Model updating–based damage detection of a concrete beam utilizing experimental damped frequency response functions

Abstract

This article evaluates the use of experimental frequency response functions for damage detection and quantification of a concrete beam with the help of model updating theory. The approach is formulated as an optimization problem that intends to adjust the analytical frequency response functions from a benchmark finite element model to match with the experimental frequency response functions from the damaged structure. Neither model expansion nor reduction is needed because the individual analytical frequency response function formulation is derived. Unlike the commonly used approaches that assume zero damping or viscous damping for simplicity, a more realistic hysteretic damping model is considered in the analytical frequency response function formulation. The accuracy and anti-noise ability of the proposed approach are first verified by the numerical simulations. Next, a laboratory reinforced concrete beam with different levels of damage is utilized to investigate the applicability in an actual test. The results show successful damage quantification and damping updating of the beam by matching the analytical frequency response functions with the experimental frequency response functions in each damage scenario.

Keywords

damage detection finite element model updating frequency response function hysteretic damping impact hammer test optimization

Introduction

Because of various environmental effects or operational accidents, damage to structures inevitably occurs during their lifetimes. Structural health monitoring (SHM) systems have been widely accepted as an effective method to monitor the behavior of structures, detect the early stage damage, and evaluate the durability of structures. With the increasing application of SHM during last two decades, increasing number of damage detection approaches has been proposed. Many approaches mainly use the dynamic vibration data to identify damage. The basic underlying principle is that the existence of damage will change any of structural stiffness, mass, or damping and thus will change the structural dynamic characteristics (such as the natural frequencies, mode shapes, and the frequency response function (FRF)) that can be extracted from vibration data. An understanding of the structural dynamic characteristics changes can help to detect, locate, and quantify the structural damage (Zhu et al., 2016). Comprehensive literature reviews on this subject can be found in Doebling et al. (1998) and Fan and Qiao (2011).

Among these works, the most commonly used structural dynamic characteristics in damage detection are the modal properties and FRFs. The modal-based damage detection technique relies on the modal properties obtained from the modal analysis (Jaishi and Ren, 2006; Vestroni and Capecchi, 2000; Zhong et al., 2014). This extraction process inherently introduces errors to the original measured vibration data. In fact, compared with the modal-based technique, the FRF-based damage detection approaches have many advantages (Esfandiari et al., 2016; Lin and Zhu, 2006; Mohan et al., 2013; Zang et al., 2003). First, the modal identification process is avoided. Thus, extraction errors are prevented. Second, for a structure with dense frequencies or large damping, accurate modal information will be difficult to obtain, and the FRF data still can be extracted without complicated calculation. Third, the damage detection process is mainly an inverse problem of structural dynamics. Therefore, the unique damage identification results require a large number of measurement data to form over-determined equations. In general, the modal test can only excite the first few modes of a structure while providing sufficient FRF data. Therefore, the use of FRFs for damage detection is very promising.

The FRF-based damage detection can be classified into two types. One type is the non-model-based approaches, which require FRF-related damage indicators. For example, Zang et al. (2003) and Sampaio et al. (2003) proposed using the correlation criterion between intact and damage FRFs as an indicator to detect damage. Sampaio et al. (1999) and Mondal et al. (2015) proposed a mode shape curvature-like indicator, that is, the FRF curvature, as the damage indicator. Reddy and Swarnamani (2012) proposed to use the FRF curvature energy to identify the damage of a plate structure. However, as discussed in Kim et al. (2006) and Shi et al. (2017), the FRF curvature method relies on the accuracy of the central difference approximation and requires a considerable number of measuring locations for sufficient spatial resolution. In fact, the most obvious drawback of using the above FRF-related damage indicators is that it is difficult to link the structure damage location to the design parameter and to quantify damage severity (Liu et al., 2009).

