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💾 ►►► DOWNLOAD FILE 🔥🔥🔥 In this paper vibration behavior of a fluid-conveying cracked pipe surrounded by a visco-elastic medium has been considered. During this work, the effect of an open crack parameters and flow velocity profile shape inside the pipe on natural frequency and critical flow velocity of the system has been analytically investigated. An explicit function for the local flexibility of the cracked pipe has been offered using principle of the fracture mechanics. Comparison between the results of the present study and the experimental data reported in the literature reveals success and high accuracy of the implemented method. It is demonstrated that the existence of the crack in the pipe, decreases the natural frequency and the critical flow velocity so that the system instability onsets at a lower flow velocity in comparison with the intact pipe. Results indicate that the flow velocity profile shape inside the pipe caused by the viscosity of real fluids, significantly affects the critical flow velocity of both intact and fluid-conveying cracked pipe. For instance, as the flow-profile-modification factor decreases from 1. Fluid-Structure Interactions. Fluid-Structure Interactions, Academic Press. Esmaeili College of Engineering, Teerthanker Mahaveer University. These pipes often lie on a foundation and would be cracked as a result of various internal and external loads while conveying fluid. So, analyzing the dynamic behavior of the cracked pipes would be practically important. Inspired by the work of Tada et al. Paris and G. Irwin Stress Analysis of Cracks Handbook. Recently, some reviews on vibration of cracked structures were reported by Dimarogonas Dimarogonas, A. Rastogi In some researches the crack has been simulated by a local reduced section or stiffness of the structure around the crack Sinha, Friswell et al. Friswell and S. Edwards The local flexibility of the structure is reduced due to the crack and this concept has been used to model the vibration behavior of the cracked pipe in some studies Xie Xie, Y. Shahsavari Honolulu, Hawaii. Sun and J. Zhou Junjie and H. Zhengjia Zhu, J. Zhou And X. Zhao Shang and L. Yin Literature review indicates that available studies about the vibration behavior of the cracked pipes can be divided into three groups;. There are fairly large amount of researches dedicated to the first two groups; Liu et al. Gurgenci and M. Veidt Ye et al. He, X. Chen, Z. Zhai, Y. Wang and Z. He Their results show that the presence of the crack in the pipe reduces the natural frequency, and its effect on the upper natural frequencies is more than the lower ones. Murigendrappa et al. Maiti and H. Srirangarajan In their works, the crack has been considered as a torsional spring and the transfer matrix method with Rayleigh's quotient has been used for vibration analysis. Naniwadekar et al. Naik and S. Maiti In contrast to the first two groups, there are fairly limited amount of investigation about the vibration behavior of the cracked fluid-conveying pipes third group. Yoon and Soo are two researchers that have contributed in most papers published about vibration behavior of the fluid-conveying cracked pipes. They investigated the effects of the open crack parameters and the moving mass on the dynamic behavior of simply supported pipe conveying fluid using Timoshenko beam model. Numerical methods based on the transfer matrix have been preferred in their works Son, Lee et al. Son, I. Lee, J. Lee and D. Son Son and J. Jin Cho and H. Yoon Son and S. Ahn Yoon, S. Kim So, available research in literature did not consider the effects of the crack parameters on the critical flow velocity which is our main objective in this paper. In present paper the vibration behavior of the cracked fluid-conveying pipe is examined via Euler-Bernoulli beam model focusing on natural frequency and critical flow velocity. Also, in order to have a more realistic model, the surrounding elastic medium is modeled via a visco-elastic foundation. Our study is carried out through an analytical method. The crack is considered using a massless torsional spring. An explicit function for obtaining the local flexibility of the cracked pipe is presented using the theory of fracture mechanics. Also, we demonstrate that how the fluid flow velocity profile would influence the critical flow velocity of the system. Dimarogonas and Papadopoulos Dimarogonas, A. Papadopoulos Following this procedure, a small vertical sectional strip is considered in the cracked section of the pipe. The strain energy of the strip is obtained using the theory of fracture mechanics. By integrating of the strain energy over the cracked section, for the local flexibility, an explicit function is obtained as a function of the crack depth. The additional strain energy due to the crack can be considered in the form of a flexibility coefficient expressed in terms of the stress intensity factor, which can be derived by Castigliano's theorem in the linear elastic range. Therefore, the local flexibility in the presence of the crack is defined by Zou et al. Chen, J. Niu and Z. Geng Where E is Young module, v is Poisson's ratio, a and b are the crack dimensions as shown in Figure 1. K I is the stress intensity factor corresponding to the first fracture mode due to the bending moment M. Figure 1: Cross section of the cracked pipe. It is seen from Eq. In order to calculate the stress intensity factor for the cracked pipes, consider a small vertical rectangular strip, having a small cross-sectional width dx and height h x with an arbitrary offset distance x , as shown in Figure 1. In Ref. Tada, Paris et al. Employing Eq. Substituting Eq. The equivalent rotational spring stiffness, K t , is defined as Dimarogonas