Intact Beam Research Articles

Damage Detection for Simply Supported Beams Using Mid-Span Displacement Indicator

This paper investigates the detection of damage for simply supported beams to ensure their safety. Based on the analysis of possible kinds of damage may occur, this paper presents a new approach which relies on the fact that any change in local stiffness caused by damage can be reflected by the change of mid-span displacement between the intact and damaged beams. Direct relationship between the change in local stiffness and the measured mid-span displacement values is developed. This approach can identify the geometric locations of damage and then inspection means are used as a complement to find the real damage phenomena within the obtained small region. A numerical example is given to illustrate the feasibility and the effectiveness of the approach. The novel content offered by authors provides a simple, convenient, cost-effective, and nondestructive damage detection approach for simply supported beams.

Structural damage identification of a cantilever beam using excitation force level control

Most studies in damage identification so far have concentrated on comparing modal parameters of a structure with an open crack with those of an intact structure. In this study, a new damage identification method for beam-like structures with a fatigue crack is proposed, which does not require comparative measurement on an intact structure but several measurements at different level of excitation forces on the cracked structure. The idea comes from the fact that dynamic behavior of a structure with a fatigue crack changes with the level of the excitation force. In other words, a beam with a real fatigue crack would behave as an intact beam at low excitation forces due to the crack closure. The 2nd spatial derivatives of frequency response functions along the longitudinal direction of a beam are used as the sensitive indicator of crack existence. Then, weighting function is employed in the averaging process in frequency domain to account for the modal participation of the differences between the dynamic behavior of beam with a fatigue crack at the low excitation and one at the high excitation. Subsequently, a damage index is defined such that the location and level of the crack may be identified. Finally, it is shown that damage identification method using the proposed damage index is very successful through experiment and finite element analysis.

710 はり構造のモード形状に基づく複数損傷の検出感度に関する検討(OS7-3 非破壊評価と構造モニタリング)

Sensitivity of damage detection for beam siructures with multiple damages is discussed. Mode shape of beam structures is estimated by strain response. The influence of damage on mode shape is shown by comparison between damaged beam and intact beam. Damages are detected from the mode shape. To detect position of the damage, continuous wavelet transform is applied to the difference in the mode shape. A peak of the wavelet transform become high with the increase of the damage level.

Analytical predictions of natural frequencies of cracked simply supported beams with a stationary roving mass

Natural frequencies of a damaged simply supported beam with a stationary roving mass are studied theoretically. The transverse deflection of the cracked beam is constructed by adding a polynomial function, which represents the effects of a crack, to the polynomial function which represents the response of the intact beam [J. Fernández-Sáez, L. Rubio, C. Navarro, Approximate calculation of the fundamental frequencies for bending vibrations of cracked beams, Journal of Sound and Vibration 225 (1999) 345–352]. By means of the boundary and kinematics conditions, approximate closed-form analytical expressions are derived for the natural frequencies of an arbitrary mode of transverse vibration of a cracked simply supported beam with a roving mass using the Rayleigh's method. The natural frequencies change due to the roving of the mass along the cracked beam. Therefore the roving mass can provide additional spatial information for damage detection of the beam. That is, the roving mass can be used to probe the dynamic characteristics of the beam by roving the mass from one end of the beam to the other. The presence of a crack causes the local stiffness of the beam to decrease which, in turn, causes a marked decrease in natural frequency of the beam when the roving mass is located in the vicinity of the crack. The magnitude of the roving mass used varied between 0% and 50% of the mass of the beam. The predicted frequencies are shown to compare very well with those obtained using the finite element method and the experimental results. Finally, the effects of crack depth, crack location and roving mass on the natural frequency of the beam are investigated. It is shown that the natural frequencies of the cracked beam decrease as the crack depth increases and as the roving mass is traversed closer to the crack location.

Modeling the Impact of Failed Members for Progressive Collapse Analysis of Frame Structures

During the past decade, increasing attention has been focused on the design of buildings to resist progressive collapse. Previously, the authors presented a nonlinear solution procedure for progressive collapse analysis of planar frame structures. In the current study, a modeling strategy to account for the impact of failed members against other structural components is developed to extend the capabilities of the initial models. Assumptions made in approximating the effects of impact on the overall behavior of frame structures are discussed. An example illustrating the importance of accounting for the effects of impact on predicting progressive collapse is also given. Results indicate that the impact velocity plays the most significant role in causing failure of intact beam elements.

A wavelet analysis-based approach for damage localization in wood beams

Free vibration testing was conducted to generate the first two mode shapes for damage detection in timbers. A wavelet transform was proposed to postprocess the mode shapes for damage pattern recognition. The wavelet used here was “db3.” The different damage severities, damage locations, and number of damaged areas were simulated by removing mass from intact beams. The results showed that the chosen wavelet db3 is suitable and that the wavelet coefficients are sufficiently sensitive to identify the existence of damage and its location in cases of different damage location, severity, and number. An edge distortion effect was apparent at the two computing edges where the wavelet coefficients were abnormally high. The wavelet coefficients showed dominant spikes around the damage locations and were zero everywhere else except the two computing edges. The dominant spikes coincided well with the damage location.

