A theoretical solution of cylindrical shells for axisymmetric plain strain elastodynamic problems

Two intersecting cylindrical shells subjected to internal pressure and external moment are of common occurrence in pressure vessel and piping industry. The highest stress intensity occurring in the vicinity of junction, which is a complex space curve when the diameter ratio d/D increases. As the new process of theoretical solution and design criteria research developed by the authors, the stress analysis based on the theory of thin shell is carried out for cylindrical shells with normally intersecting nozzles subjected to three kinds of external branch pipe moments. The thin shell theoretical solution for the main shell with cutout, on which a moment is applied, is obtained by superposing a particular solution on the homogeneous solution. The double trigonometric series solution of cylindrical shell subjected to arbitrary distributed normal and tangential forces based on Timoshenko equation is used for the particular solution and the Xue et al.’s solution, for the homogeneous solution based on the modified Morley equation instead of the Donnell shallow shell equation. The displacement function solution for the nozzle with a nonplanar end is obtained on the basis of the Goldenveizer equation instead of Timoshenko’s. The presented results are in good agreement with those obtained by experiments and by three-dimensional finite element method. The present analytical results are in good agreement with WRC Bulletin 297 when d/D is small. The theoretical solution can be applied to d/D ≤ 0.8, λ = d/DT ≤ 8 and d/D ≤ t/T ≤ 2 successfully.

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A Thin Shell Theoretical Solution for Two Intersecting Cylindrical Shells Due to External Branch Pipe Moments

Journal of Pressure Vessel Technology ◽

10.1115/1.2042471 ◽

2005 ◽

Vol 127 (4) ◽

pp. 357-368 ◽

Cited By ~ 7

Author(s):

M. D. Xue ◽

D. F. Li ◽

K. C. Hwang

Keyword(s):

Cylindrical Shells ◽

Branch Pipe ◽

Homogeneous Solution ◽

Theoretical Solution ◽

Intersection Curve ◽

3D Fem ◽

External Branch ◽

Particular Solution ◽

Shell Equation ◽

Distributed Forces

A theoretical solution is presented for cylindrical shells with normally intersecting nozzles subjected to three kinds of external branch pipe moments. The improved double trigonometric series solution is used for the particular solution of main shell subjected to distributed forces, and the modified Morley equation instead of the Donnell shallow shell equation is used for the homogeneous solution of the shell with cutout. The Goldenveizer equation instead of Timoshenko’s is used for the nozzle with a nonplanar end. The accurate continuity conditions at the intersection curve are adopted instead of approximate ones. The presented results are in good agreement with those obtained by tests and by 3D FEM and with WRC Bulletin 297 when d∕D is small. The theoretical solution can be applied to d∕D⩽0.8, λ=d∕DT⩽8, and d∕D⩽t∕T⩽2 successfully.

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An Analytical Method for Cylindrical Shells With Nozzles Due to Internal Pressure and External Loads—Part I: Theoretical Foundation

Journal of Pressure Vessel Technology ◽

10.1115/1.4001199 ◽

2010 ◽

Vol 132 (3) ◽

Cited By ~ 4

Author(s):

Ming-De Xue ◽

Qing-Hai Du ◽

Keh-Chih Hwang ◽

Zhi-Hai Xiang

Keyword(s):

