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Determination of damage location in RC beams using mode shape derivatives
Engineering Structures 28 (2006) 1566–1573 www.elsevier.com/locate/engstruct
Determination of damage location in RC beams using mode shape derivatives Z. Ismail, H. Abdul Razak ∗ , A.G. Abdul Rahman Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Received 17 May 2005; received in revised form 16 February 2006; accepted 17 February 2006 Available online 2 May 2006
Abstract This paper describes the determination of the location of damage due to single cracks and due to honeycombs in RC beams using mode shape derivatives from modal testing. The cracks were induced by application of point loads at predetermined locations on the RC beams. The load was increased in stages to obtain different crack heights to simulate the extent and severity of damage. Experimental modal analysis was performed on the beams with cracks prior to and after each load cycle, on a control beam, and beams with honeycombs. The mode shapes and the eigenvectors were used to determine the location of damage. The indicator |λ4 | was obtained by rearranging the equation for free transverse vibration of a uniform beam, and applying the fourth order centered finite-divided difference formula to the regressed mode shape data. The equation is an eigenvalue problem, and the value of |λ4 | will be a constant. Differences in the values indicate stiffness change, and the affected region indicates the general area of damage. Analysis of results using |λ4 | was able to indicate the general region of damage, the exact location being around the center of the region. Curve fitting with Chebyshev series rationals onto the mode shape also highlighted points of high residuals around the region of damage. The proposed algorithm on the mode shape can form the basis of a technique for structural health monitoring of damaged reinforced concrete structures. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Reinforced concrete; Modal test; Natural frequency; Mode shape derivative; Load carrying capacity
1. Introduction Periodic structural condition monitoring of reinforced concrete structures is necessary to ensure that they provide a continued safe service condition. Conventional assessment procedures usually rely on visual inspection and locationdependent methods. This study proposes the application of experimental modal analysis to determine the location of damage in the form of load-induced cracks and honeycombs in reinforced concrete (RC) beams. There have been several significant investigative studies carried out to determine the existence and the severity of defects in structures using one or more of their modal properties [1–9]. In most of the dynamic tests conducted on actual structures the fundamental natural frequencies have been utilized and
Abbreviations: RC: reinforced concrete, GMO: geometric mean operator, LSI: local stiffness indicator. ∗ Corresponding author. Tel. +60 3 79595233; fax: +60 3 79595233. E-mail address: [email protected] (H.A. Razak). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.02.010
found to be the most convenient parameter to be studied [10, 11]. It was found that the most easily observable change is the reduction in natural frequencies, and most investigators use this feature in one way or another. Casas [12] proposed a method of surveillance of concrete structures through monitoring the characteristics of the natural frequencies and mode shapes. Varying success has been reported where the change in modal damping has been utilized, while some work has been reported on the use of change of mode shape to detect damage. Ratcliffe [13] presented a technique for locating damage in a beam that uses a finite difference approximation of a Laplacian operator on mode shape data. In the case of damage, which is not so severe, further processing of the Laplacian output is necessary before damage location could be determined. The procedure is found to be best suited for the mode shape obtained from the fundamental natural frequency. The mode shapes obtained from higher natural frequencies may be used to verify the damage location, but they are not as sensitive as the lower modes. Yoon et al. [14] expands the ‘gapped smoothing method’ for identifying the location of structural damage in
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a beam by introducing a ‘globally optimized smooth shape’ with an analytic mode shape function and the procedure uses only the mode shapes from the damaged structure. The method can detect local stiffness losses associated with local thickness reduction of less than 1% in the case of narrow and wide damage, 13 mm and 126 mm, respectively, with finite element analysis. Instead of using mode shapes in obtaining spatial information about sources of vibration changes, an alternative method is by using the mode shape derivatives, such as curvature. It is noted that for beams, plates, and shells there is a direct relationship between curvature and bending strain. Pandey et al. [15] demonstrated that absolute changes in mode shape curvature can be a good indicator of damage for the cantilever and simply supported analytical beam structures, which they considered. The changes in the curvature increase with increase in damage. The curvature values were computed from the displacement mode shape using the central difference approximation. Stubbs et al. presented a method based on the decrease in the curvature of the measured mode shapes or the modal strain energy between two structural degrees of freedom [16]. Topole and Stubbs further showed that it was feasible to use a limited set of modal parameters to detect structural damage [17,18]. Stubbs and Kim also showed that localizing damage using this technique without baseline modal parameters was possible [19]. This approach was confirmed by Chance et al. [20] who found that numerically calculating curvature from mode shapes resulted in unacceptable errors. As a consequence measured strains were instead used to measure curvature directly, and this improved results significantly. Maeck et al. [21] used a technique to predict the location and intensity of damage directly from measured modal displacement derivatives. The technique, direct stiffness derivation, uses the basic relation that the dynamic bending stiffness, E I , in each section is equal to the bending moment, M, in that section divided by the corresponding curvature; and the dynamic torsion stiffness, G J , in each section is equal to the torsional moment, T , in that section divided by the corresponding