A statistical algorithm for comparing mode shapes of vibration testing before and after damage in timbers

Instances of local damage in timber such as knots, decay, and cracks can be translated into a reduction of service life due to mechanical and environmental loadings. In wood construction, it is very important to evaluate the weakest location and to detect damage at the earliest possible stage to avoid future catastrophic failure. In this study, modal testing was used on wood beams to generate the first two mode shapes. A novel statistical algorithm was proposed to extract a damage indicator by computing mode shapes of vibration testing before and after damage in timbers. The different damage severities, damage locations, and damage counts were simulated by removing mass from intact beams to verify the algorithm. The results showed that the proposed statistical algorithm is effective and suitable for the designed damage scenarios. It is reliable for the detection and location of local damage of different severities, location, and number. The peak values of the damage indicators computed from the first two mode shapes were sensitive to different damage severities and locations. They were also reliable for the detection of multiple cases of damage.

Modeling and analysis of a cracked composite cantilever beam vibrating in coupled bending and torsion

The coupled bending and torsional vibration of a fiber-reinforced composite cantilever with an edge surface crack is investigated. The model is based on linear fracture mechanics, the Castigliano theorem and classical lamination theory. The crack is modeled with a local flexibility matrix such that the cantilever beam is replaced with two intact beams with the crack as the additional boundary condition. The coupling of bending and torsion can result from either the material properties or the surface crack. For the unidirectional fiber-reinforced composite, analysis indicates that changes in natural frequencies and the corresponding mode shapes depend on not only the crack location and ratio, but also the material properties (fiber orientation, fiber volume fraction). The frequency spectrum along with changes in mode shapes may help detect a possible surface crack (location and magnitude) of the composite structure, such as a high aspect ratio aircraft wing. The coupling of bending and torsion due to a surface crack may serve as a damage prognosis tool of a composite wing that is initially designed with bending and torsion decoupled by noting the effect of the crack on the flutter speed of the aircraft.

Free vibration analysis of a cracked beam by finite element method

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.

Free vibrations of delaminated beam-type structures with crack bridging

The dynamic response of laminated beam-type structures presenting delaminations is characterized by stiffness degradation, natural frequency shifts and, in special cases, the occurrence of opening modes during vibration. The paper investigates how bridging mechanisms acting between the surfaces of the delaminations may affect these behaviors. Such mechanisms could be developed by a through-thickness reinforcement (stitching, z-pins) applied to the laminate in order to improve its resistance and tolerance against delamination fracture. A model based on the theory of bending of laminated plates is formulated where the bridging mechanisms are modeled as a uniform distribution of linear elastic springs that oppose longitudinal and transversal relative displacements between the surfaces of the delamination. The model is applied to predict natural frequencies and modal shapes of a delaminated cantilever beam with a rectangular cross-section. A variety of solutions and transitions in the dynamic response is found on varying the stiffness of the ligaments. A transition from damaged (unbridged) to intact beam values is predicted upon increasing the stiffness of both longitudinal and transversal springs. An application of the model to a stitched carbon-epoxy laminate for the aeronautical industry shows that low percentages of through-thickness reinforcement can substantially improve the dynamic response of delaminated structures.

VIBRATION-BASED DIAGNOSTICS OF FATIGUE DAMAGE OF BEAM-LIKE STRUCTURES

The analytical investigation of vibration of damaged structures is a complicated problem. This problem may be simplified if a structure can be represented in the form of a beam with corresponding boundary and loading conditions. In this connection, free vibrations of an elastic cantilever Bernoulli–Euler beam with a closing edge transverse crack is considered in the present work as a model of a structure with a fatigue crack. The modelling of bending vibrations of a beam with a closing crack is realized based on the solutions for an intact beam and for a beam with an open crack. The algorithm of consecutive (cycle-by-cycle) calculation of beam mode shapes amplitudes is presented. It is shown that at the instant of crack opening and closing, the growth of the so-called concomitant mode shapes which differ from the initially given mode shape takes place. Moreover, each of the half-cycles is characterized by a non-recurrent set of amplitudes of concomitant modes of vibration and these amplitudes are heavily dependent on the crack depth.The vibration characteristics of damage based on the estimation of non-linear distortions of the displacement, acceleration and strain waves of a cracked beam are investigated, and the comparative evaluation of their sensitivity is carried out.

