Increasing Crack Depth Research Articles

Analysis of the effect of V-shape and Rectangular Shape cracks on the natural frequencies of a spring steel cantilever beam

Crack is mostly found in the material which has less fatigue strength. It is progressive and limited to a small area structural damage that occurs when a material is subjected to cyclic loading. Since more than two decades, natural frequency used as an input for crack detection. In this study, the effects of multiple cracks on the dynamics of the structure are studied. The objective of this study is to see the effect of v shape and rectangular shape edge cracks on the natural frequency and static deflection on the spring steel cantilever beam. V shape and Rectangular shape crack exists in the turbines, pumps and other relevant applications in service. In the present study, free vibration analysis of a cracked cantilever beam is performed in order to see the effect of V shape and rectangular shape cracks on the natural frequency. Through the detailed study it is found that all the v shape cracked cases gives natural frequency comparatively on higher side. The variations in the natural frequency of v shape cracked cases is less than rectangular shape cracked cases with respect to natural frequency of undamaged beam. Consequently, the presence of small size rectangular shape crack in the beam can be effectively detected by vibration measurement methods than small size v shape crack. At the last location, when crack depth increases, value of natural frequency remains least affected. From static analysis found static deflection is largest for the case which has least natural frequency.

Comparative study of direct and inverse problems of cracked beams

In recent decades, the analysis and evaluation of the cracked structures were hot spots in several engineering fields and has been the subject of great interest with important and comprehensive surveys covering various methodologies and applications, in order to obtain reliable and effective methods to maintain the safety and performance of structures on a proactive basis. The presence of a crack, not only causes a local variation in the structural parameters (e.g., the stiffness of a beam) at its location, but it also has a global effect which affects the overall dynamic behavior of the structure (such as the natural frequencies). For this reason, the dynamic characterization of the cracked structures can be used to detect damage from non-destructive testing. The objective of this paper is to compare the accuracy and ability of two methods to correctly predict the results for both direct problem to find natural frequencies and inverse problem to find crack’s locations and depths of a cracked simply supported beam. Several cases of crack depths and crack locations are investigated. The crack is supposed to remain open. The Euler–Bernoulli beam theory is employed to model the cracked beam and the crack is represented as a rotational spring with a sectional flexibility. In the first method, the transfer matrix method is used; the cracked beam is modeled as two uniform sub-segments connected by a rotational spring located at the cracked section. In the second method which is based on the Rayleigh’s method, the mode shape of the cracked beam is constructed by adding a cubic polynomial function to that of the undamaged beam. By applying the compatibility conditions at crack’s location and the corresponding boundary conditions, the general forms of characteristic equations for this cracked system are obtained. The two methods are then utilized to determine the locations and depths by using any two natural frequencies of a cracked simply supported beam obtained from measurements as inputs. The two approaches are compared with a number of numerical examples for simply supported beams including one crack. The theoretical results show that the accuracy of the Rayleigh’s method to predict natural frequencies decreases for higher modes when crack depth increases. It is also found that for the inverse problem, the transfer matrix method show a good agreement with those obtained from previous works done in this field.

Comparative study of direct and inverse problems of cracked beams

In recent decades, the analysis and evaluation of the cracked structures were hot spots in several engineering fields and has been the subject of great interest with important and comprehensive surveys covering various methodologies and applications, in order to obtain reliable and effective methods to maintain the safety and performance of structures on a proactive basis. The presence of a crack, not only causes a local variation in the structural parameters (e.g., the stiffness of a beam) at its location, but it also has a global effect which affects the overall dynamic behavior of the structure (such as the natural frequencies). For this reason, the dynamic characterization of the cracked structures can be used to detect damage from non-destructive testing. The objective of this paper is to compare the accuracy and ability of two methods to correctly predict the results for both direct problem to find natural frequencies and inverse problem to find crack’s locations and depths of a cracked simply supported beam. Several cases of crack depths and crack locations are investigated. The crack is supposed to remain open. The Euler–Bernoulli beam theory is employed to model the cracked beam and the crack is represented as a rotational spring with a sectional flexibility. In the first method, the transfer matrix method is used; the cracked beam is modeled as two uniform sub-segments connected by a rotational spring located at the cracked section. In the second method which is based on the Rayleigh’s method, the mode shape of the cracked beam is constructed by adding a cubic polynomial function to that of the undamaged beam. By applying the compatibility conditions at crack’s location and the corresponding boundary conditions, the general forms of characteristic equations for this cracked system are obtained. The two methods are then utilized to determine the locations and depths by using any two natural frequencies of a cracked simply supported beam obtained from measurements as inputs. The two approaches are compared with a number of numerical examples for simply supported beams including one crack. The theoretical results show that the accuracy of the Rayleigh’s method to predict natural frequencies decreases for higher modes when crack depth increases. It is also found that for the inverse problem, the transfer matrix method show a good agreement with those obtained from previous works done in this field.