torsion rate or torsion angle per unit length. Direct calculation of the first and second derivatives from measured mode shapes results in oscillating and inaccurate values. A smoothing procedure, which is a weighted residual penalty-based technique, is applied to the measured mode shapes. The technique is further validated by Maeck and De Roeck [22] on a reinforced concrete beam, which was gradually damaged, and using instruments such as accelerometers, displacement transducers, and strain gauges. Khezel [23] performed a feasibility study on using modal testing as an inspection and surveillance tool to determine honeycombs. Mode shape data was analyzed using various modal assurance techniques, thus improving the possibility of locating the defect regions. Geometric mean operator (GMO) was proposed. The square is chosen instead of the square root for more efficient calculation of this operator since it deals with the deviation value ym from the geometric mean of the neighboring values. This operator ensures that the deviation of
Fig. 1. Experimental beam, simply supported in the modal test set-up. Table 1 Characteristics of loading cycles for beam with crack at 0.5L
No load Load cycle 1 Load cycle 2
Label
Maximum load applied (kN)
Note
Datum ML25 ML43
0 24.4 43.2
Undamaged First crack occurred Crack developed extensively
ym from the neighboring values is always positive and that it will be magnified whenever there is a deviation. This paper describes the determination of the location of damage in reinforced concrete beams due to load-induced cracks and due to honeycombs through modal testing. Modal tests on the beams were conducted to determine the modal parameters, namely frequencies and mode shapes. Modal parameters are functions of the physical properties of the structure, which are mass, damping, and stiffness; and changes in the physical properties will cause detectable changes in these modal properties. The main objective of the research study is to establish indicators for the purpose of correlating this behavior. 2. Experimental programme In this investigation, five RC beams were cast. The dimensions of the reinforced concrete beams were 150 mm wide and 250 mm deep. The beams were simply supported across an effective span of 2200 mm on concrete blocks as in Fig. 1. The first was a beam with a crack at 0.5L (Table 1) and the second was a beam with a crack at 0.7L (Table 2). A 20 mm deep saw-cut was introduced across the first beam at the soffit points at 0.5L and across the second beam at the soffit points at 0.7L. A concentrated point load was then applied at the position where the saw-cut was located. A load cycle comprised of the load applied incrementally in the following manner: zero to maximum loading at increments of approximately 0.75 kN each time and unload from maximum loading to 0 kN with decrements of approximately 0.75 kN each time. Fig. 3 depicts the set-up for the static loading test. The first beam was then subjected to maximum loading of 25 and 43 kN and modal testing carried out at each cycle while the second beam was subjected to maximum loading of 25, 41, 43, and 46 kN and
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Table 2 Characteristics of loading cycles for beam with crack at 0.7L Label
Maximum load applied (kN)
Note
No load Load cycle 1 Load cycle 2
Datum ML25 ML41
0 24.64 41.36
Load cycle 3
ML43
42.97
Load cycle 4
ML46
46.31
Undamaged First crack occurred Crack developed extensively Crack developed extensively Crack width enlarged
Table 3 Labels and dimensions of the mortar blocks
Fig. 2. Modal test set-up.
Label
Dimensions of mortar blocks (L × W × H mm)
L5 L5 × 2
440 × 90 × 90 440 × 90 × 180
Table 4 Crack width at level −230 mm from the top surface of beam with crack at 0.5L Label
Average width (mm)
Datum ML25 ML43
0.000 0.080 1.310
Table 5 Crack width at level −230 mm from the top surface of beam with crack at 0.7L Label
Average width (mm)
Datum ML25 ML41 ML43 ML46
0.000 0.073 0.383 0.700 1.076
again modal testing was carried out at each cycle. Crack widths, crack depths, and modal parameters were recorded for both beams at all cycles. Measurements for crack width of both beams were recorded from both sides of the beams at the level 230 mm from the top surface of the beams by using two crack gauges. Tables 4 and 5 list the average crack width for both beams, respectively. The final three beams were a control beam, a defect beam L5 and another defect beam L5 × 2 with various volumes of honeycombs. The dimensions of the mortar blocks are given in Table 3. The method used for creating the honeycombs in this investigation was by precasting mortar blocks with a known amount of polystyrene beads and placing them at the mid-span of the beam prior to casting the beams as shown in Fig. 4. Modal testing was also carried out on these beams. 2.1. Modal test The modal test was performed using a transfer function technique on the RC beams, which were simply supported
Fig. 3. Static load test set-up.
on concrete blocks. An accelerometer was used to pick up the response of the test beam under forced excitation. The excitation resulting from the input force was measured using a force transducer. The RC beams were randomly excited using white noise signal input to a shaker, which was permanently placed at the quarter span for all test beams. The accelerometer was moved from one coordinate point to another to pick up a total of fifty-six response signals along the length of the beam. The transfer functions were acquired through a signal analyser. Initially, the transfer function spectrum within a 5 kHz frequency span was obtained in order to locate roughly the resonant frequency peaks of all the flexural modes within the band. Subsequently, zooming within a 100 Hz span of the resonant frequency peak of a particular mode was carried out. The measurements were made using a block size of 400 lines thus giving a resolution of 0.25 Hz per spectral line. A dual-channel analyzer was used to acquire the signals and to obtain the frequency response functions (FRF) from the response and the excitation force. By using modal analysis software, the curve fitting process was performed on the transfer function spectra obtained to extract the modal parameters i.e. natural frequency, mode shape and damping. A total of ten normal bending modes were acquired in this manner. Fig. 1 shows the actual modal test set-up and Fig. 2 depicts the setup of the modal testing.
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Fig. 4. Mortar blocks to represent honeycombs.
Fig. 5. Curve fitting with Chebyshev series rational for mode 4.
