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Location sensitivity of fundamental and higher mode shapes in localization of damage within a building
Journal of Sound and Vibration 365 (2016) 244–259
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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Location sensitivity of fundamental and higher mode shapes in localization of damage within a building Gourab Ghosh, Samit Ray-Chaudhuri n Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India
a r t i c l e in f o
abstract
Article history: Received 17 May 2015 Received in revised form 1 December 2015 Accepted 4 December 2015 Handling Editor: L.G. Tham Available online 23 December 2015
It has been recently demonstrated through mathematical derivation as well as numerical studies that the fundamental mode shape and its derivatives show excellent performance in localizing damage of buildings. Although it is understood from the literature that higher mode shapes are very sensitive to local changes in stiffness, no convincing theory is available in the literature about the efficiency of higher mode shapes in localizing damage. The present work focuses on this aspect. To achieve this goal, a mathematical formulation has been derived for change in a mode shape due to damage with damage parameters. This expression shows that the efficiency of the mode shape-based approach is dependent on the location of damage within a building. To illustrate the concept, numerical studies have been performed considering shear building models, where damage conditions have been simulated by reducing the stiffness at the desired locations. Further, an experimental study has been performed considering a three-dimensional steel moment-resisting frame model. It has been found that although the higher mode shapes and their derivatives are effective in localizing damage, their efficiency may significantly get reduced depending on the location of damage, particularly for shorter buildings. Hence, this location sensitivity of fundamental and higher mode shapes may be kept in mind for effective damage localization in buildings. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Vibration-based damage detection methodologies are becoming very popular in these days due to their immense potential of becoming a better alternative to the conventional methods. Vibration-based methods work on the premise that a change in system properties (e.g., stiffness of a member) results in a change in the system's dynamic responses. Hence, by utilizing dynamic responses, damages can be detected and localized. For past few decades, various vibration-based methodologies have been developed. Doebling et al. [1] presented a comprehensive review of the vibration-based methods available for detection, localization and characterization of structural damage. They classified the methods as per the analysis procedure, required measured data and type of structures where those are applicable. Later developments of these methods can be found in Doebling et al. [2], Alvandi and Cremona [3] and Yan et al. [4]. A class of vibration-based methodologies focuses blue on change in modal properties (e.g., frequency and mode shape) to identify structural damage. A comprehensive review of such modal parameter-based damage identification methods for
n
Corresponding author. Tel.: þ91 512 259 7267; fax: þ 91 512 259 7395. E-mail addresses: [email protected] (G. Ghosh), [email protected] (S. Ray-Chaudhuri).
http://dx.doi.org/10.1016/j.jsv.2015.12.005 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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beam- or plate-type structures can be found in Fan and Qiao [5]. A review on natural frequency-based techniques by Salawu [6] suggests that although their initial application dates back to late seventies [7], they are not good candidate to accurately identify the location of damage. On the other hand, as the mode shapes and their derivatives contain local information of the structure, they become excellent choice for damage localization [8,9]. Additionally, mode shape-based damage localization methods are expected to perform well for long term health monitoring as mode shapes are less affected due to varying environmental conditions such as temperature [5,9]. The present study focuses on mode shape and its derivatives based damage detection. Pandey et al. [10] showed that the absolute change in mode shape curvature is localized at the location of damage and thus, it can be used to localize damage. Ratcliffe [11] proposed that by applying a finite difference approximation of Laplace's differential operator to the mode shape data, damage can be localized in a structure without any prior knowledge about the undamaged structure. They have concluded that the proposed methodology is best suited only for the fundamental mode and not that sensitive for the higher modes. Wahab and Roeck [12] introduced “Curvature damage factor” to localize damage in a structure. They have successfully applied this technique to localize damage in a real bridge (Z24 in Switzerland). Abdo and Hori [13] showed that the changes in the derivative of the mode shapes are more sensitive to damage localization compared to the (displacement) mode shapes. Qiao et al. [14] evaluated vibration-based damage detection methodologies for composite laminated plates using piezoelectric materials and scanning laser vibrometer. They used curvature mode shapes to detect the presence, location, and size of the delamination. It was observed that the higher mode curvatures (particularly, from the 3rd mode to 5th mode) were effective in identifying and localizing a damage. Whalen [15] concluded that higher order mode shape derivatives are more sensitive to damage compared to the mode shapes. The use of change in slope of the fundamental mode shape to detect damage in a shear building was proposed by Zhu et al. [16]. Since, this method requires information about the fundamental mode shape only, this approach can be deemed as very useful from practical aspects. Recently, Roy and Ray-Chaudhuri [17] proposed a mathematical basis to establish a correlation between structural damage and change in the structure's fundamental mode shape and its derivatives using a perturbation approach. Numerical studies considering both shear and frame buildings were performed to demonstrate the efficiency of the proposed approach for civil structures [17]. A comparison of several damage identification methods based on the analysis of mode shape curvature and related quantities (e.g., natural frequencies and modal strain energy) by evaluating their performances on a damaged Euler–Bernoulli beam can be found in a recent study by Dessi and Camerlengo [18]. In another very recent study by Ciambella and Vestroni [9], with the help of perturbation approach the usefulness of modal curvature processing is demonstrated in the context of damage localization in Euler–Bernoulli beam-type structure. It is evident from the aforementioned literature that the mode shape-based methods are popular among the researchers for damage localization. It may also be noted that most of these studies deal with beam or plate models. Another important point is that for damage localization, so far researchers have mostly focused only on the fundamental mode shape. Although Qiao et al. [14] studied the performance of higher modes, they concluded that only some of the modes (3rd mode to 5th mode) were effective in localizing damage. However, no convincing theory was provided for the success of such select modes in damage localization. Kim et al. [19] performed a numerical study considering a simply-supported beam and concluded that the second mode shape is not effective in localizing damage near the mid-span. Similar results were reported by Owolabi et al. [20] in the context of experimental studies on fixed and simply supported beams. It is worth mentioning that the results from more number of modes are always desirable as they enhance the reliability of damage localization in a structure and in a way, can help in developing a better (updated) numerical model of the structure. Hence, a systematic study is required to understand the efficiency of higher modes in damage localization. In this study, the efficiency of a first few mode shapes in damage localization within shear and frame buildings has been investigated. At first, a general mathematical formulation for the change in a mode shape of a cantilever shear beam due to damage with damage parameters has been derived based on previously published studies of Ray-Chaudhuri [21], Roy and Ray-Chaudhuri [17]. For this purpose, damage conditions have been simulated by reducing the stiffness at the desired locations. With the help of this derivation, a theory has been proposed for location dependent efficiency of the mode shapebased method, particulary for the second or higher modes. Numerical studies are then performed considering a 30-storey shear building model. Effect of mode shape smoothness is also investigated by considering a relatively shorter building (i.e., smaller number of stories). Finally, an experimental study considering a three-dimensional steel moment-resisting frame model is also performed to study the effectiveness of the proposed theory in case of short frame buildings.
2. Relation between mode shape and damage location Based on the results obtained by Ray-Chaudhuri [21] using perturbation approach, Roy and Ray-Chaudhuri [17] proposed a closed form approximate solution to illustrate the idea that changes in the fundamental mode shape and its derivatives are associated with the location of damage. To do so, they considered a multi-degree-of-freedom shear building structure with stiffness matrix ½K in the undamaged condition. For a damaged condition as shown in Fig. 1, the equation of motion for this building in free vibration can be expressed as follows: € ½MfyðtÞg þ½K^ fyðtÞg ¼ 0
(1)
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where ½M is the lumped mass matrix, ½K^ is the stiffness matrix in the damaged condition and ½ΔK being the change in stiffness due to damage, i.e., ½ΔK ¼ ½K ½K^ . ^ ðiÞ g can be related to the jth mass normalized For small value of ½ΔK, the ith mass normalized damaged mode shape fϕ ðjÞ undamaged mode shape fϕ g of the structure by the following equation [21]: ðiÞ
^ g¼ fϕ
n X
ðjÞ
fϕ gψ^ ji
(2)
j¼1
In Eq. (2), ψ^ ji is the coupling term between the damaged and undamaged orthonormal mode shapes. For a change in stiffness (damaged minus undamaged) of δkp occurring between the pth and ðp 1Þth degrees-of-freedom (DoF), RayChaudhuri [21] provided an expression for ψ^ ji in the following form: 8 ðjÞ ðjÞ ðiÞ ðiÞ > > < ðfϕp 1 g fϕp gÞðfϕp 1 g fϕp gÞ δk if j a i p ψ^ ji ¼ λj λi > > :1 if j ¼ i ðjÞ
ðjÞ
where fϕðp 1Þ g and fϕðpÞ g represent the ðp 1Þth and pth elements of the jth undamaged mode shape, respectively; λi (with i ¼ 1; 2; … ., n) being the square of the ith natural frequency of the undamaged structure. Hence, a change in ith mode shape ðiÞ ^ ðiÞ g fϕðiÞ g) can be expressed as (i.e., fδϕ g ¼ fϕ n
o
δϕðiÞ ¼
n n X
ϕðjÞ
oðϕðjÞ
p1
j ¼ 1;j a i
ðiÞ ðiÞ ϕðjÞ p Þðϕp 1 ϕp Þ δkp λi λj
(3)
which can be further simplified as n
o
δϕðiÞ ¼
n n X
o
0ðiÞ ϕðjÞ ϕ0ðjÞ p ϕp
j ¼ 1;j a i
δkp 2 h λi λj 0ðjÞ
(4) 0ðiÞ
In Eq. (4), h is the height between DoF ðp 1Þ and p as shown in Fig. 1 and ϕp and ϕp are the derivatives (evaluated by ðjÞ ðiÞ backward difference) of ϕp and ϕp , respectively. Hence, it may be noticed from Eq. (4) that the change in ith mode shape depends also on the ith mode shape derivative evaluated near the damage location.
