Finite Element Analysis of Piping Vibration with Guided Supports

Int. J. Mech. Eng. Autom. Volume 3, Number 3, 2016, pp. 96-106 Received: January 27, 2016; Published: March 25, 2016 International Journal of Mechanical Engineering and Automation Finite Element Analysis of Piping Vibration with Guided Supports Shankarachar Sutar 1, Radhakrishna Madabhushi 1 and Ramesh Babu Poosa 2 1. Design & Engineering Division, CSIR-Indian Institute of Chemical Technology, Hyderabad 500007, India 2. Department of Mechanical Engineering, College of Engineering, Osmania University, Hyderabad 500007, India Corresponding author: Radhakrishna Madabhushi (mradhakrishna@iict.res.in) Abstract: The earlier work by the authors is development of a mathematical model for dynamics and stability of pipe conveying fluid with guided supports. Simulation of the model is carried out by performing an and results are compared. The effect of end conditions on the natural frequency for an empty pipe, static and fluid flowing conditions were studied. The mathematical model is derived by using Euler-Bernoulli beam theory and Hamilton s energy expressions to get the equation of motion and fundamental transverse natural frequency of vibration is computed by Muller s Bi-section numerical method. A FORTRAN program is developed for estimation of the frequencies in non-dimensional form. The vibration of a fluid conveying pipe with guided ends are modelled by using the I-DEAS commercial software and the influence of fluid velocity on natural frequencies are analysed by ABAQUS software. The analytical results were then compared with, and found to be in good agreement. Keywords: Fluid flow, guided, mathematical model, Muller s method, natural frequency, transverse. 1. Introduction Pipes are used for transfer of fluids and are susceptible to vibrations because of fluid forces acting on pipe, which resulting flutter instability. Extensive studies were undertaken by different researchers on vibrations of piping systems subjected to different boundary conditions and loadings. The work on dynamic behaviour of a fluid conveying pipe was started in 1950 by Ashley and Havilland [1], who have first examined the vibrations of an aboveground Trans-Arabian oil pipe line, which was considered a simply supported pipe. Subsequently, Housner [2] derived the equations of motion of a fluid conveying pipe and developed an equation relating the fundamental bending frequency of a simply supported pipe to the velocity of the internal flow of the fluid. He also stated that at certain critical velocity, a statically unstable condition could exist. Long [3] presented an alternate solution to Housner s equation of motion for the simply supported ends and fixed-free end conditions. He compared the analysis with experimental results to confirm the mathematical model. His experimental results remained inconclusive since the maximum fluid velocity available for the test was low and change in bending frequency was very small. Naguleswaran and Williams [4] developed solutions for natural frequencies in axial mode for hinged-hinged, fixed-hinged, fixed-fixed boundary conditions. Stein and Tobriner [5] discussed vibration of pipes containing flowing fluid, in which the effects of foundation modulus, flow velocity and internal pressure on the dynamic stability, frequency response and wave propagation characteristics of an un-damped system was studied. Notable contributions in this area include the works of Chen [6] in most of the cases; the differential equation of motion of fluid-conveyed pipe is deduced using the Galerkin s method in Lagrange system. Subsequently, the solution of the differential

