Damage identification in beam type structures based on statistical moment using a two step method

This paper defines a novel damage index-strain statistical moment, and formulates the fourth strain statistical moment (FSSM) of beam-type structures under white noise excitation. Based on this newly defined strain statistical moment index and the least square optimization algorithm, a two-step damage identification method is proposed. This two-step method is operated like this: first use the difference curves of FSSMs before and after damage to locate damage elements; then for those identified damage elements, employ the model updating method based on the least square algorithm to assess their damage severity. Numerical studies on a simply supported beam and a two-span continuous beam are performed and the study results show that the newly defined index is effective to locate damages, even when the noise intensity is as high as 15 percent. Integrating with the least square-based model updating technique, the damage severities of beam-type structures can also be determined quantitatively. In this way, the proposed two-step method is verified and found to be capable of identifying damage positions and severities of beam-type structures and insensitive to measurement noise.     

CRACK DETECTION IN BEAM-TYPE STRUCTURES USING FREQUENCY DATA

A practical method to non-destructively locate and estimate size of a crack by using changes in natural frequencies of a structure is presented. First, a crack detection algorithm to locate and size cracks in beam-type structures using a few natural frequencies is outlined. A crack location model and a crack size model are formulated by relating fractional changes in modal energy to changes in natural frequencies due to damage such as cracks or other geometrical changes. Next, the feasibility and practicality of the crack detection scheme are evaluated for several damage scenarios by locating and sizing cracks in test beams for which a few natural frequencies are available. By applying the approach to the test beams, it is observed that crack can be confidently located with a relatively small localization error. It is also observed that crack size can be estimated with a relatively small size error.     

Stability and free vibration analyses of cantilever shear buildings with semi-rigid support conditions and multiple masses

The stability and free vibration analyses of a cantilever shear building with generalized support conditions and with multiple masses (rotational and translational) rigidly attached at both ends and along its height are presented. The proposed model includes the simultaneous effects of: (1) lateral and rotational elastic restraints at the base support; (2) a uniform distributed mass and rotary inertia plus lumped rotary and translational masses rigidly attached at both extremes and along its height; (3) linearly distributed axial load plus the concentrated vertical axial loads caused by the lumped masses; and (4) shear deformations and shear forces induced by the applied axial forces. A parametric study is carried out that shows the importance of all variables included in this work on the stability and dynamic behavior of cantilever shear buildings, particularly the effects of the attached lumped masses and the rotational and translational constraints at the base support. A comparison with results presented by other researchers in previous studies shows that the proposed method and corresponding equations can be very useful in the assessment design of cantilever shear buildings. The main objective is to present readily solutions on the static stability and free vibration of cantilever shear buildings with generalized support conditions and multiple masses rigidly attached. The proposed method and corresponding expressions for the natural frequencies and modal shapes, buckling modes and axial critical loads are extensions of those presented recently by the senior author.     

A sensitivity-based structural damage identification method with unknown input excitation using transmissibility concept

Unknown input excitation and local damages universally coexist in a practical situation. Therefore, in this paper a structural damage identification method based on the transmissibility concept in state space domain is proposed without the need for input measurements. On the basis of the transformation matrix which is computed using the system Markov parameters in state space, the relationship between two different sets of acceleration response measurements can be formulated under the same input excitation. A sensitivity-based model updating approach is applied to identify the local damages by minimizing the difference between the measured response and the reconstructed response. The sensitivity of the dynamic acceleration response with respect to the elemental stiffness factors is derived analytically in the state space domain, which accelerates the process of damage identification. A numerical cantilever beam is employed to validate that the variation of structural parameters induced by the local damages can be accurately and effectively identified without the input excitation information by the proposed method even with measurement noise considered. A laboratory test is further carried out to verify the proposed structural damage identification method based on the response reconstruction technique.     

IDENTIFICATION OF MODAL PARAMETERS FROM MEASURED OUTPUT DATA USING VECTOR BACKWARD AUTOREGRESSIVE MODEL

In this paper, a modal identification system that is based on the vector backward autoregressive (VBAR) model has been developed for the identification of natural frequencies, damping ratios and mode shapes of structures from measured output data. The modal identification using forward autoregressive approach has some problems in discriminating the structure modes from spurious modes. On the contrary, the VBAR approach provides a determinate boundary for the separation of system modes from spurious modes, and an eigenvalue filter for the selection of physical modes is existent in the proposed method. For convenience of application, the backward state equation established from VBAR model is transformed into a forward state equation, which is termed as transformed VFAR model in this paper. In addition, the extraction of equivalent system matrix of state equation of motion for structures from the transformed VFAR model has been developed, and then the normal modes can be calculated from the identified equivalent system matrix. Two examples of modal identification are carried out to demonstrate the availability and effectiveness of the proposed backward approach: (1) Numerical modal identification for a three-degree-of-freedom dynamic system with noise level in 20% of r.m.s of measured output data; (2) experimental modal identification of a cantilever beam. Finally, to show the advantage of the proposed VBAR approach on the selection of physical modes, the modal identification by stochastic subspace method was performed. The results from both methods are compared.     