Fig. 1. Schematic diagram showing damage in 30-storey shear building.
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Considering a prismatic cantilever shear beam of length L (continuous system) to approximate the expressions for the ith natural frequency and orthonormal mode shape and then discretizing the beam model into a large number of elements, the ðiÞ change between the undamaged and damaged mode shapes δϕ ðyl Þ at a location yl (from the support) was derived as 2 3 X 3 2 n 6 b ϵ πy π yp 1 πy 7 6 7 cos ð2j 1Þ sin ð2j 1Þ l 7δkp δϕðiÞ yl ¼ 2 ð2i 1Þ cos ð2i 1Þ p (5) 6 2 4 2L 2L 2L 5 c ð2i 1Þ j ¼ 1 jai ð2j 1Þ 2j 1 pffiffiffiffiffiffiffiffiffiffiffi where b is a constant (b ¼ 2=ρL with ρ being mass per unit length of the beam), and yp is the damage location. 2.1. Damage localization using fundamental mode shape Considering only the fundamental mode, i.e., i¼1 and neglecting smaller terms, Roy and Ray-Chaudhuri [17] derived the following expression for difference in a mode shape:
δϕð1Þ yl ¼
n X
μ
ð2j 1Þ j¼1
n X π μ π sin ð2j 1Þ y þyp þ sin ð2j 1Þ yl yp μf yl 2L l ð2j 1Þ 2L j¼1
with
(6)
π y π yp l cos f yl ¼ 2 sin 2L 2L
(7)
and
μ¼
3 π yp b ϵ2 cos δkp 2 2L 2c
(8) ð1Þ
One can notice from Eq. (6) that for a particular stiffness reduction (i.e., δkp ) and damage location (i.e., yp), δϕ ðyl Þ shows ð1Þ a jump around yp [17]. This jump in value of δϕ ðyl Þ leads to a steep value of slope in the difference in mode shape and a sign change or zero crossing in the second derivative of difference in mode shape (i.e., curvature of difference in mode shape) at the location of damage. This signature of mode shape difference and its derivatives has been utilized by Roy and Ray-Chaudhuri [17] for damage localization. One can notice from Eq. (8) that the term μ originates from the first derivative ð1Þ of the fundamental mode shape evaluated at yp and this term controls the magnitude of δϕ ðyl Þ. 2.2. Damage localization using higher mode shape For higher modes, i.e., for i 41, the expression for change between the undamaged and damaged mode shape cannot be simplified as in Eq. (6). This is because the term ð2i 1Þ2 =ð2j 1Þ in Eq. (5) cannot be neglected. In this case, the following expression is obtained:
δϕðiÞ yl ¼
n X j ¼ 1;j a i
ν 2
ð2j 1Þ
ð2i 1Þ 2j 1
n X π sin ð2j 1Þ yl þ yp þ 2L j ¼ 1;j a i
ν ð2j 1Þ
ð2i 1Þ 2j 1
2
π sin ð2j 1Þ yl yp 2L
(9)
with
ν¼
3 π yp b ϵ2 ð 2i 1 Þ cos ð 2i 1 Þ δkp 2L 2c2
It can be noticed that the first term of Eq. (9) converges to a constant value (assume it Eq. (9) can be written in the following form: 8 νm < n > X ν π y yp ¼ νm sin ð2j 1Þ > 2L l ð2i 1Þ2 :0 j ¼ 1;j a i ð2j 1Þ 2j 1
(10) as νm). However, the second term of :
yl 4yp
:
yl oyp
:
yl ¼ yp
Thus, the difference between the undamaged and damaged mode shapes can be expressed as: 8 2νm : yl 4 yp > < ðiÞ 0 : yl o yp δϕ ðyl Þ ¼ > : νm : yl ¼ yp
(11)
(12)
Hence, a certain amount of jump is observed in difference of the damaged and undamaged mode shapes around the location of damage, as in case of the fundamental mode. Hence, for damage localization the signature of mode shape difference and its derivatives of the fundamental mode as proposed by Roy and Ray-Chaudhuri [17] also remains valid for
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higher modes. However, the prominence of this signature is controlled by a mode shape dependent factor. In Eq. (12), the constant value m is associated with a multiplier ν, which basically comes from the first derivative of the ith mode shape for a cantilever shear beam evaluated at the damage location. So, it can easily be understood that, for a damage at the location of nodes of a particular mode shape, the multiplier has the maximum value, whereas for a damage at the antinodes, it reaches the minimum value. Thus, in case of damage at the antinodes of a mode shape, a jump cannot be easily detected from the plot of difference in mode shapes owing to its small value (i.e., for ν close to zero). In that case, a normalization of difference in mode shapes with respect to the maximum difference value may be useful, provided there is no distortion of the mode shapes due to numerical evaluation scheme used, experimental data processing etc. For example, in case of taller buildings (with large number of DoFs), identification of damage location even near (but not exactly at) the antinode of a mode shape may not be that difficult due to relatively smooth nature of the mode shape. For shorter buildings (with small number of DoFs), however, an occurrence of damage close to the antinodes of a mode shape makes it really difficult to localize the damage owing to significant distortion of the mode shapes. 2.3. Numerical illustration with shear buildings To demonstrate the efficiency of the mode shape-based approach in damage localization, at first, simulation studies have been performed considering a 30-storey shear building model. The building has been modeled considering uniform floor mass and storey stiffness distributions along the height, with typical floor mass of 1 105 kg and storey stiffness of 2 108 N=m. To get a comprehensive idea about the efficiency of mode shape-based approach in damage localization, damage scenarios have been simulated at different stories by 10 percent reduction of the corresponding storey stiffness as illustrated in Fig. 1. Eigenvalue analysis has been done to get the frequencies and mode shapes for both damaged and healthy models. Fig. 2 shows the first three mode shapes of the building. Six damage cases are considered by reducing the
Fig. 2. First three mode shapes of 30-storey shear building and damage locations for different damage cases considered.
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stiffness of the stories as shown in this figure. The locations of damage are chosen in such a way that they are either near the nodes or antinodes of the second and third mode shapes. Fig. 3(a)–(d) is plotted for damage Case-1, i.e., for damage at 11th storey. Fig. 3(a) shows the undamaged and damaged fundamental and second mode shapes. Both mode shapes are normalized in such a way that the maximum value at top DoF is unity. From Fig. 3(a), it can be observed that there is no visible difference between the damaged and undamaged mode shapes for both first and second modes. However, from Fig. 3(b), it is clear that there is a jump in mode shape at the location of damage for both modes with the fundamental mode shape showing larger jump than the second mode shape. In fact, for the second mode shape, the jump is one sided. This is because the derivative of the second mode shape at this location is minimum (close to zero). Fig. 3(c) shows the derivative of change in mode shape or slope of change in mode shape fδϕg for both fundamental and second mode evaluated using finite difference approach. One can notice from Fig. 3(c) that for both modes, the slope attains maximum at the location of damage. Fig. 3(d) provides the curvature of change in mode shape evaluated in a similar manner. A zero crossing is observed for both modes in Fig. 3(d). Also, from Fig. 3(c) and (d) one can notice that the signature as proposed by Roy and Ray-Chaudhuri [17] for the fundamental mode efficiently localizes the damage at the 11th storey level using the second mode shapes also. However, the magnitude of jump, slope or change in curvature at the damage location is larger for the fundamental mode compared to the second mode. This is because the absolute magnitude of derivative of the fundamental mode shape is higher than that of the second mode shape. Fig. 4(a)–(d) is plotted in a similar way to that of Fig. 3(a)–(d) except that it is for damage Case-2, i.e., for damage at the 21st storey. It can be observed that the signature for damage localization remains the same at the damage location. However, in this case (i.e., Case-2) contrary to Fig. 3, the magnitude of jump, slope or change in curvature at the damage location is larger for the second mode compared to the fundamental mode. This is due to the fact that at the damage location, absolute value of the derivative of second mode shape is much higher than that of the fundamental mode shape. Hence, one can conclude that at this location, second mode shape may perform better for damage localization compared to the fundamental mode. The plot of change in third mode shape, its (change in third mode shape) slope and curvature are provided in Fig. 5 for damage Case-3 (i.e., damage at the 7th storey) and Case-4 (i.e., damage at the 19th storey). Similar plots are shown in Fig. 6 for damage Case-5 (i.e., damage at the 13th storey) and Case-6 (i.e., damage at the 25th storey). It may be observed from Figs. 5 and 6 that damage is efficiently localized using the well-established signature of the fundamental mode. However, by comparing the values of X-axis in Figs. 5 and 6, one can notice that the magnitude of jump in the change in third mode
Fig. 3. Plot for Case-1 (damage at 11th storey): (a) undamaged and damaged first and second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first and second modes.
