It has been observed that the natural frequency changes significantly due to the presence of cracks depending upon inclination and depth of cracks. The results of the crack parameters have been obtained from the comparison of the results of the un-**cracked** and **cracked** cantilever **beam** during the Modal analysis using ANSYS software. It has been observed that the natural frequency changes substantially due to the presence of cracks depending upon location and size of cracks. It has been observed that when the crack positions are constant i.e. at particular crack location, the natural frequencies of a **cracked** **beam** are inversely proportional to the crack depth. The natural frequency of the **cracked** **beam** decreases with increase the crack depth. the change in frequencies is not only a function of crack depth, and crack inclination, but also of the mode number.

The **Beam** is modeled in ANSYS Software. Element SOLID45 is used for the 3-D modeling of solid structures. Material properties are provided which is briefly listed in Table 1. After that 12 models are prepared with various inclination angles for crack with the location of crack as center of **beam**. After that the **beam** is meshed. The natural frequency of the **cracked** **beam** is found by the well known Finite Element (FEM) Software ANSYS. Modal analysis is carried out using the Block Lanczos method for finding the natural frequencies. The fixed free boundary condition was applied by constraining the nodal displacement in both x and y direction. The results are tabulated in Table 1

open and closed crack leads to a model with point finite elements. In the case of **cracked** beams, the crack breathing law is quite simple, since there are only two states for the stiffness matrix: when the crack is open and when it is closed. With this behavior, the stiffness variation is assumed as step function, according to the instantaneous bending moments that is applied to the crack section, as Qian et al. [11] and Sundermeyer and Weaver [12] analyzes a simply supported **cracked** **beam**, model by two beams segments joined by a spring that represents **cracked** section. Each segment is treated as a continuous element, which obeys the differential partial equation of Euler-Bernoulli. On the other hand, Tsai and Wang [13] uses the Timoshenko’s theory in order to model the **beam** section. Qian et al. [11] formulate a method of crack location in cantilever beams, based on the change that this failure produces in the natural frequencies and mode shapes of the system. Saavedra et al. [14] presented a theoretical and experimental dynamic behaviour of different multi-beams systems containing a transverse cracks. The additional flexibility that the crack generates in its vicinity is evaluated using strain energy density function given by the linear fracture mechanics theory. Based on this flexibility, a new **cracked** finite element stiffness matrix is deduced, which can be used subsequently in the FEM analysis of crack systems. Chaiti et al. [15] addresses the problem of vibrations of a **cracked** **beam**. In general, the motion of such a **beam** can be very complex. The focus of this paper is the modal analysis of a cantilever **beam** with a transverse edge crack. The non- linearity mentioned above has been modelled as a piecewise-linear system. In an attempt to define effective

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positioning of it. The slight variations in the values are because of the various assumptions taken during numerical analyses which are slight different from the assumption taken during theoretical analysis by the authors. So the present numerical investigation proofs to be handy for such analysis of **cracked** **beam**, and it can be said that for evaluating the relative natural frequency and amplitude of the multi **cracked** **beam** numerical investigation gives sufficient information regarding the matter.

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The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analyzing the peculiar features of the non-linear response of a **cracked** **beam**. The problem of a cantilever **beam** with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional ﬁnite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modiﬁcation of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identiﬁcation. [3]

Sharma P.K., et al. (2014) were conduct the work for finite element analysis of both un-**cracked** and **cracked** cantilever **beam**. CAD design developed using CATIA software was the input file for this analysis. Totally 10 models of **cracked** **beam** having various cross sections were analysed. The results obtained from the finite element analysis were verified by theoretical method [5]. Chandradeep Kumar, et al. (2014) were conduct the Finite element analysis of a **beam** using MATLAB and ANSYS then the results are compared with theoretical calculations. Lastly harmonic analysis also performed to check the results [6].

The natural frequency decreases as the crack depth increases in a structural part. Firstly determination of natural frequency of different modes of vibration is done for un-**cracked** **beam** theoretically (then FEA analysis in ANSYS and by using FFT analyzer in experimental work. Here total 10 models have been used taking different combinations of relative crack location and relative crack depth. Certain steps are followed to carry on analysis by FFT analyzer for experimentation. It is clear from analysis that the natural frequency of different modes of vibration can be precisely obtained from these methods and tabulated in tables. A comparison is made in between theoretical values of natural frequencies with the ANSYS values of natural frequencies and experimental values of natural frequencies. The result shows that all the values obtained by three methods are closed to the agreement.

