Experimental investigation of damage identification in beam structures based on the strain statistical moment

Abstract

A new two-step damage detection technique based on the fourth strain statistical moment was recently proposed by the authors, and its sensitivity to local structural damage has been numerically demonstrated for beam-type structures. In this article, the proposed method is extended to an experimental beam to assess its feasibility and practicality. A simply supported steel beam was manufactured and subjected to Gaussian white-noise excitation before and after damage. The strain responses of each measurement point were recorded based on which fourth strain statistical moments were calculated. The proposed two-step technique was implemented to locate the damaged elements of the experimental beam, for which the damage sizes were identified based on the least-square updating algorithm. The experimental results show that the proposed fourth strain statistical moment index and the two-step damage detection technique are effective and feasible for beam-type structures.

Keywords

damage identification experimental investigation fourth strain statistical moment simply supported beam two-step technique

Introduction

Structures begin to deteriorate when they are in operation in harsh environments such as earthquakes, corrosion, or typhoons; in those cases, structural damage may be caused by the irreversible changes in structural material properties. Thus, structural damage identification is one of the most researched areas in civil, mechanical, aerospace, and related engineering fields. Many vibration-based damage detection methods have been developed in several recent decades. These methods assume that the structural dynamic properties or the responses as functions of the changes in structural physical properties will change when the structure suffers from damage (Doebling et al., 1996).

Many dynamic property–based damage detection approaches were performed to detect structural damage (Cawley and Adams, 1979; Salawu, 1997; Xu et al., 2007; Zhao et al., 2013). They mainly refer to the changes in natural frequency, mode shape (Poudel et al., 2007; Ratcliffe, 1997; Shi et al., 2000a), mode shape curvature (Abdel and De Roeck, 1999; Ndambi et al., 2002; Pandey et al., 1991), flexibility matrix (Jiashi and Ren, 2006; Li et al., 2010; Pandey and Biswas, 1994), and modal strain energy (Seyedpoor, 2012; Shi et al., 2000b). The advantages of the damage detection methods based on the natural frequency change are the relative ease of obtaining the natural frequency and its relatively higher precision than other modal parameters. However, the natural frequency has low sensitivity to local damage, which is easily affected by environment factors such as temperature and humidity. Xia et al. (2006) investigated the variations of natural frequency, mode shape, and damping with respect to temperature and humidity changes by monitoring a reinforced concrete slab in a periodical vibration test for nearly 2 years. Hua et al. (2007) proposed a method of combining the principal component analysis (PCA) and support vector regression (SVR) technique to model the temperature-caused variability of modal frequencies for structures that are instrumented with long-term monitoring systems. Compared to using natural frequencies, the advantage of using the mode shape or its derivatives as an index for damage identification is the better sensitivity of the mode shape to local damage. This advantage enables the mode shape to be directly used in damage localization because it contains local structural information. However, the disadvantage of the damage detection methods based on changes in mode shape or its derivatives is that the measured mode shapes are more prone to noise contamination than natural frequencies in practical engineering (Fan and Qiao, 2011).

Direct vibration response–based damage detection methods have achieved some degree of success in the past decades. Ma and Chang (2004) proposed a semi-model-based method to detect, locate, and quantify possible damage in a structure after a seismic event. In the method, the input signals to a designed monitor are the structural dynamic responses and excitation. The occurrence and location of the structural damage can be determined based on the display of nonzero output. Furthermore, the severity of the damage can be assessed using a time-domain system identification technique on the input–output data of the monitor. Yang et al. (2007) proposed a new approach to detect structure damage using the cross-correlation function amplitude vector of the measured vibration responses. They extended the cross-correlation function amplitude vector to the loosening of fasteners in an aircraft panel model to illustrate its feasibility. Trendafilova and Manoach (2008) introduced two viable vibration health-monitoring methods that used large-amplitude vibrations and were based on the nonlinear time series analysis. One of the methods uses the statistical distribution of state space points on the attractor of a vibrating structure, whereas the other is based on a Poincare map of the state space projected dynamic response. A new damage identification method was proposed with a new damage index based on the displacement statistical moments (Xu et al., 2009; Zhang et al., 2008). They discussed the properties of this index, such as its sensitivity to structural damage and noise robustness, and used a model-updating method based on the least-square optimization algorithm to directly identify the parameters of shear-type buildings. The method is proven to be well suited to identify the damage of a flexible tower and a steel frame structure (Zhang et al., 2011). Wang et al. (2014) and Xiang et al. (2014) defined the fourth strain statistical moment (FSSM) and numerically demonstrated its validity to detect the incipient damage of beam-type and plate structures.