3. Results and discussions For an Euler beam the equation (iv)
d4 y dx 4
− λ4 y = 0 can be
rearranged as |λ4 | = | y y |. The values of |λ4 |, which may be referred to as the local stiffness indicator (LSI) drawn as bar charts for an undamaged beam, remain constant along the length of the beam. Any changes to the natural frequencies or flexural stiffness due to occurrences of damage in the beam will show changes in the value of |λ4 |. For the beams with defects, it is approximated that the generalized equation applies. To determine the location of damage, the indicator |λ4 | is used to determine the general region of the cracks or the honeycombs. In addition, a curve-fitting technique employing the Chebyshev series rationals was performed on the experimental mode shape to prepare the data before applying
the fourth order centered finite-divided difference formula. This is done so that random errors could be eliminated. Then, the value of |λ4 | is obtained. The curve fitting technique on the mode shape data is also found to be fairly successful in highlighting points of high residuals from the curve around the region of the damage. Fig. 5 shows examples of curve fitting with Chebyshev series rationals for mode 4 on the following cases: beam with no defect, beam with load induced cracked at 0.5L after loading of 43 kN, beam with load induced crack at 0.7L after loading of 46 kN, and beam with honeycombs L5. 3.1. Load induced cracking at 0.5L and 0.7L Figs. 6 and 7 show the plots of |λ4 | for the beam with a crack at 0.5L and 0.7L, respectively, along the datum and their
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Fig. 6. Location of damage for beam with load induced crack at 0.5L.
Fig. 7. Location of damage for beam with load induced crack at 0.7L.
respective maximum loading for mode 1, mode 2, mode 3, and mode 4 after curve fitting was done to the mode shape data. For all modes, the datum case showed that for the most part of both beams the values of |λ4 | remained about constant,
except at one end of the beam where the values showed some small increases in the values of |λ4 |. Mode 1 of the beam with crack at 0.5L showed increases in the values of |λ4 | at the ends of the beam labeled ML25, and large increases at one end of the beam labeled ML41. It was
Z. Ismail et al. / Engineering Structures 28 (2006) 1566–1573
also seen that there were small increases and a wide area of large increases in the values of |λ4 | around the region of the crack for the beam labeled ML25 and ML43, respectively. For the beam labeled ML43, the maximum increases in the values of |λ4 | were at points 23 and 24. Mode 1 of the beam with crack at 0.7L showed a wide region around the crack where there are marked increases in the values of |λ4 | with a maximum at point 35 for the beam labeled ML25, at point 28 for the beam labeled ML41, at point 35 for the beam labeled ML43, and at point 33 for the beam labeled ML46. All these maximum points were fairly close to the crack location. It was also clearly seen that there were large increases in the values of |λ4 | at both ends of the beam. Mode 2 of the beams with crack at 0.5L labeled ML25 and labeled ML41 still indicated increases in the values of |λ4 | at the ends of the beam. It was also clearly seen that there were very large increases in the values of |λ4 | around the region of the crack, with a maximum value of |λ4 | at point 26 and point 24 of the beams labeled ML25 and ML43, respectively. These points were close to the crack location. Mode 2 of the beam with crack at 0.7L did not show any trend near the crack location for the beams labeled ML25, ML43 and ML46. However, it showed relatively higher values of |λ4 | near the ends of the three beams. It also showed marked increases in the values of |λ4 | at points 25 and 26 for the beams labeled ML43 and ML46. It was also clearly seen in mode 2 that the affected region around the crack was quite wide with large increases in the values of |λ4 | at point 31 and 47, which were on either side of the crack location for the beam labeled ML41. There were also high values of |λ4 | at both ends of the beam. For mode 3 of the beam with crack at 0.5L showed very large increases in the values of |λ4 | occurring at point 18 for the case of the beam labeled ML25, and there were also large increases in the values of |λ4 | around the region of the crack for the case of the beam labeled ML41. There were increases in the values of |λ4 | at both ends of the beams with crack at 0.7L labeled ML25 and ML46 in mode 3. It was also seen that the region around the crack had high values of |λ4 | with a maximum at point 36 for the beam labeled ML25 and at point 40 for the beams labeled ML41, ML43, and ML46. Mode 3 for the case of the beam labeled ML25 also showed an increase in the values of |λ4 | at point 21. Mode 4 of the beam with crack at 0.5L labeled ML25 showed very large increases in the values of |λ4 | at the ends of the beam and around the region of the crack with a maximum at point 27, which was close to the crack location. The beam labeled ML41 showed very similar behavior to the other modes with a wide area of increases in the value of |λ4 | around the crack. The maximum value was shown at point 27, which was close to the crack location. Mode 4 of the beam with crack at 0.7L showed there were increases in the values of |λ4 | at the ends for all the beams except ML43. The region around the crack showed clearly increases in the values of |λ4 | with a maximum at point 40, which was close to the crack location, for the beams labeled ML25, ML41, ML43 and ML46. The indicator |λ4 | for the datum experimental beam with cracks at 0.5L and 0.7L remained small for all modes except
