Axially Compressed Cylindrical Shells Containing Asymmetric Random Imperfections: Fourier Series Technique and ASME Section VIII Division 1 and 2 Rules

2008 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Cylindrical Shell ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Cylindrical Shells ◽  
Structural Features ◽  
Mode Analysis ◽  
Fourier Decomposition ◽  
Buckling Stress ◽  
Division 1 ◽  
Section Viii

This paper presents the comparison of reliability technique that employ Fourier series representations of random asymmetric imperfections in axially compressed cylindrical shell with evaluations prescribed by ASME Boiler and Pressure Vessel Code, Section VIII, Division 1 and 2. The ultimate goal of the reliability type technique is to predict the buckling load associated with the axially compressed cylindrical shell. Initial geometric imperfections have significant effect on the load carrying capacity of asymmetrical cylindrical shells. Fourier decomposition is used to interpret imperfections as structural features can be easily related to the different components of imperfections. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion appearing as a double Fourier series. The representation of initial geometrical imperfections in the cylindrical shell requires the determination of appropriate Fourier coefficients. The mean vector and the variance-covariance matrix of Fourier coefficients are calculated from the simulated shell profiles. Multi-mode analysis are expanded to evaluate a large number of potential buckling modes for both predefined geometries and associated asymmetric imperfections as a function of position within a given cylindrical shell. Large number of shells thus created can be used to calculate buckling stress for each shell. The probability of the ultimate buckling stress exceeding a predefined threshold stress can also be calculated.

Axially Compressed Cylindrical Shell Containing Axisymmetric Random Imperfections: Fourier Series Techniques and ASME Section VIII Division 2 Rules

2008 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Cylindrical Shell ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Cylindrical Shells ◽  
Structural Features ◽  
Geometric Imperfections ◽  
Fourier Decomposition ◽  
Buckling Stress ◽  
Section Viii ◽  
Division 2

This paper presents the comparison of reliability technique that employ Fourier series representations of random axisymmetric imperfections in axially compressed cylindrical shells with evaluations prescribed by ASME Section VIII, Division 2. The ultimate goal of the reliability type technique is to predict the buckling load associated with the axially compressed cylindrical shell. The representation of initial geometrical imperfections in the cylindrical shell requires the determination of appropriate Fourier coefficients. The buckling of cylindrical shells in any type of loading is sensitive to the form and amplitude of geometric imperfections present in the structure. Initial geometric imperfections have significant effect on the load carrying capacity of axisymmetrical cylindrical shells. Many deterministic and probabilistic techniques are there to predict shell behavior during buckling. Fourier decomposition is used to interpret imperfections as structural features can be easily related to the different components of imperfections. The mean vector and the variance-covariance matrix of Fourier coefficients are calculated from the simulated shell profiles. Recommendations for further use of Fourier coefficients through simulation by Monte Carlo Method are laid down. Large number of shells thus created can be used to calculate buckling stress for each shell. The probability of the ultimate buckling stress exceeding a predefined threshold stress can be calculated.

Buckling of Axisymmetric, Homogeneous Cylindrical Shells With Random Imperfections: Monte Carlo Method

2005 ◽  
Author(s):  
Dennis Williams
Keyword(s):  
Monte Carlo ◽  
Cylindrical Shell ◽  
Pressure Vessel ◽  
Cylindrical Shells ◽  
Large Diameter ◽  
Buckling Load ◽  
Problem Solution ◽  
Critical Buckling Load ◽  
Asme Code ◽  
Section Viii

This paper presents the second of a series of solutions to the buckling of imperfect cylindrical shells subjected to an axial compressive load. In particular, the current problem reviewed is the case of a homogeneous cylindrical shell with random axisymmetric imperfections. The problem solution for the determination of the critical buckling load utilizes a statistical approach to define the random imperfections as opposed to the deterministic methods most often employed in the pressure vessel industry. The imperfections are treated as a random function of the axial (i.e., longitudinal) position on the shell. The Monte Carlo technique is utilized to create a large sample of random shell geometries from which to eventually calculate a critical buckling load for each randomly generated shell geometry. Having matched or predefined the statistical parameters (including the co-variance) of interest as determined from actual manufacturing statistics to the Monte Carlo simulation of shell geometries, the reliability of the critical buckling load is then calculated for the set of cylindrical shells with the random axisymmetric imperfections. The ASME Boiler and Pressure Vessel Code Section VIII fabrication tolerances as supplemented by ASME Code Case 2286-1 are reviewed and addressed in light of the findings of the current study and resulting solutions with respect to the critical buckling loads. The method and results described herein are in stark contrast to the “knockdown factor” approach currently utilized in ASME Code Case 2286-1. Recommendations for further study of the imperfect cylindrical shell are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter shells designed in accordance with ASME Section VIII, Divisions 1 and 2 and ASME STS-1 in combination with the suggestions contained within Code Case 2286-1.