The objective of this study is to assess the feasibility and the accuracy of the proposed damage detection method based on the FSSM index, so an experimental investigation program was initiated and implemented. In this article, the theory of the strain statistical moment and a damage detection approach for bending-type beam structures based on the FSSM are briefly introduced. Then, an experimental study on a simply supported beam for damage identification is conducted to illustrate the proposed two-step damage identification method, and a brief discussion is subsequently addressed. Finally, the last section shows the conclusions of the experimental study.

Damage identification method based on the strain statistical moment

For a linear structure system, when the excitation is a stationary Gaussian random process, the structural response is also a stationary Gaussian random process (Zhang et al., 2008). The probability density function of the Gaussian distribution is expressed as

p (x) = \frac{1}{\sqrt{2 π} σ} e^{(- (x - μ) / 2 σ^{2})}

(1)

where x is the structural response, $p (x)$ is the probability density function of x, and $μ$ and $σ$ are the mean and variance of x, respectively.

All statistical moments from the first order to the eighth order are expressed (Meirovitch, 1975). It has been proven that a high-order statistical moment is more sensitive to structural damage than a low-order one. However, a high-order statistical moment is not stable (Zhang et al., 2008). Thus, the fourth-order statistical moment is selected as a back-up index for damage identification

M_{4} = \int_{- \infty}^{+ \infty} {(x - μ)}^{4} p (x) dx = 3 σ^{4}

(2)

The FSSM as the index to identify the structural damage is also expressed as (Wang et al., 2014)

M_{4 i}^{s} = 3 σ_{ε_{i}}^{4}

(3)

The fourth-order statistical moment of strain response ε can be calculated using the binomial theorem regardless of whether it follows a Gaussian random distribution

M_{4} = \frac{1}{N} \sum_{i = 1}^{N} x_{i}^{4} - \frac{4}{N} μ \sum_{i = 1}^{N} x_{i}^{3} + \frac{6}{N} μ^{2} \sum_{i = 1}^{N} x_{i}^{2} - 3 μ^{4}

(4)

For an N-degree-of-freedom system, its FSSM is expressed as

M_{4}^{s} = [M_{41}^{s}, M_{42}^{s}, \dots, M_{4 N}^{s}]

(5)

In this study, first, we use the FSSM index to locate the damage in an experimental steel beam. After the damage is located, a model-updating procedure based on the least-square optimization algorithm is performed to identify the accurate position and severity of the damage. The detailed process is shown in Wang et al. (2014).

Experimental study

Experimental setup

To assess the feasibility and accuracy of the proposed damage detection method based on the FSSM index, an experimental investigation procedure was initiated and implemented in this section. A simply supported beam model was designed as shown in Figure 1. The overall dimensions of the beam were 660 mm in length, 37.6 mm in width, and 5.5 mm in height. The experimental arrangement is shown in Figure 2. The simply supported beam was placed on a stable steel plate, which can be considered fully rigid ground. The beam was made of high-strength steel with a yield stress of 235 MPa and a modulus of elasticity of 200 GPa. An Agilent 33500B signal generator, a HEV-500G excitation device, a HEA-500G power amplifier, and a DH5922 dynamic signal acquisition system were used in the experiment. One Gaussian white-noise (GWN) time history with a maximum of 200 N, a frequency range of 0–200 Hz, and an acting duration of 40 s was imposed on the midspan of the steel beam for excitation using the signal generator and power amplifier. We set the time step as 1/500 s and the corresponding sampling frequency as 500 Hz. The GWN time history and its probability density distribution are shown in Figure 3. The diagrammatic sketch of the experimental arrangement is shown in Figure 4.

