Damage localization and quantification in simply supported beams using static test data

1 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 Damage localization and quantification in simply supported beams using static test data Q Ma and M Sol´ıs Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla, Spain E-mail: [email protected] Abstract. A novel simple method using static test data for damage detection, localization, and quantification in beams is presented in this paper. The method is based on the change of the deflections of the beam between a reference and a damaged state. For simply supported beams with a single damage, the maximum value of the change of deflections indicates the location of damage. Once the damage is located, one could estimate the rotational stiffness at the damaged cross section by applying a superposition scheme to isolate the effect of damage and by using basic structural analysis equilibrium equations. Afterwards, damage extent is evaluated through an existing relation between rotational stiffness and damage severity. Several static tests of a simply supported steel beam with a point load at different locations were conducted to exam the performance of the strategy. The damage is artificially introduced as an opened crack located at the bottom of the beam. The deflections of the beam were measured by using a Digital Image Correlation system. The results show that the method can accurately detect and quantify the damage. The method is non-model based and can be easily conducted. No specific loading positions are required and damage identification objective can be achieved from just one single static test. 1. Introduction The fundamental objective of damage detection is to identify the change of properies in a structure caused by damage, which includes natural frequencies, dampings, stiffness or flexibility matrix, mode shapes, and etc [1]. Many methodologies and techniques proposed by researchers are based on system identification or parameter estimation through the dynamic response of the structure. Various numerical methods are applied for damage detection and localization, such as the transimissibility function [2–4], the BAT algorithm [5], etc. Some researchers also applied similar ideas using static response data of the structure. Caddemi and Morassi [6] identified a single crack in beams with different boundary conditions using the static displacements. Lee and Eun [7] presented a method for locating damage through the change of curvature of static deflections. Bakhtiari-Nejad, Rahai and Esfandiari [8] developed an algorithm based on the change of stored strain energy in the elements using static noisy data for damage detection. Seyedpoor and Yazdanpanah [9] also illustrated a method through the change of strain energy using static noisy data. In this paper, a novel non-model based simple method for single crack damage detection and localization of beams is presented. Firstly, the theory of the method is presented to illustrate how the change of pre- and post-damaged static displacements of the beam under external 2 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 (a) (b) (c) Figure 1. States of the superposition scheme: (a) Damaged State (D) , (b) reference or Undamaged State (U) and (c) Incremental State (I). loads can be used for damage localization. From the information about the damage location, a damage quantification method based on structural analysis is introduced. Next, a series of experimental tests of a cracked simply supported beam were conducted to exam the performance of the method. 2. Theory background It is known that the prescence of a crack will cause a reduction in the local stiffness at the cracked cross section. Hence a single damaged beam could be modeled as a rotational spring at the cracked location that connects two undamaged parts of the beam [10]. The problem of a cracked beam under some external forces (P) (Damaged State,D) can be decomposed into an Undamaged State (U) plus an Incremental State (I) (Fig.1). Thus, the deformation (U) and internal forces (f) can be written as UD=UU+UIand fD=fU+fI(1) In figures Fig. 1 (a), (b) and (c), CSLand CSRare the left and right sides of the damage cross section respectively, Ktis the rotational stiffness of the spring that models the cracked cross section, mis the internal bending moment at CSLand CSR(they are equal to each other), msp is the internal torsional moment of the spring, θLand θRare the rotations at CSLand CSR, respectively, and the footnotes U,D, and Istand for the Undamaged (or reference), Damaged, and Incremental States respectively. In the Undamaged State, the rotations at CSLand CSRare set to be equal (θL,U =θR,U ), which indicates the spring is not present in the undamaged beam. It is found that this superposition is valid when the applied moment (M) in the Incremental State is equal to the internal bending moment at damage location (mU) in the Undamaged State. The damage locations are revealed in