Piping Flexibility and Stress Analysis

Influence of the spring hanger design on the piping flexibility and stress analysis is considered in the presented paper. Three aspects of this procedure are discussed and illustrated by the numerical examples: - load variation criteria: different interpretations of load variability could lead to different sizes and even types of selected springs. The key issue of this problem is definition of the cold load (theoretical installed load vs. actual cold load); - influence of the different nonlinearities (hanger’s side forces due to short rod length and support’s friction) on the values of calculated piping thermal movements and hence on the type of selected springs; - modeling and interpretation of spring hanger loads within calculation of sustained stresses. Being differently realized in the commercially available programs for piping flexibility analysis these effects are significant for flexible hot pipes and could introduce a big scattering of results.

Pipeline systems serve a crucial role in an effective transport of fluids to the designated location for medium to long span of distances. Owing to its paramount economic significance, pipeline design field have undergone extensive development over the past few years for enhancing the optimization and transport efficiency. This research paper attempts to propose a methodology for flexibility analysis of pipeline systems through employing contemporary computational tools and practices. A methodical procedure is developed, which involves modeling of the selected pipeline system in CAESAR II followed by the insertion of pipe supports and restraints. The specific location and selection of the inserted supports is based on the results derived from the displacement, stress, reaction, and nozzle analysis of the concerned pipeline system. Emphasis is laid on the compliance of the design features to the leading code of pipeline transportation systems for liquid and slurries, ASME B31.4. The ...

An analysis has been made of the behavior of flange joint components under initial stress loading (preclamping) and in service. There have been highlighted the drawbacks in calculating flange joints to be found in currently acknowledged methods that do not take into account:-the influence of the bending moment produced by the total hydrostatic end force on bolt strength and flange rotation;-the presence of cracks in the welding or in its thermal influence zone;-the residual stresses;the relaxation of the sealing gasket over time. The calculation method proposed in the paper takes into account all these particularities of loading. Furthermore, alongside the strength calculation of bolts and flanges, a proposal has been put forth for the calculation of the flange joint leak-tightness.

Quality by design in pharmaceutical manufacturing hinges on computational methods and tools that are capable of accurate quantitative prediction of the design space. This paper investigates Bayesian approaches to design space characterization, which determine a feasibility probability that can be used as a measure of reliability and risk by the practitioner. An adaptation of nested sampling-a Monte Carlo technique introduced to compute Bayesian evidence-is presented. The nested sampling algorithm maintains a given set of live points through regions with increasing probability feasibility until reaching a desired reliability level. It furthermore leverages efficient strategies 1 arXiv:2008.05917v1 [math.OC] 13 Aug 2020 from Bayesian statistics for generating replacement proposals during the search. Features and advantages of this algorithm are demonstrated by means of a simple numerical example and two industrial case studies. It is shown that nested sampling can outperform conventional Monte Carlo sampling and be competitive with flexibility-based optimization techniques in low-dimensional design space problems. Practical aspects of exploiting the sampled design space to reconstruct a feasibility probability map using machine learning techniques are also discussed and illustrated. Finally, the effectiveness of nested sampling is demonstrated on a higher-dimensional problem, in the presence of a complex dynamic model and significant model uncertainty.

This material summarizes the results regarding the analysis of the mechanical stress states for a piping system subjected to impeded thermal expansion. It were considered different load cases corresponding to the primary loads (own weight + internal pressure) and the secondary loads (induced by thermal expansion, including the thermal displacements of the equipment connection points). The stresses values determined for each load cases were compared with the allowable limits evaluated according to ASME B31.3 code, resulting that the analyzed piping system is safe.

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

This paper reviews some existing studies and numerical methods used for flow induced transverse vibrations analysis of flexible pipes. An integral method, based on the use of Green’s functions, already used for different straight beam dynamic analysis is adapted for the proposed subject. This approximate method leads to a matrix formulation and to an eigenvalue problem for free vibration analysis. The presented approach is able to estimate also the critical fluid velocities. Effects of boundary conditions and of elastic foundation characteristics, Coriolis terms and of other parameters on dynamic behavior of a pipe, can be included. Some numerical examples are also presented for comparisons with results obtained by FEM or with other data from literature. They show good agreement.

