Showing papers in "Journal of Pressure Vessel Technology-transactions of The Asme in 1998"

Strain-rate-dependent constitutive equations for concrete

Abstract: This paper summarizes the results of a comprehensive experimental study to quantify the effects of strain rate on concrete compressive and tensile strengths. Direct compression and splitting tensile tests were conducted at quasi-static rates (between 10{sup {minus}7}/s and 10{sup {minus}5}/s) in a standard MTS machine to establish the static properties. These same tests were conducted at high strain rates (between 10{sup {minus}1}/s and 10{sup 3}/s) on a split-Hopkinson pressure bar (SHPB) to determine the dynamic material properties. A statistical analysis was performed on the data and strain-rate-dependent constitutive equations, both for compression and tension, were developed. These constitutive equations were subsequently employed to modify an existing quasi-static, nonlinear concrete material model.

Limit Loads for Pipe Elbows With Internal Pressure Under In-Plane Closing Bending Moments

Abstract: The purpose of this study is to determine limit loads for pipe elbows subjected to in-plane bending moments that tend to close the elbow (i.e., decrease its radius of curvature), and the influence of internal pressure on the value of the limit load. Load-deflection curves were obtained, and from these curves plastic collapse or instability loads at various values of internal pressure were determined. This was done for different pipe bend factors (h = Rt/r 2 ) using the nonlinear finite element analysis code (ABAQUS) with its special elbow element. The limit load was found to increase and then decrease with increasing pressure for all the elbow geometries studied.

Penetration and Perforation of Composite Sandwich Panels by Hemispherical and Conical Projectiles

Abstract: The results of penetration and perforation tests carried out on composite sandwich panels with GRP skins and PVC foam cores using hemispherical-ended and conical-nosed indenters/projectiles under quasi-static, drop-weight, and ballistic impact conditions, with impact velocities up to 305 m/s, are described. Load-displacement characteristics under quasi-static loading are presented and the ballistic limits as well as perforation energies are determined. A classification of the sandwich panel responses based on the panel thickness-to-projectile diameter ratio is deduced. General empirical formulas that predict the dynamic perforation energies for fiber-reinforced plastic (FRP) laminates and composite sandwich panels loaded by hemispherical-ended projectiles are derived. The empirical equations correlate well with available experimental data. It is shown that, to a first approximation, the formulas obtained for hemispherical-ended projectiles are also applicable to conical-nosed projectiles.

Fatigue Growth Prediction of Internal Surface Cracks in Pressure Vessels

Abstract: Fatigue crack growth was numerically simulated for various internal surface cracks with initially either semi-elliptical or irregular crack fronts. The simulation was directly based on a series of three-dimensional finite element analyses from which the stress intensity factors along the front of growing cracks were estimated. The fatigue crack growth law obtained from small laboratory specimens was incrementally integrated at a set of points along the crack front, and a new crack front was then re-established according to the local advances at this set of points by using a cubic spline curve. This method enabled the crack shape to be predicted without having to make the usual assumption of semi-elliptical shape. Fatigue analysis results are presented and discussed for fatigue shape developments and deviations from the semi-elliptical shape, aspect ratio changes, stress intensity factor variations during crack growth, and fatigue life predictions. Some of the results were also compared with those obtained by two simplified methods based on one and two degree-of-freedom models, respectively.

Estimation of residual stresses in hydraulically expanded tube-to-tubesheet joints

Abstract: The knowledge of residual stresses introduced in the tubes of heat exchangers during their expansion in the tubesheet holes is important because of their effect on the structural integrity of components. This paper presents a simplified theoretical method to calculate the maximum residual stresses introduced in the transition zone of expanded tube-to-tubesheet joint. The higher positive values of tensile residual stresses and their corresponding axial locations are determined by using a standard deviation analysis. The validation of the proposed equations was accomplished by comparing their results to those obtained by the finite element method for some arbitrary cases. An upper limit has been imposed on the expansion pressure level, depending on the combination of the geometrical and material parameters that are involved in the design of the tube-to-tubesheet joints.

