m L1=0.5m L1=0.4m 0.8 0.1 0.2 0.3 0.4 Beta Value (m) Figure 4.15 Simple-clamped beam (SC); Effect of crack position to the required magnitude of applied moments (applied load position at L2 =0.7m, crack depth ratio (a/H) = 0.3) 1.8 Ma1/Mc 1.6 1.4 1.2 L1=0.6m L1=0.5m L1=0.4m 0.8 0.1 0.2 0.3 0.4 Beta Value (m) Figure 4.16 Clamped-clamped beam (CC); Effect of crack position to the required magnitude of applied moments (applied load position at L2 =0.7m, crack depth ratio (a/H) = 0.3) Chapter Conclusions and Recommendations 94 CHAPTER CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions Two main contributions are made in this study First is the detection of cracks using analytical calculations for beams of various boundary conditions and second is the repair of cracked beam In repair of cracked beam, the study is made up of two parts The first part comprises of the repair using piezoelectric patch actuators and the second part involves a FE study of the repair using two oppositely directed concentrated moments applied at either side of the crack section 5.1.1 Crack detection A method for the detection of crack using the difference in deflection which is measured from a suspected damaged beam and a corresponding healthy beam has been proposed The deflections of healthy beam of various boundary conditions are well known from readily available formulas The difference in deflection value is found to be maximum at the crack section and this information may therefore be used to detect the Chapter Conclusions and Recommendations 95 resistance of a crack including its location and the severity of the crack The proposed method is simple and easy to implement as it entails only static effect measurements 5.1.2 Repair of crack Two approaches to the repair of cracked beam are discussed Firstly, an analytical formulation of the repair of cracked beam piezoelectric patch actuators is presented Repair is carried out by placing a small piezoelectric patch directly under the crack so as to induce a local moment upon application of a suitable voltage to the piezoelectric actuators Secondly, the repair of cracked beam through applying a pair of oppositely directed concentrated moments on two sides of the crack is also presented In this case, the required magnitude of moment is found to be equal to the moment at the crack section for statically determinate beam The required moment is also not dependent on other factors When β , the distance between two concentrated applied moments, is very small, the deflected shape and rotation of repaired beam are nearly identical to the healthy one For statically indeterminate beams, the required magnitude of moment is equal to the moment at the crack section when β tends to zero However, the magnitude varies and is dependent on other factors when β 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International Journal of Structural Stability and Dynamics, 3(1), 17-33 26 Wang Q and Deng X (1999), “ Damage detection with spatial wavelets ” International Journal of Solids and Structures, 36, 3443-3468 100 References 27 Wang Q., Quek S T and Liew K M (2002), “ On the repair of cracked beam with a piezoelectric patch” Smart Materials and Structures, 11, 404-410 101 ... the repair of cracked beam with one crack is presented and the chapter also discusses the repair of Timoshenko cracked beams where shear effects are significant in thick beams In chapter the repaired... conditions of beams (crack position at L1 = 0.4m and concentrated point load 100N at L2 =0.7m) Figure 2.19 Deflection of healthy and cracked cantilever beam (crack position at L1 =0.4m and concentrated... at L1 =0.3m and L2 = 0.5m and concentrated point load 100N at L3 =0.7m) Figure2.24 Rotation of healthy and cracked SS beam with two cracks (crack position at L1 =0.3m and L2 = 0.5m and concentrated