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Research Papers: Design and Analysis

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

[+] Author and Article Information
Ming-De Xue, Keh-Chih Hwang, Zhi-Hai Xiang

Department of Engineering Mechanics, AML, Tsinghua University, Beijing 100084, P.R. China

Qing-Hai Du

China Ship Scientific Research Center, Wuxi, Jiangsu 214082, P.R. China

Some test models have fillet radii or fillet weld, and the extended part of the fillet weld of the ORNL-1 test model is removed by machine processing (36).

J. Pressure Vessel Technol 132(3), 031206 (May 19, 2010) (9 pages) doi:10.1115/1.4001199 History: Received June 03, 2009; Revised January 17, 2010; Published May 19, 2010; Online May 19, 2010

## Abstract

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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## Figures

Figure 1

The basic mechanical model

Figure 2

Five coordinate systems

Figure 3

Comparison of k among test, numerical, and analytical results for model ORNL-1 due to p

Figure 4

Comparison between test and analytical models

Figure 5

Distribution of k along the line θ=0 deg of the model due to internal pressure p (d/D=0.93, D/T=166, and t/T=1)

Figure 6

Comparison of k among the 3D FEM and theoretical results with and without additional terms around Γ in the main shell due to p (d/D=0.93, D/T=166, and t/T=1)

Figure 7

Comparison of k between the 3D FEM and theoretical results around Γ due to Pzb (d/D=0.93, D/T=166, and t/T=1)

Figure 8

Comparison of k between the 3D FEM and theoretical results around Γ due to Mxb (d/D=0.93, D/T=166, and t/T=1)

Figure 9

Comparison of k between the 3D FEM and theoretical results around Γ due to Myb (d/D=0.93, D/T=166, and t/T=1)

Figure 10

Comparison of k between the 3D FEM and theoretical results around Γ due to Mzb (d/D=0.93, D/T=166, and t/T=1)

Figure 11

Comparison of k between the 3D FEM and theoretical results around Γ due to Pxb (d/D=0.93, D/T=166, and t/T=1)

Figure 12

Comparison of k between the 3D FEM and theoretical results around Γ due to Pyb (d/D=0.93, D/T=166, and t/T=1)

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