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at some points at the ends of the beam. The curve fitting of the mode shape data for all the modes investigated also indicated that the datum beam showed no increases in the values of |λ4 | along its entire length. Imperfect boundary end conditions were manifested in increased values of the indicator. Except for the increases in the values of |λ4 | at the ends of the beam, taking the four modes together, the indicator had successfully identified a distinct region around the crack with values far exceeding the average values |λ4 | along the beam. 3.2. Honeycombs Fig. 8 shows the plots of |λ4 | for the control beam and beams with honeycombs for mode 1 to mode 4 after curve fitting was done to the mode shape data. Mode 1 did not show any trend of increases in the values of |λ4 | along the entire length of all beams. For mode 2, mode 3, and mode 4 of the control beam, the most part of it showed constant low values of |λ4 | except at points 5, 17, and 44 in mode 4. Mode 2 showed increases in the values of |λ4 | around the region of the honeycombs with a maximum at point 26, which was about the center of the honeycombs, for the beams labeled L5 and L5 × 2. The increase was more prominent in the beam label L5 × 2. Mode 3 showed increases of values of |λ4 | about the ends of the honeycombs region with a maximum at point 18 and at point 35, which represented the extremities of the honeycombs for the beam labeled L5. There were also increases in values of |λ4 | with a maximum at point 35, which represented one extremity of the honeycombs for the beam labeled L5×2. There were increases in values of |λ4 | at the ends of the beams labeled L5 and L5 × 2 shown in mode 4. Mode 4 also showed clearly increases in the values of |λ4 | around the region of the honeycombs for both beams with a maximum at point 26, which was about the center of the honeycombs. Taking the four modes together, the indicator |λ4 | for the control beam remained small and constant for all modes except at a few points along the beam in mode 4. The region around the honeycombs showed clearly increases in the values of the indicator |λ4 | with a maximum at point 26, which was about the center of the honeycombs for the beams labeled L5 and L5 × 2. The analysis of the experimental results using LSI was able to indicate the general region of the crack and honeycombs, the exact location of the damage being around the center of the region; but the severity of the damage could not be easily distinguished. LSI also indicated problems with the mode shapes at the ends of the beam. This was because it was very difficult to get good mode shapes from a simply supported system using rollers due to the constraining effects at the ends. 4. Conclusions A curve fitting technique was applied to experimental data to examine kinks in the curve, which could be due to the response to structural damage. In the case of the damaged beams, curve fitting the mode shape data using the Chebyshev series rationals produced curves with higher residual values around the damage location as compared to the other parts of the beam. Beams with a load-induced crack at 0.5L and 0.7L, and honeycombs L5
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Fig. 8. Location of damage for beam with honeycombs.
and L5 × 2 produced curves with low residual values along the entire length. Non-ideal boundary end conditions produced increased values of the residuals. Taking the patterns of the residuals for all the modes as a whole and the intensity of the increased residuals, it can be seen that there were irregularities in mode shape around the crack locations and about the region of the honeycombs. An indicator |λ4 | or LSI is proposed to be used in determining the location of the damage by examining the intensity of increases in the values of the indicator around the damage region. It was found that the maximum increases were close to the crack location or around the middle of the honeycombs. The severity of the damage and the resulting loss of structural carrying capacity of the beams due to the damage were also indicated by a higher intensity of perturbations of the indicator. There was a trend shown in the loss of structural stiffness from the changes in the modal parameters such as natural frequencies and mode shapes. The technique does not depend on undamaged or datum state data, thus it can easily be used for structural assessment purposes of RC structures as an alternative to more conventional approaches which are more labor intensive and with higher costs due to more elaborate procedures and longer system shutdown time. For actual structural systems the procedure could be applied to individual structural elements since the damage is normally localized. In the procedure, the generalized solution was used, and the results showed that this is still applicable to handle the dynamic behavior of damaged RC beams. Also in all cases the procedures produced poor results in the vicinity of the supports implying that the techniques cannot detect damage close to the supports.
Acknowledgements The authors gratefully acknowledge the financial assistance provided by the National Council for Scientific Research and Development through a research grant under the Intensification of Research in Priority Areas (IRPA) programme under project 03-02-03-0610. References [1] Chen Y, Swamidas ASJ. Dynamic characteristics and modal parameters of a plate with a small growing surface crack. In: Proc. of 12th international modal analysis conference. 1994, p. 1155–61. [2] Dong C, Zhang PQ, Feng WQ, Huang TC. The sensitivity study of the modal parameters of a cracked beam. In: Proc. of 12th international modal analysis conference. 1994, p. 98–104. [3] Kam TY, Lee TY. Detection of cracks in structures using modal test data. Engineering Fracture Mechanics 1992;42(2):381–7. [4] Lim TW. Structural damage detection using modal test data. AIAA Journal 1991;29(12):2271–4. [5] Liu PL. Identification and damage detection of stresses using modal data. Journal of Structural Engineering 1995;121(4):599–608. [6] Narkis Y. Identification of crack location in vibrating simply supported beams. Journal of Sound and Vibration 1994;172(4):549–58. [7] Nwosu DI, Swamidas ASJ, Guigne JY, Olowokere DO. Studies on influence of cracks on the dynamic response of beams. In: Proc. of 13th international modal analysis conference. 1995, p. 1122–8. [8] Penny JET, Wilson DAL, Friswell MI. Damage location in structures using vibration data. In: Proc. 11th international modal analysis conference. 1993, p. 861–7. [9] Rizos PF, Aspragathos N, Dimarogonas AD. Identification of crack location and magnitude in a cantilever from the vibration modes. Journal of Sound and Vibration 1990;138(3):381–8.
