Experimental study of structural damage identification based on modal parameters and decay ratio of acceleration signals

Abstract

A novel damage assessment method based on the decay ratio of acceleration signals (DRAS) was proposed. Two experimental tests were used to show the efficiency. Three beams were gradually damaged, and then the changes of dynamic parameters were monitored from initial to failure state. In addition, a new method was compared with the linear modal-based damage assessment using wavelet transform (WT). The results clearly show that DRAS increases in linear elasticity state and microcrack propagation state, while DRAS decreases in macrocrack propagation state. Preliminary analysis was developed considering the beat phenomenon in the nonlinear state to explain the turn point of DRAS. With better sensibility of damage than modal parameters, probably DRAS is a promising damage indicator in damage assessment.

Keywords

damage assessment / decay ratio of acceleration signals (DRAS) / wavelet transform (WT) / modal analysis / reinforced concrete beam / beat phenomenon

Introduction

Huge constructions, under complicated loads such as earthquakes and typhoons, are prone to all kinds of damage, such as tensile fracture to high arch dams, which may pose a serious threat to their security, reliability and durability. For significant structures, an accurate and reliable method for detection is essential. Vibration-based methods are sensitive, to some extent, to identify the damage evolution, and even the microcracks are situated in hidden or internal zones [1]. These detecting techniques are based on the fact that damage alters the dynamic properties of the structure, like natural frequencies, damping, stiffness, mode shapes and quantities derived from these parameters. Therefore, measuring these changes can provide an identification of damage.

However, most currently used vibration techniques are linear techniques, based on the variation of modal parameters. Generally, the linearity hypothesis of the vibration is not accurate enough [2], and low sensitivity might be a problem [3–6]. Recently, some new methods focusing on the nonlinearity or amplitude dependence of the material’s complex module have been developed [1,7,8]. Thanks to the nonlinear feature of RC structure in essence, it is superior to using nonlinear techniques than linear hypothesis to identify concrete damage. Furthermore, the complexity of nonlinear performance of RC, study on using the nonlinear amplitude-dependent vibration properties of RC to detect damage has concentrated on experimental work [7]. However, there is not much literature existing on identification of RC damage using dynamic test.

In this paper, a novel damage assessment method using the decay ratio of acceleration signal (DRAS) index is presented. One can obtain a consistent relationship between the DRAS and the damage severity during crack propagation (linear elasticity state, microcrack propagation state and macrocrack propagation state). Two tests including three RC beams in laboratory were conducted to show the efficiency. In addition, the new method was compared with the means of modal-based damage assessment using wavelet transform (WT). The results demonstrate that DRAS can identify the severity of damage of structures more effectively.

Description of experiment

Test samples and setup for the tests

There were two tests with different main failure types. The first test containing one brittle beam was damaged as shear failure, while the second test including two flexible beams were damaged as bending failure.

In test one, the specimen of beam 1 was a 0.1 m × 0.16 m × 1.4 m reinforced concrete beam under four-point loading (see Figs. 1 and 2). Besides, three standard concrete cubes (150 mm × 150 mm × 150 mm) were made to measure the concretes’ compressive strength. The yield strength of reinforced steel bars and the compressive strength of concrete were f_y = 210 MPa and f_cu,k = 13.3 MPa. Two YD-65 accelerometers were used to measure the dynamic response (vertical acceleration at different points; see Fig. 2) of the beam, sampled at 512 Hz each channel. The experimental setup test was schematically shown in Fig. 3(a).

In test two, the specimens of beam 2 and beam 3 were both 0.2 m × 0.16 m × 2.7 m reinforced concrete beams under four-point loading (see Figs. 4(a) and (b)). Beam 3 was initially damaged by cutting two gaps, which were specimen 2-2 and specimen 3-3 (see Fig. 5).

The yield strength of reinforced steel bars and the compressive strength of concrete were the same with the test one; however, four YD-65 accelerometers were used to measure the dynamic response (vertical acceleration at different points; see Fig. 4) of the structure, sampled at 2056 Hz each channel. The setup of test two was schematically shown in Fig. 3(b).

Test procedure and experimental observations

According to beams’ bearing capacity, beam 1 and beam 2 were damaged to failure in nine steps (see Table 1), while, beam 3 was damaged to failure in six steps (see Table 2). Loads increased monotonically until beams were destroyed completely. As a hypothesis, damage level corresponded to the load stage (see Table 1). When the beam was subjected to a load stage for ten minutes, the concrete’s up-surface’s strain was measured at 1/3, 1/2, and 2/3 span lengths (C1, C2, C3), as well as tension reinforcements’ strain at mid-span, namely, R1 and R2 (see Figs. 6–8). However, up-surface’s strain was measured only at mid-span on beam 1. Additionally, cracks were recorded (see Tables 2–4).