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Fig. 4. Plot for Case-2 (damage at 21st storey): (a) undamaged and damaged first and second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first and second modes.
shape, maximum value of slope of change in this mode shape and curvatures of change in this mode shape at damage location is one order higher for damage Case-5 and Case-6 when compared with damage Case-3 and Case-4. This implies that a damage at nodes is more sensitive to the change in third mode shape and its derivatives when compared to a damage at antinodes. This is in agreement with the observations of the second mode. To study how mode shape smoothness affect the efficiency of damage localization in difficult locations (near antinodes), another shear building with the same mass and stiffness distributions along the height is considered but with only 10-stories. Hence, the mode shapes for this building will be less smoother (when adjacent DoFs are joined by a straight line) due to lower number of DoFs when compared to the 30-storey building. Fig. 7(a) shows the damaged and undamaged second mode shapes for damage at 11th storey of the 30-storey building (i.e., damage Case-1) and for damage at 4th storey for the 10-storey building. In all plots of Fig. 7, the height of the building is normalized to unity for convenience of comparison. It may be observed from Fig. 7(a) that although the undamaged mode shape of the 10-storey building slightly deviates from that of the 30-storey building (due to change in number of DoFs), the undamaged and damaged mode shapes almost match for each building. However, from Fig. 7(b)–(d), it is clear that for the 10-storey building, the damage signature is not at all visible, while for the 30-storey building, a clear damage signature is observed. This is due to the fact that at this location, the change in mode shape follows a gently sloping curve (a slowly varying curve) and hence, the slope of this curve evaluated with a larger (finite difference) step size becomes lower compared to that with a smaller step size. Fig. 8 provides similar plots but with the third mode shape for damage at the 7th storey of the 30-storey building (i.e., damage Case-3) and for damage at the 2nd storey for the 10-storey building. It is observed that damage signatures are present for both buildings in change in mode shape, and slope and curvature of change in mode shapes. It may however be noted that for the 30-storey building identification of damage location becomes easier compared to the 10-storey building. For 10-storey building, by comparing Fig. 7 with Fig. 8, one can note that identification of damage becomes easier for the third mode compared to the second mode. This is because, the absolute derivative of the third mode shape near the damage location (shown by ——- line in Fig. 8) is higher than that of the second mode at damage location shown in Fig. 7.
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Fig. 5. Plot for Case-3 (damage at 7th storey) and Case-4 (damage at 19th storey): (a) difference in mode shapes, (b) slope of difference in mode shapes and (c) curvature of difference in mode shapes for third mode.
3. Experimental study with moment frame Usage of moment frames is very common for buildings around the world. It is well-known that the dynamic behavior of moment-frame buildings can be represented as the combination of shear and Euler–Bernoulli beams, though in short to middle height buildings, the behavior is primarily dominated by shear beam-type characteristics (Miranda and Taghavi [22]). In this section, the efficiency of mode shape-based methods in localizing damage has been studied experimentally considering a six-storey three-dimensional steel moment-resisting frame building model. Impact hammer tests were performed to evaluate the dynamic characteristics (frequencies and mode shapes) of the model. The model was fabricated using steel and aluminium sections. A schematic diagram of the frame model is shown in Fig. 9. Fig. 9(a) and (b) shows the elevation from front and plan from top of the undamaged structure, respectively. Total height of the structure was 780 mm and each storey height was 130 mm. The floor plan of the model was 200 mm 200 mm. Each storey had two steel beams of length 200 mm each and two aluminium plates each of length 200 mm placed along two perpendicular directions. Columns were made of steel and the height of each one was 130 mm. Cross sectional dimension of the steel beams were 8 mm 8 mm and that of the columns were 10 mm 10 mm, whereas the aluminium plates had a cross sectional dimension of 40 mm 5 mm. A strong column-weak beam approach was followed while preparing the experimental model. The connections were designed in such a way that by opening a few screws, a beam or a column element can be replaced for simulating damage conditions. In this scaled model, the mass and stiffness were uniformly distributed in all the stories. It may be noted that the intended direction of vibration (i.e., for which the dynamic properties are evaluated) is perpendicular to the plane of the paper in Fig. 9(a) and along the length of the paper in Fig. 9(b).
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Fig. 6. Plot for Case-5 (damage at 13th storey) and Case-6 (damage at 25th storey): (a) difference in mode shapes, (b) slope of difference in mode shapes and (c) curvature of difference in mode shapes for third mode.
3.1. Damage scenarios Since the structure is short, only two damage scenarios were simulated to experimentally verify the efficiency of mode shape curvature-based approach in localizing damage in the frame structure. Damage cases are simulated by replacing the steel beams of a particular storey with another set of beam of same length but with notches being cut at the ends of the beams. Snapshots of the undamaged model and of a particular damage case has been shown in Figs. 11 and 12, respectively. The damage cases are shown in Fig. 10, where the direction of vibration is also shown by an arrow at the top of the schematics. In Case-I, two steel beams at the second floor level were replaced whereas in Case-II, similar replacement was performed at the fourth floor level. It may be clear from Fig. 12 that the effect of damage in steel beams will result in change in eigenproperties of the model along the direction of vibration as a result of stiffness reduction along this direction. 3.2. Instrumentation and test procedure Impact hammer test was performed with a PCB086D20 [23] type short-sledge impulse hammer (sensitivity of 0.23 mV/ N). For measuring the acceleration response, PCB393B04 [23] seismic, miniature (50 gm), ceramic flexural ICP accelerometer with a sensitivity of 1 V/g was used. Agilent35670A [24] 4 channel FFT Dynamic Signal Analyzer was used for data acquisition. Channel one is the input channel and the remaining channels are used as output channel. The input excitation voltage for each channel was controlled to ensure perfect functioning of the instrument. The measured data was saved (in SDF format) in the internal disk. Then using the Agilent Datalink [24] software the stored data was transferred to a computer for postprocessing. The hitting was done at the top corner of the frame along the direction of steel beams. The accelerometer was attached at the middle of a particular floor with the help of an L-shaped plate (can be observed in Fig. 12). The accelerometer was connected in such a way that the sensing direction coincided with the direction of intended acceleration measurement. For
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Fig. 7. Effect of mode shape smoothness: (a) undamaged and damaged second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for second mode in case of damage at 4th storey of 10-storey shear building and damage at 11th storey of 30-storey shear building.
each floor, 10 FRFs (frequency response function) were recorded and the average was calculated using the dynamic signal analyzer. Subsequently, the accelerometer was connected to the next floor and the same procedure was repeated with the hitting being done at the same point, i.e., at the top corner. The test was concluded by collecting FRF data from all floors. While getting the FRFs from the experiment, it was ensured that the coherence was above 0.99. 3.3. Results The recorded FRFs were then transferred to a computer using Agilent Datalink [24] software and analyzed to obtain the frequencies and mode shapes with ME'scopeVES [25] software. For this purpose, a wire frame model of the six-storey frame was created in ME'scopeVES [25] software and then the FRF data was imported. By overlaying all traces, it was observed that these traces are in good agreement. Then using the global polynomial curve fitting method, residues, damping and mode shapes were obtained. Noting the peaks in the FRFs, modal frequencies were identified and from the table of shapes, respective mode shape values along with their phases were recorded. Fig. 13 shows overlaid traces in the ME'scopeVES [25] as well as the wire-frame model. The mode shapes were normalized in such a way so that the value corresponding to the top DoF is equal to unity. Subsequently, their derivatives (up to the second order) were calculated using finite difference method in MATLAB [26]. Table 1 provides the experimentally obtained natural frequencies of the fundamental and second modes for the undamaged structure and structure with Case-I and Case-II damages. It may be noted that the structure is very stiff with the fundamental frequency of 24.5 Hz and only two modes were obtained by the test. It may also be noted that the maximum reduction in frequency due to damage in Case-I for the fundamental mode and Case-II for the second mode. Also, the percentage reduction in frequency occurred for a damage causes higher percentage reduction in the fundamental mode compared to the second mode. These results are in agreement with the findings of Ray-Chaudhuri [21]. In addition, the maximum percentage reduction in frequency is about 6 percent for the fundamental mode and 1.5 percent for the second mode. To check the validity of the experimental results, a numerical model of the experimental specimen has been created in SAP2000 [27]. Frame sections were assigned to the beam and columns of the 3D frame model. The base of the structure was considered to be fixed for modeling purpose and self-weight was considered as mass source.