Finite Element Analysis of Piping Vibration with Guided Supports 97 equation is obtained by considering many numerical methods such as transfer matrix, finite element, perturbation, Runge-Kutta and differential quadrature. Weaver and Unny [7] studied the dynamic stability of finite length of pipe conveying fluid using Flugge-Kempner equation and found the critical flow velocities. Paidoussis and Issid [8] presented a general transverse frequency equation with gravitational force, pressure effects, material damping and viscous damping effects. Wang, et al. [9] had conducted research on solid liquid coupling dynamics of pipe conveying fluid, where the influence of flowing velocity, pressure, solid-liquid coupling damping and solid-liquid coupling stiffness on natural frequency for simply supported ends was studied. Zhang, et al. [10] studied the vibration of pipes conveying fluid and developed dynamic equilibrium matrix using Lagrange Principle for discretized pipe element flowing fluid for simply supported, fluid conveying pipe subjected to initial axial tensions are compared with experimentally obtained results. Oz and Boyaci [11] developed a mathematical model for transverse vibrations of tensioned pipes conveying fluid with time dependent velocity. The principal parametric resonance cases are studied, stability boundary conditions are determined analytically. Numerical results are presented for the first two modes. New closed-form solutions for the natural frequency of a clamped-guided beam are derived for their investigation [12]. By postulating the mode shape of the clamped-guided beam, whose material density and stiffness are taken as polynomial functions, a closed-form solution was calculated for the natural frequency. Wiggert and Tijsseling [13] reviewed the fluid structure interaction in piping highlighting the effects of fluid forces on piping and its support structure and explained how transfer of momentum and forces between piping and the contained liquid during unsteady flow, excitation mechanisms may be caused by rapid changes in flow and pressure or initiated by mechanical action of the piping. Sinha, et al. [14] developed a finite element model for cantilever pipe conveying fluid. The instability in the pipe is due to change in the natural frequencies for its dynamic behavior. The results of FE model of a cantilever pipe was compared with experimental data. Zhang, et al. [15] developed a FE model to predict the vibration of cylindrical shells conveying fluid and compared the results with published experimental results to validate the developed model. Seo, et al. [16] studied frequency analysis for cylindrical shells using FEM. The influence of fluid velocity on the frequency response function was illustrated. Chellapilla, et al. [17] studied two parameter foundation effects for fluid conveying pipes resting on soil media and found the frequencies. Huang [18] used the Galerkin s method to obtain the natural frequencies for fluid conveying pipeline for different boundary conditions. The four variables mass, stiffness, length and flow velocity were studied in detail to estimate the effect of flow velocity on the natural frequency. Al-Hilli et al. [19] developed a mathematical model for dynamic behaviour of pipe under general boundary conditions. Considering the supports as compliant material with linear and rotational springs, studied for simply, free built, guide and found the natural frequencies for each conditions. Guided pipe supports were always treated as free ends by the previous authors. The present work is therefore carried out to understand the vibrations of pipes that convey fluid with guided type of pipe supports. The pipe is considered as Euler-Bernoulli beam and the equation of motion is derived by using the Hamilton s variation approach, and an exact solution is obtained for guided boundary conditions to get transcendental frequency equation. Section 2 considers the fourth-order partial differential equation and the terms in the equation are represented in non-dimensional form. Section 3 includes derivation of natural frequency equation for