Figure 1.

A simply supported beam model.

Figure 2.

Experimental setup.

Figure 3.

The properties of Gaussian white-noise excitation: (a) time history and (b) probability density distribution.

Figure 4.

Diagrammatic sketch of the experimental arrangement.

Damage scenarios

In the experiment, the experimental beam was divided into 11 uniform elements. To simulate the beam damage, the beam width was reduced from both sides at a designated element, as shown in Figure 5. Four damage scenarios were designed in this experiment. In damage scenario 1 (D1), a 1.85-mm reduction from each side of element 5 was implemented to reduce the element width by 10%. In damage scenario 2 (D2), a 4-mm reduction from each side of element 5 was implemented to reduce the element width by 23%. In damage scenario 3 (D3), a 6.1-mm reduction from each side of element 5 was implemented to reduce the element width by 33%. In damage scenario 4 (D4), a 6.5-mm reduction from each side of element 8 was implemented to reduce the element width by 35%, in addition to a 33% reduction in the width of element 5. Figure 6(a) to (d) and Table 1 show the damage scenarios of the beam model. D1, D2, and D3 represent different single-damage scenarios with damage at the fifth element of the beam model. D4 represents a dual-damage scenario with damage at two elements of the beam.

Figure 5.

Beam damage.

Figure 6.

Schematic damage location and size: (a) 1.85-mm reduction from both sides of the beam at the fifth element, (b) 4-mm reduction from both sides of the beam at the fifth element, (c) 6.1-mm reduction from both sides of the beam at the fifth element, and (d) 6.1-mm reduction from both sides of the beam at the fifth element and 6.5-mm reduction from both sides of the beam at the eighth element.

Table 1.

Damage scenarios for the simply supported beam.

Damage scenarios	$α_{5} (%)$	$α_{8} (%)$
D1	10	−
D2	23	−
D3	33	−
D4	33	35

Finite element modeling

A numerical beam structure model with identical overall dimensions with the experimental beam mode is also addressed in this section. The numerical beam was divided into 11 elements, as shown in Figure 1. Compared to the three-story shear frame structure (Xu et al., 2009), the beam structure is much more complex because it has 12 nodes in total, each of which has 3 degrees of freedom. The relevant parameters of the simply supported beam model are identical to those of the aforementioned experimental beam. Rayleigh damping assumption was considered, and the damping ratio of the first two orders was set as 0.03. The mass and stiffness matrices in the finite-element model are shown in equations (6) and (7)

M_{e} = \frac{ρ A l_{0}}{420} (\begin{matrix} 140 & 0 & 0 & 70 & 0 & 0 \\ 0 & 156 & 22 l_{0} & 0 & 54 & - 13 l_{0} \\ 0 & 22 l_{0} & 4 l_{0}^{2} & 0 & 13 l_{0} & - 3 l_{0}^{2} \\ 70 & 0 & 0 & 140 & 0 & 0 \\ 0 & 54 & 13 l_{0}^{2} & 0 & 156 & - 22 l_{0} \\ 0 & - 13 l_{0} & - 3 l_{0}^{2} & 0 & - 22 l_{0} & 4 l_{0}^{2} \end{matrix})

(6)