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Fig. 8. Effect of mode shape smoothness: (a) undamaged and damaged third mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for third mode in case of damage at 2nd storey of 10-storey shear building and damage at 7th storey of 30-storey shear building.
Dynamic properties of the model was evaluated by performing an eigenvalue analysis. It is found from this analysis that the fundamental frequencies corresponding to the undamaged, damaged Case-I and Case-II are 26.6, 25, 25.5 Hz, respectively, implying a reasonable agreement with those obtained from the experimental data. Small but consistent Á over-estimations observed ðA10 percentÞ in frequencies for these cases are due to various modeling assumptions. One probable reason includes higher stiffness of the analysis model due to rigid-joint assumption in modeling, which may not be completely valid due to the way members are connected in the experimental model. To check the efficiency of the fundamental and second mode shapes in damage localization, two damage cases are considered as shown in Fig. 10. Fig. 14 shows the results for fundamental mode shape. In Fig. 14(a), the fundamental mode shapes in undamaged and two damaged conditions are shown. It can be observed from this figure that the damage at the 2nd floor beam (Case-I) causes more changes compared to the damage at the 4th floor beam (Case-II). This can also be verified from Fig. 14(b), where changes in fundamental mode shape due to damage for both Case-I and Case-II are shown. Fig. 14(c) and (d) provide the slope and curvature of change in fundamental mode shape due to damage. It can be observed from these figures that the damage can be localized in both cases using the signature of the fundamental mode shape, i.e., maximum slope and zero crossing at the location of damage. Fig. 15(a)–(d) provides the plots similar to that of Fig. 14(a)–(d) except that Fig. 15 is for the second mode. One can notice from this figure that the damage at the 4th floor beam can be identified more easily (more pronounce features) compared to the case of damage at the 2nd floor beam. This is because a damage at the 2nd floor and 4th floor beams are near the antinode and node of the second mode, respectively. As argued earlier (see Section 2.2), the 2nd floor damage location is close to the antinode of the second mode shape and hence, it is difficult to detect a jump in the difference of mode shapes. A slight distortion in the second mode shape and its derivatives (due to a low number of degrees-of-freedom coupled with the location of damage being close to the base) further complicates the detection of damage. Hence, based on these observations, one can conclude that the experimental results support the location sensitivity of mode shape curvature-based methods in damage localization, as proposed earlier.
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Fig. 9. Schematic diagram of the experimental model. (a) Evaluation. (b) Plan.
Fig. 10. Damage scenarios considered for frame building model (the direction of vibration is shown by an arrow at the top of the schematic).
4. Conclusions In this study, a mathematical formulation between the difference in mode shape due to damage and damage parameters has been derived. From this derivation, it has been found that the efficiency of mode shape-based damage localization approach is dependent on the location of damage in a building. In particular, it has been found that the change in mode
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Fig. 11. Snapshot of the experimental model showing details.
Fig. 12. Snapshot of (a) damaged structure and (b) damaged second floor beam.
shape due to damage is related to the derivative of that mode shape evaluated at the location of damage. Numerical studies have been performed considering shear building models to show this location sensitivity of mode shapes for damage localization. Damage conditions have been simulated by reducing the stiffness at desired locations. Further, the effect of mode shape smoothness on damage localization has also been studied by considering building with relatively less number of stories. An experimental study has also been performed considering a three-dimensional six-storey steel moment-
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Fig. 13. Snapshot of (a) overlaid traces and (b) wire frame model of experimental specimen.
Table 1 Experimentally obtained natural frequencies (in Hz). Mode
Undamaged
Case-I
Case-II
Fundamental Second
24.5 84.4
23 83.4
23.8 82.5
resisting frame model to verify the analytical findings for short frame buildings. The major findings of this study can listed as follows: 1. The damage signature proposed by Roy and Ray-Chaudhuri [17] for the fundamental mode is also valid for higher modes. 2. It has been found that although the higher mode shapes are effective in localizing damage, their efficiency may significantly get reduced depending on the location of damage. 3. If a damage is located near the nodes of a particular mode, the damage can be localized most efficiently using that mode. On the other hand, if a damage is located near the antinodes of a particular mode, it may be difficult to localize such damage by using that particular mode. 4. The efficiency of damage localization also gets affected by the smoothness of a higher mode shape (i.e., number of DoFs), especially when the damage is located near the antinodes of that particular mode. 5. The location sensitivity of mode shapes in damage localization is found to be valid for short moment-resiting building frames. It may be noted that the findings of this study are valid for small change in stiffness and for shear beam or buildings, which show dominant shear-type behavior. For Euler–Bernoulli beam or other cases, a formulation has to be carried out to establish the location dependency of mode shapes. This will help in explaining the observations of Kim et al. [19], Qiao et al. [14] and Owolabi et al. [20]. It may also be noted that identification of an exact damage location in a beam or in a column of a typical frame structure using mode shape-based signatures as proposed herein is a challenging task. In absence of a dense array of additional sensors (which may be prohibitively expensive), post-event local inspection may be good idea to further
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Fig. 14. Plot of (a) undamaged and damaged mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first mode in case of damage at second floor and fourth floor beams.
Fig. 15. Plot of (a) undamaged and damaged mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for second mode in case of damage at second floor and fourth floor beams.
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localize the damage. Nonetheless, the mode shape-based damage signatures similar to those proposed by Panikkaveettil et al. [28] can be used to narrow down whether the damage is in a beam or in a column. Acknowledgments The authors would like to thank Dr. Koushik Roy and Ms. Bhavana Valeti (former graduate student) for designing the experimental model. The help from the workers of the Structural Engineering Laboratory, IIT Kanpur during the experiment is also greatly appreciated.