Then, the load was removed, and the beam was beaten by an impact hammer (LC-02A, the max impact force is 5 kN). Meanwhile, a free vibration accelerate signal was measured. The procedure was then repeated for the next loading stage. The signals generated by the accelerometers were amplified by a charge amplifier YE5858B and recorded by Signal Acquisition and Analysis Software (CRAS).

As mentioned before, beam 1, whose first frequency was approximately 110 Hz, was a brittle beam; in addition, the impact force (no more than 5 kN) was so limited that it produced limiting energy; therefore, it was not able to excite higher modes. In test 1, it produced only the first mode. However, beam 2 and beam 3, whose first frequency was about 20 Hz, were flexible beams. Due to the decrement of the structure’s natural frequency, they could produce higher modes. In test 2, it produced the first three modes. Therefore, the first mode of beam 1 was analyzed, while the first three modes of beam 2 and 3 were analyzed.

Experimental analysis on modal techniques

Identification of modal parameters using wavelet transform

The free response of a linear time-invariant vibration system with one degree of freedom is given by

(1)x(t)=A_{\mathrm{0}}{\mathrm{e}}^{-\zeta w_{\mathrm{n}}t}\cos (w_{\mathrm{d}}t-{\varphi }_{\mathrm{0}}),

where A₀ and θ₀ are the initial magnitude and the phase angel, respectively. And w_d = w_n(1-ζ²)^1/2 is the damped natural frequency. Both A₀ and θ₀ depend on the initial condition.

Here we choose the well-known Morlet wavelet, which is quite natural to view information in terms of harmonics instead of scales, as given by

(2)g(t)={\mathrm{e}}^{i{w}_{\mathrm{0}}t}{\mathrm{e}}^{-t^{\mathrm{2}}/\mathrm{2}},

where w₀ is a wavelet tunable frequency, and w₀ = 2πf₀, where f₀ is a central frequency. The Morlet wavelet transform (MWT) of the system response (1) can be expressed as

(3)W_g(a,b)={\displaystyle \frac{\mathrm{1}}{\sqrt{a}}}{\displaystyle {\int }_{-\infty }^{+\infty }x(t)}{g}^{·}\left({\displaystyle \frac{t-b}{a}}\right)\mathrm{d}t,

where b is the translation parameter; a is the scale translation parameter. g* is the complex conjugate of g. Finally, W_g (a, b) can be gained by

(4)W_g(a,b)\approx {\displaystyle \frac{\sqrt{\mathrm{2}\mathrm{\pi }a}}{\mathrm{2}}}{A}_{\mathrm{0}}{\mathrm{e}}^{\alpha +\mathrm{i}\beta },

where

(5)\{\begin{array}{l}\alpha =-\zeta w_{\mathrm{n}}b-\mathrm{0.5}[(\mathrm{1}-\mathrm{2}{\zeta }^{\mathrm{2}})w_{\mathrm{n}}^{\mathrm{2}}{a}^{\mathrm{2}}-\mathrm{2}{w}_{\mathrm{0}}{w}_{\mathrm{d}}a+w_{\mathrm{0}}^{\mathrm{2}}],\hfill \\ \beta =w_{\mathrm{d}}b-{\varphi }_{\mathrm{0}}+\zeta w_{\mathrm{0}}{w}_{\mathrm{n}}a-\zeta w_{\mathrm{n}}{w}_{\mathrm{d}}{a}^{\mathrm{2}}.\hfill \end{array}

Therefore, fixing the scaling factor, i.e., taking a = a₀, one can obtain

(6)\ln \left|W_g(a,b)\right|\approx -\zeta w_{\mathrm{n}}b+c_{\mathrm{1}},

(7)\angle W_g(a,b)\approx w_{\mathrm{d}}b+c_{\mathrm{2}},

where c₁ and c₂ are independent of the translation factor b.

It is obvious that damping ratio ζ and natural frequency w_n can be estimated by Eqs. (6) and (7) from the slopes of the logarithm of modulus and phase of the wavelet transform with respect to parameter b, that is, k₁ and k₂,

(8)\{\begin{array}{l}{k}_{\mathrm{1}}=-\zeta w_{\mathrm{n}},\hfill \\ k_{\mathrm{2}}=w_{\mathrm{n}}\sqrt{\mathrm{1}-{\zeta }^{\mathrm{2}}}.\hfill \end{array}

Therefore, one can easily obtain damping ratio ζ and natural frequency w_n.

Results of assessment of modal parameters

After analyzing the data with MWT, the variation relationship between the modal parameters (frequency and damping ratio) and damage severity is obtained. By curve fitting with a least-square algorithm, one can get the slopes of magnitude and phase. Then, w_n and ζ can be easily estimated from Eq. (8).