K_{e} = (\begin{matrix} \frac{EA}{l_{0}} & 0 & 0 & - \frac{EA}{l_{0}} & 0 & 0 \\ 0 & \frac{12 EI}{l_{0}^{2}} & \frac{6 EI}{l_{0}^{2}} & 0 & - \frac{12 EI}{l_{0}^{2}} & \frac{6 EI}{l_{0}^{2}} \\ 0 & \frac{6 EI}{l_{0}^{2}} & \frac{4 EI}{l_{0}} & 0 & - \frac{6 EI}{l_{0}^{2}} & \frac{2 EI}{l_{0}} \\ - \frac{EA}{l_{0}} & 0 & 0 & - \frac{EA}{l_{0}} & 0 & 0 \\ 0 & - \frac{12 EI}{l_{0}^{2}} & - \frac{6 EI}{l_{0}^{2}} & 0 & \frac{12 EI}{l_{0}^{2}} & - \frac{6 EI}{l_{0}^{2}} \\ 0 & \frac{6 EI}{l_{0}^{2}} & \frac{2 EI}{l_{0}} & 0 & - \frac{6 EI}{l_{0}^{2}} & \frac{4 EI}{l_{0}} \end{matrix})

(7)

where $ρ$ , A, and $l_{0}$ are the density, cross-sectional area, and length of the element, respectively; E and I are Young’s modulus and the moment of inertia of the cross section of the element, respectively.

The numerical natural frequencies, the numerical strain responses at the “measurement” points, and the FSSM index values of the beam are calculated according to the finite-element model and are compared with the corresponding experimental results in this article.

Experimental procedure and results

In this experiment, a test was first conducted on the undamaged beam model. A force hammer was used to provide a random excitation to the undamaged simply supported beam. Then, the vertical acceleration at each element node of the undamaged beam was measured using DH187 acceleration sensors. The natural frequency information of the undamaged beam was also obtained using a DH5922 dynamic acquisition instrument as listed in Table 2. Subsequently, the undamaged beam was subjected to a GWN excitation as shown in Figure 3. To guarantee that the beam was operated in the elastic stage during excitation, the standard deviation of the excitation force was limited to 7 N. In this test, 10 strain sensors were arranged at the element nodes on the underside of the beam, as shown in Figure 7, and the strain responses of all measurement points were obtained during the exciting process. The strain responses at measurement point 5 are shown in Figure 8.

Table 2.

The experimental and numerical natural frequencies in the cases of D0, D1, D2, D3, and D4.

f	D0			D1			D2			D3			D4
f	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)
$f_{1}$	29.297	28.899	2.3	29.132	28.635	1.74	29.297	28.316	3.47	28.414	27.919	1.77	24.844	27.231	8.76
$f_{2}$	113.281	115.631	1.7	117.256	115.257	1.73	107.422	114.841	6.46	122.070	114.335	6.77	107.422	110.866	3.10
$f_{3}$	251.953	260.185	2.7	263.556	259.061	1.73	253.906	257.725	1.48	249.023	256.109	2.7	248.047	255.363	4.34

Figure 7.

Sensor arrangement.

Figure 8.

The experimental and numerical results of the strain responses at measurement point 5: (a) D0, (b) D1, (c) D2, (d) D3, and (e) D4.

Similarly, the first three natural frequencies of the damaged beam in different damage scenarios (D1, D2, D3, and D4) were obtained, as shown in Table 2. The strain responses of all measurement points on the damaged beam were obtained during the exciting process, and the strain responses at measurement point 5 are shown in Figure 8.

Furthermore, the numerical natural frequencies in the undamaged and damaged scenarios were computed and compared with those experimentally obtained, which are listed in Table 2. The comparison shows significant differences between the numerical and experimental natural-frequency results, and the maximal error was 8.6%. The reason should be the boundary conditions, the geometry, and the material properties of the experimental beam, which are often different from those of the idealized numerical model.

The numerical strain responses at all measurement points were computed in the undamaged and damaged scenarios. The numerical strain responses at measurement point 5 were compared with those experimentally obtained, as shown in Figure 8. The experimental strain responses were slightly larger than those numerically obtained. The main reason may be the supported devices of the experimental beam because a perfect simply supported beam is difficult to accurately achieve in the experiment.

Initial finite-element model update

An initial finite-element model update is necessary to obtain a much more accurate finite-element model to identify the damage extent in the next step of the proposed approach. Thus, an initial finite-element model update procedure was performed to obtain an improved finite-element model of the intact steel beam using the experimental strain data of the intact beam.