Unified constraint parameter solutions for axial and circumferential surface cracks in pressurized pipes under creep condition

Unified creep constraint parameter Ac for axial and circumferential surface cracks in pressurized pipes have been calculated and investigated by three-dimensional finite element method. It has been shown that the parameter Ac can effectively characterize in-plane and out-of-plane creep constraints in pressurized pipes. The overall constraint levels of pipe cracks increase with increasing crack depth, length, pipe wall thickness and radius-thickness ratio. The parameter Ac solutions have been obtained for different pipe geometries and crack sizes. In creep life assessments of pressurized pipes, the overall constraint effect can be incorporated by the unified constraint parameter Ac. For shallower and shorter surface cracks in pipes, more benefits can be gained by incorporating constraint effect.

Periodic nonlinear waves resulting from the contact interaction of a crack

When two different inputs of distinct low and high frequencies are applied to a medium, the linear responses are composed of waves of two dominant frequencies. However, microcracks such as fatigue cracks generate nonlinear waves by modulating the characteristics of the incident waves. Although this phenomenon has been observed and used to detect microcracks, the underlying principles have not been thoroughly elucidated. The hysteresis properties were introduced to describe the nonlinear relationship between the stress and strain to explain these phenomena [Van Den Abeele et al., Res. Nondestruct. Eval. 12, 17 (2000) and Nazarov et al., Acoust. Phys. 49, 344 (2003)]. The generation of harmonics was explained by superimposing stress–strain relations that vary with crack width and excitation magnitude. As the crack depth increases, the ratio of magnitudes of the second harmonic to the first harmonic increases, but the increment becomes smaller [Kawashima et al., Ultrasonics 40, 611 (2002)]. Here, we show that the waves affected by the contact motion of the crack surfaces cultivate the nonlinearity in waveforms, resulting in high frequency off-band signals. With the hypothesis that the clapping of cracks might generate nonlinear components close to the high excitation frequency, we prove that the generation of the high frequency off-band peaks is directly affected by the clapping contact interaction of the crack surfaces. The amount of energy transmitted is closely related to the size of the crack width and the magnitudes of low and high frequency excitations.

Comparative vibration study of EN 8 and EN 47 cracked cantilever beam

Since earlier days, most of the failures encountered by the structures or machines are mainly due to material fatigue. The dynamic behaviour of the beam may change when cracks begin to appear in it. Knowledge of these changes in the dynamic individualism is important in crack detection as well as in structure or in machine design. This paper deals with systematic study on the free vibration of Euler-Bernoulli beam containing open edge transverse cracks. In this study, two springs steel materials (EN 8 and EN 47) are considered. The effect of the top side cracks and bottom side cracks on the natural frequency of a cantilever beam is discussed. The natural frequency of a cracked case cantilever beam is investigated numerically using FE analysis software ANSYS. Experimental work is done by using DeweFRF to investigate the natural frequency of cracked beams for strong validation of the numerical results. The results of this study suggest that the average value of natural frequencies for all top side cracked beams are identical to the average value of natural frequency for all bottom side cracked beams. This is true for both EN 47 and EN 8. Hence, it is clear that the dynamic characteristic (natural frequency) is not changing, when same configuration of cracks is either on top or bottom side of the beam. The natural frequencies for EN 8 material are comparatively on higher side than EN 47 material for the same crack configurations. In most of the cracked cases, the damping effect of EN 47 is greater than EN 8. It is also found that as crack location increases at constant crack depth, then natural frequency increases. At the last location, as crack depth increases, natural frequencies almost remain same. It is observed that, the presence of top side crack and bottom side crack of the same configuration in the cantilever beam is not a function of natural frequency, when cantilever beam is of a square cross section.