HEALTH-MONITORING METHOD FOR BRIDGES UNDER ORDINARY TRAFFIC LOADINGS

A method for damage estimation of a bridge structure is presented using ambient vibration data caused by the traffic loadings. The procedure consists of identification of the operational modal properties and the assessment of damage locations and severities. An experimental study is carried out on a bridge model with a composite cross-section subjected to vehicle loadings. Vertical accelerations of the bridge deck are measured while vehicles are running. The modal parameters are identified from the free-decay signals extracted using the random decrement method. The damage assessment is carried out based on the estimated modal parameters using the neural networks technique. As input to the neural networks, the ratios of the resonant frequencies between before and after damages and the mode shapes after the damages are used to take into account the mass effect of the traffic on the bridge. The identified damage locations and severities agree reasonably well with the inflicted damages on the structure.     

The dynamic analysis of a cracked Timoshenko beam by the spectral element method

The aim of this paper is to introduce a new finite spectral element of a cracked Timoshenko beam for modal and elastic wave propagation analysis. The proposed approach deals with the spectral element method. This method is suitable for analyzing wave propagation problems as well as for calculating modal parameters of the structure. In the paper, the results of the change in modal parameters due to crack appearance are presented. The influence of the crack parameters, especially of the changing location of the crack, on the wave propagation is examined. Responses obtained at different points of the beam are presented. Proper analysis of these responses allows one to indicate the crack location in a very precise way. This fact is very promising for the future work in the damage detection field.     

Structural damage evolution assessment using the regularised time step integration method

This paper presents an approach to identify both the location and severity evolution of damage in engineering structures directly from measured dynamic response data. A relationship between the change in structural parameters such as stiffness caused by structural damage development and the measured dynamic response data such as accelerations is proposed, on the basis of the governing equations of motion for the original and damaged structural systems. Structural damage parameters associated with time are properly chosen to reflect both the location and severity development over time of damage in a structure. Basic equations are provided to solve the chosen time-dependent damage parameters, which are constructed by using the Newmark time step integration method without requiring a modal analysis procedure. The Tikhonov regularisation method incorporating the L-curve criterion for determining the regularisation parameter is then employed to reduce the influence of measurement errors in dynamic response data and then to produce stable solutions for structural damage parameters. Results for two numerical examples with various simulated damage scenarios show that the proposed method can accurately identify the locations of structural damage and correctly assess the evolution of damage severity from information on vibration measurements with uncertainties.     

Damage identification in plates using finite element model updating in time domain

A response sensitivity-based approach is presented for identifying the local damages in isotropic plate structures from the measured structural dynamic responses. The local damage is simulated by a reduction in the elemental Young's modulus of the plate. In the forward analysis, the forced vibration responses of the plate under external force are obtained from Newmark direct integration. In the inverse analysis, a response sensitivity-based finite element model updating approach is used to identify local damages of the plate in time domain. The damage identification results are obtained iteratively with the penalty function method with Tikhonov regularization using the measured structural dynamic responses. Two numerical examples are investigated to illustrate the correctness and efficiency of the proposed method. Both single damage and multiple damages cases are studied. The effects of measurement noise and measurement point on the identification results are investigated. Studies in this paper indicate that the proposed method is efficient and robust for both single and multiple damages for plate structures. Good identified results can be obtained from the short time histories of a few number of measurement points.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

Damage localisation in plate like-structures using the two-dimensional polynomial annihilation edge detection method

The topic of non-destructively detecting localised damage in plates is addressed in this article. Since the presence of a crack or a delamination causes a discontinuity in the mode shape first derivatives, a numerical method for detecting discontinuities in smooth piecewise functions and their derivatives, based on a polynomial-annihilation technique is presented. The method, already proposed for beam-type structures, has been extended to enable the detection and localisation of damage in plate-like structures for which only post-damage mode shapes are available. Applying finite element analysis, the mode shapes of an isotropic plate with a saw-cut and a multi-layered composite plate with a delamination have been calculated and the performance of the approach evaluated for increasing amounts of noise. Encouraging results indicate that further development of the technique for non-destructive testing of plate-like structures would be highly worthwhile.     