In the initial finite-element model update, we set the stiffness coefficients of all finite elements as the variable to be optimized and let all initial values of the stiffness coefficients be 1. The experimental strain data of the intact beam were used to update the finite-element model of the intact beam. The updated results of the stiffness coefficients for all elements were obtained, as shown in Figure 9. Figure 9 shows that the updated stiffness coefficients of elements 1, 10, and 11 near the supports are notably larger than those of other elements, which indirectly reflects the effect of the boundary restrictions on the structural natural frequencies and responses.

Figure 9.

Identification results of the element stiffness coefficients using the initial finite-element model update.

The natural frequencies of the updated beam model were calculated and are listed in Table 3. Table 3 shows that the differences between numerical and experimental natural frequencies decreased after the initial finite-element model update was performed and the maximal error was 3%. In particular, all differences of the first natural frequencies were less than 1% in different damage cases.

Table 3.

The experimental and numerical natural frequencies in the cases of D0, D1, D2, D3, and D4 after the initial model update.

f	D0			D1			D2			D3			D4
f	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)	Exp. (Hz)	Num. (Hz)	Errors (%)
$f_{1}$	29.297	29.143	0.526	29.132	29.165	0.113	29.297	29.3085	0.039	28.414	28.3085	0.371	24.844	25.013	0.680
$f_{2}$	113.281	114.345	0.939	117.256	119.0181	1.501	107.422	110.0256	2.423	122.070	120.4601	1.318	107.422	109.347	1.791
$f_{3}$	251.953	255.768	1.5	263.556	265.471	0.727	253.906	258.136	1.665	249.023	255.4552	2.58	248.047	253.736	2.248

Damage identification based on experimental strain data

In this section, the damage in the experimental beam is identified based on the proposed two-step identification technique in section “Damage identification method based on the strain statistical moment.” The first step is to locate the damage using the difference in FSSMs before and after the damage; the second step is to assess the damage severity using the model-updating procedure with the least-square optimization algorithm. The two-step identification technique decreases the identification scope, minimizes the number of variables, saves computing time, and enhances the identification precision.

Damage location results

After the strain responses of the experimental beam in the undamaged and damaged scenarios were obtained, the corresponding FSSM indices were computed for the four damage scenarios and the undamaged scenario. To make a notable comparison, the numerical FSSM indices before and after the initial finite-element model update was implemented were also computed for the experimental beam model. Both experimental and numerical FSSM results are shown in Figure 10. Figure 10 shows that the FSSM curve in each scenario was not smooth, including the undamaged scenario, where abrupt changes appeared at measurement points 5, 6, 7, and 8. The abrupt changes at the measurement points do not correspond to the preset damage locations, so the FSSM index cannot be used to locate the damage. However, the value of the FSSM index increases with the increase in damage severity because the decrease in stiffness because of the damage amplifies the strain responses at each measurement point, particularly those at the midspan of the experimental beam. In addition, Figure 10 shows that in different scenarios, the numerical FSSM curves after the implementation of the initial finite-element model update are more comparable to the experimental FSSM curves than those before the implementation.

Figure 10.

FSSM curves for a simply supported beam in different scenarios: (a) D0, (b) D1, (c) D2, (d) D3, and (e) D4.

Then, the difference curves of the experimental FSSMs before and after the damage were calculated and are shown in Figure 11. The FSSM difference curves were no longer as irregular as the FSSM curves; the abrupt peaks (marked by a small circle in the figure) at measurement points 5 and 8 accurately correspond to the preset damage locations, and the absolute peak value of the FSSM difference increases with the increase in damage severity. The FSSM difference curves before and after the damage are well suited to locate the damage of the experimental beam, although there are some errors in measuring the strain responses.

Figure 11.

Difference curves of experimental FSSMs before and after the damage of a simply supported beam in different scenarios: (a) D1, (b) D2, (c) D3, and (d) D4.