Process Plant can be operated safely and efficiently with the help of good design of equipments and piping systems connecting to the various equipments like tanks, heat exchangers, pumps etc. Design of piping system includes the pipe and fitting sizing, thickness calculation, equipment layout, pipe routing, support type, support location finalization and stress analysis. This study explains the stress analysis of piping system as per process piping code ASME B 31.3 using 3D software tool CAESAR II. Major requirements in piping stress analysis are to provide adequate flexibility for absorbing thermal expansion, code compliance for stresses incurred in piping system, safe nozzle loads and displacement. The design is said to be safe if all these are in allowable range as per code. In this study, the criterion of selection of piping system for flexibility analysis is explained analytically. The two piping systems are stress analyzed, compared and the effect of flexibility of piping system on nozzle loads and stresses developed are observed. The software output is discussed and safer piping system is identified.

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

Pipeline systems serve a crucial role in an effective transport of fluids to the designated location for medium to long span of distances. Owing to its paramount economic significance, pipeline design field have undergone extensive development over the past few years for enhancing the optimization and transport efficiency. This research paper attempts to propose a methodology for flexibility analysis of pipeline systems through employing contemporary computational tools and practices. A methodical procedure is developed, which involves modeling of the selected pipeline system in CAESAR II followed by the insertion of pipe supports and restraints. The specific location and selection of the inserted supports is based on the results derived from the displacement, stress, reaction, and nozzle analysis of the concerned pipeline system. Emphasis is laid on the compliance of the design features to the leading code of pipeline transportation systems for liquid and slurries, ASME B31.4. The ...

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

Fatigue failure of a hydraulic hose systems, caused by violent vibrations, has become a critical factor creating operational and maintenance cost for the end user of rock drill equipment. Similar behavior is also appearing in, for example, forestry machines. Hoses are used as parts of the energy feeding system in machines such as the ones use for mining and civil construction operations. This work aims to create an understanding of the dynamic behavior of a selected hydraulic hose. The numerical modeling approach selected includes a boundary element method approach in the fluid-elastic analysis of the dynamics of a pressurized hose with conveying fluid. Experimental modal analysis was used to validate the numerical model. Pre-tension and pressure-induced tension were monitored with an in-house-developed strain gauge–based load cell. The analysis and experiments show that a complex coupling, of pure structural bending modes, appears when the hose is subjected to internal flow. Some o...

Design of piping system requires a systematic consideration of various factors as addressed by the codes and standards. This research paper aims to provide a method for flexibility analysis of a selected area of process piping at an industrial plant. Analysis is done for the purpose of accommodating a spare heat exchanger in the process layout. The analysis follows a systematic procedure, with preparation of a tentative model of the system on CAESAR II software followed by insertion of different pipe supports. The selection and location of these supports is based on the results obtained from displacement, stress, reaction and equipment nozzle analysis of the piping system. The design is in accordance with ASME B31.3, which is the standard code for process piping. The proposed method can be adapted for piping configuration of any industrial plant. With the provision of a systematic procedure, the method ensures time saving and efficient flexibility analysis of any piping system.

Site inspections of Hot Reheat Piping system have indentifed cracking at a number locations at butt welds between pipe runs and elbows.Furthermore,it was discovered that pipe runs are instaled with wall thickness of SCH60(ANSI/ASME code[3]),and elbows with wall thickness of SCH30 what is not according to design documentation.This wall thickness discrepancy creates a stress enhancement at the connections between elbows and piping runs.Also,because the elbows are more flexible,there could be a propensity for a creep strains to be concentrated in the elbows as the piping system stresses will be relaxed at these lower strength elbows rather than being distributed throughout the main piping runs.So,a detailed finite element analysis,including elastic and creep analysis, is performed to determine if the secondary stresses during relaxation,or if the stress enhacement of the wall thickness discrepancy,play significant roles in the cracking detected in the piping system.