Z. Ismail et al. / Engineering Structures 28 (2006) 1566–1573 [10] Javor T. Damage classification of concrete structures. The state of the art report of RILEM technical committee 104-DCC. Material and Structures 1991;24:253–9. [11] Konig G, Giegerich G. Identification of structural properties using dynamic tests. In: Proc. of the IABSE symposium on durability of structures 1989, p. 835–40. [12] Cawley F, Adams RD. The locations of defects in structures from measurements of natural frequencies. Journal of Strain Analysis 1979; 14(2):49–57. [13] Ratcliffe CP. Damage detection using a modified laplacian operator on mode shape data. Journal of Sound and Vibration 1997;204(3):505–17. [14] Yoon MK, Heider D, Gillespsie JW Jr, Ratcliffe CP, Crane RM. Local damage detection using a global fitting method on mode shape data. In: Proceeding of 19th international modal analysis conference. 2001, p. 1358–62. [15] Pandey AK, Biswas M, Samman MM. Damage detection from changes in curvature mode shapes. Journal of Sound and Vibration 1991;145(2): 321–32. [16] Stubbs N, Kim JT, Topole KG. An efficient and robust algorithm for damage localization in offshore platforms. In: Proc. of ASCE 10th structures congress. 1992, p. 543.
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[17] Topole KG, Stubbs N. Nondestructive damage evaluation of a structure from limited modal parameters. Earthquake Engineering and Structural Dynamics 1995;24(11):1427–36. [18] Topole KG, Stubbs N. Nondestructive damage evaluation in complex structures from a minimum of modal parameters. International Journal of Analytical and Experimental Modal Analysis 1995;10(2):95–103. [19] Stubbs N, Kim JT. Damage localization in structures without baseline modal parameters. AIAA Journal 1996;34(8):1644–9. [20] Chance J, Tomlinson GR, Worden K. Simplified approach to the numerical and experimental modeling of the dynamics of a cracked beam. In: Proc. of 12th international modal analysis conference. 1994, p. 778. [21] Maeck J, Abdel Wahab M, De Roec G. Damage localization in reinforced concrete beams by dynamic stiffness determination. In: Proc. 17th international modal analysis conference. 1999, p. 1289–95. [22] Maeck J, De Roeck G. Damage assessment of a gradually damaged RC beams by dynamic stiffness determination. In: Proceeding of 17th international modal analysis conference. 2002, p. 1289–95. [23] Khezel IS. The effect of honeycombing on modal response of reinforced concrete beams. Thesis. Kuala Lumpur (Malaysia): M.Eng. Sc., Faculty of Engineering, University of Malaya; 2002.
Determination of damage location in RC beams using mode shape derivatives Z. Ismail, H. Abdul Razak ∗ , A.G. Abdul Rahman Department of Civil Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Received 17 May 2005; received in revised form 16 February 2006; accepted 17 February 2006 Available online 2 May 2006
Abstract This paper describes the determination of the location of damage due to single cracks and due to honeycombs in RC beams using mode shape derivatives from modal testing. The cracks were induced by application of point loads at predetermined locations on the RC beams. The load was increased in stages to obtain different crack heights to simulate the extent and severity of damage. Experimental modal analysis was performed on the beams with cracks prior to and after each load cycle, on a control beam, and beams with honeycombs. The mode shapes and the eigenvectors were used to determine the location of damage. The indicator |λ4 | was obtained by rearranging the equation for free transverse vibration of a uniform beam, and applying the fourth order centered finite-divided difference formula to the regressed mode shape data. The equation is an eigenvalue problem, and the value of |λ4 | will be a constant. Differences in the values indicate stiffness change, and the affected region indicates the general area of damage. Analysis of results using |λ4 | was able to indicate the general region of damage, the exact location being around the center of the region. Curve fitting with Chebyshev series rationals onto the mode shape also highlighted points of high residuals around the region of damage. The proposed algorithm on the mode shape can form the basis of a technique for structural health monitoring of damaged reinforced concrete structures. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Reinforced concrete; Modal test; Natural frequency; Mode shape derivative; Load carrying capacity
1. Introduction Periodic structural condition monitoring of reinforced concrete structures is necessary to ensure that they provide a continued safe service condition. Conventional assessment procedures usually rely on visual inspection and locationdependent methods. This study proposes the application of experimental modal analysis to determine the location of damage in the form of load-induced cracks and honeycombs in reinforced concrete (RC) beams. There have been several significant investigative studies carried out to determine the existence and the severity of defects in structures using one or more of their modal properties [1–9]. In most of the dynamic tests conducted on actual structures the fundamental natural frequencies have been utilized and
Abbreviations: RC: reinforced concrete, GMO: geometric mean operator, LSI: local stiffness indicator. ∗ Corresponding author. Tel. +60 3 79595233; fax: +60 3 79595233. E-mail address: [email protected] (H.A. Razak). c 2006 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2006.02.010
found to be the most convenient parameter to be studied [10, 11]. It was found that the most easily observable change is the reduction in natural frequencies, and most investigators use this feature in one way or another. Casas [12] proposed a method of surveillance of concrete structures through monitoring the characteristics of the natural frequencies and mode shapes. Varying success has been reported where the change in modal damping has been utilized, while some work has been reported on the use of change of mode shape to detect damage. Ratcliffe [13] presented a technique for locating damage in a beam that uses a finite difference approximation of a Laplacian operator on mode shape data. In the case of damage, which is not so severe, further processing of the Laplacian output is necessary before damage location could be determined. The procedure is found to be best suited for the mode shape obtained from the fundamental natural frequency. The mode shapes obtained from higher natural frequencies may be used to verify the damage location, but they are not as sensitive as the lower modes. Yoon et al. [14] expands the ‘gapped smoothing method’ for identifying the location of structural damage in