References [1] S.W. Doebling, C.R. Farrar, M.B. Prime, D.W. Shevitz, Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in their Vibration Characteristics: A Literature Review, Technical Report, Los Alamos National Lab., New Mexico (United States), 1996. [2] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, Shock and Vibration Digest 30 (2) (1998) 91–105. [3] A. Alvandi, C. Cremona, Assessment of vibration-based damage identification techniques, Journal of Sound and Vibration 292 (1) (2006) 179–202. [4] Y.J. Yan, L. Cheng, Z.Y. Wu, L.H. Yam, Development in vibration-based structural damage detection technique, Mechanical Systems and Signal Processing 21 (5) (2007) 2198–2211. [5] W. Fan, P. Qiao, Vibration-based damage identification methods: a review and comparative study, Structural Health Monitoring 10 (1) (2011) 83–111. [6] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Engineering Structures 19 (9) (1997) 718–723. [7] P. Cawley, R.D. Adams, A vibration technique for non-destructive testing of fibre composite structures, Journal of Composite Materials 13 (2) (1979) 161–175. [8] M.M.F. Yuen, A numerical study of the eigenparameters of a damaged cantilever, Journal of Sound and Vibration 103 (1985) 301–310. [9] J. Ciambella, F. Vestroni, The use of modal curvatures for damage localization in beam-type structures, Journal of Sound and Vibration 340 (2015) 126–137. [10] A.K. Pandey, M. Biswas, M.M. Samman, Damage detection from changes in curvature mode shapes, Journal of Sound and Vibration 145 (2) (1991) 321–332. [11] C.P. Ratcliffe, Damage detection using a modified Laplacian operator on mode shape data, Journal of Sound and Vibration 204 (3) (1997) 505–517. [12] M.M.A. Wahab, G.D. Roeck, Damage detection in bridges using modal curvatures: application to a real damage scenario, Journal of Sound and Vibration 226 (2) (1999) 217–235. [13] M.A.B. Abdo, M. Hori, A numerical study of structural damage detection using changes in the rotation of mode shapes, Journal of Sound and Vibration 251 (2) (2002) 227–239. [14] P. Qiao, K. Lu, W. Lestari, J. Wang, Curvature mode shape-based damage detection in composite laminated plates, Composite Structures 80 (3) (2007) 409–428. [15] T.M. Whalen, The behavior of higher order mode shape derivatives in damaged, beam-like structures, Journal of Sound and Vibration 309 (3) (2008) 426–464. [16] H. Zhu, L. Li, X.Q. He, Damage detection method for shear buildings using the changes in the first mode shape slopes, Computers & Structures 89 (9) (2011) 733–743. [17] K. Roy, S. Ray-Chaudhuri, Fundamental mode shape and its derivatives in structural damage localization, Journal of Sound and Vibration 332 (21) (2013) 5584–5593. [18] D. Dessi, G. Camerlengo, Damage identification techniques via modal curvature analysis: overview and comparison, Mechanical Systems and Signal Processing 52–53 (2015) 181–205. [19] J.T. Kim, Y.S. Ryu, H.M. Cho, N. Stubbs, Damage identification in beam-type structures: frequency-based method vs mode-shape-based method, Engineering Structures 25 (1) (2003) 57–67. [20] G.M. Owolabi, A.S.J. Swamidas, R. Seshadri, Crack detection in beams using changes in frequencies and amplitudes of frequency response functions, Journal of Sound and Vibration 265 (1) (2003) 1–22. [21] S. Ray-Chaudhuri, Change in instantaneous eigenproperties due to yielding of a structure, Journal of Sound and Vibration 312 (4–5) (2008) 754–768. [22] E. Miranda, S. Taghavi, Approximate floor acceleration demands in multistory buildings. I: formulation, ASCE Journal of Structural Engineering 131 (2) (2005) 203–211. [23] PCB. PCB Piezotronics, PCB Piezotronics Inc, 3425 Walden Avenue Depew, New York, USA, 2013, pp. 14043–2495. [24] Agilent Technologies, Agilent, Agilent Technologies Inc, Santa Clara, California, 2013. [25] ME'scopeVES, ME'scopeVES5.0, Vibrant Technology Inc., 5 Erba Lane, Suite B, Scotts Valley, California, 95066, 2013. [26] MATLAB, version 8.0.0.783 (R2012b), The MathWorks Inc., Natick, Massachusetts, 2012. [27] SAP2000, SAP2000 V14, Computers and Structures Inc., 1646 N. California Blvd., Suite 600, Walnut Creek, California, 94596, USA, 2000. [28] H. Panikkaveettil, K. Roy, S. Ray-Chaudhuri, Damage characterization in frame structures using output-only modal and feature-based techniques, in: Advances in Control and Optimization of Dynamical Systems, IIT Kanpur, India, vol. 3, 2014, pp. 973–980.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Location sensitivity of fundamental and higher mode shapes in localization of damage within a building Gourab Ghosh, Samit Ray-Chaudhuri n Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur, UP 208016, India
a r t i c l e in f o
abstract
Article history: Received 17 May 2015 Received in revised form 1 December 2015 Accepted 4 December 2015 Handling Editor: L.G. Tham Available online 23 December 2015
It has been recently demonstrated through mathematical derivation as well as numerical studies that the fundamental mode shape and its derivatives show excellent performance in localizing damage of buildings. Although it is understood from the literature that higher mode shapes are very sensitive to local changes in stiffness, no convincing theory is available in the literature about the efficiency of higher mode shapes in localizing damage. The present work focuses on this aspect. To achieve this goal, a mathematical formulation has been derived for change in a mode shape due to damage with damage parameters. This expression shows that the efficiency of the mode shape-based approach is dependent on the location of damage within a building. To illustrate the concept, numerical studies have been performed considering shear building models, where damage conditions have been simulated by reducing the stiffness at the desired locations. Further, an experimental study has been performed considering a three-dimensional steel moment-resisting frame model. It has been found that although the higher mode shapes and their derivatives are effective in localizing damage, their efficiency may significantly get reduced depending on the location of damage, particularly for shorter buildings. Hence, this location sensitivity of fundamental and higher mode shapes may be kept in mind for effective damage localization in buildings. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Vibration-based damage detection methodologies are becoming very popular in these days due to their immense potential of becoming a better alternative to the conventional methods. Vibration-based methods work on the premise that a change in system properties (e.g., stiffness of a member) results in a change in the system's dynamic responses. Hence, by utilizing dynamic responses, damages can be detected and localized. For past few decades, various vibration-based methodologies have been developed. Doebling et al. [1] presented a comprehensive review of the vibration-based methods available for detection, localization and characterization of structural damage. They classified the methods as per the analysis procedure, required measured data and type of structures where those are applicable. Later developments of these methods can be found in Doebling et al. [2], Alvandi and Cremona [3] and Yan et al. [4]. A class of vibration-based methodologies focuses blue on change in modal properties (e.g., frequency and mode shape) to identify structural damage. A comprehensive review of such modal parameter-based damage identification methods for
n
Corresponding author. Tel.: þ91 512 259 7267; fax: þ 91 512 259 7395. E-mail addresses: [email protected] (G. Ghosh), [email protected] (S. Ray-Chaudhuri).
http://dx.doi.org/10.1016/j.jsv.2015.12.005 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
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beam- or plate-type structures can be found in Fan and Qiao [5]. A review on natural frequency-based techniques by Salawu [6] suggests that although their initial application dates back to late seventies [7], they are not good candidate to accurately identify the location of damage. On the other hand, as the mode shapes and their derivatives contain local information of the structure, they become excellent choice for damage localization [8,9]. Additionally, mode shape-based damage localization methods are expected to perform well for long term health monitoring as mode shapes are less affected due to varying environmental conditions such as temperature [5,9]. The present study focuses on mode shape and its derivatives based damage detection. Pandey et al. [10] showed that the absolute change in mode shape curvature is localized at the location of damage and thus, it can be used to localize damage. Ratcliffe [11] proposed that by applying a finite difference approximation of Laplace's differential operator to the mode shape data, damage can be localized in a structure without any prior knowledge about the undamaged structure. They have concluded that the proposed methodology is best suited only for the fundamental mode and not that sensitive for the higher modes. Wahab and Roeck [12] introduced “Curvature damage factor” to localize damage in a structure. They have successfully applied this technique to localize damage in a real bridge (Z24 in Switzerland). Abdo and Hori [13] showed that the changes in the derivative of the mode shapes are more sensitive to damage localization compared to the (displacement) mode shapes. Qiao et al. [14] evaluated vibration-based damage detection methodologies for composite laminated plates using piezoelectric materials and scanning laser vibrometer. They used curvature mode shapes to detect the presence, location, and size of the delamination. It was observed that the higher mode curvatures (particularly, from the 3rd mode to 5th mode) were effective in identifying and localizing a damage. Whalen [15] concluded that higher order mode shape derivatives are more sensitive to damage compared to the mode shapes. The use of change in slope of the fundamental mode shape to detect damage in a shear building was proposed by Zhu et al. [16]. Since, this method requires information about the fundamental mode shape only, this approach can be deemed as very useful from practical aspects. Recently, Roy and Ray-Chaudhuri [17] proposed a mathematical basis to establish a correlation between structural damage and change in the structure's fundamental mode shape and its derivatives using a perturbation approach. Numerical studies considering both shear and frame buildings were performed to demonstrate the efficiency of the proposed approach for civil structures [17]. A comparison of several damage identification methods based on the analysis of mode shape curvature and related quantities (e.g., natural frequencies and modal strain energy) by evaluating their performances on a damaged Euler–Bernoulli beam can be found in a recent study by Dessi and Camerlengo [18]. In another very recent study by Ciambella and Vestroni [9], with the help of perturbation approach the usefulness of modal curvature processing is demonstrated in the context of damage localization in Euler–Bernoulli beam-type structure. It is evident from the aforementioned literature that the mode shape-based methods are popular among the researchers for damage localization. It may also be noted that most of these studies deal with beam or plate models. Another important point is that for damage localization, so far researchers have mostly focused only on the fundamental mode shape. Although Qiao et al. [14] studied the performance of higher modes, they concluded that only some of the modes (3rd mode to 5th mode) were effective in localizing damage. However, no convincing theory was provided for the success of such select modes in damage localization. Kim et al. [19] performed a numerical study considering a simply-supported beam and concluded that the second mode shape is not effective in localizing damage near the mid-span. Similar results were reported by Owolabi et al. [20] in the context of experimental studies on fixed and simply supported beams. It is worth mentioning that the results from more number of modes are always desirable as they enhance the reliability of damage localization in a structure and in a way, can help in developing a better (updated) numerical model of the structure. Hence, a systematic study is required to understand the efficiency of higher modes in damage localization. In this study, the efficiency of a first few mode shapes in damage localization within shear and frame buildings has been investigated. At first, a general mathematical formulation for the change in a mode shape of a cantilever shear beam due to damage with damage parameters has been derived based on previously published studies of Ray-Chaudhuri [21], Roy and Ray-Chaudhuri [17]. For this purpose, damage conditions have been simulated by reducing the stiffness at the desired locations. With the help of this derivation, a theory has been proposed for location dependent efficiency of the mode shapebased method, particulary for the second or higher modes. Numerical studies are then performed considering a 30-storey shear building model. Effect of mode shape smoothness is also investigated by considering a relatively shorter building (i.e., smaller number of stories). Finally, an experimental study considering a three-dimensional steel moment-resisting frame model is also performed to study the effectiveness of the proposed theory in case of short frame buildings.