Another type of the FRF-based damage detection is the well-known finite element model (FEM) updating approach. This approach can be used for not only damage localization but also quantification, and it is easy to link structure damage to the change of the physical parameters in the FEM. Lin and Ewins (1994) and Lin and Zhu (2006) proposed using an element-level model updating approach for damage detection. In addition, Wang et al. (1997) verified the applicability of the model updating approach in Lin and Ewins (1994) for damage detection of a frame structure. Esfandiari et al. (2016) employed the measured FRFs through a quasi-linear sensitivity equation to estimate the damage location and severity of a beam structure. Although this approach has been based on the FRF data, modal extraction was not seriously avoided because the measured frequencies and mode shapes are still required in the FRF formulation. However, the above model updating approaches must derive sensitivity equations; such derivation is complicated and requires a complete measurement at all degrees of freedom (DOFs), making it impractical. Therefore, the data approximation technique (such as model reduction and FRF data expansion) is required for incomplete measurements. Thus, the data approximation technique will add uncertainties and errors to the original data. Moreover, the use of the analytical sensitivities emphasizes the accuracy of the FEM and the quality of the measured data (Sipple and Sanayei, 2014). To overcome these limitations, Mohan et al. (2013), Sipple and Sanayei (2014), and Hong et al. (2016, 2018) decomposed the FRF formulation from the matrix level to the DOF level; thus, damage identification can be realized directly through comparison of the individual intact FRF curve with the damage FRF curve. The commercial optimization techniques can be utilized for solving the inverse problem.

Another challenge in FRF-based damage detection approaches is the damping assumption, which cannot be ignored because the experimental FRFs are always complex. Lin and Zhu (2006) used a simulation study to prove that the use of real modal properties or real FRF data without considering damping will cause failure to detect the real damages. It is well known that damping mechanism in a real structure is very complicated, and it remains the least-known aspect compared with stiffness and mass. However, the damping correction in a damage detection or FEM updating study will improve the physical parameter detection accuracy (Hong et al., 2018). Among all damping models, viscous damping is most widely used because of its convenience in structural design. In fact, the viscous damping hypothesis has a serious drawback. The energy loss per cycle depends on the frequency; however, this result is inconsistent with many experimental results (Chopra, 2001). Another damping assumption, hysteretic damping, can more accurately describe the energy dissipation. Although it is difficult to translate the hysteretic damping mechanism into the time domain, it is promising to use the hysteretic damping assumption in the frequency domain.

This research differs from the conventional FRF-based FEM updating work; an exact formulation for individual FRF considering hysteretic damping is presented. Thus, no reduction or expansion technique is required in the incomplete measurement case. State-of-the-art optimization algorithms are utilized for solving the inverse problem instead of forming the complicated analytical sensitivity equation. Unlike most damage detection experiments, which tend to use preset damage (pre-cutting the structure or adding mass to the structure) to validate the accuracy of those approaches, to achieve a more realistic damage pattern, the experimental part of this research used a jack to introduce mild to severe damage to an reinforced concrete (RC) beam. First, numerical simulations were conducted to demonstrate the accuracy of the method and to address the selection of frequency range. Next, a laboratory beam was utilized to investigate the applicability of the presented approach in an actual case. The measured FRF was utilized for obtaining a benchmark FEM of the beam and identifying the stiffness and damping parameters change after each loading stage.

Mathematical background

Analytical formulation of individual FRF considering hysteretic damping

For an n-DOF structure with hysteretic damping, the FRF for displacement can be written as

H (ω) = {(- ω^{2} M + K + j D)}^{- 1}

(1)

The FRF in equation (1) is defined as receptance. Here, $M \in R^{n \times n}$ is the mass matrix; $K \in R^{n \times n}$ is the stiffness matrix; $D \in R^{n \times n}$ is the hysteretic damping matrix; $j = \sqrt{- 1}$ is the imaginary unit; ω is an angular frequency point. To consider a general form of proportional hysteretic damping, the Caughey damping model is utilized here. Caughey damping is expressed as a series expression related to mass and stiffness matrices (Caughey and O’Kelly, 1965)

D = \sum_{l = 1}^{n} a_{l} M {[M^{- 1} K]}^{l - 1}

(2)

Next, take the inverse of both sides of equation (1) and pre-multiply and post-multiply both sides by $Φ^{T}$ and $Φ$ , respectively

Φ^{T} H {(ω)}^{- 1} Φ = Φ^{T} (- ω^{2} M + K + j D) Φ

(3)

where $Φ = [ϕ_{1}, \dots, ϕ_{i}, \dots, ϕ_{n}]$ is the mass-normalized mode shape matrix from the eigenvalue problem shown in equation (4)

ω_{i}^{2} M ϕ_{i} = K ϕ_{i}

(4)

Using modal orthogonality, $Φ^{T} M Φ = I$ , $Φ^{T} K Φ = S$ , $Φ^{T} D Φ = \sum_{l = 1}^{n} a_{l} S^{l - 1} = κ$ (Lin and Zhu, 2006) to simplify equation (3)

Φ^{T} H {(ω)}^{- 1} Φ = - ω^{2} I + S + j κ

(5)

where $S = diag (ω_{1}^{2}, \dots, ω_{i}^{2}, \dots, ω_{n}^{2})$ and $κ = diag (κ_{1}, \dots, κ_{i}, \dots, κ_{n})$ are diagonal matrices.