On the separation of internal and boundary damage in slender bars using longitudinal vibration frequencies and equivalent linearization of damaged bolted joint response

The problem of detecting localized large-scale internal damage in structures with imperfect bolted joints is considered. The proposed damage detection strategy utilizes the structural damping and an equivalent linearization of the bolted lap joint response to separate the combined boundary damage from localized large-scale internal damage. The frequencies are found approximately using asymptotic analysis and a perturbation technique. The proposed approach is illustrated on an example of longitudinal vibrations in a slender elastic bar with both ends clamped by bolted lap joints with different levels of damage. It is found that while the proposed method allows for the estimation of internal damage severity once the crack location is known, it gives multiple possible crack locations so that other methods (e.g., mode shapes) are required to obtain a unique crack location.     

Damage detection in beams using vibratory power estimated from the measured accelerations

While there have been several analytical studies to estimate the vibratory power of damaged structures, only a few attempts have been tried to identify the damage for practical implementations. In order to understand the characteristics of the vibratory power in damaged structures, it is necessary to trace the time histories of the instantaneous power in the vicinity of the damage. The spatial distribution of the vibratory power should also be investigated, and a proper damage index is required to diagnose the damage. In this paper, a practicable local damage detection method is proposed using the vibratory power estimated from the accelerations measured on the damaged beam structure. A new damage index is defined based on the proposed damage detection method and is applied to identify the structural damage. Numerical simulation and experiment are carried out on a beam to confirm the validity of the proposed method. In the experiments, the damage considered as an open crack such as slit inflicted on the top surface of the beam. Changes in the vibratory power of the damaged beam are investigated, and the results show that the proposed method identifies successfully the structural damage in the beam.     

Improved damage detection method based on Element Modal Strain Damage Index using sparse measurement

An improved damage detection method based on the concept of Element Modal Strain Damage Index is introduced. The proposed methods attempts to address some of the weaknesses of the damage detection method based on modal curvatures. The use of numerical differentiation procedures is identified as the main cause for the poor performance of the modal curvature method under sparse and noisy measurement. An improved damage index that does not rely on numerical differentiation is then formulated. The proposed damage index can be calculated using only modal displacement and modal rotation. A penalty-based minimization approach is then used to find the unknown modal rotation using sparse and noisy modal displacement measurement. Numerical simulation and experiment validation confirm the relative advantage of the proposed method compared with modal curvature-based approaches.     

On the tuning of variable piezoelectric transducer circuitry network for structural damage identification

The concept of using piezoelectric transducer circuitry with tunable inductance has been recently proposed to enhance the performance of frequency-shift-based damage identification method. While this approach has shown promising potential, a piezoelectric circuitry tuning methodology that can yield the optimal damage identification performance has not been synthesized. This research aims at advancing the state-of-the-art by exploring the characteristics of inductance tuning such that the enrichment of frequency measurements can be effectively realized to highlight the damage occurrence. Analysis shows that when the inductance is tuned to accomplish eigenvalue curve veering, the change of system eigenvalues induced by structural damage will vary significantly with respect to the change of inductance. Therefore, by tuning the inductance near the curve-veering range, one may obtain a family of frequency response functions that could effectively reflect the damage occurrence. When multiple tunable piezoelectric transducer circuitries are integrated to the mechanical structure, multiple eigenvalue curve veering can be simultaneously accomplished, and a series of inductance tunings can be formed by accomplishing curve veering between different pairs of system eigenvalues. It will then be shown that, to best characterize the damage occurrence, the favorable inductance tuning sequence should be selected as that leads to a “comprehensive” set of eigenvalue curve veering, i.e., all measurable natural frequencies undergo curve veering at least once. An iterative second-order perturbation-based algorithm is used to identify the locations and severities of the structural damages based on the frequency measurements before and after the damage occurrence. Numerical analyses on benchmark beam and plate structures have been carried out to examine the system performance. The effects of measurement noise on the effectiveness of the proposed damage identification method are also evaluated. It is demonstrated that the damage identification results can be significantly improved by using the variable piezoelectric transducer circuitry network with the favorable inductance-tuning scheme proposed in this research.     