98 Finite Element Analysis of Piping Vibration with Guided Supports which the exact solution is obtained by using the Matrix method. A finite element model for the guided-guided pipe supports were developed using the ABAQUS structural and fluid modules and the results are presented in Section 4. Section 5 deals with the steps involved in development of a FE model and Tables 1-3 are presented for empty pipe, fluid static and fluid flow condition with three different materials. Sections 6, presents results and discusses the comparison of analytical with finite element analysis with percentage error. Conclusions are presented in Section 7. 2. Theoretical Model The guided boundary condition is as shown in Fig. 1. Table 1 Vibration analysis of empty pipe with three materials supported in guided slots. Mode No Eigen value Natural frequency of pipe Eigen Natural frequency of pipe foreigen Natural frequency of pipe (λ 1 ) for thermocol supported in guided slots (Hz) value (λ 2 ) foam supported in guided slots (Hz) value (λ 3 ) for spring supported in guided slots (Hz) Analytical Analytical Analytical 1 1.0e3 3.7 5.0 1.0e3 3.7 5.0 0.9e3 3.7 5.0 2 3.7e4 33.5 30.6 7.7e4 45.5 44.3 3.6e4 33.5 30.5 3 9.3e4 45.5 48.6 1.5e5 59.5 62.6 4.4e4 33.5 33.7 4 1.9e5 59.5 70.8 2.0e5 75.3 72.7 1.6e5 64.1 64.8 5 2.2e5 75.3 74.9 7.8e5 157.1 140.6 2.0e5 75.3 72.5 6 3.4e5 92.9 93.1 1.2e5 182.2 180.6 3.4e5 92.9 93.0 7 5.2e5 112.5 115.3 - - - 5.3e5 112.5 116.6 Table 2 Vibration analysis of fluid filled (static condition) pipe with three materials supported in guided slots. Mode No Eigen Natural frequency of pipe Eigen Natural frequency of pipe Eigen Natural frequency of pipe for value (λ 1 ) for thermocol supported in value (λ 2 ) for foam supported in value (λ 3 ) spring supported in guided guided slots (Hz) guided slots (Hz) slots (Hz) Analytical Analytical Analytical 1 1.0e3 3.5 5.0 1.0e3 3.5 5.0 1.0e3 3.48 5.0 2 7.8e4 42.6 44.5 1.5e4 21.7 19.9 2.5e4 21.7 25.5 3 1.4e5 55.6 59.9 7.8e4 42.6 44.6 5.3e4 42.6 37.0 4 2.8e5 86.9 84.2 1.6e5 70.4 65.5 2.3e5 86.9 77.7 5 4.5e5 105.2 106.7 2.3e5 70.4 76.7 2.6e5 86.9 82.3 6 5.8e5 125.2 122.0 2.4e5 86.7 79.5 4.7e5 105.2 109.0 Table 3 Vibration analysis of fluid conveying pipe with guided-guided supports. Mode No Vel. (m/s) Natural frequency of pipe Vel. Natural frequency of pipe for Vel. Natural frequency of pipe for for thermocol supported in (m/s) guided slots (Hz) foam supported in guided slots (Hz) (m/s) spring supported in guided slots (Hz) Analytical Analytical Analytical 1 20.1 14.8 15.5 20.5 8.0 10.3 20.1 3.5 5.0 2 21.2 3.7 5.3 21.3 21.7 26.1 21.2 8.0 10.6 3 23.3 3.7 5.4 22.3 21.7 20.6 23.3 8.0 10.6 4 24.6 14.8 15.7 23.4 3.5 5.4 23.6 3.5 5.2

Finite Element Analysis of Piping Vibration with Guided Supports 99 Fig. 1 Guided-guided pipe support. The guided slot allows the pipe to move freely in the vertical direction, and thereby not supporting a shear force. Also it prevents the pipe end from rotation to keep slope at zero [9]. The linear fourth order partial differential equation is as mentioned below [10] + +2 + + =0 (1) where EI bending stiffness of a pipe, A cross-sectional area of pipe, L length of pipe, U fluid velocity, m p + m f pipe plus fluid mass, w lateral deflection of pipe, V non-dimensional fluid velocity and ω n circular frequency, and λ is wavelength. Eq. (1) is simplified by assuming the terms for the following