To eliminate the inverse of $H (ω)$ , first pre-multiply each side of the equation by $Φ^{- T}$ and post-multiply each side by $Φ^{- 1}$ , and then take the inverse of equation (5)

H (ω) = Φ^{T} {[- ω^{2} I + S + j κ]}^{- 1} Φ

(6)

Equation (6) is a matrix form; however, it is inefficient to compare the complete FRF matrix from the intact structure and the damaged structure. Moreover, it is not feasible for measuring all DOFs in a structural testing. To satisfy a more frequent situation, namely, incomplete measurement case, decompose equation (6) to obtain an individual FRF formulation

H_{r, e}^{disp} (ω) \approx \sum_{i = 1}^{m} \frac{ϕ_{r, i} ϕ_{e, i}}{- ω^{2} + ω_{i}^{2} + j κ_{i}}

(7)

The analytical calculation form of FRF is written in a superposition form. $H_{r, e}^{disp}$ represents the FRF for excitation at DOF-e and the corresponding displacement response at DOF-r; the subscripts r and e represent the response DOF and excitation DOF, respectively. $ϕ_{r, i}$ and $ϕ_{e, i}$ are the rth and the eth entry of the ith mass-normalized mode shape, respectively; $ω_{i}$ is the ith natural angular frequency; $κ_{i}$ is the ith hysteretic modal damping; $m$ is the total number of modes used in the superposition. Usually, the first few low modal orders are the main concern in structural engineering. In this regard, m can be set much smaller than the total DOF number n to save calculation cost. According to Sipple and Sanayei (2014), when m is set three times as large as the number of the modes of interest, acceptable accuracy can be achieved.

In addition to the analytical receptance expression, according to different response data types, the analytical expressions of the mobility (velocity FRF) and accelerance (acceleration FRF) are derived, as shown in equations (8) and (9), respectively

H_{r, e}^{vel} (ω) = \frac{V_{r} (ω)}{F_{e} (ω)} \approx j ω H_{r, e}^{disp} (ω)

(8)

H_{r, e}^{acc} (ω) = \frac{A_{r} (ω)}{F_{e} (ω)} \approx - ω^{2} H_{r, e}^{disp} (ω)

(9)

Equations (7) to (9) are deduced for a single FRF under a certain frequency point. Therefore, the analytical FRF can be directly compared with the experimental FRF at incomplete measurement case, and there is no need to utilize the DOF matching technique. However, only one FRF point cannot truly reflect the damage pattern of a structure. A sufficient number of FRFs getting from different measurement locations and excitation locations are required for robust damage detection and solution uniqueness. A long FRF vector is formulated as

H^{Ana} = {H_{a, b} (ω_{1}, \dots, ω_{n_{ω}}), \dots, H_{h, g} (ω_{1}, \dots, ω_{n_{ω}})}

(10)

The subscripts $a$ , $b$ , $h$ , and $g$ represent different DOF locations. $ω_{i} (i = 1, \dots, n_{ω})$ is a selected frequency point, and $n_{ω}$ is the total number of selected frequency points.

In the damage detection process, the magnitude of FRFs is often recommended for adoption (Hong et al., 2016)

{\bar{H}}^{Ana} = | H^{Ana} |

(11)

FEM for the damaged beam structure

This study assumes the linear dynamic behavior of the structure throughout the test, and damage is set locally to an element

K^{d} = \sum_{r = 1}^{n} (1 - α_{r}) K_{e, r}

(12)

where $K_{e, r}$ is the rth undamaged element stiffness matrix; $α_{r}$ is the damage ratio associated with rth element, which is always between 0 and 1, such that 0 denotes no damage, and 1 denotes completely damage in the element.