Since the first commercial operation in 1983, Wolsung unit 1 has experienced several aging phenomena, especially in pressure tubes and feeder pipes. In case of pressure tubes, in-service inspections (ISI) revealed that a major portion of inspected tubes was in contact with calandria tubes. The likelihood of blister formation was a safety concern because it is a potential threat to pressure tube integrity. Wolsung unit 1 was, therefore, subjected to the SLAR (Spacer Location And Reposition) work to separate the contacted pressure tubes from calandria tubes. In this paper, experience of in-service inspection to the pressure tubes will be discussed including the irradiation creep elongation and CIGAR (Channel Inspection and Gauging Apparatus for Reactors) examination of pressure tubes. On the other hand, the problem of wall thinning in the feeder pipes became of great concern since 1996. Inspections, in compliance with CSA N285.4, were conducted in the upper portions of the feeders clo...

The dynamic stability of elastically restrained pipe conveying fluid is investigated in this study. The frequency expression is derived for classical boundary conditions by considering the supports as compliant material with linear and rotational stiffness. A new transcendental frequency equation is developed by using Euler-Bernoulli beam theory; the equation of motion is derived from energy expressions using the Hamilton's Principle. The natural frequencies are presented for a wide range of restraint parameters. Cases are studied for different boundary conditions-Linearly and Rotationally Restrained, Rotational-Linearly & Linearly Restrained, Rotational-Linearly and Fixed, Rotationally Restrained-Guided, are computed and it is noted that as the flow velocity increases the first mode frequency decreases and by varying the mass ratio the frequency increases.

The dynamic stability of elastically restrained pipe conveying fluid is investigated in this study. The frequency expression is derived for classical boundary conditions by considering the supports as compliant material with linear and rotational stiffness. A new transcendental frequency equation is developed by using Euler-Bernoulli beam theory; the equation of motion is derived from energy expressions using the Hamilton's Principle. The natural frequencies are presented for a wide range of restraint parameters. Cases are studied for different boundary conditions-Linearly and Rotationally Restrained, Rotational-Linearly & Linearly Restrained, Rotational-Linearly and Fixed, Rotationally Restrained -Guided, are computed and it is noted that as the flow velocity increases the first mode frequency decreases and by varying the mass ratio the frequency increases.

The dynamics and stability of flow induced vibration of flow conveying in pipes particularly in case of high velocity flow may lead to severe damage. Predicting the circular natural frequencies and critical fluid velocities is an important tool in design and prevent system failures. In this study transverse dynamic response of simply supported pipe with variable tubular cross sectional area carrying fluid with a constant flow rate is investigated. Euler Bernoulli's beam theory is used to model the pipe. Hamilton's principle will be used to produce the governing equation of motion for the system. The resulting partial differential equation is solved using Galerkin's technique. The impact of the flow velocity and non-uniform variable cross section on the natural frequencies of the system, critical flow velocity and system stability is presented.

This paper presents a numerical change in the mechanical vibrations of conditions. Finite Element Analysis models with their attached inlet and outlet feeder pipes noise forcing function and the corresponding to compare the vibration response with the support point (Off-bearing). Two types of excitations are considered; f fitting excitations. The finite element calculate the ratio between the feeder vibration journal ring support. It is observed that levels is 20% with the feeder bend excitation the end-fitting excitation. It is also observed that the bends excitation did not significantly change excitation results showed a reduction in the calculated INTRODUCTION Fuel channels (FC) on typical rings to allow for axial creep elongation reactor faces to control the axial position of the fuel channels and feeders model showing the end the support points on the east and west end locked using a welded stop collar allowing the fuel channel to elongate axially in the east dire prevent east...

The study presents a numerical analysis of the impact of creep in thermoplastic stub ends on the flange clamping force of a thermoplastic pipe flange connection. The analysis uses available creep modulus values that depend on the stress level, temperature and time. A method that takes the creep modulus into account is proposed and applied to four different detailed variants with the aid of the finite element method. The calculations were performed for selected times at a constant temperature and with static load. The obtained results show that the influence of creep in thermoplastic stub ends causes a substantial reduction in flange clamping force in less than one year.