1567
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a beam by introducing a ‘globally optimized smooth shape’ with an analytic mode shape function and the procedure uses only the mode shapes from the damaged structure. The method can detect local stiffness losses associated with local thickness reduction of less than 1% in the case of narrow and wide damage, 13 mm and 126 mm, respectively, with finite element analysis. Instead of using mode shapes in obtaining spatial information about sources of vibration changes, an alternative method is by using the mode shape derivatives, such as curvature. It is noted that for beams, plates, and shells there is a direct relationship between curvature and bending strain. Pandey et al. [15] demonstrated that absolute changes in mode shape curvature can be a good indicator of damage for the cantilever and simply supported analytical beam structures, which they considered. The changes in the curvature increase with increase in damage. The curvature values were computed from the displacement mode shape using the central difference approximation. Stubbs et al. presented a method based on the decrease in the curvature of the measured mode shapes or the modal strain energy between two structural degrees of freedom [16]. Topole and Stubbs further showed that it was feasible to use a limited set of modal parameters to detect structural damage [17,18]. Stubbs and Kim also showed that localizing damage using this technique without baseline modal parameters was possible [19]. This approach was confirmed by Chance et al. [20] who found that numerically calculating curvature from mode shapes resulted in unacceptable errors. As a consequence measured strains were instead used to measure curvature directly, and this improved results significantly. Maeck et al. [21] used a technique to predict the location and intensity of damage directly from measured modal displacement derivatives. The technique, direct stiffness derivation, uses the basic relation that the dynamic bending stiffness, E I , in each section is equal to the bending moment, M, in that section divided by the corresponding curvature; and the dynamic torsion stiffness, G J , in each section is equal to the torsional moment, T , in that section divided by the corresponding torsion rate or torsion angle per unit length. Direct calculation of the first and second derivatives from measured mode shapes results in oscillating and inaccurate values. A smoothing procedure, which is a weighted residual penalty-based technique, is applied to the measured mode shapes. The technique is further validated by Maeck and De Roeck [22] on a reinforced concrete beam, which was gradually damaged, and using instruments such as accelerometers, displacement transducers, and strain gauges. Khezel [23] performed a feasibility study on using modal testing as an inspection and surveillance tool to determine honeycombs. Mode shape data was analyzed using various modal assurance techniques, thus improving the possibility of locating the defect regions. Geometric mean operator (GMO) was proposed. The square is chosen instead of the square root for more efficient calculation of this operator since it deals with the deviation value ym from the geometric mean of the neighboring values. This operator ensures that the deviation of
Fig. 1. Experimental beam, simply supported in the modal test set-up. Table 1 Characteristics of loading cycles for beam with crack at 0.5L
No load Load cycle 1 Load cycle 2
Label
Maximum load applied (kN)
Note
Datum ML25 ML43
0 24.4 43.2
Undamaged First crack occurred Crack developed extensively
ym from the neighboring values is always positive and that it will be magnified whenever there is a deviation. This paper describes the determination of the location of damage in reinforced concrete beams due to load-induced cracks and due to honeycombs through modal testing. Modal tests on the beams were conducted to determine the modal parameters, namely frequencies and mode shapes. Modal parameters are functions of the physical properties of the structure, which are mass, damping, and stiffness; and changes in the physical properties will cause detectable changes in these modal properties. The main objective of the research study is to establish indicators for the purpose of correlating this behavior. 2. Experimental programme In this investigation, five RC beams were cast. The dimensions of the reinforced concrete beams were 150 mm wide and 250 mm deep. The beams were simply supported across an effective span of 2200 mm on concrete blocks as in Fig. 1. The first was a beam with a crack at 0.5L (Table 1) and the second was a beam with a crack at 0.7L (Table 2). A 20 mm deep saw-cut was introduced across the first beam at the soffit points at 0.5L and across the second beam at the soffit points at 0.7L. A concentrated point load was then applied at the position where the saw-cut was located. A load cycle comprised of the load applied incrementally in the following manner: zero to maximum loading at increments of approximately 0.75 kN each time and unload from maximum loading to 0 kN with decrements of approximately 0.75 kN each time. Fig. 3 depicts the set-up for the static loading test. The first beam was then subjected to maximum loading of 25 and 43 kN and modal testing carried out at each cycle while the second beam was subjected to maximum loading of 25, 41, 43, and 46 kN and
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Table 2 Characteristics of loading cycles for beam with crack at 0.7L Label
Maximum load applied (kN)
Note
No load Load cycle 1 Load cycle 2
Datum ML25 ML41
0 24.64 41.36
Load cycle 3
ML43
42.97
Load cycle 4
ML46
46.31
Undamaged First crack occurred Crack developed extensively Crack developed extensively Crack width enlarged
Table 3 Labels and dimensions of the mortar blocks
Fig. 2. Modal test set-up.