FRF-based model updating

In this research, the main goal is to detect, locate and quantify the damage of an RC beam through the FEM updating approach. Thus, the objective function given by equation (13) is the difference between the FRFs from FE benchmark model and the experimental FRFs from the damaged RC beam

f (α, κ) = {\bar{H}}^{Ana} (α, κ) - {\bar{H}}^{Exp}

(13)

For example, the acceleration FRF-based model updating approach can be formulated as a constrained optimization problem as follows

\begin{matrix} \underset{α, κ}{minimize} ‖ f (α, κ) ‖_{2}^{2} \\ subject to [K (α) - ω_{n, i}^{2} M] ϕ_{i} = 0 \\ H_{r, e}^{acc} (ω) \approx \sum_{i = 1}^{m} \frac{- ω^{2} ϕ_{r, i} ϕ_{e, i}}{- ω^{2} + ω_{i}^{2} + j κ_{i}} \\ α_{i}^{l} \leq α_{i} \leq α_{i}^{u} \\ κ_{i}^{l} \leq κ_{i} \leq κ_{i}^{u} \end{matrix}

(14)

where $‖ ‖_{2}^{2}$ is the two-norm square function; $α_{i}$ and $κ_{i}$ are the ith selected optimization variables. The lower bound (superscript $l$ ) and upper bound (superscript $u$ ) are set for stiffness updating variables to ensure the change is always reasonable.

To better explain the damage detection through FEM updating process, the iterative approach is drawn in Figure 1.

Figure 1.

Iterative damage detection process through the FRF-based FEM updating approach.

Optimization procedures

Because FRF is a highly nonlinear function of structural parameters, as with the objective function, a constrained nonlinear multivariable function solver nonlinear least-square is adopted for solving the non-convex optimization problem. “lsqnonlin” in MATLAB (The MathWorks, Inc., 2015) seeks a minimum value for the objective function through the Levenberg–Marquardt (LM) algorithm (Ranganathan, 2004) or trust-region-reflective algorithm (Coleman and Li, 1996). The LM algorithm interpolates between the Gauss-Newton algorithm and the steepest descent algorithm. A non-negative damping factor is utilized to decide the contribution of the Gauss–Newton algorithm and the steepest descent algorithm. When the objective function value at the current step successfully reduces compared with the former step, the damping factor will decrease, so that the LM algorithm is more similar to the Gauss–Newton algorithm. In contrast, when the objective function value at current step increases compared with the former step, the damping factor will increase, and it is more similar to a steepest descent algorithm. At each iteration, the search direction is a solution from the linearization of the objective function with respect to the corresponding optimization parameters. The trust-region-reflective algorithm is an improvement of the interior-reflective Newton method. The basic idea of the trust-region-reflective is to approximate the objective function with a quadratic function in a neighborhood around the current points. Each iteration involves the decision regarding the size of the trust region and a step direction.

However, both LM and trust-region-reflective implemented in MATLAB have some limitations. The LM algorithm does not allow bound constraints; however, it can solve the underdetermined system of equations. In contrast, the trust-region-reflective can handle bound constraints; however, it requires the over-determined system of equation, that is, the number of equations must be greater than or equal to the number of updating parameters. For the FRF-based approach, it is unnecessary be concerned about the insufficient data to form an over-determined system of equations. Moreover, for a convex optimization problem, it is better to set upper and low bounds to ensure a meaningful result.

To increase the chance of finding the global optimum, the minimum search is recommended to start with different initial values. The smallest value among all the local optimums is determined as the best solution in the final. The “MultiStart” optimization solver in MATLAB is designed for this purpose. Starting points are generated by uniform distribution within bounds. “lsqnonlin” is then adopted to find a local minimum from each starting point. The more starting points are set to the “MultiStart” solver, the higher chance that the final solution approaches the global optimum. The associated computational effort will increase accordingly. Therefore, the selection of a number of starting points is usually based on the complexity of the objective function and the quality of the data.

Numerical simulation

Simulated experimental FRF from a damaged beam

To validate the accuracy and anti-noise ability of the proposed method for damage detection, a case of randomly distributed damage along a simply supported beam is simulated. In this article, a 2.8 m simply supported beam is utilized to simulate noise-contaminated FRFs, as shown in Figure 2. The dimensions and material properties of the beam in the “intact stage” are also listed in the same figure. The FEM of the beam is modeled with 2D beam element in the MATLAB environment. The axial DOF was ignored; thus, each node has only vertical DOF and rotational DOF. In total, the FEM has 14 elements with 15 nodes and 28 DOFs and is similar to the actual beam in the next section. There are three vertical excitation locations and seven vertical accelerometers, as shown in Figure 2. An example FRF plot is shown in Figure 3.

Figure 2.

FEM of simple supported beam.

Figure 3.

An observation of simulated noisy FRFs in the mild damage case.