Https://marz.kau.edu.sa/Files/320/Researches/70650_43625.pdf

A crack is a type of damage that can lead to catastrophic failure when it grows in a structure. Thus prediction of cracks is a very important problem that has to be addressed. Some studies have been done on non-destructive techniques to identify cracks. Research interest in this area has been growing up quickly. In this study, a beam made of aluminum was used for detecting cracks using vibration characteristics. A method to identify cracks in an aluminum beam was derived from the equation of the dynamic stiffness. Rotational springs were used to model the cracks in the beam and the frequency response function was calculated using a spectral element model using Finite Element Model (FEM). This procedure provides a relationship between the frequency response and crack structure. The inverse problem was solved repetitively for crack positions and sizes using the Newton-Raphson method. The finite element model was formulated to simulate the results and to provide vibration overall amplitude measurements. The results showed at certain crack size, the amplitude increases with increasing crack depth for all crack positions. At crack size of 12 mm, an increase of 4.5% in amplitude was obtained for 60 mm crack position at a depth of 1.6 mm compared to its value at a depth of 0.16 mm. Similarly, it was 8.4% at crack size of 16 mm. Also, the amplitude varies inversely with position. Furthermore, at certain crack position, the amplitude increases with increasing crack depth for all crack sizes. At crack position of 75 mm, an increase of 12.6% in amplitude was obtained for 20 mm crack size at a depth of 1.6 mm compared to its value at a depth of 0.16 mm. Similarly, it was 5.4% at crack position of 150 mm. Finally, the amplitude increases as crack size increases at a certain depth.

Study on wear and rolling contact fatigue behaviors of wheel/rail materials under different slip ratio conditions

The objective of this study is to evaluate the effect of slip ratio on the wear and rolling contact fatigue (RCF) of wheel/rail materials using a rolling–sliding wear testing apparatus. The results indicate that two wear types are presented in terms of wear rate: type I (mild wear) and type II (severe wear). In type I wear, cracks propagate parallel to the surface. While in type II, the peeling is aggravated and spalling can be observed. With the slip ratio increasing, the wear mechanism of rollers transforms from slight oxidation wear and peeling to severe fatigue wear and spalling. Due to the mild wear and light plastic deformation in type I, the angle and depth of cracks show no obvious differences between the wheel and rail rollers. The crack depth and angle increase in type II wear owing to severe plastic deformation, while the depth is smaller on the wheel rollers. The size of flake wear debris presents an increasing trend and the main composition is Fe2O3 and metallic iron, and the content of iron diminishes with increasing oxidation.

Energy dissipation caused by fatigue crack in beam-like cracked structures

In this study, free vibration of cracked beam in the first vibration mode was considered taking into account both the structural damping and the damping due to the presence of crack. Increased flexibility due to the presence of crack was modeled by a massless spring and the increased damping was modeled by a massless damper at crack location. Using continuous wavelet transform and based on exponential decay of the power of the wavelet transform, the damping coefficient and damping ratio of the beam was calculated. Applying the mechanical energy balance method, the time response of the cracked beam was calculated, and compared with the time response of experimentally tested beams; an optimization method was used to estimate the damping coefficient of crack and the share of the presence of crack in total energy dissipation was calculated. Results showed that the damping coefficient of crack increases by increase in crack depth as well as crack dimensionless location. Also, the share of crack in total energy dissipation depends on crack depth, crack location, and structural damping.