Damage detection of beams using operational deflection shapes

This work is an in-depth study of a boundary effect detection (BED) method for pinpointing locations of small damages in beams using operational deflection shapes (ODSs) measured by a scanning laser vibrometer. The BED method requires no model or historical data for locating structural damage. It works by decomposing a measured ODS into central and boundary-layer solutions using a sliding-window least-squares curve-fitting technique. For high-order ODSs of an intact beam, boundary-layer solutions are non-zero only at structural boundaries. For a damaged beam, its boundary-layer solutions are non-zero at the original boundaries and damage locations because damage introduces new boundaries. At a damage location, the boundary-layer solution of slope changes sign, and the boundary-layer solution of displacement peaks up or dimples down. The theoretical background is shown in detail. Noise and different types of damage are simulated to show how they affect damage locating curves. Experiments are performed on several different beams with different types of damage, including surface slots, edge slots, surface holes, internal holes, and fatigue cracks. Experimental results show that this damage detection method is sensitive and reliable for locating small damages in beams.

Matrix cracking in transverse layers of cross-ply beams subjected to bending and its effect on vibration frequencies

Abstract The solution for the matrix crack distribution in transverse layers of a cross-ply beam subjected to bending is developed based on the model of Han and Hahn (Compos Sci Technol 35 (1989) 377). This solution accounts for a bimodulus-type of the material response. As a result, it is possible to predict the extent and distribution of matrix cracks as well as the stiffness distribution in the beam subjected to bending, accounting for residual thermal stresses. The results of the solution of the bending problem are employed in the analysis of free and forced vibrations of a cross-ply beam with matrix cracks in transverse layers. The solution differs from the case of vibrations of an intact beam due to a difference in response of damaged layers under tension and compression. The results are shown for the natural frequency of a representative ceramic matrix composite (CMC) beam.

Neural net-based monitoring of steel beams

An artificial neural network (ANN) approach was used to characterize vibration data for nondestructive evaluation purposes. Acoustic signatures were obtained from clamped–clamped metal beams of rectangular cross section. The beams were either intact, or had one (small) slot in them. The digitized data were used to train the ANN to predict future samples of the measured time series given past and present samples. The trained ANNs were used in two ways. In the first method, an ANN was trained with vibration data from intact beams. Once the ANN could adequately predict the training signal, vibration signals obtained from beams with slots were presented. Significant differences between prediction errors for the intact beam and beams with slots as shallow as 0.1 in. were found. Furthermore, the resulting prediction errors gradually increased as the slots in the beams grow deeper, suggesting that this method is useful to estimate defect size. In the second method, the connection weights of the ANNs trained on vibration data from intact beams were compared to the corresponding weights of ANNs retrained on the test data. Again, this approach was very sensitive, but no useful relationship with the slot depth was found.

Detecting delaminations in a composite beam using anti-optimization

The present study proposes a detection technique for delaminations in a laminated beam. The proposed technique optimizes the spatial distribution of harmonic excitation so as to magnify the difference in response between the delaminated and intact beam. The technique is evaluated by numerical simulation of two-layered aluminum beams. Effects of measurement and geometric noise are included in the analysis. A finite element model for a delaminated composite, based on a layer-wise laminated plate theory is used in conjunction with a step function to simulate delaminations

Boundary Stresses in a Plate having an Elliptic-Hypotrochoidal Hole

Applying the elliptic-hypotrochoidal co-ordinates recently introduced by Prof. Y. Watanabe, the author investigated the stress distribution in a plate, or in a deep beam h wing an elliptic-hypotrochoidal hole (representing such as hatchway, deck-opening, cargo-port etc.), small enough in comparison with the size of the plate or the beam, under (1) uniform tension, (2) simple shear and (3) simple bending, respectively.Although the decisive conclusion should be retained owing to the lack of experimental verifications, the author states the augumentation of the stresses in the following degree in the presence of the hole in a plate or a beam.For a plate or a beam having a rectangular hole with moderately rounded corners, and with the ratio of the breadth to the length of the hole being 20.4.a) When it is subjected to a uniform tension T parallel to one side of the hole, the max. pull stress occurs i a the neighbourhood of corners and amounts to 3.53 times the mean pull stress at the vicinity clear of the hole. The values of pull stresses at the mid-point of the sides parallel to the pull direction, are about 2.31.2T. Whatever the form of corner-rounding may be, the more the hole lengthens in the pull direction, the more the stress will diminish.b) When it is subjected to a shearing stress S, parallel to one side of the hole, the max. tension or the max. compression occurs at the vicinity of corners of the plate, as in the case (a), and is of the value of 8 S thereabout. The more the hole tends to a square shape rather than a rectangular one, the more the max. stress will diminish.c) When a deep beam having such a hole in the web, as its edges being parallel to the axis of beam, is subjected to a uniform bending, the stress at the mid-point of the edge of the hole amounts to 11.5 ββ0, where ββ0 is the bending stress at the same point of the intact beam, and as the hole lengthens in the direction of the neutral axis of beam, the stress will diminish.When the edge of a hole is inclined and its centre lies out of the neutral axis of a beam, the calculation may be somewhat complicated, but it can be carried out in a similar manner. It is also observed that the general expressions (7), after slight modifications, may be applied to the cylindrical or anticastic bending of a plate having especially a circular hole or an elliptic one.