2. Relation between mode shape and damage location Based on the results obtained by Ray-Chaudhuri [21] using perturbation approach, Roy and Ray-Chaudhuri [17] proposed a closed form approximate solution to illustrate the idea that changes in the fundamental mode shape and its derivatives are associated with the location of damage. To do so, they considered a multi-degree-of-freedom shear building structure with stiffness matrix ½K in the undamaged condition. For a damaged condition as shown in Fig. 1, the equation of motion for this building in free vibration can be expressed as follows: € ½MfyðtÞg þ½K^ fyðtÞg ¼ 0
(1)
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where ½M is the lumped mass matrix, ½K^ is the stiffness matrix in the damaged condition and ½ΔK being the change in stiffness due to damage, i.e., ½ΔK ¼ ½K ½K^ . ^ ðiÞ g can be related to the jth mass normalized For small value of ½ΔK, the ith mass normalized damaged mode shape fϕ ðjÞ undamaged mode shape fϕ g of the structure by the following equation [21]: ðiÞ
^ g¼ fϕ
n X
ðjÞ
fϕ gψ^ ji
(2)
j¼1
In Eq. (2), ψ^ ji is the coupling term between the damaged and undamaged orthonormal mode shapes. For a change in stiffness (damaged minus undamaged) of δkp occurring between the pth and ðp 1Þth degrees-of-freedom (DoF), RayChaudhuri [21] provided an expression for ψ^ ji in the following form: 8 ðjÞ ðjÞ ðiÞ ðiÞ > > < ðfϕp 1 g fϕp gÞðfϕp 1 g fϕp gÞ δk if j a i p ψ^ ji ¼ λj λi > > :1 if j ¼ i ðjÞ
ðjÞ
where fϕðp 1Þ g and fϕðpÞ g represent the ðp 1Þth and pth elements of the jth undamaged mode shape, respectively; λi (with i ¼ 1; 2; … ., n) being the square of the ith natural frequency of the undamaged structure. Hence, a change in ith mode shape ðiÞ ^ ðiÞ g fϕðiÞ g) can be expressed as (i.e., fδϕ g ¼ fϕ n
o
δϕðiÞ ¼
n n X
ϕðjÞ
oðϕðjÞ
p1
j ¼ 1;j a i
ðiÞ ðiÞ ϕðjÞ p Þðϕp 1 ϕp Þ δkp λi λj
(3)
which can be further simplified as n
o
δϕðiÞ ¼
n n X
o
0ðiÞ ϕðjÞ ϕ0ðjÞ p ϕp
j ¼ 1;j a i
δkp 2 h λi λj 0ðjÞ
(4) 0ðiÞ
In Eq. (4), h is the height between DoF ðp 1Þ and p as shown in Fig. 1 and ϕp and ϕp are the derivatives (evaluated by ðjÞ ðiÞ backward difference) of ϕp and ϕp , respectively. Hence, it may be noticed from Eq. (4) that the change in ith mode shape depends also on the ith mode shape derivative evaluated near the damage location.
Fig. 1. Schematic diagram showing damage in 30-storey shear building.
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Considering a prismatic cantilever shear beam of length L (continuous system) to approximate the expressions for the ith natural frequency and orthonormal mode shape and then discretizing the beam model into a large number of elements, the ðiÞ change between the undamaged and damaged mode shapes δϕ ðyl Þ at a location yl (from the support) was derived as 2 3 X 3 2 n 6 b ϵ πy π yp 1 πy 7 6 7 cos ð2j 1Þ sin ð2j 1Þ l 7δkp δϕðiÞ yl ¼ 2 ð2i 1Þ cos ð2i 1Þ p (5) 6 2 4 2L 2L 2L 5 c ð2i 1Þ j ¼ 1 jai ð2j 1Þ 2j 1 pffiffiffiffiffiffiffiffiffiffiffi where b is a constant (b ¼ 2=ρL with ρ being mass per unit length of the beam), and yp is the damage location. 2.1. Damage localization using fundamental mode shape Considering only the fundamental mode, i.e., i¼1 and neglecting smaller terms, Roy and Ray-Chaudhuri [17] derived the following expression for difference in a mode shape:
δϕð1Þ yl ¼
n X
μ
ð2j 1Þ j¼1
n X π μ π sin ð2j 1Þ y þyp þ sin ð2j 1Þ yl yp μf yl 2L l ð2j 1Þ 2L j¼1
with
(6)
π y π yp l cos f yl ¼ 2 sin 2L 2L
(7)
and
μ¼
3 π yp b ϵ2 cos δkp 2 2L 2c
(8) ð1Þ
One can notice from Eq. (6) that for a particular stiffness reduction (i.e., δkp ) and damage location (i.e., yp), δϕ ðyl Þ shows ð1Þ a jump around yp [17]. This jump in value of δϕ ðyl Þ leads to a steep value of slope in the difference in mode shape and a sign change or zero crossing in the second derivative of difference in mode shape (i.e., curvature of difference in mode shape) at the location of damage. This signature of mode shape difference and its derivatives has been utilized by Roy and Ray-Chaudhuri [17] for damage localization. One can notice from Eq. (8) that the term μ originates from the first derivative ð1Þ of the fundamental mode shape evaluated at yp and this term controls the magnitude of δϕ ðyl Þ. 2.2. Damage localization using higher mode shape For higher modes, i.e., for i 41, the expression for change between the undamaged and damaged mode shape cannot be simplified as in Eq. (6). This is because the term ð2i 1Þ2 =ð2j 1Þ in Eq. (5) cannot be neglected. In this case, the following expression is obtained:
δϕðiÞ yl ¼
n X j ¼ 1;j a i
ν 2
ð2j 1Þ
ð2i 1Þ 2j 1
n X π sin ð2j 1Þ yl þ yp þ 2L j ¼ 1;j a i
ν ð2j 1Þ
ð2i 1Þ 2j 1
2
π sin ð2j 1Þ yl yp 2L
(9)
with
ν¼
3 π yp b ϵ2 ð 2i 1 Þ cos ð 2i 1 Þ δkp 2L 2c2
It can be noticed that the first term of Eq. (9) converges to a constant value (assume it Eq. (9) can be written in the following form: 8 νm < n > X ν π y yp ¼ νm sin ð2j 1Þ > 2L l ð2i 1Þ2 :0 j ¼ 1;j a i ð2j 1Þ 2j 1
(10) as νm). However, the second term of :
yl 4yp
:
yl oyp
:
yl ¼ yp
Thus, the difference between the undamaged and damaged mode shapes can be expressed as: 8 2νm : yl 4 yp > < ðiÞ 0 : yl o yp δϕ ðyl Þ ¼ > : νm : yl ¼ yp
(11)
(12)
Hence, a certain amount of jump is observed in difference of the damaged and undamaged mode shapes around the location of damage, as in case of the fundamental mode. Hence, for damage localization the signature of mode shape difference and its derivatives of the fundamental mode as proposed by Roy and Ray-Chaudhuri [17] also remains valid for
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higher modes. However, the prominence of this signature is controlled by a mode shape dependent factor. In Eq. (12), the constant value m is associated with a multiplier ν, which basically comes from the first derivative of the ith mode shape for a cantilever shear beam evaluated at the damage location. So, it can easily be understood that, for a damage at the location of nodes of a particular mode shape, the multiplier has the maximum value, whereas for a damage at the antinodes, it reaches the minimum value. Thus, in case of damage at the antinodes of a mode shape, a jump cannot be easily detected from the plot of difference in mode shapes owing to its small value (i.e., for ν close to zero). In that case, a normalization of difference in mode shapes with respect to the maximum difference value may be useful, provided there is no distortion of the mode shapes due to numerical evaluation scheme used, experimental data processing etc. For example, in case of taller buildings (with large number of DoFs), identification of damage location even near (but not exactly at) the antinode of a mode shape may not be that difficult due to relatively smooth nature of the mode shape. For shorter buildings (with small number of DoFs), however, an occurrence of damage close to the antinodes of a mode shape makes it really difficult to localize the damage owing to significant distortion of the mode shapes. 2.3. Numerical illustration with shear buildings To demonstrate the efficiency of the mode shape-based approach in damage localization, at first, simulation studies have been performed considering a 30-storey shear building model. The building has been modeled considering uniform floor mass and storey stiffness distributions along the height, with typical floor mass of 1 105 kg and storey stiffness of 2 108 N=m. To get a comprehensive idea about the efficiency of mode shape-based approach in damage localization, damage scenarios have been simulated at different stories by 10 percent reduction of the corresponding storey stiffness as illustrated in Fig. 1. Eigenvalue analysis has been done to get the frequencies and mode shapes for both damaged and healthy models. Fig. 2 shows the first three mode shapes of the building. Six damage cases are considered by reducing the
Fig. 2. First three mode shapes of 30-storey shear building and damage locations for different damage cases considered.