the overall deformed shape of UIsince the external forces will introduce slope (rotation) discontinuities at damaged cross sections (Fig. 1 (c)). For a single cracked simply-supported beam, the shape of UIis piecewise linear and it is independent from 3 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 the loading position. Its magnitude depends on the magnitude of external loads, the severity of damage and the relative position of the load and the damage. A finite element model of a simply-supported Timoshenko beam with a 1200mm length (L) and a 100 ×20mm rectangle cross section was built in ANSYS (mesh size 120mm). A spring with a rotational stiffness (Kt) of 1.8e5N/m2was used to model the crack at 0.4Lfrom the left end. A concentrated load, 1kN, was applied at 0.6Lfrom the left end. The deflections of the beams for the Damaged and Undamaged States are shown in Fig 2 (a). The deflection under a self-equilibrated bending moment mUcorresponding to the Incremental State (UI) and the difference between the displacements of the Undamaged and Damaged States (∆U) are shown in Fig. 2 (b). It is shown that UIis equal to ∆U. The discontinuity in the slope indicates the damage location precisely. The slight difference between ∆Uand UIis due to numerical errors. 0 200 400 600 800 1000 1200 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 Position (mm) U (mm) UU UD (a) 0 200 400 600 800 1000 1200 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 Position (mm) U (mm) ∆U UI (b) Figure 2. Deflection results of the finite element model: (a) UDand UUof simply-supported beam; (b) ∆Uand UIof simply-supported beam Once the damage is localized, the rotational stiffness of the cracked beam could be estimated from the Incremental State using the following expression: Kt=msp,I (θL,I −θR,I )=msp,I ∆θI (2) The rotation discontinuity (∆θI) can be directly computed from ∆U(UI) and the moment absorbed by the spring (msp,I ) can be automatically calculated for a statically determinate beam since the reactions of the beam for the Incremental State are null and therefore msp,I equals M (and mU, as indicated previously). Once the rotational stiffness of the damaged cross-section is determined, the extent of damage can be estimated by comparing it with an existing correlation between damage size and rotational stiffness. 3. Experimental Test of A Simply-Supported Beam 3.1. Test Setup An experimental test of a simply-supported steel beam was conducted to test the performance of the methodology. The dimension of the beam was 1200 ×100 ×20mm. A notch was cut at the bottom of the beam at 0.35L(425mm) from the left end. The depth of the notch was set to be 7mm (35% of the beam height). A Digital Imagine Correlation(DIC) system (Fig. 3 (a), (b) and (c)) was used for measuring the deflection of the beam under loading. A total number of 241 measuring points (damage at the 86th) were marked along the beam with an equal spacing of 5mm. A concentrated force was applied on the beam vertically through hanging a 120kg mass on it. 21 tests were performed by putting the mass at 21 equally distributed positions along the beam. The scheme of the test is shown in Fig. 3 (d). 4 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 (a) (b) (c) (d) Figure 3. (a) Experimental setup, (b) DIC measuring system, (c) Tested beam and (d) Test scheme 3.2. Implementation of the methodology Due to the effect of noise on the measured data, a trend estimate function named l1 Trending Filter is used to estimate the overall shape of ∆U. The l1 Trending Filter produces trend estimation that is piecewise linear through minimize the objective function in Eqn. (3), where λis a nonnegative parameter. xtis the estimated trend and ytis the signal [11]. This trending filter automatically identifies the turning point along the piecewise shape data. (1/2) n X t=1 (yt−xt)2+λ n−1 X t=2 |xt−1−2xt+xt+1|2(3) The results of ∆Uand the application of l1 Filter to ∆U(∆Ul1) are displayed in Fig. 4 and 5. For all 21 loading positions, the shape of ∆Uwas estimated correctly. The effect of noise only takes a relative high influence when the loading positions are close to supports of the beam (at positions 1, 20 and 21). The position where the maximum value of ∆Ul1takes place is considered as the damage location. The predicted results are listed in table 1. All the predicted damage locations fall into a small range from the correct location. The furthest predition is at point 93 (for loading position 20), which is 35mm to the right of the real damage. Therefore, it is shown that the methodology can successfully localize the damage for this damage scenario. For a notch type opened cracked on an elastic beam with rectangular cross-section, the equivalent rotational stiffness of the damaged cros-section (Kt) proposed by Rizos, Aspragathos, and Dimarogonas [12] (Eqn. (4) and (5)) is used in this paper for damage extent estimation: 5 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 0 200 400 600 800 1000 