and Papadopoulos, Dimarogonas, A. Taking into account the coordinate system illustrated in Figure 1 , h x and b calculated as:. Substitution of Eqs. For the flexibility coefficient, the dimensionless form introduced as:. By numerical integration of Eq. It should be noted that according to the main assumption in the fracture mechanics, Eq. This explicit function may be applied for any pipe with specified mechanical properties and geometrical dimensions. Considering Eqs. In order to verify the accuracy of this model, the predicted values of K t and the experimental results reported by Murigendrappa et al. The comparison shows that there is a good agreement between the equivalent stiffness obtained from the proposed method and the experimental data. So, the maximum difference between the predicted results and the experimental data of Murigendrappa et al. Thumbnail Table 1: The local stiffness due to crack, Comparison between the calculated results and the experimental data of Murigendrappa et al. The effect of the crack depth on the dimensionless local flexibility coefficient is shown in Figure 2. The figure indicates that an increase in the crack depth enhances the flexibility coefficient of the cracked pipe because of the reduction in equivalent stiffness Eq. Figure 2: Effect of the crack depth on the dimensionless local flexibility coefficient. Figures 3a and 3b illustrate schematically a fluid-conveying cracked pipe resting on a visco-elastic foundation and its corresponding mathematical model, respectively. Using the Euler-Bernoulli theory, the modified governing equation of motion for the uniform fluid-conveying pipe which is subjected to an external force, F ext , can be written as Guoa et al. Zhanga and M. Paidoussis Considering the power-law profile for the time-average flow velocity profile Streeter et al. Wylie and K. Bedford Figure 3: a Schematic view of the cracked clamped-clamped pipe, b Mathematical model. Where n is dependent on the Reynolds number. In the present work, the external force related to the visco-elastic foundation modeled using two-parameter Kelvin-Voigt model. Based on this model F ext can be written as:. K s is an additional parameter defining the foundation, usually termed as the shear constant of the foundation Kargarnovin et al. Younesian, D. Thompson and C. Jones Duhamel Part II: Infinite beam under moving harmonic loads. Esmailzadeh and D. Younesian Kameswara Rao By substituting of Eq. It is assumed that the solution of Eq. The dimensionless variables are defined as:. The solution of the Eq. Substituting this expression in Eq. The Eq. The explicit algebraic formulas for a general solutions of Eq. Mathematical Handbook. The general solution of Eq. Consider the corresponding mathematical model of the cracked pipe conveying fluid resting on a visco-elastic medium as shown in Figure 3b. For analyzing the free vibration of the cracked pipe, the entire pipe is divided from the cracked section into the two pipe segments. The crack is represented by a torsional mass-less spring that its stiffness was calculated previously in section 2. The two pipe segments can be treated separately. The equations of motion for two mentioned intact segments are obtained from Eq. The compatibility of displacement, moment and shear force of both segments at the dimensionless crack location i. General boundary conditions can be applied for the present model. In this paper, clamped-clamped supported pipe is considered. Thus the four boundary conditions may be written as:. Substituting Eqs. The system of equations 32 has the nontrivial solution if and only if the determinant of the coefficients becomes zero. So the characteristic equation of the system will be:. Now, by solving the Eq. To the best knowledge of the authors, there is no analytical or experimental result in the literature for the vibration analysis of a problem that is completely the same of one stated here. So in order to validate the aforementioned theoretical model and analytical procedure, we use available data for a similar problem. For instance, comparison made between the present work results and the experimental data of Mahjoob and Shahsavari Mahjoob, M. The implemented mechanical and geometrical properties required to calculate the results are listed in Table 2. The maximum difference between the experimental data of Mahjoob and Shahsavari Mahjoob, M. Therefore, this comparison indicates an excellent agreement between present study and the experimental data. Thumbnail Table 2: Mechanical properties and geometrical dimensions of the investigated pipe, fluid and foundation. Comparison between results of present analytical method and the experimental data of Mahjoob and Shahsavari Mahjoob, M. Having insured the accuracy of the present model, it is used to examine the vibration behavior of the cracked pipe conveying fluid resting on a visco-elastic foundation. Gazetas Marjani and E. Esmailzadeh It is found that the crack parameters affect the vibration behavior of the cracked pipe. It is obvious that the frequencies of a system generally decrease due to the crack. Results show that when the crack location approaches to the fixed ends of the pipe, the effect of the crack on the frequency reduction is increased. This tendency is intensified with an increase in the crack depth. Indeed, the mentioned locations are the inflection points for the first vibration mode of a clamped-clamped pipe. The same behavior is observed for the alteration of the second natural frequency shown in Figure 5. Figure 4: First dimensionless natural frequency of the fluid-conveying cracked pipe as a function of the dimensionless crack location. Figure 5: Second dimensionless natural frequency of the fluid-conveying cracked pipe as a function of the dimensionless