Effects of Repair Weld Residual Stresses on Wide-Panel Specimens Loaded in Tension

Abstract: As a part of the welding fabrication procedure development for the next generation space shuttle external tank, aluminum-lithium wide-panel specimens were used to assess the interactions between repair weld residual stresses and external loading conditions. The detailed residual stress development in the wide panel specimens with a repair weld was analyzed using an advanced finite element procedure. External tension loading effects were then incorporated in the residual stress model to study the interactions between the residual stress field and external tensile loading. Wide-panel tensile tests were also performed to extract photo strain and strain-gage results. A good agreement between the finite element and experimental results was obtained. The results demonstrate that the presence of high tensile residual stresses within a repair weld has a drastic impact on the stress/strain distribution in the wide panel specimens subjected to external loading. Its implications on structural integrity are discussed in light of the wide-panel results. The effects of post-welding mechanical treatment such as planishing were also examined.

Tensile Plastic Instability of Axisymmetric Pressure Vessels

Abstract: This paper examines the calculated pressure at a tensile plastic instability of a pressure vessel and its relationship to burst test results. It is proposed that the instability pressure be accepted as an upper bound to the pressure at which a vessel bursts, and that a strength reduction factor be used to predict the burst. The paper also presents a suitable mathematical model for the calculation of the instability pressures for thin-walled axisymmetric vessels. The proposition is tested by applying the model to a pressurized diaphragm, four cylindrical shells, and two torispherical heads, for which experimental burst data are available. It is found that the ratio of the test burst pressure to the calculated pressure at the tensile plastic instability, expressed in percent, ranges from 71 to 96 percent. The highest ratio occurs for a pressurized diaphragm with no significant defects. The lowest ratios occur for cylindrical shells with longitudinal welds, suggesting that the presence of the welds had a detrimental effect on the burst strength. These results may be useful when designing a pressure vessel with respect to its ultimate strength.

Effects of LWR Coolant Environments on Fatigue Lives of Austenitic Stainless Steels

Abstract: Fatigue tests have been conducted on Types 304 and 316NG stainless steels to evaluate the effects of various material and loading variables, e.g., steel type, strain rate, dissolved oxygen (DO) in water, and strain range, on the fatigue lives of these steels. The results confirm significant decreases in fatigue life in water. Unlike the situation with ferritic steels, environmental effects on Types 304 and 316NG stainless steel are more pronounced in low-DO than in high-DO water. Experimental results have been compared with estimates of fatigue life based on a statistical model. The formation and growth of fatigue cracks in air and water environments are discussed.

Accelerated Stress Rupture Testing for Creep Life Prediction—Its Value and Limitations

Abstract: Accelerated stress rupture testing has become a common method for determination of the remaining life of in-service components subject to creep damage at elevated temperatures. Stress and temperature increases have both been used to cause accelerated test failures, although the temperature accelerated tests have been preferred during the last decade. Remaining life estimation schemes have essentially involved extrapolation of results of the accelerated tests to the operating conditions. The accelerated test approach has found such widespread acceptance that most users today are unaware of its limitations and have started using the results in a definitive and quantitative way as a direct prediction of the remaining life of the component. EPRI (Electric Power Research Institute) investigators have examined the application and validity of a number of aspects of accelerated testing, as well as the underlying assumptions in the extrapolation procedures. This paper reviews the current practice, describes results from a number of research activities, and provides general guidelines for accelerated rupture testing.

Vibration damping in multispan heat exchanger tubes

Abstract: Heat exchanger tubes can be damaged or fail if subjected to excessive flow-induced vibration, either from fatigue or fretting-wear. Good heat exchanger design requires that the designer understands and accounts for the vibration mechanisms that might occur, such as vortex shedding, turbulent excitation or fluidelastic instability. To incorporate these phenomena into a flow-induced vibration analysis of a heat exchanger requires information about damping. Damping in multispan heat exchanger tubes largely consists of three components: viscous damping along the tube, and friction and squeeze-film damping at the supports. Unlike viscous damping, squeeze-film damping and friction damage are poorly understood and difficult to measure. In addition, the effect of temperature-dependent fluid viscosity on tube damping has not been verified. To investigate these problems, a single vertical heat exchanger tube with multiple spans was excited by random vibration. Tests were conducted in air and in water at three different temperatures (25, 60, and 90 C). At room temperature, tests were carried out at five different preloads. Frequency response spectra and resonant peak-fitted damping ratios were calculated for all tests. Energy dissipation rates at the supports and the rate of excitation energy input were also measured. Results indicate that damping does not change overmore » the range of temperatures tested and friction damping is very dependent on preload.« less

Fretting Wear Damage of Heat Exchanger Tubes: A Proposed Damage Criterion Based on Tube Vibration Response