Stability Evaluation of Filled-Subgrade with Cracks Formed after Earthquake

Analysis model for stability of post-earthquake embankment (filled-subgrade) with cracks was established to determine the impact significance of influencing factors. Influence law of subgrade stability was studied which focused on characteristics of cracks conditions. The results showed that the safety factor reduced with increase of crack depth, i.e. subgrade stability worsening with crack reach downwards.

Application of laser ultrasonic technique for non-contact detection of structural surface-breaking cracks

Abstract Based on the finite element method (FEM), the surface-breaking cracks have been investigated by using the laser-generated Rayleigh wave. The features of laser-generated Rayleigh wave interaction with cracks are analyzed in time and frequency domain. The simulation results show that the surface acoustic wave induced by the pulsed laser is sensitive to the surface-breaking cracks. As the crack depth increases, the transmission coefficients almost linearly decrease and the reflection coefficients show a dip. The corresponding experimental results have verified the feasibility of numerical calculation and reached a good agreement with simulation results. The research findings would provide a potential application for testing surface-breaking cracks of aircraft parts.

Effects of gear crack propagation paths on vibration responses of the perforated gear system

This paper investigates the dynamic behaviors of a perforated gear system considering effects of the gear crack propagation paths and this study focuses on the effects of a crack propagating through the rim on the time-varying mesh stiffness (TVMS) and vibration responses. Considering the effects of the extended tooth contact, a finite element (FE) model of a gear pair is established based on ANSYS software. TVMS of the perforated gear with crack propagating through tooth and rim are calculated by using the FE model. Furthermore, a lumped mass model is adopted to investigate the vibration responses of the perforated gear system. The results show that there exist three periods related to slots of the gear body in a rotating period of the perforated gear. Gear cracks propagating through tooth and rim both reduce the gear body stiffness and lead to reduction of TVMS besides the crack tooth contact moment, and the TVMS weakening for the former is less than that for the latter. Moreover, the results also show that the gear crack propagating through the rim (CPR) has a greater effect on vibration responses than the gear crack propagating through the tooth (CPT) under the same crack level. Vibration level increases with the increasing crack depth, especially for the gear with CPR.

Effects of edge crack on the vibration characteristics of delaminated beams

Delaminations and cracks are common failures in structures. They may significantly reduce the stiffness of the structure and affect their vibration characteristics. In the present study, an analytical solution is developed to study the effect of an edge crack on the vibration characteristics of delaminated beams. The rotational spring model, the „free mode‟ and „constrained mode‟ assumptions in delamination vibration are adopted. This is the first study on how an edge crack affects the vibration characteristic of delaminated beams and new nondimensional parameters are developed accordingly. The crack may occur inside or outside the delaminated area and both cases are studied. Results show that the effect of delamination length and thickness-wise location on reducing the natural frequencies is aggravated by an increasing crack depth. The location of the crack also influences the effect of delamination, but such influence is different between crack occurring inside and outside the delaminated area. The difference of natural frequencies between „free mode‟ and „constrained mode‟ increases then decreases as the crack moves from one side of the delaminated region to the other side, peaking at the middle. The analytical results of this study can serve as the benchmark for FEM and other numerical solutions.

Numerical Studies of Reinforced Concrete Beam Damage Detection Based on the Piezoelectric Impedance and Neural Network Technology

Aiming at the problem of reinforced concrete structure crack damage detection, an algorithm of crack damage monitoring based on the piezoelectric impedance technology was presented in this paper. With a piezoelectric coupling beam model was established, the electrical impedance equation of piezoelectric coupling beam was derived. The finite element model of reinforced concrete beam pasting with piezoelectric ceramic piece was established. The effect of different crack depth on the electrical impedance of coupling beam was analyzed. The results show that with the increase of crack depth, impedance peak decreases, spectrum shift to the left; neural network technology data processing method was used to the location identification of reinforced concrete beam crack depth. It proves that the piezoelectric impedance technology can successfully identify the degree of reinforced concrete beam damage to some extent.