Label
Dimensions of mortar blocks (L × W × H mm)
L5 L5 × 2
440 × 90 × 90 440 × 90 × 180
Table 4 Crack width at level −230 mm from the top surface of beam with crack at 0.5L Label
Average width (mm)
Datum ML25 ML43
0.000 0.080 1.310
Table 5 Crack width at level −230 mm from the top surface of beam with crack at 0.7L Label
Average width (mm)
Datum ML25 ML41 ML43 ML46
0.000 0.073 0.383 0.700 1.076
again modal testing was carried out at each cycle. Crack widths, crack depths, and modal parameters were recorded for both beams at all cycles. Measurements for crack width of both beams were recorded from both sides of the beams at the level 230 mm from the top surface of the beams by using two crack gauges. Tables 4 and 5 list the average crack width for both beams, respectively. The final three beams were a control beam, a defect beam L5 and another defect beam L5 × 2 with various volumes of honeycombs. The dimensions of the mortar blocks are given in Table 3. The method used for creating the honeycombs in this investigation was by precasting mortar blocks with a known amount of polystyrene beads and placing them at the mid-span of the beam prior to casting the beams as shown in Fig. 4. Modal testing was also carried out on these beams. 2.1. Modal test The modal test was performed using a transfer function technique on the RC beams, which were simply supported
Fig. 3. Static load test set-up.
on concrete blocks. An accelerometer was used to pick up the response of the test beam under forced excitation. The excitation resulting from the input force was measured using a force transducer. The RC beams were randomly excited using white noise signal input to a shaker, which was permanently placed at the quarter span for all test beams. The accelerometer was moved from one coordinate point to another to pick up a total of fifty-six response signals along the length of the beam. The transfer functions were acquired through a signal analyser. Initially, the transfer function spectrum within a 5 kHz frequency span was obtained in order to locate roughly the resonant frequency peaks of all the flexural modes within the band. Subsequently, zooming within a 100 Hz span of the resonant frequency peak of a particular mode was carried out. The measurements were made using a block size of 400 lines thus giving a resolution of 0.25 Hz per spectral line. A dual-channel analyzer was used to acquire the signals and to obtain the frequency response functions (FRF) from the response and the excitation force. By using modal analysis software, the curve fitting process was performed on the transfer function spectra obtained to extract the modal parameters i.e. natural frequency, mode shape and damping. A total of ten normal bending modes were acquired in this manner. Fig. 1 shows the actual modal test set-up and Fig. 2 depicts the setup of the modal testing.
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Fig. 4. Mortar blocks to represent honeycombs.
Fig. 5. Curve fitting with Chebyshev series rational for mode 4.
3. Results and discussions For an Euler beam the equation (iv)
d4 y dx 4
− λ4 y = 0 can be
rearranged as |λ4 | = | y y |. The values of |λ4 |, which may be referred to as the local stiffness indicator (LSI) drawn as bar charts for an undamaged beam, remain constant along the length of the beam. Any changes to the natural frequencies or flexural stiffness due to occurrences of damage in the beam will show changes in the value of |λ4 |. For the beams with defects, it is approximated that the generalized equation applies. To determine the location of damage, the indicator |λ4 | is used to determine the general region of the cracks or the honeycombs. In addition, a curve-fitting technique employing the Chebyshev series rationals was performed on the experimental mode shape to prepare the data before applying
the fourth order centered finite-divided difference formula. This is done so that random errors could be eliminated. Then, the value of |λ4 | is obtained. The curve fitting technique on the mode shape data is also found to be fairly successful in highlighting points of high residuals from the curve around the region of the damage. Fig. 5 shows examples of curve fitting with Chebyshev series rationals for mode 4 on the following cases: beam with no defect, beam with load induced cracked at 0.5L after loading of 43 kN, beam with load induced crack at 0.7L after loading of 46 kN, and beam with honeycombs L5. 3.1. Load induced cracking at 0.5L and 0.7L Figs. 6 and 7 show the plots of |λ4 | for the beam with a crack at 0.5L and 0.7L, respectively, along the datum and their
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Fig. 6. Location of damage for beam with load induced crack at 0.5L.
Fig. 7. Location of damage for beam with load induced crack at 0.7L.
respective maximum loading for mode 1, mode 2, mode 3, and mode 4 after curve fitting was done to the mode shape data. For all modes, the datum case showed that for the most part of both beams the values of |λ4 | remained about constant,
except at one end of the beam where the values showed some small increases in the values of |λ4 |. Mode 1 of the beam with crack at 0.5L showed increases in the values of |λ4 | at the ends of the beam labeled ML25, and large increases at one end of the beam labeled ML41. It was
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also seen that there were small increases and a wide area of large increases in the values of |λ4 | around the region of the crack for the beam labeled ML25 and ML43, respectively. For the beam labeled ML43, the maximum increases in the values of |λ4 | were at points 23 and 24. Mode 1 of the beam with crack at 0.7L showed a wide region around the crack where there are marked increases in the values of |λ4 | with a maximum at point 35 for the beam labeled ML25, at point 28 for the beam labeled ML41, at point 35 for the beam labeled ML43, and at point 33 for the beam labeled ML46. All these maximum points were fairly close to the crack location. It was also clearly seen that there were large increases in the values of |λ4 | at both ends of the beam. Mode 2 of the beams with crack at 0.5L labeled ML25 and labeled ML41 still indicated increases in the values of |λ4 | at the ends of the beam. It was also clearly seen that there were very large increases in the values of |λ4 | around the region of the crack, with a maximum value of |λ4 | at point 26 and point 24 of the beams labeled ML25 and ML43, respectively. These points were close to the crack location. Mode 2 of the beam with crack at 0.7L did not show any trend near the crack location for the beams labeled ML25, ML43 and ML46. However, it showed relatively higher values of |λ4 | near the ends of the three beams. It also showed marked increases in the values of |λ4 | at points 25 and 26 for the beams labeled ML43 and ML46. It was also clearly seen in mode 2 that the affected region around the crack was