Abstract: A simple criterion is proposed to estimate fretting wear damage in heat exchanger tubes with clearance supports. The criterion is based on parameters such as vibration frequency, midspan vibration amplitude, span length, tube mass, and an empirical wear coefficient. It is generally accepted that fretting wear damage is proportional to a parameter called work rate. Work rate is a measure of the dynamic interaction between a vibrating tube and its supports. Due to the complexity of the impact-sliding behavior at the clearance supports, work rate calculations for heat exchanger tubes require specialized nonlinear finite element codes. These codes include contact models for various clearance support geometries. Such nonlinear finite element analyses are complex, expensive and time consuming. The proposed criterion uses the results of linear vibration analysis (i.e., vibration frequency and mid-span vibration amplitude due to turbulence) and does not require a nonlinear analysis. It can be used by nonspecialists for a quick evaluation of the expected work rate, and hence, the fretting wear damage of heat exchanger tubes. The proposed criterion was obtained from an extensive parametric study that was conducted using a nonlinear finite element program. It is shown that, by using the proposed work rate criteria, work rate can be estimated within a factor of two. This result, however, requires further testing with more complicated flow patterns.

A Reliability Assessment Method in Strain-Based Fatigue Life Analysis

Abstract: For the purpose of fatigue reliability assessment based on strain-life analysis, a family of reliability defined {var_epsilon}-N{sub f} curves, called R-{var_epsilon}-N{sub f} curves, is constructed by considering the interference model of fatigue strain capacity and applied strain history. The main effort of this work is to define reliability factors which are used to modify the conventional {var_epsilon}-N{sub f} curve into a family of R-{var_epsilon}-N{sub f} curves. A major contribution of this paper is to define two unique reliability factors, one for elastic-strain-life relation and the other for plastic-strain-life relation, for a certain reliability by using an empirical {var_epsilon}-N{sub f} curve. A numerical example is presented to demonstrate the method.

A sensitive ultrasonic approach to NDE of tightly closed small cracks

Abstract: A sensitive ultrasonic technique for the quantitative nondestructive evaluation of small closed cracks is developed. The use of an oblique longitudinal wave with small angle of incidence upon the specimen surface is emphasized. Firstly, ultrasonic testing is performed with open cracks in order to predict the optimum angle of evaluation, and an empirical calibration equation is then derived for the same as a basis to deal with the analysis of closed cracks. From the measurement on real fatigue cracks, the parameters of interest, specifically the crack size and the closure stress, are determined simultaneously by analyzing the inverse problem The method developed here is fully automated and computer-controuea. Evaluated results are found to be in good conformity with the destructive measurements, and the corresponding superiority of the method is realized, especially for tightly closed small fatigue cracks, when compared with the method using a beam of longitudinal wave incident normally and/or with a large angle.

Experimental Data on Seismic Response of Piping Components

Abstract: Experimental data from the EPRI Piping and Fitting Dynamic Reliability Program are evaluated. High-amplitud, eismic time history loads were applied to 32 piping components in a cantilevered configuration. Components included elbow tees, reducers, a reinforced fabricated tee, a nozzle, and lugs. The test levels were about 2 to 8 times Level D. The seismic capability of elbows is shown to be remarkable. The seismic performance of lugs is judged to be poor. The tests demonstrate that collapse is a potential failure mode, particularly in low-frequency systems, The tests also demonstrate that a fatigue failure in a single, high-level, seismic event is possible. Of the 15 nonelbow tests that failed by fatigue, six (40 percent) failed during the first high-level seismic test. Design and fabrication details are crucial to seismic performance. Trends in the data need to be assessed to determine appropriate Section III stress limits for seismic loads. In particular, the mpact of component frequency on seismic capability has to be quantified.

NDE of a 3-D Surface Crack Using Closely Coupled Probes for DCPD Technique

Abstract: A procedure of applying the d-c potential drop technique using the closely coupled probes to NDE of a 3-D surface crack is newly developed. The calibration equation for three sensors which differ in the distance between the probes is derived. Experiments validated the use of the calibration equation for the NDE of cracks. The method to use the three sensors properly based on the measuring sensitivity is shown.