Analytical Solution ◽

Internal Pressure ◽

Stress Analysis ◽

Pressure Vessel ◽

Cylindrical Shells ◽

Branch Pipe ◽

Theoretical Solution ◽

External Branch ◽

Complex Valued ◽

External Loads

An improved version of the analytical solutions by Xue, Hwang and co-workers (1991, “Some Results on Analytical Solution of Cylindrical Shells With Large Opening,” ASME J. Pressure Vessel Technol., 113, 297–307; 1991, “The Stress Analysis of Cylindrical Shells With Rigid Inclusions Having a Large Ratio of Radii,” SMiRT 11 Transactions F, F05/2, 85–90; 1995, “The Thin Theoretical Solution for Cylindrical Shells With Large Openings,” Acta Mech. Sin., 27(4), pp. 482–488; 1995, “Stresses at the Intersection of Two Cylindrical Shells,” Nucl. Eng. Des., 154, 231–238; 1996, “A Reinforcement Design Method Based on Analysis of Large Openings in Cylindrical Pressure Vessels,” ASME J. Pressure Vessel Technol., 118, 502–506; 1999, “Analytical Solution for Cylindrical Thin Shells With Normally Intersecting Nozzles Due to External Moments on the Ends of Shells,” Sci. China, Ser. A: Math., Phys., Astron., 42(3), 293–304; 2000, “Stress Analysis of Cylindrical Shells With Nozzles Due to External Run Pipe Moments,” J. Strain Anal. Eng. Des., 35, 159–170; 2004, “Analytical Solution of Two Intersecting Cylindrical Shells Subjected to Transverse Moment on Nozzle,” Int. J. Solids Struct., 41(24–25), 6949–6962; 2005, “A Thin Shell Theoretical Solution for Two Intersecting Cylindrical Shells Due to External Branch Pipe Moments,” ASME J. Pressure Vessel Technol., 127(4), 357–368; 2005, “Theoretical Stress Analysis of Two Intersecting Cylindrical Shells Subjected to External Loads Transmitted Through Branch Pipes,” Int. J. Solids Struct., 42, 3299–3319) for two normally intersecting cylindrical shells is presented, and the applicable ranges of the theoretical solutions are successfully extended from d/D≤0.8 and λ=d/(DT)1/2≤8 to d/D≤0.9 and λ≤12. The thin shell theoretical solution is obtained by solving a complex boundary value problem for a pair of fourth-order complex-valued partial differential equations (exact Morley equations (Morley, 1959, “An Improvement on Donnell’s Approximation for Thin Walled Circular Cylinders,” Q. J. Mech. Appl. Math. 12, 89–91; Simmonds, 1966, “A Set of Simple, Accurate Equations for Circular Cylindrical Elastic Shells,” Int. J. Solids Struct., 2, 525–541)) for the shell and the nozzle. The accuracy of results is improved by some additional terms to the expressions for resultant forces and moments in terms of complex-valued displacement-stress function. The theoretical stress concentration factors due to internal pressure obtained by the improved expressions are in agreement with previously published test results. The theoretical results discussed and presented herein are in sufficient agreement with those obtained from three dimensional finite element analyses for all the seven load cases, i.e., internal pressure and six external branch pipe load components involving three orthogonal forces and the respective three orthogonal moments.

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A Stress Analysis Method for Cylindrical Shells With Nozzles Subjected to Internal Pressure, External Forces and Moments

Volume 3: Design and Analysis ◽

10.1115/pvp2006-icpvt-11-94053 ◽

2006 ◽

Cited By ~ 1

Author(s):

Ming-De Xue ◽

Qing-Hai Du ◽

Dong-Feng Li ◽

Keh-Chih Hwang

Keyword(s):

Internal Pressure ◽

Stress Analysis ◽

Thin Shell ◽

Cylindrical Shells ◽

Branch Pipe ◽

Three Dimensional ◽

Theoretical Solution ◽

Analysis Method ◽

External Forces ◽

Three Dimensional Finite Element

An identical stress analysis method based on the thin shell theory is carried out for cylindrical shells with normally intersecting nozzles subjected to internal pressure and six kinds of external branch pipe loads involving axial tension, two kinds of transverse shear forces, longitudinal and circumferential bending and torsion moments. The thin shell theoretical solution is obtained based on the Morley equation instead of the Donnell shallow shell equation. The accurate continuity conditions at the intersecting curve, which is a complicated space curve, are adopted. The presented results are verified by three-dimensional finite element method (FEM). The theoretical solution can be applied to d/D ≤ 0.8, λ = d/DT ≤ 12 and d/D ≤ t/T ≤ 2 successfully. The solutions are in good agreement with WRC Bulletin 297 when diameter ratio is small. In the paper some typical design curves calculated by the theoretical solutions are presented and their applicable ranges are greatly expanded in comparison with current design methods.

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