1200 −0.06 −0.04 −0.02 0 0.02 0.04 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (a) 0 200 400 600 800 1000 1200 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (b) 0 200 400 600 800 1000 1200 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (c) 0 200 400 600 800 1000 1200 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (d) 0 200 400 600 800 1000 1200 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (e) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (f) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (g) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (h) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (i) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (j) 0 200 400 600 800 1000 1200 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (k) 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (l) Figure 4. ∆Uand ∆Ul1with loading at positions 1 (a) to 12 (l). 6 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 0 200 400 600 800 1000 1200 −0.2 −0.15 −0.1 −0.05 0 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (a) 0 200 400 600 800 1000 1200 −0.15 −0.1 −0.05 0 0.05 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (b) 0 200 400 600 800 1000 1200 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (c) 0 200 400 600 800 1000 1200 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (d) 0 200 400 600 800 1000 1200 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (e) 0 200 400 600 800 1000 1200 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (f) 0 200 400 600 800 1000 1200 −0.08 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (g) 0 200 400 600 800 1000 1200 −0.06 −0.04 −0.02 0 0.02 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (h) 0 200 400 600 800 1000 1200 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 Position (mm) ∆ U (mm) ∆ U ∆ Ul1 (i) Figure 5. ∆Uand ∆Ul1with loading at positions 13 (a) to 21 (i). Kt= 1/c c = 6πhJ(ξ)/EI (4) where his the height of the beam, Ethe elastic modulus of the material of the beam, Ithe inertia of the cross-section and Jis the following function of the ratio (ξ) between the notch depth and the height of the beam. J(ξ)=1.86(ξ)2−3.95(ξ)3+ 16.375(ξ)4−37.226(ξ)5 + 76.81(ξ)6−126.9(ξ)7+ 172.5(ξ)8−143.97(ξ)9+ 66.56(ξ)10 (5) By using this empirical relationship, damage severity can be estimated from the experimentally identified rotational stiffness (Kt) in Eqn.(2). The rotation discontinuity (∆θI) can be evaluated using a piecewise linear regression function of UI, whereas msp,I (equivalent to mU) can be easily computed from equilibrium equations of the undamaged beam. The estimated damage extent for each test are also listed in table 1. All of the estimated damage severities are bigger than the real value. This may indicate some discrepancy between the analytical model of 7 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 842 (2017) 012007 doi :10.1088/1742-6596/842/1/012007 Table 1. Estimated damage location and severity and their errors Localization Quantification Loading Measuring Location Deviation Severity Error Error Position Point (mm) (mm) (mm) (mm) (%) 1 85 415 5 7.5 0.5 7 2 89 440 15 7.7 0.7 10 3 89 440 15 7.7 0.7 10 4 88 435 10 7.9 0.9 13 5 88 435 10 7.8 0.8 11 6 88 435 10 7.8 0.8 11 7 86 425 0 7.8 0.8 11 8 86 425 0 7.8 0.8 11 9 87 430 5 7.8 0.8 11 10 85 415 5 7.9 0.9 13 11 87 430 5 7.8 0.8 11 12 87 430 5 7.7 0.7 10 13 87 430 5 7.7 0.7 10 14 87 430 5 7.8 0.8 11 15 89 440 15 7.7 0.7 10 16 87 430 5 8.0 1.0 14 17 91 450 25 7.8 0.8 11 18 89 440 15 7.8 0.8 11 19 90 445 20 7.9 0.9 13 20 93 460 35 7.7 0.7 10 21 87 430 5 7.7 0.7 10 the crack as a rotational spring and the actual behavior of the damaged cross section. However, for all the loading positions, the method provides predictions with high accuracy even for those with low signal to noise ratio. 4. Conclusion A non-model based damage detection and localization methodology based on the static displacements is presented in this paper. No specific loading positions are needed for the experimental test and structural identification is not required. Experimental results of a simplysupported steel beam with a single crack are provided. The methodology successfully predicts the crack location with a very high accuracy for all loading positions. From the predicted damage locations, the damage extent can be estimated using an existing analytical correlations between damage extent and rotational stiffness of the damaged cross section. The method provides results with high accuracy as well. In summary, the paper proves the efficiency and simplicity of the method for practical purpose. Acknowledgments This work was supported by the Consejer´ıa de Econom´ıa, Innovaci´on, Ciencia y Empleo of Andaluc´ıa (Spain) under project P12-TEP-2546. The financial support is gratefully acknowledged. 8 1234567890 12th International Conference on Damage Assessment of Structures IOP Publishing IOP Conf. Series: Journal of Physics: Conf. 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