J-R curves from circumferentially through-wall-cracked pipe tests subjected to combined bending and tension : Part II : Experimental and analytical validation

Abstract: In Part I (Miura and Wilkowski, 1998) of this paper, the theory of the two η-factor solutions for circumferentially through-wall-cracked pipes subjected to combined bending and tension due to internal pressure was presented. These solutions seemed to give reasonable predictions by comparing with the existing simplified J -estimation scheme. It was also ascertained that the J would be underestimated if the effect of the internal pressure was not properly considered. Consequently, this paper presents the application of these solutions to full-scale pipe tests. The tests were performed at 288°C (550°F) under combined bending and internal pressure. The materials used for the tests were both carbon steel and stainless steel. The effect of combined loading on the J-R curves was determined and compared to C(T) specimen J-R curves. The solutions were then verified by using three-dimensional finite element analysis.

Flow-Induced Wear in Steam Generator Tubes—Prediction Versus Operational Experience

Abstract: Many nuclear steam generators have accumulated more than 10 effective-full-power-years of operation. Eddy-current inspections revealed that a number of these steam generator tubes, notably those located in high local cross-flow regions, have indications of wear at some support plate elevations after 5 to 10 yr of effective-full-power operations. In the last 5 yr, a number of technical papers on nonlinear tube bundle dynamics has been published to address the effect of tube and support plate interactions. At the same time, test data relating wear and tube wall thickness losses for different material combinations and different support plate geometries became available. Based on the available data in the literature, as well as data obtained in the author's affiliation, this paper assesses the cumulative tube wall wear after 5, 10, and 15 effective-full-power years of operation of a typical commercial nuclear steam generator, using different wear models. It is hoped that this study will shed some light on the probable mechanism that caused the observed wear in today's operating nuclear steam generators.

Mechanical Behaviors of Bolted Joint in Various Clamping Configurations

Abstract: In a bolted joint, failures usually initiate at the first root of the bolt thread. However, rupture around the bolt head is sometimes reported for a tap bolt because of high stresses produced by tightening torque applied to the bolt head. It is also well known that manufacturing errors of internal threads in a tapped hole are generally much larger than those of external threads, thus leading to the failures concerned. In this paper, mechanical behaviors of bolted joints in various clamping configurations are analyzed using FEM as multi-body elastic contact problem, and the effects of nominal diameter, friction and pitch error upon stress concentrations are evaluated for through bolts, studs, and tap bolts. It is then quantitatively estimated on the effectiveness of “recessed internal threads” for reducing the stress concentration occurred around the far end of bolt hole. In addition, the tightening process and strength of a bottoming stud, which have seldom been studied despite favorable performance in preventing stress concentration at the runout of threads, are also investigated.

Cracks Emanating From an Erosion in a Pressurized Autofrettaged Thick-Walled Cylinder—Part I: Semi-Circular and Arc Erosions

Abstract: Erosion geometry effects on the mode I stress intensity factor (SIF) for a crack emanating from the erosion’s deepest point in an autofrettaged, pressurized, thick-walled cylinder are investigated. The problem is solved via the FEM method and knowledge of the asymptotic behavior of short cracks. Autofrettage, based on von Mises yield criterion, is simulated by thermal loading and SIFs are determined by the nodal displacement method. SIFs are evaluated for a variety of relative crack lengths, a0 /W = 0.01 – 0.45, emanating from the tip of erosions of different geometries. In Part I of this paper, two configurations are considered: (a ) semi-circular erosions of relative depths of 5 percent of the cylinder’s wall thickness, W ; and (b ) arc erosions for several dimensionless radii of curvature, r′/W = 0.05 – 0.4. While deep cracks are almost unaffected by the erosion, the effective SIF for relatively short cracks is found to be significantly enhanced by the presence and geometry of the erosion and might reduce the vessel’s fatigue life.

A Unified 3-D Homogenization Model of Beam Bundle in Fluid

Abstract: A 3-D homogenization model is developed to predict the overall dynamic property of a beam bundle immersed in an acoustic fluid. It is shown that the existing two models, given by Benner and Schumann (1981) and Hammami and Brochard (1991), respectively, are its 2-D application.

Experimental and Numerical Study of Multi-Pass Welding Process of Pipe-Flange Joints

Abstract: Residual deformations of pipe-flange joints due to thermal strains caused by the multi-pass welding procedure in a manufacturing process are investigated both experimentally and numerically (FEM). To avoid any risk of leakage between two mounted pipes using flange joints, geometrical distortions (twisting) of the flange must be prevented. How the flange is distorted depends both on the groove shape and the weld sequence used. However, to be spared from machine work after the welding process, the distortions have to be minimized. A casting-technique has been developed in order to measure the twisting of the flange after each weld pass. Also, measurements of the residual radial deflection of the pipe have been performed.