The development of a method for crack-depth estimation in concrete by the electric response parameters to pulse mechanical excitation

A method is proposed for contactless testing of cracks in concrete by the electric response parameters to pulse mechanical excitation. Theoretical research is conducted on the electric response to pulsed elastic impact of the material with piezoelectric quartz inclusions. Numerical results are confirmed by experimental research on concrete samples. It is shown that the presence and growth of a crack in concrete lead to a change in measured electrical signal frequency response. Increase in crack depth leads to a shift of dominant spectral peaks to the lower-frequency region and decreases the cross-correlation coefficient between spectra from the sample containing defects and a solid sample.

Effect of Load Angle on Limit Load of Polyethylene Miter Pipe Bends1

The aim of this paper is to investigate the effect of crack depth a/W = 0–0.4 and load angle (30 deg, 45 deg, and 60 deg) on the limit load of miter pipe bends (MPB) under out-of-plane bending moment with a crosshead speed 500 mm/min. The geometry of cracked and un-cracked multi miter pipe bends are: bend angle, α = 90 deg, pipe bend factor, h = 0.844, standard dimension ratio, SDR = 11, and three junctions, m = 3. The material of the investigated pipe is a high-density polyethylene (HDPE), which is applied in natural gas piping systems. Butt-fusion welding is used to produce the welds in the miter pipe bends. An artificial crack is produced by a special cracking device. The crack is located at the crown side of the miter pipe bend, such that the crack is collinear with the direction of the applied load. The crack depth ratio, a/W = 0, 0.1, 0.2, 0.3, and 0.4 for out-of-plane bending moment “i.e., loading angle ϕ = 0 deg”. For each out-of-plane bending moment and all closing and opening load angles the limit load is obtained by the tangent intersection method (TI) from the load deflection curves produced by the specially designed and constructed testing machine at the laboratory (Mechanical Design Department, Faculty of Engineering, Mataria, Helwan University, Cairo/Egypt). For each out-of-plane bending moment case, the experimental results reveals that increasing crack depth leads to a decrease in the stiffness and limit load of MPB. In case of combined load (out-of-plane and in-plane opening; mode) higher load angles lead to an increase in the limit load. The highest limit load value appears at a loading angle equal, ϕ = 60 deg. In case of combined load (out-of-plane and in-plane closing; mode) the limit load decreases upon increasing the load angle. On the other hand, higher limit load values appear at a specific loading angle equal ϕ = 30 deg. For combined load opening case; higher values of limit load are obtained. Contrarily, lower values are obtained in the closing case.

Computational Analysis of Fatigue Crack Growth based on Stress Intensity Factor Approach in Axial Flow Compressor Blades

Gas turbine compressor blades are subjected to centrifugal, gas bending and vibratory loads. This repeated loading and unloading can reduce the life of compressor blades. This research aims at the estimation of fatigue crack growth and fatigue life of various three dimensional crack in a compressor blade considering the stress intensity factor calculations for a half-elliptical crack, subjected to centrifugal loading. Static stress analysis was carried out to ascertain the critical region or crack zone of the blade. The maximum Von - Mises stress was found at the fillet region near the root of the blade, the predicted state of stress has helped in identifying the region of singularity which may lead to crack initiation. Finite Element Method was used to evaluate the range of stress intensity factor solutions in the blade with a half-elliptical flaw. Semi-elliptical crack lengths ranging from 0.2mm to 6mm were considered in the crack zone. Stress intensity factor was evaluated for different rotational velocity of 5000, 10000 and 20000rpm. The stress intensity factor range obtained is employed to predict fatigue crack growth behavior and fatigue crack life estimation by using Paris law. The maximum Mode I stress intensity factor of 61.4 Mpa√m was found at the surface interception point, for a crack length of 6mm and crack depth of 2.4mm. With the increase in rotational velocity of 5000rpm, 10000rpm and 20000rpm, fatigue crack length growth rate was estimated to be and 1.95 x 10-09, 1.65 x 10-07 m/cycle and 4.15 x 10-05 m/cycle and the fatigue crack propagation life was estimated to be of 4.3 x 108 cycles, 7.3 x 106 cycles and 5 x 103 cycles respectively. It was concluded that at the surface crack interception point, fatigue crack growth rate increases with increase in crack depth and the fatigue crack propagation life tends to decrease with increase in the rotational velocity of the compressor blade.

