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.

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.

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.

Design Equation for Minimum Required Thickness of a Cylindrical Shell Subject to Internal Pressure Based on Von Mises Criterion

2019 ◽  
Author(s):  
James Lu ◽  
Barry Millet ◽  
Kenneth Kirkpatrick ◽  
Bryan Mosher
Keyword(s):  
Cylindrical Shell ◽  
Internal Pressure ◽  
Yield Criterion ◽  
Design Equation ◽  
Shell Wall ◽  
Section Viii ◽  
Von Mises Yield Criterion ◽  
Von Mises ◽  
Division 2 ◽  
Shell Wall Thickness

Abstract Design equation (4.3.1) for the minimum required thickness of a cylindrical shell subjected to internal pressure in Part 4 “design by rule (DBR)” of the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 [1] is based on the Tresca Yield Criterion, while design by analysis (DBA) in Part 5 of the Division 2 Code is based on the von Mises Yield Criterion. According to ASME PTB-1 “ASME Section VIII – Division 2 Criteria and Commentary”, the difference in results is about 15% due to use of the two different criteria. Although the von Mises Yield Criterion will result in a shell wall thickness less than that from Tresca Yield Criterion, Part 4 (DBR) of ASME Division 2 adopts the latter for a more convenient design equation. To use the von Mises Criterion in lieu of Tresca to reduce shell wall thickness, one has to follow DBA rules in Part 5 of Division 2, which typically requires detailed numeric analysis performed by experienced stress analysts. This paper proposes a simple design equation for the minimum required thickness of a cylindrical shell subjected to internal pressure based on the von Mises Yield Criterion. The equation is suitable for both thin and thick cylindrical shells. Calculation results from the equation are validated by results from limit load analyses in accordance with Part 5 of ASME Division 2 Code.

Calculation of Working Pressure for Cylindrical Vessel Under External Pressure

2010 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Monte Carlo ◽  
Cylindrical Shell ◽  
Pressure Vessel ◽  
External Pressure ◽  
Working Pressure ◽  
Geometric Imperfections ◽  
The Monte Carlo Method ◽  
Cylindrical Pressure Vessel ◽  
Section Viii ◽  
Initial Geometric Imperfections

Initial geometric imperfections have a significant effect on the load carrying capacity of asymmetrical cylindrical pressure vessels. This research paper presents a comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a defined cylindrical pressure vessel subjected to external pressure. Evaluations as prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 rules are also presented and discussed in light of the proposed reliability technique presented herein. The ultimate goal of the reliability type technique is to statistically predict the buckling load associated with the cylindrical pressure vessel within a defined confidence interval. The example cylindrical shell considered in this study is a fractionating tower for which calculations have been performed in accordance with the ASME B&PV Code. The maximum allowable external working pressure of this tower for the shell thickness of 0.3125 in. is calculated to be 15.1 psi when utilizing the prescribed ASME B&PV Code, Section VIII, Division 1 methods contained within example L-3.1. The Monte Carlo method as developed by the current authors and published in the literature is then used to calculate the maximum allowable external working pressure. Fifty simulated shells of geometry similar to the example tower are generated by the Monte Carlo method to calculate the nondeterministic buckling load. The representation of initial geometric imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion that appears in the form of a double Fourier series. Multi-mode analyses are expanded to evaluate a large number of potential buckling modes for both predefined geometries and the associated asymmetric imperfections as a function of position within a given cylindrical shell. The method and results described herein are in stark contrast to the dated “knockdown factor” approach currently utilized in ASME B&PV Code.