dimensionless parameters: = ; = + w(x) represents the shape of the deformed configuration and ω n the circular frequency of vibration of that particular deformed configuration. + (V ) +2γU λ w=0 (2) where ξ = natural boundary condition, γ = mass ratio. = ; = + ( ) =0 (3) 2 = ( ); = When the natural frequency of the pipe approaches zero the critical flow velocity has been computed for all the end conditions. When the fluid flow velocity is equal to the critical velocity, the pipe bows out and buckles as the forces required to make the fluid deform to the pipe curvature are greater than the stiffness of the pipe. The term Coriolis force represents the damping of the system and its effect on the frequency of vibration is negligible and so is omitted, as the present work aims to obtain upper bounds for the frequencies of vibration of the pipe conveying fluid. The damping term is omitted and Eq. (2) is a non-dimensional partial differential equation of higher order with boundary problem. Now let ( ) =. (4) ( ) =c.s.e ; =c.s.e (5). +2.. +.. =0 +2. + =0 where c and s are constants. ( ) +2. +( ) =0 (6) The roots of Eq. (10) is given by = ± ( ) ; = ± = ; =; = ; = Considering first two roots = and =. The boundary conditions for guided-guided pipe is as given below (, ) = 0 at = 0 (7) (, ) = 0 at = 0 (8) (, ) = 0 at = (9) (, ) =0 at = (10) 3. Derivation of Equation for Natural Frequency Let the solution of the general Eq. (1) is given as ( ) = sinh + cosh + + sin + cos (11) Then, ( ) = cosh+sinh+ +cos sin (12) ( ) = α sinh + cosh Then, ( ) sin β cosβx (13) = A cosh + B sinh cos + sin (14) = + + = + + (15) where and are the roots of equation and A, B, C

100 Finite Element Analysis of Piping Vibration with Guided Supports and D are the arbitrary constants. Differentiating Eqs. (11)-(14) and on substitution of the boundary conditions in Eqs. (7)-(10), we get 0 0 = 0 0 =0 (16) where = cosh() ; = sinh() ; = cos() ; = sin(). Similarly, = cosh () ; = sinh() ; = cos(); = sin(). 0 0 = =0 (17) 0 0 = =0 (18) After simplification of two matrices (17)-(18), we get the frequency equation of guided-guided pipe support condition is as given below: [( + ) ( )]sinh(). sin() = 0 (19) 4. Simulation of a Pipe Conveying Fluid The simulation of the pipe is modelled by using I-DEAS software and the analysis is done by ABAQUS Software, which includes the standard and CFD modules to couple and analyse fluid structure interaction. Figs. 2-4 show a pipe of span 3 m with guided supports at both ends, the pipe material density is = 7850 kg/m 3, Young s modulus is E = 200 Gpa, Poisson ratio is = 0.28, the outer diameter of pipe is = 34 mm and inner diameter is = 30.44 mm, the thickness of pipe wall is 3.34 mm. The fluid density conveying in the pipe is = 1000 kg/m 3, the mass of pipe is 8.0 kg and the fluid mass is 2.0 kg inside the pipe; water at 20 C has dynamic viscosity of 0.001002 Pa s. Case 1: pipe is supported on springs with a stiffness of 54 N/mm; Case 2: the pipe is rested on thermocol; Case 3: the pipe is placed on Foam. Fig. 2 Schematic diagram of guided-guided piping support. Material properties: (1) Steel: density = 7850 kg/m 3, Young s modulus = 200 GPa, Poisson s ratio = 0.28. (2) Thermocol: density = 20 kg/m 3, compressive strength = 1.2 kg/cm, cross breaking strength = 1.8 kg/cm, tensile strength = 4 kg/cm, thermal conductivity = 0.029 Kcalm/hr mc. (3) Foam: density = 100 kg/m 3, E xx = 0.35 GPa, E yy = 0.35 GPa, E zz = 0.35 GPa, G xy = 0.035 GPa, G yz = 0.035 GPa, G zx = 0.035 GPa, Gxy = 0.035 GPa, G yz = 0.035 GPa, G zx = 0.035 GPa, υ xy = 0.3, υ yz = 0.3, υ zx = 0.3. (4) Fluid water properties: density = 1000 kg/m 3, water at 20 C, dynamic viscosity of 0.001002 Pa s, bulk modulus = 2250 MPa.