Free Vibration of Delaminated Beams with an Edge Crack

Delaminations and cracks are two common modes of failure in structures. They may significantly reduce the stiffness of the structure and affect their vibration characteristics. In the present study, an analytical solution is developed to study the effect of edge crack on the vibration characteristics of delaminated beams. The rotational spring model, the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. This is the first study on how edge crack affects the vibration characteristic of delaminated beams and new nondimensional parameters are developed accordingly. Results show that the effect of delamination length and thickness-wise location on reducing the natural frequencies is aggravated by an increasing crack depth. The location of the crack also influences the effect of delamination, but such influence is different between crack occurring inside and outside the delaminated area. The difference of natural frequencies between ‘free mode’ and ‘constrained mode’ increases then decreases as the crack moves from one side of the delaminated region to the other side, peaking at the middle. The analytical results of this study can serve as the benchmark for FEM and other numerical solutions.

Analytical estimation of natural frequencies and mode shapes of a beam having two cracks

Abstract In this paper, an analytical estimation based on the Rayleigh's method is extended for a beam having one or two cracks to find natural frequencies and mode shapes in order to overcome weakness of solving eigenvalue problem. Weakness of solving eigenvalue problem to obtain exact natural frequencies and mode shapes is that, an algebraic equation must be solved numerically and then coefficients of trigonometric and hyperbolic terms in mode shapes will be found using matrices obtained from compatibility conditions at each point of cracks and boundary conditions. So this method does not show effects of crack size and location in the explicit form. The advantage of analytical estimation based on the Rayleigh's method over the eigen analysis method is that, the Rayleigh method obtains explicit expression for both natural frequencies and mode shapes in which effect of parameters such as crack size and location on natural frequencies and mode shapes can be investigated analytically. In the analytical estimation method, mode shapes of cracked beam are constructed by adding a polynomial function which shows effect of cracks, to mode shape of undamaged beam. The coefficients of the polynomial function are obtained by using boundary conditions and compatibility equations at the point of cracks. However in this paper it is shown that the accuracy of this estimation decreases when crack depth increases. Therefore, this paper also investigates on the upper limit of crack depth in which natural frequencies have error less than 5% and mode shapes have error less than 7% obtained by analytical estimation in compare to the exact solution. In the literature, it is shown that, analytical estimation based on the Rayleigh's method can predict only the first natural frequency of a simply supported cracked beam with sufficient accuracy, but no more investigation reported for finding the reliable range of crack depth natural mode shapes or higher natural frequencies. So, in this study, upper limit of crack depth has been found for both cases of beam with one or two cracks for first three natural frequencies and mode shapes.

Plastic Load of Precracked Polyethylene Miter Pipe Bends Subjected to In-Plane Bending Moment1

The main purpose of the present paper is to investigate the effect of crack depth on the plastic load (collapse load) of miter pipe bends (MPB) under in-plane bending moment. The experimental work is conducted to investigate multimiter pipe bends, with a bend angle 90 deg, pipe bend factor h = 0.844, standard dimension ratio (SDR) = 11, and number of welding junctions m = 3 under a crosshead speed 500 mm/min. The material of the investigated pipe is a high-density polyethylene (HDPE), which is used in natural gas (NG) piping systems. The welds in the miter pipe bends are produced by butt-fusion method. The crack depth varies from intrados to extrados location according to the in-plane opening/closing bending moment, respectively. For each in-plane bending moment, the plastic load is obtained by the tangent intersection (TI) method from the load–deflection curves produced by the testing machine specially designed and constructed in the laboratory.5 The study reveals that increasing the crack depth leads to a decrease in the stiffness and plastic load of MPB for both in-plane closing and opening bending moment. Higher values of the plastic load are reached in case of opening bending moment. This behavior is true for all investigated crack depths. A circumferential external crack has an obvious effect on the behavior of load–deflection curve. The linear elastic region in both mode of loading decreases with increasing crack depth.