Two different damage severities (the mild damage case and the severe damage case) are created randomly to simulate the possible damage pattern in the experiment section (serious damage may occur within the loading area). For both damage cases, the bending rigidity of each element (in total 14) and first fifteen hysteretic modal dampings (the first five modes are enough for comparison in the simulations, so m in equation (7) is set to be 15) will be considered for updating. Figures 4(a) and 5(a) show bending rigidity reduction (on the “intact stage”) of each element along beam length as low-level and high-level damage, respectively. The value of hysteretic modal damping of each mode, $κ_{i}$ , in the “intact stage” is calculated by assuming Rayleigh damping (Chopra, 2001) for simplicity. The coefficients associated with mass and stiffness are set to 0.01 and 0.02, respectively. For “damage stage,” to make damping no longer satisfy the Rayleigh rule, different change ratios are set to different hysteretic modal dampings, as shown in Figures 4(c) and 5(c).

Figure 4.

Mild damage case: (a) actual damage and average predicted damage for stiffness parameters and (b) COV of predicted damage for stiffness parameters and (c) actual change and average predicted change for hysteretic modal dampings and (d) COV of predicted change for hysteretic modal dampings.

Figure 5.

Severe damage case: (a) actual damage and average predicted damage for stiffness parameters and (b) COV of predicted damage for stiffness parameters and (c) actual change and average predicted change for hysteretic modal dampings and (d) COV of predicted change for hysteretic modal dampings.

The simulated measured FRFs for both cases are calculated using the analytical form of the FRF in equation (9). To check the anti-noise performance and add more challenges to this damage detection approach, white Gaussian noise has been added to the clean “damaged FRFs” to make the signal-to-noise ratio (SNR) reach 30 dB, as shown in Figure 3. To eliminate the random effect of noise on the damage identification results, a Monte Carlo simulation is performed for 100 iterations to observe different measured FRFs in each damage case. The number of starting points is maintained at 50 throughout the numerical and experiment sections. In this research, the convergence limits for objective function value and each optimization variable are 10⁻⁶ and 10⁻⁸, respectively.

Selection of the frequency range

As mentioned before, damage detection or model updating is the inverse problem of structural dynamics and requires a large amount of experimental data to achieve a robust result. As shown in Figure 3, the valley parts of the FRF are easily contaminated by noise, which should not be included in the calculation. In contrast, the resonance ranges of the FRF are comparatively pure and smooth. In addition, the resonance range can more clearly describe the change in the unknown structural parameters than other parts (Sipple and Sanayei, 2014); moreover, the damping controls the amplitude of FRFs at the resonance regions (Esfandiari et al., 2016). From the above reasons, it is recommended to choose resonant FRF points from a wide frequency range for a good observation (Hong et al., 2016). In this simulation part, intervals of −3 dB around the resonances (the red part in Figure 3) were used for damage detection and damping identification. It should be noted that the number of FRF points in the resonances of lower modes is less than that in the higher modes. Proper weighting factors are needed to balance the contribution of each mode. This article suggests normalizing the FRF points in each mode by its two-norm so that all modes have the same weight in model updating.

Accuracy of damage detection

Figure 4(a) and (c) exhibits the good performances of the proposed algorithm for mild damage case. Figure 4(b) and (d) shows the coefficients of variation (COVs) for each updating parameter to investigate the dispersion of the detection results from the Monte Carlo simulation. Figure 5(a) and (c) exhibits the good performances for severe damage case. Figure 5(b) and (d) shows the COVs for each updating parameter.

From the above cases, it can be observed that there is a close match between the average predicted stiffness damage and the simulated real stiffness damage, and the low COVs indicate a robust detection of element stiffness reduction. Regarding hysteretic modal damping change, both cases show that only the first five hysteretic modal damping changes can be accurately identified, possibly because the high-order hysteretic damping coefficients are not sensitive to the objective function as the high-order modes contribute little to the low-order modes. Similarly, the COVs for damping-related parameters in higher modes are much larger than those in low order. Generally, the results from the numerical simulations indicate the potential of the proposed approach in a noisy environment.

Experimental validation

Introduction of the test structure

An experiment is a good method to evaluate theoretical method (Gou et al., 2017). To investigate the ability of the proposed method to predict extensive damages as well as its robustness against measurement errors, a RC beam was designed. The length of the RC beam was 3 m, and the cross-section area was 0.3 × 0.2 m. The test elastic modulus and density of the concrete were around 2.85 × 10⁴ MPa and 2410 kg/m³, respectively. The test beam was reinforced with six whole linked steel bars and 14 mm in diameter, with three steel bars in each of the upper layer and the lower layer. The stirrups were 8 mm in diameter and spaced 200 mm apart. Figure 6 shows the test RC beam with experimental setups.