Finite Element Analysis of Piping Vibration with Guided Supports 101 Fig. 3 Finite element mesh model of guided-guided pipe support. (1) Structure mesh statistics: element type = S4, total number of nodes = 13678, total number of elements = 13392, element shape = quadrilateral, element order = linear. (2) Fluid mesh statistics: element type = FC3D8, total number of nodes = 4131, total number of elements = 2904, element shape = hexahedral fluid elements, element order = linear. Fig. 4 Structure mesh for guided-guided pipe support. The entire pipe is modelled in slots of two L-angles of 50 mm 60 mm with a length of 600 mm which is fixed to 10mm thick base plate. The pipe frequencies are computed for the above three conditions. Case 1 computes frequencies when pipe is empty. Case 2 is simulated with fluid in static condition and Case 3 is a situation where the fluid is flowing with four different velocities, by varying the valve position from one-fourth opening to fully open condition. ABAQUS/CFD with ABAQUS/Standard are combined and simulated to run the fluid structure interaction problem. The pipe stiffness was extracted for particular velocity and this stiffness is used to evaluate the frequency by performing the harmonic analysis in linear dynamics domain. The 3-D CAD model was generated using I-DEAS and the model is exported to ABAQUS for

102 Finite Element Analysis of Piping Vibration with Guided Supports considering it to be a FSI (fluid structure interaction) problem. Meshing for structural analysis was done by considering 13678 nodes and 13392 elements. The shape of the element is quadrilateral, linear and shell type (S4). The CFD analysis was carried out for fluid flow in a pipe by meshing the model using hexahedral fluid elements (4131 elements and 2904 nodes); Element order is linear type used for fluid mesh is FC3D8. A convergent solution is obtained by bridging structural and fluid elements. The average aspect ratio at the holes, corners and fillet is good and is found to be 1:4. For the fluid static condition in the pipe, the coupled acoustic frequency analysis is used to extract the natural frequencies of the fluid filled pipe. Harmonic analysis (sine sweep) was carried out on the structure by applying harmonic force of 1g acceleration to plot the acceleration vs. frequency (to find the frequency peaks on the pipe), the first mode shape was excited at 5Hz as shown in Fig. 5. This is a transient (implicit method) non-linear FSI analysis run for 2 s to calculate the stress and displacement response. Below contour plot indicates the peak displacement at a particular time increment as shown in Fig. 6. The study indicates that at high frequencies the displacement is lower and at lower frequencies, the fluid velocity will be very high which lead to turbulence in the pipe. At critical fluid flow velocity, the frequency approaches zero and pipe starts buckling and result in an unstable condition. 5. Fluid Structure Interaction by Simulation The FSI technique was used using ABAQUS/Standard and ABAQUS/CFD to couple the fluid structure interaction where fluid flow is at different velocities and turbulence of the fluid is also considered; Turbulence k-epsilon model was considered in ABAQUS/CFD to model the turbulence of the fluid; The dynamic analysis was done for 2secs.The Implicit dynamic analysis method was used to calculate the response of the structure with the fluid (water); After extracting the fluid forces on the structure at maximum deflection in FSI. The base state at the maximum deflection was used in the frequency analysis to extract the frequencies; Harmonic analysis (sine sweep) was done on the same structure by applying base excitation of 1g acceleration force to plot the displacement vs. frequency; As the velocity increases, the natural frequency of the pipe decreases. Fig. 5 Harmonic analysis (sine sweep) for guided-guided pipe.

Finite Element Analysis of Piping Vibration with Guided Supports 103 Fig. 6 Vibration analysis of fluid conveying pipe for guided-guided support. Tables 1, 2 and 3 show the natural frequency of guided-guided pipe with three conditions: empty pipe, fluid filled and fluid flow with different velocities: comparison of results for theoretical and are given. Table 1 and Fig. 7 represent the results for an empty pipe resting on three pipe support materials: Thermocol, Foam and Steel Spring. Eigen values change with corresponding natural frequency of the pipe as shown in Table 1. Frequencies and Eigen values are computed for all the three supports and found that thermocol and spring supported pipe at 6th mode of force excitation value shows 93 Hz whereas for Foam support it is 182 Hz. Table 2 and Fig. 8 present the results of vibration analysis of fluid filled pipe (static condition) with guided-guided supports for three different materials. It is found that the natural frequency for thermocol and Fig. 7 Non-dimensional natural frequency vs. eigenvalue for empty pipe supported with three different materials supported in guided slots. Fig. 8 Non-dimensional natural frequency vs. eigenvalue for pipe filled with water (static condition) guided-guided supports.