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stiffness of the stories as shown in this figure. The locations of damage are chosen in such a way that they are either near the nodes or antinodes of the second and third mode shapes. Fig. 3(a)–(d) is plotted for damage Case-1, i.e., for damage at 11th storey. Fig. 3(a) shows the undamaged and damaged fundamental and second mode shapes. Both mode shapes are normalized in such a way that the maximum value at top DoF is unity. From Fig. 3(a), it can be observed that there is no visible difference between the damaged and undamaged mode shapes for both first and second modes. However, from Fig. 3(b), it is clear that there is a jump in mode shape at the location of damage for both modes with the fundamental mode shape showing larger jump than the second mode shape. In fact, for the second mode shape, the jump is one sided. This is because the derivative of the second mode shape at this location is minimum (close to zero). Fig. 3(c) shows the derivative of change in mode shape or slope of change in mode shape fδϕg for both fundamental and second mode evaluated using finite difference approach. One can notice from Fig. 3(c) that for both modes, the slope attains maximum at the location of damage. Fig. 3(d) provides the curvature of change in mode shape evaluated in a similar manner. A zero crossing is observed for both modes in Fig. 3(d). Also, from Fig. 3(c) and (d) one can notice that the signature as proposed by Roy and Ray-Chaudhuri [17] for the fundamental mode efficiently localizes the damage at the 11th storey level using the second mode shapes also. However, the magnitude of jump, slope or change in curvature at the damage location is larger for the fundamental mode compared to the second mode. This is because the absolute magnitude of derivative of the fundamental mode shape is higher than that of the second mode shape. Fig. 4(a)–(d) is plotted in a similar way to that of Fig. 3(a)–(d) except that it is for damage Case-2, i.e., for damage at the 21st storey. It can be observed that the signature for damage localization remains the same at the damage location. However, in this case (i.e., Case-2) contrary to Fig. 3, the magnitude of jump, slope or change in curvature at the damage location is larger for the second mode compared to the fundamental mode. This is due to the fact that at the damage location, absolute value of the derivative of second mode shape is much higher than that of the fundamental mode shape. Hence, one can conclude that at this location, second mode shape may perform better for damage localization compared to the fundamental mode. The plot of change in third mode shape, its (change in third mode shape) slope and curvature are provided in Fig. 5 for damage Case-3 (i.e., damage at the 7th storey) and Case-4 (i.e., damage at the 19th storey). Similar plots are shown in Fig. 6 for damage Case-5 (i.e., damage at the 13th storey) and Case-6 (i.e., damage at the 25th storey). It may be observed from Figs. 5 and 6 that damage is efficiently localized using the well-established signature of the fundamental mode. However, by comparing the values of X-axis in Figs. 5 and 6, one can notice that the magnitude of jump in the change in third mode
Fig. 3. Plot for Case-1 (damage at 11th storey): (a) undamaged and damaged first and second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first and second modes.
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Fig. 4. Plot for Case-2 (damage at 21st storey): (a) undamaged and damaged first and second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first and second modes.
shape, maximum value of slope of change in this mode shape and curvatures of change in this mode shape at damage location is one order higher for damage Case-5 and Case-6 when compared with damage Case-3 and Case-4. This implies that a damage at nodes is more sensitive to the change in third mode shape and its derivatives when compared to a damage at antinodes. This is in agreement with the observations of the second mode. To study how mode shape smoothness affect the efficiency of damage localization in difficult locations (near antinodes), another shear building with the same mass and stiffness distributions along the height is considered but with only 10-stories. Hence, the mode shapes for this building will be less smoother (when adjacent DoFs are joined by a straight line) due to lower number of DoFs when compared to the 30-storey building. Fig. 7(a) shows the damaged and undamaged second mode shapes for damage at 11th storey of the 30-storey building (i.e., damage Case-1) and for damage at 4th storey for the 10-storey building. In all plots of Fig. 7, the height of the building is normalized to unity for convenience of comparison. It may be observed from Fig. 7(a) that although the undamaged mode shape of the 10-storey building slightly deviates from that of the 30-storey building (due to change in number of DoFs), the undamaged and damaged mode shapes almost match for each building. However, from Fig. 7(b)–(d), it is clear that for the 10-storey building, the damage signature is not at all visible, while for the 30-storey building, a clear damage signature is observed. This is due to the fact that at this location, the change in mode shape follows a gently sloping curve (a slowly varying curve) and hence, the slope of this curve evaluated with a larger (finite difference) step size becomes lower compared to that with a smaller step size. Fig. 8 provides similar plots but with the third mode shape for damage at the 7th storey of the 30-storey building (i.e., damage Case-3) and for damage at the 2nd storey for the 10-storey building. It is observed that damage signatures are present for both buildings in change in mode shape, and slope and curvature of change in mode shapes. It may however be noted that for the 30-storey building identification of damage location becomes easier compared to the 10-storey building. For 10-storey building, by comparing Fig. 7 with Fig. 8, one can note that identification of damage becomes easier for the third mode compared to the second mode. This is because, the absolute derivative of the third mode shape near the damage location (shown by ——- line in Fig. 8) is higher than that of the second mode at damage location shown in Fig. 7.
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Fig. 5. Plot for Case-3 (damage at 7th storey) and Case-4 (damage at 19th storey): (a) difference in mode shapes, (b) slope of difference in mode shapes and (c) curvature of difference in mode shapes for third mode.
3. Experimental study with moment frame Usage of moment frames is very common for buildings around the world. It is well-known that the dynamic behavior of moment-frame buildings can be represented as the combination of shear and Euler–Bernoulli beams, though in short to middle height buildings, the behavior is primarily dominated by shear beam-type characteristics (Miranda and Taghavi [22]). In this section, the efficiency of mode shape-based methods in localizing damage has been studied experimentally considering a six-storey three-dimensional steel moment-resisting frame building model. Impact hammer tests were performed to evaluate the dynamic characteristics (frequencies and mode shapes) of the model. The model was fabricated using steel and aluminium sections. A schematic diagram of the frame model is shown in Fig. 9. Fig. 9(a) and (b) shows the elevation from front and plan from top of the undamaged structure, respectively. Total height of the structure was 780 mm and each storey height was 130 mm. The floor plan of the model was 200 mm 200 mm. Each storey had two steel beams of length 200 mm each and two aluminium plates each of length 200 mm placed along two perpendicular directions. Columns were made of steel and the height of each one was 130 mm. Cross sectional dimension of the steel beams were 8 mm 8 mm and that of the columns were 10 mm 10 mm, whereas the aluminium plates had a cross sectional dimension of 40 mm 5 mm. A strong column-weak beam approach was followed while preparing the experimental model. The connections were designed in such a way that by opening a few screws, a beam or a column element can be replaced for simulating damage conditions. In this scaled model, the mass and stiffness were uniformly distributed in all the stories. It may be noted that the intended direction of vibration (i.e., for which the dynamic properties are evaluated) is perpendicular to the plane of the paper in Fig. 9(a) and along the length of the paper in Fig. 9(b).
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Fig. 6. Plot for Case-5 (damage at 13th storey) and Case-6 (damage at 25th storey): (a) difference in mode shapes, (b) slope of difference in mode shapes and (c) curvature of difference in mode shapes for third mode.