Figure 6.

Test beam with experimental setup: (a) static loading test and (b) impact hammer test.

Two identical concrete columns were prefabricated to support the RC beam. The section size of each column was 0.6 × 0.6 m, and the height was 0.5 m. The concrete columns were anchored to the rigid floor of the laboratory through the pre-stressed reinforcement. Rubber supports were designed for both sides of the RC beam. To create flat contact surfaces and fix the rubber bearings with the columns, a thin cement mortar layer was placed on each contact surface, as shown in Figure 7(b). The RC beam test started after the two surfaces fully bonded.

Figure 7.

(a) FEM of the RC beam and (b) rubber bearing details.

The initial FEM of the RC beam was built using two-dimensional and two nodded elements (the axial DOF is ignored). Due to the use of rubber bearings, two springs at the supports were considered to model the stiffness of the rubber supports. An initial value of 50 MN/m is roughly considered for the stiffness of each spring (the initial value was approximated according to experience). There are 18 elements in total. The FEM of the RC beam is shown in Figure 7(a). The RC beam was finely manufactured with high accuracy. The geometric error of the test beam was within 2 mm, the design values for the geometric parameters were directly assigned to the initial FEM, and the test elastic modulus and density were used.

Roving hammer test and static test

The RC beam test is divided into two parts: the roving hammer test and the static loading test. The test procedure is given below:

Step 1: Before the first hammer test conducted, a 3 kN pre-loading was applied to the beam to ensure there was no gap between the beam and rubber bearing. After removing the loading device, hammer tests on the RC beam started at the intact stage. The measured FRF data were utilized for calibrating the initial FEM.

Step 2: After obtaining a benchmark FEM of the intact beam, the RC beam underwent the first stage of static loading. The static force was provided through a jack and then was transferred by a steel beam to a symmetrical two-point on the top surface of the beam, as shown in Figure 6(a). The distance between the two loading points was 0.8 m. The load was gradually added to 10 kN and then held for a while. After removing the load, the loading devices were removed from the beam, and then a new set of hammer impacts were performed, as shown in Figure 6(b). The measured FRF data were compared with the FRF data from the benchmark FEM to identify the location and extent of damage.

Step 3: At second damage stage, the beam was loaded with a force of 20 kN, as in Step 2. After unloading, another set of hammer tests were performed on the RC beam.

Step 4: At the third damage stage, the RC beam was loaded with a force of 35 kN. The procedure of static loading and roving hammer test was exactly the same as that in Step 2.

For the hammer test, there were nine accelerometers (Lance LC0109D, frequency range: 0.5–5000 Hz) installed on the top of the beam (shown in Figure 6(b)) to collect vertical vibration. To obtain more experimental FRF data from different locations to capture the bending stiffness loss of different elements, three excitation locations were set. Please note that only the measured FRF curves with high quality would be selected so that the model updating or damage detection can yield the most accurate results.

The impact hammer (B&K 8207) was successively applied on the beam along the excitation location number. To reduce measurement error and noise effect, each excitation location was hit five times to allow the measured FRF data to be averaged in the frequency domain. Please note that next excitation cannot be implemented until the last beam vibration died out. Data acquisition system form B&K (LAN-XI 3160) was used. The sampling frequency was 4096 Hz and the acceleration response was captured for 4 s. In the model updating and damage detection process, the FRFs of the first six modes were used.

Model updating for an intact beam

Figure 8(a) and (b) shows data from an example hammer impact at Node 13 and measured acceleration response at Node 6, respectively. The measured FRF for the corresponding excitation and response location is calculated through the H1 estimator (Schoukens and Pintelon, 1990). As shown in Figure 8(c), an averaged FRF from five measurements for H _6,13 is shown. The coherence of the example hammer impact and the acceleration response is presented in Figure 8(d). The plot shows that the coherence value is greater than 0.98 in most frequency ranges, and there is a strong correlation between test signals; this correlation can be used for choosing less noisy frequency points for model updating.

Figure 8.

(a) An example of hammer impact on Node 13 and (b) corresponding response at Node 6 and (c) accelerance H _6,13 and (d) coherence of the example hammer impact and corresponding response.