J-R Curves From Circumferentially Through-Wall-Cracked Pipe Tests Subjected to Combined Bending and Tension—Part I: Theory and Numerical Simulation

Abstract: One of the key aspects in leak-before-break analyses is to predict the maximum load-carrying capacity of a circumferentially cracked pipe uch analyses require the fracture resistance of the material using the J-integral parameter, typically using small-scale laboratory specimens, such as compact tension, C(T), or three-point bend specimens. To evaluate the similitude between the laboratory specimens and a circumferentially through-wall-cracked pipe, the toughness can be evaluated directly from the pipe using an analysis typically called an η-factor approach. The fracture resistance from the pipe tests can then be compared to laboratory specimen toughness values to assess similitude issues. Additionally, several analysis methods (i.e., LBB.NRC, LBB.ENG, LBB.GE, etc.) that predict maximum load capability of through-wall-cracked pipes have η-factor analyses embedded in them. Hence, the evaluation of the J-R curve accuracy or consistency with small-scale specimens is a verification of one step in such predictive analyses. This paper presents extensions to the earlier η-facto r solutions for circumferentially through-wall-cracked pipes where the previous analyses were for cracks in pipes under either pure bending or pure tension. The improvements investigated account for loading under combined bending and tension due to internal pressure. The application of these methods to full-scale pipe tests is presented in Part II (Miura and Wilkowski, 1998) of this paper.

A Simple Upper-Bound Method for Calculating Approximate Shakedown Loads

Abstract: A simple approach for calculating upper-bound shakedown loads is described. The method is based on a series of iterative elastic finite element analyses (the elastic compensation procedure) applied to Koiter's upper-bound shakedown theorem. The method is demonstrated for a typical pressure vessel application; an axisymmetric nozzle in a spherical shell. Several geometrical configurations are investigated. The calculated upper-bound shakedown loads are compared with lower-bound results obtained by the authors, simple shakedown criteria, and various results given in the literature.

Experimental Study on Fatigue Strength of Small-Diameter Socket-Welded Pipe Joints

Abstract: The authors conducted fully reversed four-point bending fatigue tests on socket-welded joints 20 to 50 mm in nominal diameter, and rotating bending fatigue tests on socket-welded joints 20 mm in nominal diameter. S-N curves for 33 series of different types of specimens were obtained. Examination was made of the effects of various parameters listed in the forthcoming on fatigue strength such as steel types (carbon and stainless steels), diameter, pipe thickness (Sch), fillet shape, slip-on gap, and root defects. Bending fatigue test results indicated fatigue strength for socket-welded joints to be less for longer life regions than reported in the literature by Markl and George (1950). Fatigue strength for socket joints of 50 mm nominal diameter at 10 7 cycles of fatigue life was 46 MPa for carbon steel and 60 MPa for stainless steel with nominal bending stress on the pipe surface. Cracks generally originated from the toe when stress amplitude was high with shorter fatigue life and from the root when amplitude was small with longer life. Fatigue strength was greater for smaller diameter, larger Sch (thicker pipe wall), final welding pass on the toe of pipe side, and in the absence of a slip-on gap. From fatigue test results of socket joints with weld defects at the roots, an empirical equation for the relation of defect size with decrease in fatigue strength was established. Fatigue strength was found to decrease to 60 percent the original level for defect size 25 percent of leg length.

An Improved Split-Ring Method for Measuring the Level of Autofrettage in Thick-Walled Cylinders

Abstract: A simple, yet improved, experimental method for measuring the level of autofrettage in thick-walled cylinders, such as gun barrels, is proposed. A representative ring cut from the barrel serves as the test specimen. The ring is split by cutting it radially, while measuring the released hoop strain at the inner and outer surfaces diametrically opposite from the split line. The opening angle resulting from the spring apart from the ring is also monitored. An analysis based on Hill's residual stress field yields an approximate relation which readily enables the determination of the prevailing level of autofrettage from strain measurements as well as from the value of the opening angle. This relation is found to be practically universal for all relevant cylinder configurations b/a = 1.6-2.2 and all levels of autofrettage of up to Φ = 100 percent.