A Study of Stresses in Cylindrical Shells Having Tapped Holes Parallel to the Axis of the Cylinder

2016 ◽  
Author(s):  
Dipak K. Chandiramani ◽  
Shyam Gopalakrishnan ◽  
Ameya Mathkar
Keyword(s):  
Cylindrical Shell ◽  
Internal Pressure ◽  
Shell Thickness ◽  
Pressure Vessels ◽  
Inside Surface ◽  
Shell Wall ◽  
Allowable Stresses ◽  
Metal Thickness ◽  
Division 1 ◽  
Section Viii

Clause UG-27 of ASME Section VIII Division 1 [1] provides rules for calculating the thickness of shells under internal pressure. Mandatory Appendix-2 of Code [1] provides rules for design of bolted flanged connections. In certain high pressure and high thickness pressure vessels having a cylindrical shell with bolted cover flange, Manufacturers avoid a separate end flange welded to the shell, as the construction becomes bulky. Instead of the same, Manufacturers provide tapped holes in shell wall parallel to axis of the cylindrical shell. The cover is directly bolted to these tapped holes provided in the shell. This type of construction may be economical as compared to welding a conventional flange to the end of the shell. However this type of construction is not covered in the Code [1]. When such tapped holes are provided in the cylindrical shell, generally the total metal thickness provided at the tapped hole location meets UG-27 requirement of the Code [1]. However due to the tapped holes, the thickness from inside surface of vessel to inside surface of tapped hole is less than the required thickness of UG-27. It is therefore required to analyze the stresses due to these tapped holes in the shell thickness to ensure that Code [1] allowable stresses are not exceeded. The work reported in this paper was undertaken to determine the effect of internal pressure on the stresses in a cylindrical shell having tapped holes parallel to axis of the cylindrical shell.

scholarly journals The Hilbert Space of Double Fourier Coefficients for an Abstract Wiener Space

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 389
Author(s):  
Jeong-Gyoo Kim
Keyword(s):  
Hilbert Space ◽  
Fourier Series ◽  
Fourier Coefficients ◽  
Double Sequence ◽  
Intermediate Stage ◽  
Wiener Space ◽  
Abstract Wiener Space ◽  
Abstract Space ◽  
Double Sequences ◽  
Two Parameter

Fourier series is a well-established subject and widely applied in various fields. However, there is much less work on double Fourier coefficients in relation to spaces of general double sequences. We understand the space of double Fourier coefficients as an abstract space of sequences and examine relationships to spaces of general double sequences: p-power summable sequences for p = 1, 2, and the Hilbert space of double sequences. Using uniform convergence in the sense of a Cesàro mean, we verify the inclusion relationships between the four spaces of double sequences; they are nested as proper subsets. The completions of two spaces of them are found to be identical and equal to the largest one. We prove that the two-parameter Wiener space is isomorphic to the space of Cesàro means associated with double Fourier coefficients. Furthermore, we establish that the Hilbert space of double sequence is an abstract Wiener space. We think that the relationships of sequence spaces verified at an intermediate stage in this paper will provide a basis for the structures of those spaces and expect to be developed further as in the spaces of single-indexed sequences.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

Limit Pressures of Cylindrical and Spherical Shells

2000 ◽  
Vol 123 (3) ◽  
pp. 288-292 ◽  
Author(s):  
Arturs Kalnins ◽  
Dean P. Updike
Keyword(s):  
Geometric Parameter ◽  
Cylindrical Shells ◽  
Length Effect ◽  
Spherical Shells ◽  
Simply Supported ◽  
Limit Pressure ◽  
Section Viii ◽  
Division 2

Tresca limit pressures for long cylindrical shells and complete spherical shells subjected to arbitrary pressure, and several approximations to the exact limit pressures for limited pressure ranges, are derived. The results are compared with those in Section III-Subsection NB and in Section VIII-Division 2 of the ASME B&PV Code. It is found that in Section VIII-Division 2 the formulas agree with the derived limit pressures and their approximations, but that in Section III-Subsection NB the formula for spherical shells is different from the derived approximation to the limit pressure. The length effect on the limit pressure is investigated for cylindrical shells with simply supported ends. A geometric parameter that expresses the length effect is determined. A formula and its limit of validity are derived for an assessment of the length effect on the limit pressures.