The differential equations of motion are obtained by using Hamilton’s principle. The considered problem is investigated within the Euler-Bernoulli **beam** theory by using finite element method. The **cracked** **beam** is modeled as an assembly of two sub-beams connected through a massless elastic rotational spring. Material properties of the **beam** change in the thickness direction according to exponential distributions. In the study, the effects of the location of crack, the depth of the crack and different material distributions on the natural frequencies and the mode shapes of the functionally graded beams are investigated in detail. Also, some of the present results are compared with the previously published results to establish the validity of the present formulation.

crack is modeled as piecewise linear system with bilinear natural frequency while geometric nonlinearities are incorporated in a cubic Duffing’s term. The reduced order model was able to match the original FEM data to desired accuracy with only first two POD modes of the system and capture the change in frequency introduced by the damage. Robustness of the macromodel is checked under different loading conditions viz. changed forcing frequency, pressure loading and damping. Natural frequency of the **cracked** **beam** reduces due to presence of local flexibility in the form of breathing crack and is observed from FFT of the forced vibration response.

Due to limited fatigue strength, the fatigue cracks ours in the material under service conditions. Cracks are also found inside the material due to poor manufacturing processes. Single sided cracks are produced in the material as a result of fluctuating loads. Crack generally may be of two types, transverse cracks and oblique cracks. The magnitude and orientation of the manufacturing defect decides the origin of either transverse cracks or oblique cracks in the **beam**. Hence it is very essential to study the effect of top side and bottom side oblique cracks on the **beam**. Out of two cracks i.e. top side or bottom side cracks, one crack will be comparatively more critical, hence it requires much attention. Crack get propagated in the material due to the action of fatigue load and at the end, it gives catastrophic failure. Understanding the dynamics of the **cracked** **beam** is of most importance because various vibration parameters like natural frequency, resonant amplitude of uncracked and **cracked** cases of a **beam** used as a basic criteria in the crack detection by vibration methods. In this study, most practical spring steel material (EN 47) is considered for the cantilever **beam**. ANSYS software used to find the natural frequency and zero frequency deflection of **cracked** cases of beams. Stiffness of defective beams is calculated by a conventional formula (Load / deflection). In this study, it is found that the value of stiffness and natural frequencies for top side **cracked** cases are comparatively on lower side than bottom side **cracked** cases when crack angle equal to 20 0 . It is also found that up to 10 0 crack angle, the algebraic sum of stiffness of top side **cracked** cases is equal to the algebraic sum of stiffness of bottom side **cracked** cases. This condition is true also for natural frequency. It is also observed that, when crack angle is 20 0 , then presence of top side crack and bottom side crack of the same configuration in the cantilever **beam** is a function of natural frequency, when cantilever **beam** is of a square cross section.

different papers on multiple cracks, respective influences, and identification methods in some structures such as beams, pipes, rotors etc. Lee [11] used FEM to solve forward problem in a multi-**cracked** **beam**. In this paper, an inverse problem was solved iteratively for the locations and depths of the cracks using the Newton- Raphson method. Patil and Maiti [12] identified multiple cracks using frequency measurements. Their procedure presented an explicit linear relationship between the changes of natural frequencies and damage parameters. Mazanoglu et al. [13] performed the vibration analysis of multi-**cracked** variable cross-section beams using the Rayleigh–Ritz approximation method. Binici [14] presented a parametric study on the effect of cracks and axial force levels on the eigenfrequencies. A new method for natural frequency analysis of beams with an arbitrary number of cracks was developed by Khiem and Lien [15]. Cam et al. [16] studied the vibrations of **cracked** **beam** as a result of impact shocks to obtain information about location of cracks and depth of beams.

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Dynamic individuality of **cracked** and intact materials is dissimilar; it means that the reliability of both the materials is not same. So, the vibration analysis of a **cracked** **beam** or shafts is one of the most severe problems in various machinery. The investigation of **cracked** beams for various vibration parameters are very much needed because of its practical importance. Measurement of natural frequencies, vibration modes of **cracked** and un- **cracked** **beam** is used as a basic criterion in crack detection. Most of the cracks may be a of fatigue type and occurs in the **beam** in service due to the limited fatigue strength. In the literature many studies deal with the structural safety of beams. Kocharla et al. [1] studied that presence of crack in a structure changes the vibration properties of the **beam** like natural frequency and mode shape. These change in vibration properties used in inverse problem for the crack detection. In this study, they treated the turbine blades as a cantilever **beam**. Vibration analysis of a cantilever **beam** extended successfully to develop online crack detection methodology in turbine blade. Cantilever **beam** is modeled with two U-notches and observed the influence of one U-notch on the other for natural frequencies and mode shapes. By using central difference approximation, the curvature mode shapes were calculated from the displacement mode shapes. The depth and location corresponding to any peak on this curve becomes possible notch parameters. Babu et al. [2] treated turbine blade as a cantilever **beam** and a shaft as a simply supported **beam**. Vibration analysis of a cantilever **beam** and simply supported **beam** is extended successfully to

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K. H. Barad et al. [4] detected the crack presence on the surface of **beam**-type structural element using natural frequency. First two natural frequencies of the **cracked** **beam** have been obtained experimentally and used for detection of crack location and size. Detected crack locations and size are compared with the actual results. The effect of crack location and depth on natural frequency is presented and compared.