Functionally gradient materials and small-scale pipes have great importance in the industry because of its wide applications in many engineering fields such as fluid transport in fluidic devices. In this article, an analytical solution for the stability of functionally graded (FG) material micro-pipe fluid-conveying is offered. The properties of the material are changed constantly across the micro-pipes thickness and depend on power law distribution. Hamilton's principle, Euler beam model, and modified coupled stress theory are utilized to get an equation of motion for conservative FG micro-pipe (Simply supported and clamped-clamped). The vibration equation of FG micro-pipe is converted from 4th order to a 2nd order using the Galerkin's method. Solving the Eigenvalue problem results in a distinct equation depicting the frequency relationship of the parameters. By plotting the root locus of the characteristic equations, the main stability characteristics such as stability, instability, and flutter are studied graphically. The effect of different parameters such as gradient index, length scale, fluid velocity, micro flow parameter, and mass ratio on FG micro-pipe was studied. The results showed that each parameter had a significant impact on the behavior of stability as the chain of stability could change significantly. 1. Introduction Pipes convey fluid have wide applications in engineering and industrial fields. More recently, this topic has been developed in the study and analysis of Fluid-Structure Interaction (FSI) which has great applications in aerodynamics, aerospace, medical engineering, ship movement, etc. Pipes fluid-conveying are classified as gyroscopic system according to a gyroscopic effect due to the relative rotational movement of the fluid element, which vibrates sideways to adapt to the movement of the tubes. In general, pipe fluid-conveying systems operate at a relatively low (pre-critical) velocity. At some critical speeds, the pipes may lose their stability either by oscillating the growth of the ace (flutter) or by the static convergence (buckle). When more high-velocity (post-critical) pipes are made in a different way, they can still stabilize in the same position (flutter or buckle), restore stability or change their instability. The assessment of the stability zones at the parameters of the different pipes is important in the design of such systems to ensure the desirability of safe operation of specific parameters for liquids and pipes. Simply support, clamped and clamped-pinned pipes belong to this category. Several researchers have analyzed and studied their persistence in the pipe fluid-conveying such as Weaver and Unny [1], Bishop et al [2] and Si-Ung et al [3]. Their investigations focused on assessing the critical fluid velocity of instability and stability properties for simply supported, clamped and clamped-pinned pipes.

This paper presents vibration analysis and mathematical model using Euler-Bernoulli and Hamilton's energy expressions for fluid conveying welded Galvanized Iron pipe with clamped-clamped boundary condition. A 3D CAD Model was developed in NX-ideas for empty pipe, pipe with fluid flow and welded pipe with fluid flow. Vibration analysis was performed on developed models for 3 mode shapes to generate frequency data using ABAQUS. The developed models were imported in to ABAQUS for generating frequency data. For the welded pipe, effects of weldment at girth welds on the vibration characteristics and stability of pipe was investigated. Results acquired from ABAQUS were compared with theoretical results of natural frequencies and observed the variation of 3% error.

A new method is proposed for calculating the aircraft equivalent single vulnerable area using Monte-Carlo simulation technique. The method simulates the kill events of single vulnerable components and multiply vulnerable components or redundant components to 'Model of Filling Boxes with Balls', in which the area of each box is equal to the component's vulnerable area. By randomly and uniformly sampling the threat hit locations, the expected number of hits required to kill an aircraft can be given and the aircraft equivalent singly vulnerable area can be attained. Examples demonstrate the correctness and feasibility of the proposed method. Analysis show that the developed method overcomes two shortcomings of currently used methods.

This paper aims to investigate the relationship between maximum stresses produced and cycle life of different shaped bellows expansion joints. Flexible pipe bellows which have been selected in the present study are made of stainless (STS 304) and (STS 316) material based on two types of (axial, and axial with control ring) with three shapes of bellow section (U-shape, Ω-shape and disc-shape). Our calculation is done using simulation model written by MATLAB. The simulation results show that, U-shaped bellow has smaller internal pressure-induced stress, longer fatigue life, and is more suited for higher internal pressure situations. The most essential bellow design factor is the correct specification of the bellow movement requirements.