crack location. When the crack depth,a c , increases, the local flexibility of the pipe reduces according to Eq. Figure 6: First natural frequency of the fluid-conveying cracked pipe. As seen from frequency equation Eq. Figure 7 shows the natural frequency of the intact pipe versus the dimensionless flow velocity for the first and second vibration modes. Langer This equals to the lowest critical flow velocity that causes instability in the system. Figure 7: Variation of the natural frequency of the clamped-clamped fluid-conveying cracked pipe against the dimensionless flow velocity. Experiments of Nikuradse revealed that fluid velocity profile in the circular pipes changes based on the quality of wall roughness. In order to take into account the effect of the mentioned change in the velocity profile, on vibration behavior of the fluid-conveying pipes, Guoa et al. The expression, a , appears in the centrifugal term of the modified equation Eq. As depicted in Figure 8 , the flow-profile-modification factor would influence the critical flow velocity of both intact and cracked fluid-conveying pipe. As Guoa et al. This trendency corresponds to increase in critical flow velocity according to Figure 8. Thus it can be deduced that the instability in a typical fluid-conveying pipe would be delayed in turbulence regime in comparison with the laminar one. For an instance, the dimensionless critical flow velocity of intact pipe increases from 5. Also, it affects the critical flow velocity of an intact clamped-clamped pipe right vertical axis. In the case of a cracked fluid-conveying pipe, the crack depth influences significantly the impact of flow-profile-modification factor on critical flow velocity. An interesting behavior observed in Figure 8 is about the effect of crack and its depth on the system instability. This behavior shows that the crack and the instability would practically reinforce each other in a pipe system causing its destruction. Figure 9 shows the effect of the foundation parameters on the critical velocity as a function of the dimensionless damping factor, c, for the different foundation parameters. Figure 9: Variation of the dimensionless critical flow velocity of the clamped-clamped cracked pipe against the dimensionless damping factor for different values of the dimensionless stiffness, k s. The results indicate that any augmentation in the foundation stiffness constant is followed by increasing the natural frequencies. This means that increasing the elastic constant makes the pipe stiffer. Also, effect of the foundation stiffness is more pronounced in the lower natural frequencies than the upper ones. Figure Effect of the foundation stiffness, k m , on the first three natural frequencies of the fluid-conveying cracked pipe. In the current study, the vibration behavior of a cracked pipe conveying fluid resting on a visco-elastic foundation was investigated analytically. A new approach for calculating the local flexibility caused by a crack was developed. The mentioned approach is applicable for various pipes with different mechanical properties. Comparison between the analytical results and available experimental data in literature shows good agreement for a wide range of the crack parameters. The results indicate that increasing the crack depth improves the flexibility, and therefore the local stiffness reduces at the crack location. This leads to reduction in the both natural frequency and critical flow velocity of the cracked pipe system. This feature can be utilized for crack detection based on the vibration analysis. Also, when the crack location approaches to the fixed ends of the pipe, reduction in the natural frequency becomes more considerable. Furthermore, the results show that a decrease in the natural frequency reduces when the crack gets closer to the inflection points until it completely vanishes in the inflection points. The fluid flow inside the pipe and its profile shape influences the natural frequency of the cracked pipe significantly. Moreover, the crack decreases the critical flow velocity so that the system instability onsets at a lower flow velocity. The authors would like to appreciate Ahar Branch, Islamic Azad University for the financial support of this research, which is based on a research project contract. Open menu Brazil. Latin American Journal of Solids and Structures. Open menu. Text EN Text English. Vahid A. About the authors. Abstract In this paper vibration behavior of a fluid-conveying cracked pipe surrounded by a visco-elastic medium has been considered. Literature review indicates that available studies about the vibration behavior of the cracked pipes can be divided into three groups; Having no fluid inside, Filled with fluid without flow zero flow velocity Having fluid flow inside fluid-conveying pipe. Table 1: The local stiffness due to crack, Comparison between the calculated results and the experimental data of Murigendrappa et al. Table 2: Mechanical properties and geometrical dimensions of the investigated pipe, fluid and foundation. Acknowledgement The authors would like to appreciate Ahar Branch, Islamic Azad University for the financial support of this research, which is based on a research project contract. References Ansari, M. Bai, Q. Basu, D. Dimarogonas, A. Doare, O. Gerolymos, N. Ghaitani, M. Hu, J. Ibrahim, R. Kargarnovin, M. Kumar, C. Liu, D. Mahjoob, M. Murigendrappa, S. Naniwadekar, M. Nguyen, V. Sinha, J. Tada, H. Xie, Y. Ye, J. Yoon, H. Yumin, H. Zou, J. Publication Dates Publication in this collection Jan This is an open-access article distributed under the terms of the Creative Commons Attribution License. Figures 10 Tables 3 Formulas Stay informed of issues for this journal through your RSS reader. PDF English. Google Google Scholar.