Damage severity identification results

After the damage elements were localized, we began the second step to assess the damage size using a model-updating procedure based on the least-square algorithm. The numerical FSSM values $M_{4}^{s, N} (α)$ of the intact beam before and after the initial finite-element model update was implemented were set as the reference values, and the damage coefficient α of the identified damage elements was set as the variable to be optimized. Then, we computed the experimental FSSM values $M_{4}^{s, E} (α)$ and residual vector $F (α)$ between $M_{4}^{s, E} (α)$ and $M_{4}^{s, N} (α)$ . The least-square optimization algorithm was introduced to minimize the residual vector. By obtaining the element damage coefficients, we can directly assess the damage severity of the structure. The final identification results of damage severities before and after the initial finite-element model update was implemented were obtained using the least-square optimization algorithm, as listed in Table 4. The values in brackets are the errors of the identification results compared with the preset damage sizes in Table 1.

Table 4.

Identified results of damage severities for the experimental beam.

Damage cases	$α_{5}$		$α_{8}$
Damage cases	Before updating	After updating	Before updating	After updating
D1	13.9% (3.9%)	9.51% (0.49%)	−	−
D2	34.1% (11.1%)	24.18% (1.18%)	−	−
D3	43.1% (10.1%)	38.4% (5.4%)	−	−
D4	44.7% (11.7%)	39.6% (6.6%)	46% (11%)	39.2% (4.2%)

The identified results of damage severities in Table 3 show that before the initial finite-element model update was implemented, the lowest identification error was 3.9% in scenario D1, and the maximal one was 11.7% in scenario D4. Compared with the preset damage sizes in Table 1, all corresponding identified damage severities were relatively larger, which is directly attributed to the oversize strain measurements in the experiment. The presence of oversize strain responses in the experiment should result from two factors: the experimental beam is not a perfect simply supported beam, and the beam may be subjected to pre-stress from the exciter, which results in additional deformation of the beam.

Table 3 also shows that after the initial finite-element model update was implemented, the lowest identification error of the damage severities was 0.49% in scenario D1, and the maximal one was 6.6% in scenario D4; for all damage scenarios, the identification errors were at most 7%. Compared with the identified results before the initial model update was implemented, the identification errors dramatically decreased after the initial model was updated. Thus, the initial finite-element model update significantly affects the identification results of damage severities.

Discussion

In the experimental study, to detect the damage location and severity of the steel beam, an identical white-noise excitation was applied for the test with undamaged and damage scenarios. However, in many real situations, it is not possible to obtain the strain measurements under the exactly identical white-noise excitation levels from two tests. Hence, the proposed damage detection technique has been successfully performed to identify the damage of a beam structure in only a relatively ideal situation. In future studies, we will explore the feasibility of detecting the damage location and severity of the beam structure by applying different white-noise excitation levels to test the undamaged and damage beams. We think that the proposed damage detection technique will potentially solve the issue of different excitation levels after it is adequately improved. When the excitation levels of white-noise are different for the undamaged and damage beam structure, we can normalize the strain statistical moments of different beam elements for undamaged and damage scenarios to remove the effect of different excitation levels.

Conclusion

A recently developed structural damage detection method based on the FSSMs of the dynamic responses of a beam structure was experimentally examined in this article. After a brief introduction to the theory of FSSM, the two-step method was applied to an experimental simply supported beam to identify the positions and severities of the damage. The newly defined index has a desirable damage-localization capability with the difference of FSSMs before and after the damage, although there are some measurement errors for the experimental strain responses. After an initial finite-element model update, the damage severities can be well determined based on the two-step damage detection procedure using the least-square optimization algorithm; the identification precision is affected by the accuracy of the finite-element model and the measurement errors of the experimental strain responses. The proposed damage detection method has been experimentally verified to effectively identify single damage or multi-damage in bending beam-type structures.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to extend their thanks to the joint financial support by the National Natural Science Fund of China (51278215), National Basic Research Program of China (973 Program: 2011CB013800), and Fundamental Research Funds for the Central Universities (HUST: 2015MS058).

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