T he Significance of, an d Suggested L im its for, the S tress in P ipe Lines D u e to th e C o m ­ bined Effects of P re ssu re a n d E x p an sio n By D. B. ROSSHEIM1 and A. R. C. MARKL,2 NEW YORK, N. Y. T h is p a p er h a s b e e n p r e p a r ed a s a v e h ic le fo r d is c u s s io n o f t h e fo llo w in g b a s ic p r o b le m s , o n w h ic h a g r e e m e n t m u s t be r e a c h e d t o e s t a b lis h a s a tis f a c to r y w o r k in g-s tr e s s b a s is for p ip e l i n e s : 1 P ro p er a llo w a b le s t r e s s e s fo r c o m b in e d p r e ssu r e a n d e x p a n s io n e ffe c ts. 2 I n flu e n c e o f a n d l i m i t s for lo c a liz e d s tr e s s e s u n d e r s t a t i c a n d r e p e a te d lo a d in g. 3 C a p a c ity o f b o lte d j o i n t s t o w it h s t a n d e x p a n s io n e ffe c ts w it h o u t le a k a g e o r d a m a g e t o fla n g e s, b o lt s , or g a s k e ts. 4 E ffect o f p r e sp r in g in g , s e lf-s p r in g in g , c r e e p , a n d y ie ld in g o n o p e r a tin g a n d o f f-s t r e a m s t r e s s e s. A n a t t e m p t h a s b e e n m a d e t o p o in t o u t v a r io u s a s p e c t s o f e a c h is s u e , r a th e r t h a n l im i t t h e p r e s e n t a t io n t o t h e p e r so n a l v iew s o f t h e a u t h o r s. D URING the last decade, economic considerations and new processes in the power, oil-refinery, and chemical industries have produced a trend toward large-scale units and high operating temperatures and pressures. With the attendant in­ crease in line sizes and wall thicknesses, the subdivision of pipe lines into convenient runs with expansion bends soon proved entirely inadequate; instead, today most piping for severe service is carefully analyzed for forces and stresses, full advan­ tage being taken of the inherent flexibility by minimizing the num­ ber of anchors, guides, or other restraints. A natural consequence of this improved accuracy in evaluating thermal effects is the necessity for a review of stress limitations as they apply to ex­ pansion stresses alone and in combination with internal pressure. This problem has received the active consideration of Subgroup No. 3 on Expansion and Flexibility, Subcommittee No. 8 on Fabrication Details of the Code for Pressure Piping (ASA B31) who, functioning with the Applied Mechanics and Power Divi­ sions of the A.S.M.E., have sponsored this symposium. This paper has been prepared as a vehicle for discussion of the following basic problems, on which agreement must be reached to establish a statisfactory working-stress basis for pipe lines. An attempt has been made to point out various aspects of each issue, rather than limit the presentation to the personal views of the authors. 1 Proper allowable stresses for combined pressure and expan­ sion effects. 2 Influence of and limits for localized stresses under static and repeated loading, such as in bends and corrugated pipe. 3 Capacity of bolted joints to withstand expansion effects without leakage or damage to flanges, bolts, or gaskets. 1 M echanical Engineer, M. W. K ellogg Co. M em. A .S .M .E. 2 A ssistant M echanical Engineer, M. W. K ellogg Co. Contributed b y the Pow er D ivision and presented at the Annual M eeting, Philadelphia, P a., Decem ber 4-8 , 1939, of T h e A m e r i c a n S o c i e t y o f M e c h a n i c a l E n g i n e e r s. N o t e : S tatem en ts and opinions advanced in papers are to be under­ stood as individual expressions of their authors, and n ot those of the Society. 4 Effect of prespringing, self-springing, creep, and yielding on operating and off-stream stresses R esume of Simple Stresses I nvolved A pipe line is essentially a pressure vessel, the internal pressure causing a radial stress at the inner face which is converted into circumferential stress on the way through the wall, leaving zero radial stress at the outer surface; at any point the sum of the radial and hoop stresses varies inversely as the radius, while their difference is the same at all points. These are the laws of Lam<5 which involve the assumption of a uniform or zero longitudinal stress. Following this reasoning, in a closed cylinder the longi­ tudinal stress equals the product of the pressure and the internal area divided by the cross-sectional area of the wall, and has the same value throughout the thickness. While most runs of piping are not technically closed cylinders, the same longitudinal stresses occur due to pressure effects on the projections of elbows, there being a complete absence of longitudinal pressure stress only in straight runs between vessels of infinite rigidity and such including frictionless expansion joints. Under temperature changes with free expansion no stresses are introduced. However, ordinarily, expansion is restrained by the equipment to which the pipe line is attached, as well as anchors, guides, solid hangers, or supports. In calculating a line for expansion, the ends are commonly assumed completely fixed; this connotes three forces and an equal number of moments in the case of a problem in space, resulting in longitudinal bending stresses in two mutually perpendicular planes and torsional stress about the pipe axis, as well as two shears and a normal stress. With the assumption of hinged ends, the end moments reduce to zero, a condition which would be fully realized only in a frictionless ball-and-socket joint. Ac­ tually, an intermediate condition of restraint will prevail in practice due to sympathetic deflections, rotations, or distortions of the equipment to which the pipe is attached. On the other hand, cases will arise where external movements of the ends tend to increase rather than decrease their degree of fixity; in addition, guides, solid or spring supports, and hangers often inhibit free deformation of the line and thus add to the stresses. To avoid additional complexity, the weight of the piping is commonly neglected in flexibility calculations. In heavy pipe lines spring hangers are often used to balance the dead-load effects when the line is at working temperature, and solid hangers can be similarly employed in neutral locations. Properly de­ signed supports appear to offer the most suitable means for handling this effect, which can then be disregarded in the stress analysis of most lines. Where piping is not insulated, the heat loss from the exposed surface produces a temperature drop through the metal thick­ ness causing longitudinal and circumferential stresses of equal magnitude which may be evaluated by the formulas of Lorenz (l).3 Since the outside surface is relatively colder, tensile 3 N um bers in parentheses refer to the Bibliography at the end of the paper.