Investigation of Reduction in Buckling Capacity of Cylindrical Shells Under External Pressure Due to Partially Cut Ring Stiffeners

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
Snehankush Chikode ◽  
Nilesh Raykar
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Cylindrical Shells ◽  
External Pressure ◽  
Geometrical Parameters ◽  
Section Shape ◽  
Clear Guideline ◽  
Section Viii ◽  
Special Circumstances ◽  
Division 2

Circumferential ring stiffeners are commonly used to improve the buckling strength of cylindrical shells. Under special circumstances, stiffener ring needs to be partially cut in order to avoid interference with vessel attachments or surrounding structures. No clear guideline is available in rule-based method to deal with such case. This paper investigates the extent of reduction in buckling capacity for a range of cylindrical shell geometries with stiffener rings having different cross sections and different extents of circumferential cut. Finite-element (FE)-based analysis as per ASME Section VIII, Division 2, Part 5 has been employed to determine the permissible external pressure in each of the cases. Effects of ring cross section and extent of circumferential cut of stiffening ring on the maximum permissible external pressure have been presented. A total of 63 combinations of shell-stiffening ring configurations of different L/D, D/t ratios, cross section shape, and extent of cut have been investigated. Geometrical parameters for these combinations under study are so chosen that normal working range in industries is covered. The results obtained provide guidelines to design shells with partially cut stiffening rings.

Fourier Series Analysis of Cylindrical Pressure Vessel Subjected to External Pressure

2009 ◽  
Author(s):  
Gurinder Singh Brar ◽  
Yogeshwar Hari ◽  
Dennis K. Williams
Keyword(s):  
Fourier Series ◽  
Pressure Vessel ◽  
Large Diameter ◽  
External Pressure ◽  
Pressure Vessels ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Cylindrical Pressure Vessel ◽  
Section Viii ◽  
Division 2

This paper presents the comparison of a reliability technique that employs a Fourier series representation of random asymmetric imperfections in a cylindrical pressure vessel subjected to external pressure. Comparison with evaluations prescribed by the ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 Rules for the same shell geometries are also conducted. The ultimate goal of the reliability type technique is to predict the critical buckling load associated with the chosen cylindrical pressure vessel. Initial geometric imperfections are shown to have a significant effect on the load carrying capacity of the example cylindrical pressure vessel. Fourier decomposition is employed to interpret imperfections as structural features that can be easily related to various other types of defined imperfections. The initial functional description of the imperfections consists of an axisymmetric portion and a deviant portion, which are availed in the form of a double Fourier series. Fifty simulated shells generated by the Monte Carlo technique are employed in the final prediction of the critical buckling load. The representation of initial geometrical imperfections in the cylindrical pressure vessel requires the determination of appropriate Fourier coefficients. Multi-mode analyses 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. The probability of the ultimate buckling stress that may exceed a predefined threshold stress is also calculated. The method and results described herein are in stark contrast to the “knockdown factor” approach as applied to compressive stress evaluations currently utilized in industry. Recommendations for further study of imperfect cylindrical pressure vessels are also outlined in an effort to improve on the current design rules regarding column buckling of large diameter pressure vessels designed in accordance with ASME Boiler and Pressure Vessel Code, Section VIII, Division 2 and ASME STS-1.

Comparison of the Requirements of the British R5, French RCC-MRx, Proposed New Rules in ASME Section VIII, and API 579 Codes in the Design of a Cylinder Subjected to Pressure With Thermal Gradient at Elevated Creep Temperatures

2014 ◽  
Author(s):  
David Anderson ◽  
Nadarajah Chithranjan ◽  
Maan Jawad ◽  
Antoine Martin
Keyword(s):  
Finite Element Analysis ◽  
Cylindrical Shell ◽  
Thermal Gradient ◽  
External Pressure ◽  
Sample Problem ◽  
Element Analysis ◽  
Buckling Strength ◽  
Creep Fatigue ◽  
Section Viii ◽  
Division 2

The authors analyze two sample problems using four different international codes in the evaluation. The first is the British R5 code, the second is the French RCC-MRx code, third is the ASME Section VIII, Division 2, code using proposed new simplified rules taken from the ASME nuclear code section NH, and the fourth is the API 579 code. The requirements, assumptions, and limitations of each of the four codes as they pertain to the sample problems are presented. The first sample problem is for creep-fatigue analysis of a cylindrical shell subjected to internal pressure with a linear thermal gradient through the wall. The second sample problem is evaluating the critical buckling strength of the cylindrical shell under external pressure in accordance with proposed new rules in ASME Section VIII, Division 2, API 579, and a finite element analysis. Paper published with permission.

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.

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.

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