104 Finite Element Analysis of Piping Vibration with Guided Supports spring type supports is more than the foam support natural frequencies. Table 3 and Fig. 9 represent the non-dimensional fluid velocities vs. natural frequency computed for three different pipe support materials like, thermocol, foam and spring respectively. In this case it is found that the frequencies for thermocol and spring are closer than the foam type. 6. Results and Discussion The results of piping vibration for guided-guided ends are obtained theoretically and are validated by simulation using ABAQUS software. The percentage error in frequencies for all the 3 conditions is listed in Tables 4-6. From the results presented above, it is observed that for foam type of support the frequencies tend to vary (follow a cyclic pattern of increase and decrease) with increase in flow velocities. However, for other type of supports, i.e., thermocol and spring, it is seen that the frequencies drastically reduce for the fluid flow condition only when compared to the other two conditions as shown in Tables 4-6. It is also observed that when the pipe rests on foam type of support, the frequency is higher when pipe is empty than when in static and flow conditions. 7. Conclusions Fig. 9 Fluid velocity vs. natural frequency of fluid conveying pipe with three different materials supported in guided slots. The equation of motion for pipe conveying fluid is derived from energy expressions using the Hamilton s Principle. A new transcendental frequency equation is derived for guided end conditions by using separation of variables method to obtain natural frequencies of fluid conveying pipe for different support conditions. A FORTRAN program is developed to solve the transcendental frequency equation. The results obtained are validated with and are found to be in good agreement with the analytical results. The results of the guided end conditions show interesting facts, which was not dealt before. As the fluid velocity increases, both analytical and results presents the variation in frequencies. Table 4 Comparison of analytical results vs. of empty pipe. Mode No Natural frequency of pipe for thermocol Natural frequency of pipe for foam Natural frequency of pipe for spring supported in guided slots (Hz) supported in guided slots (Hz) supported in guided slots (Hz) Analytical % Error of Theo. vs. Analytical % Error of Theo. vs. Analytical % Error of Theo. vs. 1 3.7 5.0 25.8 3.72 5.0 25.6 3.7 5.0 25.6 2 33.5 30.6 8.5 45.55 44.3 2.7 33.5 25.5 23.8 3 45.5 48.6 6.3 59.49 62.6 5.0 33.5 37.0 9.5 4 59.5 70.8 15.9 75.30 72.7 3.4 64.1 77.7 17.5 5 75.3 74.9 0.5 157.1 140.6 10.5 75.3 82.3 8.5 6 92.7 93.1 0.2 182.2 180.6 0.8 92.7 109.0 14.7

Finite Element Analysis of Piping Vibration with Guided Supports 105 Table 5 Comparison of analytical results vs. for fluid filled pipe in static condition. Mode No Natural frequency of pipe for thermocol Natural frequency of pipe for foam Natural frequency of pipe for spring supported in guided slots (Hz) supported in guided slots (Hz) supported in guided slots (Hz) Analytical % Error analytical vs. Analytical %Error analytical vs. Analytical % Error of analytical vs. 1 3.48 5.0 30.4 3.48 5.0 30.4 3.48 5.0 30.4 2 42.61 44.5 4.24 21.74 19.9 8.46 21.74 25.5 14.74 3 55.66 59.9 7.07 42.61 44.6 4.46 42.61 37.0 13.16 4 86.96 84.2 3.17 70.44 65.5 7.01 86.96 77.7 10.64 5 105.2 106.7 1.40 70.44 76.7 8.16 86.96 82.3 5.35 6 125.2 122.0 2.55 86.96 79.5 8.58 105.22 109.0 3.47 Table 6 Comparison of analytical results vs. for fluid flow condition. Mode Vel. Natural frequency of pipe for Vel. No (m/s) thermocol supported in guided slots (Hz) (m/s) Analytical % Error analytical. vs. Natural frequency of pipe for Vel. foam supported in guided slots (m/s) (Hz) Analytical % Error analytical. vs. Natural frequency of