ABSTRACT
Pipes are widely used for transfer of fluids from one point to another and are susceptible to vibrations. Flow-induced vibration analysis of pipes conveying fluid is thus important in structural dynamics. It is well known that pipeline systems may undergo divergence and flutter type of instability due to fluid-structure interaction. Over the last sixty years, extensive studies have been carried out on dynamic analysis of piping systems subjected to different boundary conditions and loadings.

Notable contributions in this area include the works of Chen et al and Paidoussis et al. 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 equation is obtained by considering many numerical methods such as transfer matrix, finite element, perturbation, Runge-Kutta and differential quadrature.

The paper presents the work carried out in the development of a mathematical model to study the vibrations and stability of fluid conveying pipe. It includes validation of the model by conducting experiments in a pipe test facility. Also, a computational simulation of the piping is done by using ABAQUS software.

The experimental test setup consists of piping, centrifugal pump / motor, water meter, globe type angle valve, clamps, linear spring supports and civil foundation structure. Experimental study and analysis of flow induced vibration is to be carried out by studying the effect of turbulence of fluid flow in the piping system. Mathematical model of elastically restrained pipe carrying fluid has been developed. Governing differential equation of motion for pipe conveying fluid of real system has been derived. The critical fluid velocity has been calculated by analytical approach.

The mathematical model is derived by using Euler-Bernoulli beam theory and Hamilton’s energy expressions to get the equation of motion for transverse vibration of fluid conveying pipe. The exact solution of the differential equation is expressed as a polynomial function. The boundary conditions are substituted in fourth order derivatives of the polynomial function and by using the Gaussian elimination rule the transcendental frequency equation is obtained by Mueller's Bisection method. The natural frequencies for the first mode of vibration are computed by using the Muller’s Bi-section numerical method. A FORTRAN computer program is developed for estimation of the frequencies in non-dimensional form. The vibration structure of pipe is modelled by using I-DEAS software as per specifications and the effect of turbulence of fluid flow in the piping element is analysed by using Abaqus Software. Dynamic behaviour of fluid flow is studied. Modal analysis has been carried out and found the natural frequency of 4 mode shapes. Experimental studies/flow trials have been carried out to validate the analytical results.

The dynamic stability of elastically restrained
pipe conveying fluid is investigated in this study. The
frequency expression is derived for classical boundary
conditions by considering the supports as compliant material
with linear and rotational stiffness. A new transcendental
frequency equation is developed by using Euler-Bernoulli
beam theory; the equation of motion is derived from energy
expressions using the Hamilton's Principle. The natural
frequencies are presented for a wide range of restraint
parameters. Cases are studied for different boundary
conditions- Linearly and Rotationally Restrained, Rotational-
Linearly & Linearly Restrained, Rotational-Linearly and
Fixed, Rotationally Restrained –Guided, are computed and it
is noted that as the flow velocity increases the first mode
frequency decreases and by varying the mass ratio the
frequency increases.