Figure 9(b) compares the initial FRF with the experimental FRF. The large difference indicates the necessity of calibrating the FEM first to obtain a benchmark model. For the initial model calibration (or the later damage detections), the selection of frequency points should base on the selection principle recommended in the previous simulation case. Based on experience, there is no need to calibrate the measured density of a concrete beam because the density is usually consistent along the beam length. The elastic modulus of all individual elements and the axial stiffness of the spring elements are simultaneously updated using the proposed FRF-based model updating approach.

Figure 9.

Intact stage: (a) comparison of the updated FRFs H _6,13 from the initial model and the experimental FRFs and (b) updated elastic modulus of the RC beam.

The stiffness of the springs was found to be 33.096 and 44.370 MN/m, respectively. The updated elastic modulus of each beam element is shown in Figure 9(a). Overall, a similar rigidity of each element of the intact beam can be seen, indicating the good manufacturing of the RC beam. Moreover, Figure 9(b) shows the updated FRF plot matches well with the experimental FRF. The frequency domain assurance criterion (FDAC) value (Sampaio et al., 2003) between the analytical FRF and measured FRF improved from 0.819 to 0.951, that is, high FDAC value indicates an accurate benchmark model. The small difference between the amplitude of the updated FRF and the experimental FRF in Figure 9(b) shows that the damping of the structure has been updated with good accuracy. To explain the updated hysteretic modal dampings in an intuitive manner, the damping loss factors (Chopra, 2001) of the first six modes are calculated, as shown in Table 1. The natural frequencies of the initial analytical model, the intact beam, and the updated FEM model are compared in Table 2. Except for a comparatively large frequency error (4.5%) in the third mode (which may cause by the use of rubber supports), other frequency errors between the measured and updated frequency are very small.

Table 1.

Updated damping loss factors of each mode.

Dampingloss factor	Intactstage	Damagestage 1	Damagestage 2	Damage stage 3
Mode 1	0.094	0.076	0.082	0.078
Mode 2	0.132	0.178	0.219	0.156
Mode 3	0.082	0.081	0.112	0.130
Mode 4	0.102	0.102	0.059	0.050
Mode 5	0.041	0.031	0.027	0.029
Mode 6	0.034	0.033	0.026	0.028

Table 2.

Measured and analytical natural frequencies, intact stage.

Mode no.	Measured	Initial	Frequencyshift	Updated	Error (%)
1	35.91	36.36	0.45	35.88	−0.09
2	97.83	105.78	7.95	95.91	−1.97
3	139.8	160.88	21.08	146.09	4.50
4	239.01	248.81	9.8	241.51	1.05
5	424.24	426.62	2.38	424.96	0.17
6	669.85	679.27	9.42	673.76	0.58

Damage detection after the first loading stage

After getting the benchmark FEM for the intact RC beam, the first loading stage was started. The beam was gradually loaded to 10 kN. During the first loading stage, no visible crack was found. Following a new set of hammer impacts, the measured FRFs from the RC beam after the first damage stage were compared with the FRFs from the benchmark FEM, in Figure 10(b). To detect the damage location and quantification, another model updating was performed. At this stage and later damage stages, the stiffness of elements 1 and 16 were used as the value in the benchmark FEM because they are located beyond the supports and would not contribute to the FRF changes. Due to the large compressive strength of the rubber bearings (greater than 70 MPa), even the largest static load would not cause any damage to the support, so the stiffness identification of the springs is exempted from all damage detections.

Figure 10.

Damage stage 1: (a) damage identification result and (b) experimental and updated FRFs: H _10,13.

The damage ratio to each beam element in equation (12) can be detected by the bending stiffness reduction regarding the benchmark model. However, if the bending stiffness of each element can vary independently in the model updating process, the realistic damage pattern is hard to be guaranteed. The non-symmetric damage function in Wahab et al. (1999) was adopted to describe the damage pattern which caused by two concentrated loads in the middle of a beam. The theoretical background of the damage function is beyond the scope of this article and detail expression of the damage function can be found in Wahab et al. (1999).

The percentage of element rigidity reduction regarding the benchmark model is plotted in Figure 10(a). The largest stiffness reduction percentage after the first loading stage is approximately 5% compared to the intact stage. The damage pattern shows the damage is gradually diminishing from the middle loading area to the two ends. The comparison between the measured damaged FRF and updated FRF is plotted in Figure 10(b), which shows a good agreement. The FDAC value between the updated FRF and measured FRF in Figure 10(b) is 0.989, and the updated damping loss factors of the first six modes are shown in Table 1. Table 3 summarizes the natural frequencies of updated model and damaged beam after the first loading stage. As the intact stage, the frequency error in third mode is slightly larger than other modes.