pipe for spring supported in guided slots (Hz) Analytical % Error analytical. vs. 1 20.1 14.8 15.5 4.0 20.5 8.0 10.2 22.0 20.1 3.5 5.0 30.8 2 21.2 3.7 5.3 30.0 21.3 21.7 26.1 16.7 21.2 8.0 10.6 24.7 3 23.3 3.7 5.4 31.3 22.3 21.7 20.6 5.2 23.3 8.0 10.6 24.7 4 24.6 14.8 15.7 5.2 23.4 3.4 5.4 35.4 23.6 3.5 5.2 33.3 Acknowledgments The authors gratefully acknowledge the Management of CSIR-Indian Institute of Chemical Technology, Hyderabad, India and College of Engineering, Osmania University, Hyderabad, India for providing the infrastructural facilities. References [1] H. Ashley, G. Haviland, Bending Vibration of a pipeline containing flowing fluid, Transactions of the ASME Journal of Applied Mechanics 72 (1950) 229-232. [2] G.W. Housner, Bending vibration of a pipe line containing flowing fluid, Transactions of the ASME Journal of Applied Mechanics 19 (6) (1952) 205-208 [3] R.H. Long, Jr., Experimental and theoretical study of transverse vibration of a tube containing flowing fluid, Transactions of the ASME Journal of Applied Mechanics 22 (1) (1955) 65-68. [4] S. Naguleswaran, C.J.H. Williams, Lateral vibration of a pipe conveying a fluid, Journal of Mechanical Engineering Science 10 (3) (1968) 228-238. [5] R.A. Stein, M.W. Tobriner, Vibration of pipes containing flowing fluid, Transactions of the ASME Journal of Applied Mechanics 92 (1970) 906-916. [6] S.-S. Chen, Free vibration of a coupled fluid/structural system, Journal of Sound and Vibration 21 (4) (1972) 387-398. [7] D.S. Weaver, T.E. Unny, On the dynamic stability of fluid conveying pipes, Transactions of the ASME Journal of Applied Mechanics 40 (1973) 48-52. [8] M.P. Paidoussis, Dynamic stability of pipes conveying fluid, Journal of Sound and Vibration 33 (3) (1974) 267-294. [9] S. Wang, Y. Liu, W. Huang, Research on solid-liquid coupling dynamics of pipe conveying fluid, Journal of Applied Mathematics and Mechanics 19 (11) (1998) 1065-1071. [10] Y.L. Zhang, D.G. Gorman, J.M. Reese, Analysis of the vibration of pipes conveying fluid, in: Proc. Instn. Mech. Engineers, Vol. 213, Part C, 1999, pp. 849-860. [11] H.R. Oz, H. Boyaci, Transverse vibrations of tensioned pipes conveying fluid with time-dependent velocity, Journal of Sound and Vibration 236 (2) (2000) 259-276.

106 Finite Element Analysis of Piping Vibration with Guided Supports [12] R. Becquet, I. Elishakoff, Class of analytical closed-form polynomial solutions for clamped guided in-homogeneous beams, Chaos, Solitons & Fractals 12 (9) (2001) 1657-1678. [13] D.C. Wiggert, A.S. Tijsseling, Fluid transients and fluid structure interaction in flexible liquid-filled piping, American Society of Mechanical Engineers Journal 54 (5) (2001) 455-481. [14] J.K. Sinha, S. Singh, A Rama Rao, Finite element simulation of dynamic behaviour of an open-ended cantilever pipe conveying fluid, Journal of Sound and Vibration 240 (1) (2001) 189-194. [15] Y.L. Zhang, J.M. Reese, D.G. Gorman, Finite element analysis of the vibratory characteristics of cylindrical shells conveying fluid, Comut. Methods Appl. Mech. Engineering 191 (2002) 5207-5231. [16] Y.-S. Seo, W.-B. Jeong, W.-S. Yoo, Frequency response analysis of cylindrical shells conveying fluid using finite element method, Journal of Mechanical Engineering Science 19 (2) (2005) 625-633. [17] K.R. Chellapilla, H.S. Simha, Critical velocity of fluid conveying pipes resting on two parameter foundation, Journal of Sound and Vibration 302 (2007) 387-397. [18] Y.-M. Huang, Y.-S. Liu, B.-H. Li, Y.-J. Li, Z.-F. Yue, Natural frequency analysis of fluid conveying pipeline with different boundary conditions, Nuclear Engineering and Design 240 (2010) 461-467. [19] A.H. Al-Hilli, T.J, Ntayeesh, Free vibration characteristics of elastically supported pipe conveying fluid, Nahrain University College of Engineering Journal 16 (1) (2013) 9-19.