The estimation results are summarized in Figs. 9-11. f1, f2, and f3 depict the first frequency, the second frequency and the third frequency, respectively, and so does the damping ratio description, u1, u2, u3. As can be shown, the frequency successively decreased due to consecutive loading stages. However, the decline trends are not obvious. The maximum frequency reduction of beam 1 is from the initial value 114.5 Hz to the final value 104.9 Hz, reaching only 8.4%, and beam 2 and beam 3 are 23.0%, 20.5%, respectively. On the other hand, the damping ratio scatters during the damaged procedure, though the first two damping ratios of beam 2 and beam 3 were increasing with fluctuation. Therefore, it is not an ideal index for damage estimation in this experiment.

Experimental analysis on decay ratio of acceleration signals

Novel damage estimation strategy

In this section, the strategy of the new damage assessment method will be introduced. The algorithm in detail is as follows:

Step 1 Convert measured acceleration signal data to free decay signal x(t) at different damage levels by random decrement technique (RDT), which can be used to transform the initial signal to the free vibration response.

Step 2 Compute the modal frequency f_k of signal data by means of WT by Eq. (8) or FFT (fast Fourier transformation) at different damage levels.

Step 3 Obtain a signal S_k containing a single mode, whose frequency is between f_kmin (f_kmin = frequency at the last damage level) and f_kmax (f_kmax = frequency at the first damage level) by filtration in order to exclude signals containing other modes. To improve the accuracy of the band-pass filter, the amplification 1/10 is employed, which is enough to separate signal with a particular frequency. Therefore, the filtration scope is set from 0.9f_kmin to 1.1f_kmax.

Step 4 Select the initial amplitude A_k and number of cycles T_k;

Step 5 Compute the time span:

(9)\mathrm{\Delta }{t}_{ki}=T_k/f_{ki}.

Step 6 Integrate the data to obtain the acceleration response’s decay amplitude’s accumulation E_ki, that is

(10)E_{ki}={\displaystyle {\int }_{t_{\mathrm{0}}(i)}^{t_{\mathrm{1}}(i)}{\ddot{x}}_i{(t)}^{\mathrm{2}}\mathrm{d}t}.

Step 7 Compute the acceleration signal’s decay power, that is

(11)P_j={\displaystyle \frac{E_{ki}}{\mathrm{\Delta }{t}_{ki}}}.

Step 8 Compute the decay ratio of acceleration signal index (DRAS), that is

(12)D_j={\displaystyle \frac{P_{\mathrm{1}}-P_j}{P_{\mathrm{1}}}}\times \mathrm{100}\%.

Consequently, one can gain a relationship between the DRAS and the damage severity. The implementation of the above algorithm is expeditious and convenient. The procedure of DRAS is schematically shown in Fig. 12.

Results of the new index

The number of cracks of beam 1 in test 1 became larger as the load increased since load stage 4. Meanwhile, the cracks became thicker and longer, especially after load stage 5; it was the same on beam 2. The further observation showed that macrocracks did not appear until load stage 5 from Tables 3 and 4. As can be shown, the number of cracks of beam 3 rose sharply from 2 to 10 at load stage 3; besides, the width of the largest crack went up to 0.2 from microcracks at load stage 4.

Figure 13 clearly illustrates the variation of the DRAS during the damage process. The influence of different location of impacting to DRAS value in the same damaged level was considered by impacting locations 2 to 12, as shown in Fig. 2. We investigated 14 samples and analyzed the co-relationships between DRAS and damage severity. The mean values represent 14 samples well, therefore they are underlined the solid red circle and plotted. The increasing excitation to a damage level increased the mean DRAS value by about 62.9% from the load stages 1 to 5 until macrocracks appeared. After that, the DRAS reduced to nearly 19.6% at the load stage 9 when the beam was totally damaged.

It is also worth mentioning that there was the same phenomenon on beam 2 and beam 3 (see Figs. 14 and 15). When macrocracks were emerging at load stage 6 on beam 2, at load stage 3 on beam 3, the DRAS index decreased among three modes except the first modal DRAS on beam 3. When we look at the FFT of the first mode of beam 3, we find that the energy of the first mode was relatively minor; therefore, it probably was influenced by other noisy frequencies. In Figs. 14 and 15, DRAS1, DRAS2, DRAS3 are DRAS of the first mode, the second mode and the third mode, respectively.

In brief, three tests have similar changing trends of DRAS during the crack propagation.