explained that, fault detection in a single **cracked** **beam** has been worked out. To identify location and the depth of crack in a **beam** containing single transverse crack is done through conceptual and experimental analysis respectively. It has come to noticed that a crack in a **beam** has great effect on dynamic characteristics of **beam**. The strain energy density function also applied to examine the few more flexibility produced because of the presence of crack. Considering the flexibility an additional stiffness matrix is taken away and it is used to obtain the natural frequency and mode shape of the **cracked** **beam** of different end conditions. The difference of mode shapes of cantilever **beam**, simply supported **beam** and Clamped – Clamped **beam** in between the first three mode shapes of **cracked** and un-**cracked** respectively **beam** with its amplified view at the location of the crack are studied. The theoretical analyses are carried out of the crack structure. Finally for the validation of result are matched with the both theoretical and experimental analysis. It is found that the agreed between their results is excellent. The comparisons of result in both methodologies written above are performed. The future work on the problem of fault recognition of a **cracked** **beam** can be carried by using more advanced hybrid techniques with the help of finite element method and artificial intelligence technique.

It is already known that the natural frequency decreases as the crack depth increases in a structural part. Firstly determination of natural frequency of different modes of vibration is done for un-**cracked** **beam** theoretically (solving Euler’s Equation for **Beam** in vibration analysis),then FEA analysis in ANSYS and by using FFT analyser in experimental work. Here total 10 models have been usedtaking different combinations of relative crack location and relative crack depth. Several steps have been shown to develop a natural frequency modal based on FEA which is explained through an example and all the frequency values are tabulated in chapter 5 along with the knowledge of finite element analysis. Certain steps are followed to carry on analysis by FFT analyser for experimentation. Several decisions are made to carry out experimental work in chapter 6. The brief information has been obtained due to experimentation. It is clear from analysis that the natural frequency of different modes of vibration can be precisely obtained from these methods and tabulated in table 8.1. The variation of natural frequency with crack depth and location is shown in Fig. 8.1 and 8.2

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The third specimen RCB3 was loaded the same way as RCB2 and observed similar progress in ini- tial crack propagation. The cracks were repaired with epoxy injection and allowed for a week to cure the resin and to develop a good bond. Then the specimen was prestressed to 150 kN, similar to RCB2, and reloaded. An interesting result was ob- served in the crack propagation. A new shear crack was initiated that lead to failure of the **beam** (Figure 8). The repaired crack did not open-up again during the subsequent loading. This proved that the epoxy repair was properly done. Furthermore, it has in- creased the capacity of the member to 310 kN, 58%

Abstract : Early detection of damage is of special concern for engineering structures. A comparatively recent development for the diagnosis of structural crack location and size by using the finite element method and Fuzzy logics techniques has improved. The traditional methods of damage detection includes visual inspection or instrumental evaluation .A method based on measurement of natural frequencies is presented for detection of the location and size of a crack in a cantilever **beam**. Numerical and programming in MATLAB is used for solving the Euler equation for un-crack **beam** to obtain first three natural frequencies of different modes of vibration considering boundary conditions for the **beam**.

Kamble and Chavan [5] identified the crack in cantilever **beam** by using experiment and wavelet analysis. In this study crack was modeled by rotational spring and equation was developed for non-dimensional spring stiffness. Now by taking first three natural frequencies by vibration measurement, curves of crack equivalent stiffness were plotted and the intersection of the three curves indicated the crack location and size. The experiment on cantilever was done with single crack at different position and different depth size by FFT Analyzer and the natural frequency obtained was compared with ANSYS package. The time- amplitude data obtained was further used in the wavelet analysis to obtain time-frequency data. The above data played vital role to find the small crack parameters which affect the dynamic properties of the system.

Numerical modelling of RC **beam** model was done using ANSYS 16.2 WORKBENCH, a finite element software for mathematical modelling and analysis. The dimensions of all the specimens are same. width of **beam** is 200mm, a depth is 450mm and the length is 4000mm.The **beam** were under- reinforced using three 12mm deformed steel bars as the internal tension reinforcement and two 12mm deformed steel bars as the internal compression reinforcement, and 10mm deformed steel bars with a centre to-centre spacing of 100mm as the stirrups and M35 grade of concrete is used. For concrete, steel and GFRP, Analysis requires input data for material properties are shown in Table 1 and Table 2. Figure 1 showing modelled view of RC **beam** and Figure 2 showing modelled view of reinforcement.

The theoretical and software based simulation work for simple bending problem were performed by considering example of cantilever **beam**. Numerical simulations for simple and **cracked** cantilever beams using finite element stiffness method, analytical **beam** theory and finite element package (ANSYS) were evaluated and then results were verified with code generated in C++ language. The results gathered by mathematical modeling are very beneficial. They enable a better understanding and ability to predict deformation process; however, when using the mathematical modeling assumptions may have to be made. In order to compensate, a factor of safety is always added into the equations when the results are going to be used in applications.

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