Cracks Emanating From an Erosion in a Pressurized Autofrettaged Thick-Walled Cylinder—Part II: Erosion Depth and Ellipticity Effects

Abstract: In Part I of this paper, the effects of constant depth erosion on the mode I stress intensity, factor ( SIF) were determined for a crack emanating from the erosion deepest point in a pressurized, autofrettaged, thick-walled cylinder. The erosion geometries investigated included semi-circular erosions and several a erosions of ari radii of curvature. Due to the trends found in that portion of the study, erosion depth and ellipticity are believed to have equally important impact on the SIFs. The present paper delves further into these two parameters using the following configurations. (a) semi-circular erosions of relative depths of 1-10 percent of the cylinder's wall thickness, W; and (b) semi-elliptical erosions with ellipticities of d/h = 0.3 - 2.0. Deep cracks are found to be practically unaffected by the erosion, similar to the results presented in Part I of the paper. The effective SIF for relatively short cracks is found to be dramatically enhanced by the stress concentration factor (SCF), which encompasses the depth of the erosion as well as its radius of curvature at the tip. As a result of the increased effective SIF, a significant decrease in the vessel's fatigue life of up to an order of magnitude may occur.

On Generation of Ultra-High Pressure by Converging of Underwater Shock Waves

Abstract: The characteristics of a new assembly for the shock consolidation of difficult-to-consolidate powders, such as inter-metallic compounds or ceramic materials, were investigated by both the experimental method and numerical simulation method. The assembly consists of an explosive container, a water chamber, and a powder container. Once the explosive is detonated, a detonation wave occurs and propagates, and then impinges on the water surface of the water chamber. After that, there occurs immediately an underwater shock wave in the water chamber. The underwater shock wave interacts with the wall of the chamber during its propagation so that its strength is increased by the converging effect. We used the usual shadow graph system to photograph the interaction process between detonation wave and water. We also used a Manganin piezoresistance gage to measure the converged pressure of the conical water chamber. Finally, we numerically investigated, in detail, the converging effects of the various conical water chambers on the underwater shock waves. The experimental results and the correspondingly numerical results agree quite well with each other.

Analysis of Liquid Slug Motion in a Voided Line

Abstract: A quasi-two-dimensional two-phase flow cylindrical model of slug motion in a voided line is developed that can reasonably predict the change of flow pattern of the slug, air entrainment, holdup and the distribution of axial velocity. However, when using the theory of incompressible momentum transfer to estimate the pressure-time history of slug at the elbow, the calculated results are not in good agreement with those of the experiments. Further analysis of the experimental results indicate that an acoustic, or waterhammerlike response may occur immediately upon impact of the high-speed slug with the elbow, and subsequently, the waveform exhibits momentum transfer due to the acceleration of the slug at the elbow

Statistical Analysis of the ASME KIc Database

Abstract: The American Society of Mechanical Engineers (ASME) K{sub Ic} curve is a function of test temperature (T) normalized to a reference nil-ductility temperature, RT{sub NDT}, namely, T-RT{sub NDT}. It was constructed as the lower boundary to the available K{sub Ic} database. Being a lower bound to the unique but limited database, the ASME K{sub Ic} curve concept does not discuss probability matters. However, a continuing evolution of fracture mechanics advances has led to employment of the Weibull distribution function to model the scatter of fracture toughness values in the transition range. The Weibull statistic/master curve approach was applied to analyze the current ASME K{sub Ic} database. It is shown that the Weibull distribution function models the scatter in K{sub Ic} data from different materials very well, while the temperature dependence is described by the master curve. Probabilistic-based tolerance-bound curves are suggested to describe lower-bound K{sub Ic} values.

Free Vibrations of Transversely Isotropic Cylinders and Cylindrical Shells

Abstract: Three displacement functions are introduced to decompose three displacement components so that the three-dimensional equations of motion of a transversely isotropic body are uncoupled. Expanding these functions in terms of orthogonal ries, the equations of free vibration problem of a transversely isotropic cylindrical shell with ends simply supported are further simplified to be readily dealt with. As contrast to previous works, modified Bessel function solution with complex arguments is directly adopted, for the case of complex eigenvalues. Moreover, for solid cylinders, ( modified) Bessel functions of the first kind even with pure imaginary arguments, are used because of their peculiar properties. Two numerical examples are given to check the correctness of the present method by comparing the results with those of others. Because no assumption is introduced in the paper, the method developed is completely three-dimensionally exact.