quite wide with large increases in the values of |λ4 | at point 31 and 47, which were on either side of the crack location for the beam labeled ML41. There were also high values of |λ4 | at both ends of the beam. For mode 3 of the beam with crack at 0.5L showed very large increases in the values of |λ4 | occurring at point 18 for the case of the beam labeled ML25, and there were also large increases in the values of |λ4 | around the region of the crack for the case of the beam labeled ML41. There were increases in the values of |λ4 | at both ends of the beams with crack at 0.7L labeled ML25 and ML46 in mode 3. It was also seen that the region around the crack had high values of |λ4 | with a maximum at point 36 for the beam labeled ML25 and at point 40 for the beams labeled ML41, ML43, and ML46. Mode 3 for the case of the beam labeled ML25 also showed an increase in the values of |λ4 | at point 21. Mode 4 of the beam with crack at 0.5L labeled ML25 showed very large increases in the values of |λ4 | at the ends of the beam and around the region of the crack with a maximum at point 27, which was close to the crack location. The beam labeled ML41 showed very similar behavior to the other modes with a wide area of increases in the value of |λ4 | around the crack. The maximum value was shown at point 27, which was close to the crack location. Mode 4 of the beam with crack at 0.7L showed there were increases in the values of |λ4 | at the ends for all the beams except ML43. The region around the crack showed clearly increases in the values of |λ4 | with a maximum at point 40, which was close to the crack location, for the beams labeled ML25, ML41, ML43 and ML46. The indicator |λ4 | for the datum experimental beam with cracks at 0.5L and 0.7L remained small for all modes except
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at some points at the ends of the beam. The curve fitting of the mode shape data for all the modes investigated also indicated that the datum beam showed no increases in the values of |λ4 | along its entire length. Imperfect boundary end conditions were manifested in increased values of the indicator. Except for the increases in the values of |λ4 | at the ends of the beam, taking the four modes together, the indicator had successfully identified a distinct region around the crack with values far exceeding the average values |λ4 | along the beam. 3.2. Honeycombs Fig. 8 shows the plots of |λ4 | for the control beam and beams with honeycombs for mode 1 to mode 4 after curve fitting was done to the mode shape data. Mode 1 did not show any trend of increases in the values of |λ4 | along the entire length of all beams. For mode 2, mode 3, and mode 4 of the control beam, the most part of it showed constant low values of |λ4 | except at points 5, 17, and 44 in mode 4. Mode 2 showed increases in the values of |λ4 | around the region of the honeycombs with a maximum at point 26, which was about the center of the honeycombs, for the beams labeled L5 and L5 × 2. The increase was more prominent in the beam label L5 × 2. Mode 3 showed increases of values of |λ4 | about the ends of the honeycombs region with a maximum at point 18 and at point 35, which represented the extremities of the honeycombs for the beam labeled L5. There were also increases in values of |λ4 | with a maximum at point 35, which represented one extremity of the honeycombs for the beam labeled L5×2. There were increases in values of |λ4 | at the ends of the beams labeled L5 and L5 × 2 shown in mode 4. Mode 4 also showed clearly increases in the values of |λ4 | around the region of the honeycombs for both beams with a maximum at point 26, which was about the center of the honeycombs. Taking the four modes together, the indicator |λ4 | for the control beam remained small and constant for all modes except at a few points along the beam in mode 4. The region around the honeycombs showed clearly increases in the values of the indicator |λ4 | with a maximum at point 26, which was about the center of the honeycombs for the beams labeled L5 and L5 × 2. The analysis of the experimental results using LSI was able to indicate the general region of the crack and honeycombs, the exact location of the damage being around the center of the region; but the severity of the damage could not be easily distinguished. LSI also indicated problems with the mode shapes at the ends of the beam. This was because it was very difficult to get good mode shapes from a simply supported system using rollers due to the constraining effects at the ends. 4. Conclusions A curve fitting technique was applied to experimental data to examine kinks in the curve, which could be due to the response to structural damage. In the case of the damaged beams, curve fitting the mode shape data using the Chebyshev series rationals produced curves with higher residual values around the damage location as compared to the other parts of the beam. Beams with a load-induced crack at 0.5L and 0.7L, and honeycombs L5
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Fig. 8. Location of damage for beam with honeycombs.
and L5 × 2 produced curves with low residual values along the entire length. Non-ideal boundary end conditions produced increased values of the residuals. Taking the patterns of the residuals for all the modes as a whole and the intensity of the increased residuals, it can be seen that there were irregularities in mode shape around the crack locations and about the region of the honeycombs. An indicator |λ4 | or LSI is proposed to be used in determining the location of the damage by examining the intensity of increases in the values of the indicator around the damage region. It was found that the maximum increases were close to the crack location or around the middle of the honeycombs. The severity of the damage and the resulting loss of structural carrying capacity of the beams due to the damage were also indicated by a higher intensity of perturbations of the indicator. There was a trend shown in the loss of structural stiffness from the changes in the modal parameters such as natural frequencies and mode shapes. The technique does not depend on undamaged or datum state data, thus it can easily be used for structural assessment purposes of RC structures as an alternative to more conventional approaches which are more labor intensive and with higher costs due to more elaborate procedures and longer system shutdown time. For actual structural systems the procedure could be applied to individual structural elements since the damage is normally localized. In the procedure, the generalized solution was used, and the results showed that this is still applicable to handle the dynamic behavior of damaged RC beams. Also in all cases the procedures produced poor results in the vicinity of the supports implying that the techniques cannot detect damage close to the supports.
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