Damage localization in linear-form structures based on sensitivity investigation for principal component analysis

This paper addresses the problem of damage detection and localization in linear-form structures. Principal component analysis (PCA) is a popular technique for dynamic system investigation. The aim of the paper is to present a damage diagnosis method based on sensitivities of PCA results in the frequency domain. Starting from frequency response functions (FRFs) measured at different locations on the structure; PCA is performed to determine the main features of the signals. Sensitivities of principal directions obtained from PCA to structural parameters are then computed and inspected according to the location of sensors; their variation from the healthy state to the damaged state indicates damage locations. It is worth noting that damage localization is performed without the need of modal identification. Influences of some features as noise, choice of parameter and number of sensors are discussed. The efficiency and limitations of the proposed method are illustrated using numerical and real-world examples.     

Laser ultrasonic anomalous wave propagation imaging method with adjacent wave subtraction: Application to actual damages in composite wing

Laser ultrasonic wave propagation imaging methods have great potential for integrated structural health management and non-destructive evaluation. However, application of these techniques to complex structures in the field is difficult because they give rise to complicated wave propagation patterns. We developed an anomalous wave propagation imaging method with adjacent wave subtraction using laser ultrasonic scanning to solve this problem. The proposed method is suitable for non-destructive evaluation of complex structures because it highlights the propagation of anomalous waves related to structural discontinuities, and suppresses complex incident waves without the need of pre-stored reference data. In this study, the method was applied to a real composite wing subjected to bending and impact tests. The method enhanced the visibility of the anomalous waves related to damages such as stringer tip debonding, skin-spar debonding, and invisible impact damage. Based on these anomalous waves, variable time window amplitude mapping was performed to show the damage location, size, and shape resemble to the actual damage. Comparisons showed that the methods performed better than the ultrasonic A-scan in terms of damage detection and sizing accuracy. The presence of structural elements such as spars, stringers, ribs, and surface-mounted PZT elements did not adversely affect the inspection. The proposed wing test setup with a built-in ultrasonic propagation imaging system for automatic NDE could be easily expanded throughout a hanger for maintenance inspection.     

A frequency shift curve based damage detection method for cylindrical shell structures

A novel damage detection method based on frequency shift curve (FSC) is developed for cylindrical shell structures. The FSC is caused by auxiliary mass containing both the natural frequencies and mode shapes information. According to axis-symmetry, the FSC is flat when there is no damage. However, it shows obvious periodic peaks when localized imperfections or damages occur. Furthermore, for the +2nd FSC, the trough with minimum value indicates the circumferential location of the damage and the difference between the lowest trough value and the values of the other three troughs represents the severity of the local damage. Through changing the location of the accelerometer, which can be considered as an auxiliary mass itself, around the cylindrical shell circumference, the FSCs can be measured and then the damage can be detected and located. Moreover, the difference between the averages of ±2nd FSCs also reflects the severity of damages. Numerical simulation and experimental tests have confirmed the finding. Compared with other vibration based methods, the proposed method is fast, sensitive and feasible to implement in practice as the measured frequency is more accurate than the mode shapes, and only a single accelerometer is required in the tests.     

Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response

Numerical evaluation of the flexural forced vibration of a cantilever beam having a transverse surface crack extending uniformly along the width of the beam was performed to relate the nonlinear resonances to the crack presence, location, and depth. To this end, the qualitative characteristics, namely phase portrait distortions, sub- and super-harmonic components in the Fourier spectrum, and curved shape of the modal line were considered. Furthermore, quantitative parameters, such as the eccentricity and the excursion of the orbit, and the harmonic amplitude in the spectrum were measured. Then, an identification procedure was proposed which was based on the intersection of constructed surfaces which allowed to identify the structural damage. The acceleration record of the beam tip was sufficient to detect the existence of the crack and to identify crack depth and site.     

MULTIPLE DAMAGE LOCATION WITH FLEXIBILITY CURVATURE AND RELATIVE FREQUENCY CHANGE FOR BEAM STRUCTURES

The purpose of this paper is to seek efficient methods for multiple damage location in beam structures. Two methods are studied in terms of finite element model (FEM) simulations on beam structure. Firstly, a finite element model of reinforced concrete beam is established to compute changes in flexibility and flexibility curvature of the beam with various damage patterns, and sensitivities of flexibility and flexibility curvature for closely distributed damage patterns of various extents are also compared. Due to its high sensitivity to closely distributed damages, flexibility curvature is recommended for multiple damage location. Secondly, the full correlation between the relative frequency changes and the analytical frequency changes are confirmed with FEM analyses: therefore, the relative frequency changes, rather than the frequency changes themselves, are used in the multiple damage location assurance criterion (MDLAC) to increase the location efficiency. Numerous examples are presented to verify the two methods.