Acoustic Radiation From Thick Laminated Cylindrical Shells With Sparse Cross Stiffeners

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Xiongtao Cao ◽  
Chao Ma ◽  
Hongxing Hua
Keyword(s):  
Cylindrical Shell ◽  
Radial Displacement ◽  
Acoustic Radiation ◽  
Periodic Structures ◽  
Cylindrical Shells ◽  
Acoustic Power ◽  
Acoustic Energy ◽  
Parallel Plates ◽  
Wavenumber Domain ◽  
Laminated Cylindrical Shells

A general method for predicting acoustic radiation from multiple periodic structures is presented and a numerical solution is proposed to find the radial displacement of thick laminated cylindrical shells with sparse cross stiffeners in the wavenumber domain. Although this method aims at the sound radiation from a single stiffened cylindrical shell, it can be easily adapted to analyze the vibrational and sound characteristics of two concentric cylindrical shells or two parallel plates with complicated periodic stiffeners, such as submarine and ship hulls. The sparse cross stiffeners are composed of two sets of parallel rings and one set of longitudinal stringers. The acoustic power of large cylindrical shells above the ring frequency is derived in the wavenumber domain on the basis of the fact that sound power is focused on the acoustic ellipse. It transpires that a great many band gaps of wave propagation in the helical wave spectra of the radial displacement for stiffened cylindrical shells are generated by the rings and stringers. The acoustic power and input power of stiffened antisymmetric laminated cylindrical shells are computed and compared. The acoustic energy conversion efficiency of the cylindrical shells is less than 10%. The axial and circumferential point forces can also produce distinct acoustic power. The radial displacement patterns of the antisymmetric cylindrical shell with fluid loadings are illustrated in the space domain. This study would help to better understand the main mechanism of acoustic radiation from stiffened laminated composite shells, which has not been adequately addressed in its companion paper (Cao et al., 2012, “Acoustic Radiation From Shear Deformable Stiffened Laminated Cylindrical Shells,” J. Sound Vib., 331(3), pp. 651-670).

scholarly journals Elastic Response of Acoustic Coating on Fluid-Loaded Rib-Stiffened Cylindrical Shells

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
Christopher Gilles Doherty ◽  
Steve C. Southward ◽  
Andrew J. Hull
Keyword(s):  
Cylindrical Shell ◽  
Cylindrical Shells ◽  
Coupled System ◽  
Elastic Response ◽  
System Response ◽  
Elastic Cylindrical Shell ◽  
Fluid Environment ◽  
Reinforcing Ribs ◽  
Double Index ◽  
Stress Boundary Conditions

Reinforced cylindrical shells are used in numerous industries; common examples include undersea vehicles, aircraft, and industrial piping. Current models typically incorporate approximation theories to determine shell behavior, which are limited by both thickness and frequency. In addition, many applications feature coatings on the shell interior or exterior that normally have thicknesses which must also be considered. To increase the fidelity of such systems, this work develops an analytic model of an elastic cylindrical shell featuring periodically spaced ring stiffeners with a coating applied to the outer surface. There is an external fluid environment. Beginning with the equations of elasticity for a solid, spatial-domain displacement field solutions are developed incorporating unknown wave propagation coefficients. These fields are used to determine stresses at the boundaries of the shell and coating, which are then coupled with stresses from the stiffeners and fluid. The stress boundary conditions contain double-index infinite summations, which are decoupled, truncated, and recombined into a global matrix equation. The solution to this global equation results in the displacement responses of the system as well as the exterior scattered pressure field. An incident acoustic wave excitation is considered. Thin-shell reference models are used for validation, and the predicted system response to an example simulation is examined. It is shown that the reinforcing ribs and coating add significant complexity to the overall cylindrical shell model; however, the proposed approach enables the study of structural and acoustic responses of the coupled system.

Sign in / Sign up

Export Citation Format

Share Document