Table 3.

Measured and analytical natural frequencies, damage stage 1.

Mode no.	Measured	Initial	Frequencyshift	Updated	Error (%)
1	35.56	35.88	0.32	35.12	−1.24
2	95.82	95.91	0.09	95.37	−0.47
3	136.95	146.09	9.14	145.2914	6.09
4	236.9	241.51	4.61	239.0728	0.92
5	423.2	424.96	1.76	422.2949	−0.21
6	665.68	673.76	8.08	663.5998	−0.31

Damage detection after the second loading stage

At the second loading stage, the beam was loaded with a force with 20 kN. During the second loading stage, a few cracks can be detected at the bottom of the beam near the loading areas. The percentage of element stiffness reduction regarding the benchmark FEM is plotted in Figure 11(a). Damage stage 2 shows approximately 7.25% damage compared to the intact stage. After calibration, the updated analytical FRF can show a good match with the measured FRF in Figure 11(b). The FDAC value between the updated FRF and measured FRF in Figure 11(b) is 0.986; the updated damping loss factors of the first six modes are shown in Table 1. Table 4 summarizes the natural frequencies of updated model and damaged beam after the second loading stage.

Figure 11.

Damage stage 2: (a) damage identification result and (b) experimental and updated FRFs: H _10,13.

Table 4.

Measured and analytical natural frequencies, damage stage 2.

Mode no.	Measured	Initial	Frequencyshift	Updated	Error (%)
1	35.16	35.88	0.79	34.84	−0.92
2	94.65	95.91	1.26	95.23	0.61
3	133.83	146.09	12.26	143.83	7.47
4	236.13	241.51	5.38	237.57	0.61
5	419.70	424.96	5.26	419.70	0.00
6	666.95	673.76	6.81	659.34	−1.14

Damage detection after the third loading stage

At the final loading stage, the beam was loaded with a force of 35 kN. During the third loading stage, the old cracks gradually increased and more cracks can be detected between the loading points. The percentage of element stiffness reduction regarding the benchmark FEM is plotted in Figure 12(a). The final damage stage shows approximately 12% damage compared to intact stage. After calibration, the updated analytical FRF can show a good correlation with the measured FRF in Figure 12(b). The FDAC value between the updated FRF and measured FRF in Figure 12(b) is 0.981; the updated damping loss factors are shown in Table 1. Table 5 summarizes the natural frequencies of updated model and damaged beam after the third loading stage.

Figure 12.

Damage stage 3: (a) damage identification result and (b) experimental and updated FRFs: H _10,13.

Table 5.

Measured and analytical natural frequencies, damage stage 3.

Mode no.	Measured	Initial	Frequencyshift	Updated	Error (%)
1	34.59	35.88	1.29	34.01	−1.67
2	89.81	95.91	6.1	87.15	−2.96
3	127.25	146.09	18.84	132.47	4.11
4	229.65	241.51	11.86	231.80	0.94
5	416.15	424.96	8.81	418.63	0.60
6	662.53	673.76	11.23	660.80	−0.26

Conclusion and future work

The conclusions of this work are provided as follows:

The proposed FRF-based damage detection approach was investigated through numerical simulations and a laboratory RC beam. The analytical FRF matrix formulation considering hysteretic damping was decomposed to the individual FRF. Model expansion and reduction were found to be avoided using the proposed approach.

The proposed approach can be implemented with state-of-the-art optimization techniques and does not strictly require the quality of the FEM. MATLAB optimization solvers “lsqnonlin” and “Multistart” were carefully introduced.

The Monte Carlo simulation was performed to add random noise to the “measured” FRFs. The accuracy of the predicted results and low COVs indicate the anti-noise ability and robustness of the proposed method. The criterion for choosing appropriate frequency ranges was addressed in this part.

The successful damage detection of the proposed FRF-based model updating approach in the concrete beam revealed its potential for damage detection on more realistic structures. The results showed that the stiffness reductions are more severe within the loading area.

The most significant problem of the FRF-related method is that the calculation of the experimental FRF requires the accurate measurement of the excitation force; such measurement is difficult to implement in a real project, especially for those in-service structures. In the future, we plan to study the power spectral density function of structural response for model updating or damage detection; this power spectral density function is easy to obtain from an ambient test.

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) received no financial support for the research, authorship, and/or publication of this article.

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