Comparison of results from modal and DRAS analysis

According to the results of assessment of modal parameters, frequency declines monotonically as damage evolves in both two tests. While it only dropped 3.0% when macrocracks appeared at load stage 5 and decreased 8.4% when beam 1 was totally damaged (see Table 5). On beam 2 and beam 3, there are larger drops which are about 26.1%, 23.2%, and 19.8% (averaging equals 23.0%); 27.3%, 21.8%, and 12.5% (averaging equals 20.5%), respectively, in the first three modal frequencies. However, it has a low decline when macrocracks appeared (averaging equals 11.3% on beam 2; averaging equals 12.3% on beam 3). Therefore, it is not a sensitive parameter for damage assessment especially for brittle RC structures.

Although the first two damping ratios of beam 2 and beam 3 were increasing with fluctuation, the damping ratio scatters during the damaged procedure in a whole. Therefore, it is not an ideal index for damage estimation in this experiment.

In comparison with low sensitivity of frequency and scatter of the damping ratio, the DRAS index has a much more sensitivity and a clear identification of damage propagation. The two tests demonstrate that the DRAS rises in beams’ linear elasticity state and the microcrack propagation state, while DRAS declines when beams are in macrocrack propagation state as the damage propagation. However, the style of cracks (microcracks or macrocracks) cannot be judged by one single DRAS value because one single DRAS value has two damage levels. But when macrocracks are emerging, the beat phenomenon of signals was shown clearly (see Fig. 16 and Fig. 17). Therefore, the DRAS index can be used to judge the style of cracks with the help of the beat phenomenon. Besides, it shows the turning point of the damage development, which is critical to structural safety when we know microcracks are evolving forward to macrocracks.

Analysis in nonlinear state after macrocracks

Explanation for the beat phenomenon of signals

It is necessary to consider the nonlinear behavior when structures are severely damaged. The most evident qualitative difference is the presence of the beat in the cracked RC structures (see Figs. 16 and 17). Zhu et al. [8] suggested that the response of a cracked beam to an impulsive load changes in time with the crack opening and closure. When the beating signal was present, the resonance peak (on the response FFT) split in the frequency domain. Zonta et al. [9] demonstrated that it was possible to explain the frequency splitting phenomenon through a linear general model, that is,

(13)m\ddot{x}+(c+di)\dot{x}+(k+hi)x=\mathrm{0},

where m, c, and k are the mass, damping and oscillator stiffness. The coefficients h and d are the hysteretic damping and the system dispersion.

Equation (13) has considered not only damping, but also imaginary damping (referred to as dispersion) and hysteretic damping (well known in the literature, for example, in the work by Ewin [10]). Therefore, Eq. (13) may be able to represent damaged motion equation.

Possible explanation for the decline of DRAS

When structures are damaged lightly, the cracks are slight, free response decays as an exponential decay style. According to Eqs. (10) and (11), the acceleration signal’s decay power decreases when free response decays as the exponential decay style; therefore, DRAS declines as damage developing due to the decreasing of the frequency and increasing of the damping ratio. However, macrocracks emerge when structures are damaged seriously, which causes the beat phenomenon. Meanwhile, the characteristic frequency splits to two close frequencies with different amplitudes. Equation (10) overlaps other close-frequency signals. It is clearly shown that the original signal is strengthened by another signal with extra frequency. Sequentially, the acceleration response’s decay amplitude’s accumulation E_k stops decreasing and then turns to rise when macrocracks emerge. Therefore, Eq. (12) decreases due to the rise of Eq. (11).

Conclusions

The impacting methods with different impacting frequencies and impacting amplitudes representing different input energy probably influence the variation of the DRAS index. It is better if we are able to use a sine vibration exciter with different frequencies and various amplitudes to impact the response of structures than using an impact hammer. Due to the shortage of the sine exciter in the laboratory, in this paper, three different kinds of reinforced concrete beams with different frequencies were designed. The results of three beams subjected to two different types of failure demonstrate that the DRAS increases in linear elasticity state and the microcrack propagation state, while DRAS decreases in macrocrack propagation state; meanwhile, the damage detection results were also compared with those obtained by frequency and damping methods based on MWT. In comparison with modal parameters, the DRAS index is more sensitive to damage.

As a time-domain analysis method, the DRAS index intuitively reflects the changes of free response signals due to the influence of damage. It uses only an output signal, with an assistance of RDT and FFT, without further treatment. Moreover, it is not required to undertake a modal analysis. With the help of the beat phenomenon, different crack styles can be judged by the DRAS index.

Therefore, DRAS is probably a promising damage indicator in structural health monitoring (SHM) which may be applicable to not only lightly damaged but also severely damaged systems during the process of damage.

It is also stressed that this work is a starting point for damage assessment based on DRAS. Further studies, however, may be needed to validate the relationship between DRAS and damage severity, for instance, studying the effect of noise on DRAS to real measurement data and analyzing the turning point of DRAS during the development of cracks based on damage mechanics and numerical simulation.

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