3.1. Damage scenarios Since the structure is short, only two damage scenarios were simulated to experimentally verify the efficiency of mode shape curvature-based approach in localizing damage in the frame structure. Damage cases are simulated by replacing the steel beams of a particular storey with another set of beam of same length but with notches being cut at the ends of the beams. Snapshots of the undamaged model and of a particular damage case has been shown in Figs. 11 and 12, respectively. The damage cases are shown in Fig. 10, where the direction of vibration is also shown by an arrow at the top of the schematics. In Case-I, two steel beams at the second floor level were replaced whereas in Case-II, similar replacement was performed at the fourth floor level. It may be clear from Fig. 12 that the effect of damage in steel beams will result in change in eigenproperties of the model along the direction of vibration as a result of stiffness reduction along this direction. 3.2. Instrumentation and test procedure Impact hammer test was performed with a PCB086D20 [23] type short-sledge impulse hammer (sensitivity of 0.23 mV/ N). For measuring the acceleration response, PCB393B04 [23] seismic, miniature (50 gm), ceramic flexural ICP accelerometer with a sensitivity of 1 V/g was used. Agilent35670A [24] 4 channel FFT Dynamic Signal Analyzer was used for data acquisition. Channel one is the input channel and the remaining channels are used as output channel. The input excitation voltage for each channel was controlled to ensure perfect functioning of the instrument. The measured data was saved (in SDF format) in the internal disk. Then using the Agilent Datalink [24] software the stored data was transferred to a computer for postprocessing. The hitting was done at the top corner of the frame along the direction of steel beams. The accelerometer was attached at the middle of a particular floor with the help of an L-shaped plate (can be observed in Fig. 12). The accelerometer was connected in such a way that the sensing direction coincided with the direction of intended acceleration measurement. For
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Fig. 7. Effect of mode shape smoothness: (a) undamaged and damaged second mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for second mode in case of damage at 4th storey of 10-storey shear building and damage at 11th storey of 30-storey shear building.
each floor, 10 FRFs (frequency response function) were recorded and the average was calculated using the dynamic signal analyzer. Subsequently, the accelerometer was connected to the next floor and the same procedure was repeated with the hitting being done at the same point, i.e., at the top corner. The test was concluded by collecting FRF data from all floors. While getting the FRFs from the experiment, it was ensured that the coherence was above 0.99. 3.3. Results The recorded FRFs were then transferred to a computer using Agilent Datalink [24] software and analyzed to obtain the frequencies and mode shapes with ME'scopeVES [25] software. For this purpose, a wire frame model of the six-storey frame was created in ME'scopeVES [25] software and then the FRF data was imported. By overlaying all traces, it was observed that these traces are in good agreement. Then using the global polynomial curve fitting method, residues, damping and mode shapes were obtained. Noting the peaks in the FRFs, modal frequencies were identified and from the table of shapes, respective mode shape values along with their phases were recorded. Fig. 13 shows overlaid traces in the ME'scopeVES [25] as well as the wire-frame model. The mode shapes were normalized in such a way so that the value corresponding to the top DoF is equal to unity. Subsequently, their derivatives (up to the second order) were calculated using finite difference method in MATLAB [26]. Table 1 provides the experimentally obtained natural frequencies of the fundamental and second modes for the undamaged structure and structure with Case-I and Case-II damages. It may be noted that the structure is very stiff with the fundamental frequency of 24.5 Hz and only two modes were obtained by the test. It may also be noted that the maximum reduction in frequency due to damage in Case-I for the fundamental mode and Case-II for the second mode. Also, the percentage reduction in frequency occurred for a damage causes higher percentage reduction in the fundamental mode compared to the second mode. These results are in agreement with the findings of Ray-Chaudhuri [21]. In addition, the maximum percentage reduction in frequency is about 6 percent for the fundamental mode and 1.5 percent for the second mode. To check the validity of the experimental results, a numerical model of the experimental specimen has been created in SAP2000 [27]. Frame sections were assigned to the beam and columns of the 3D frame model. The base of the structure was considered to be fixed for modeling purpose and self-weight was considered as mass source.
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Fig. 8. Effect of mode shape smoothness: (a) undamaged and damaged third mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for third mode in case of damage at 2nd storey of 10-storey shear building and damage at 7th storey of 30-storey shear building.
Dynamic properties of the model was evaluated by performing an eigenvalue analysis. It is found from this analysis that the fundamental frequencies corresponding to the undamaged, damaged Case-I and Case-II are 26.6, 25, 25.5 Hz, respectively, implying a reasonable agreement with those obtained from the experimental data. Small but consistent Á over-estimations observed ðA10 percentÞ in frequencies for these cases are due to various modeling assumptions. One probable reason includes higher stiffness of the analysis model due to rigid-joint assumption in modeling, which may not be completely valid due to the way members are connected in the experimental model. To check the efficiency of the fundamental and second mode shapes in damage localization, two damage cases are considered as shown in Fig. 10. Fig. 14 shows the results for fundamental mode shape. In Fig. 14(a), the fundamental mode shapes in undamaged and two damaged conditions are shown. It can be observed from this figure that the damage at the 2nd floor beam (Case-I) causes more changes compared to the damage at the 4th floor beam (Case-II). This can also be verified from Fig. 14(b), where changes in fundamental mode shape due to damage for both Case-I and Case-II are shown. Fig. 14(c) and (d) provide the slope and curvature of change in fundamental mode shape due to damage. It can be observed from these figures that the damage can be localized in both cases using the signature of the fundamental mode shape, i.e., maximum slope and zero crossing at the location of damage. Fig. 15(a)–(d) provides the plots similar to that of Fig. 14(a)–(d) except that Fig. 15 is for the second mode. One can notice from this figure that the damage at the 4th floor beam can be identified more easily (more pronounce features) compared to the case of damage at the 2nd floor beam. This is because a damage at the 2nd floor and 4th floor beams are near the antinode and node of the second mode, respectively. As argued earlier (see Section 2.2), the 2nd floor damage location is close to the antinode of the second mode shape and hence, it is difficult to detect a jump in the difference of mode shapes. A slight distortion in the second mode shape and its derivatives (due to a low number of degrees-of-freedom coupled with the location of damage being close to the base) further complicates the detection of damage. Hence, based on these observations, one can conclude that the experimental results support the location sensitivity of mode shape curvature-based methods in damage localization, as proposed earlier.
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Fig. 9. Schematic diagram of the experimental model. (a) Evaluation. (b) Plan.
Fig. 10. Damage scenarios considered for frame building model (the direction of vibration is shown by an arrow at the top of the schematic).
4. Conclusions In this study, a mathematical formulation between the difference in mode shape due to damage and damage parameters has been derived. From this derivation, it has been found that the efficiency of mode shape-based damage localization approach is dependent on the location of damage in a building. In particular, it has been found that the change in mode
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Fig. 11. Snapshot of the experimental model showing details.
Fig. 12. Snapshot of (a) damaged structure and (b) damaged second floor beam.
shape due to damage is related to the derivative of that mode shape evaluated at the location of damage. Numerical studies have been performed considering shear building models to show this location sensitivity of mode shapes for damage localization. Damage conditions have been simulated by reducing the stiffness at desired locations. Further, the effect of mode shape smoothness on damage localization has also been studied by considering building with relatively less number of stories. An experimental study has also been performed considering a three-dimensional six-storey steel moment-
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Fig. 13. Snapshot of (a) overlaid traces and (b) wire frame model of experimental specimen.
Table 1 Experimentally obtained natural frequencies (in Hz). Mode
Undamaged
Case-I
Case-II
Fundamental Second
24.5 84.4
23 83.4
23.8 82.5
resisting frame model to verify the analytical findings for short frame buildings. The major findings of this study can listed as follows: 1. The damage signature proposed by Roy and Ray-Chaudhuri [17] for the fundamental mode is also valid for higher modes. 2. It has been found that although the higher mode shapes are effective in localizing damage, their efficiency may significantly get reduced depending on the location of damage. 3. If a damage is located near the nodes of a particular mode, the damage can be localized most efficiently using that mode. On the other hand, if a damage is located near the antinodes of a particular mode, it may be difficult to localize such damage by using that particular mode. 4. The efficiency of damage localization also gets affected by the smoothness of a higher mode shape (i.e., number of DoFs), especially when the damage is located near the antinodes of that particular mode. 5. The location sensitivity of mode shapes in damage localization is found to be valid for short moment-resiting building frames. It may be noted that the findings of this study are valid for small change in stiffness and for shear beam or buildings, which show dominant shear-type behavior. For Euler–Bernoulli beam or other cases, a formulation has to be carried out to establish the location dependency of mode shapes. This will help in explaining the observations of Kim et al. [19], Qiao et al. [14] and Owolabi et al. [20]. It may also be noted that identification of an exact damage location in a beam or in a column of a typical frame structure using mode shape-based signatures as proposed herein is a challenging task. In absence of a dense array of additional sensors (which may be prohibitively expensive), post-event local inspection may be good idea to further
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Fig. 14. Plot of (a) undamaged and damaged mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for first mode in case of damage at second floor and fourth floor beams.
Fig. 15. Plot of (a) undamaged and damaged mode shapes, (b) difference in mode shapes, (c) slope of difference in mode shapes and (d) curvature of difference in mode shapes for second mode in case of damage at second floor and fourth floor beams.
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localize the damage. Nonetheless, the mode shape-based damage signatures similar to those proposed by Panikkaveettil et al. [28] can be used to narrow down whether the damage is in a beam or in a column. Acknowledgments The authors would like to thank Dr. Koushik Roy and Ms. Bhavana Valeti (former graduate student) for designing the experimental model. The help from the workers of the Structural Engineering Laboratory, IIT Kanpur during the experiment is also greatly appreciated.
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