Numerical Analysis Pv Heads

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/0 /0 l:2q/1 Cff3 A NUMERICAL PROCEDUREFOR THE ANAL YSIS OF PRESSURE,VESSEL HEADS lIetz-R'eference Roem .0.1 viI EY'..ginee:rlng "Departmen, B106 C. E. Building University of Urbana.. Illinoia6l801 TECHNICAL REPORT to the OFFICE OF NAVAL RESEARCH Contract N60ri-71, Task Order VI Project NR-035-183 By T. Au, l. E. Goodman and" N. M. Newmark DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF ILLINOIS URBANA, ILLINOIS A NUMERICAL PROCEDURE FOR THE ANALYSIS OF PBESStJBE VESSEL READS. By T. Au, L. E. Goodman and N. M. Newmark A Technical Report of Q Cooperative Investigation Sponsored by TEE OFFICE OF NAVAL RESEARCH, DllPARTMEmT OF THE NAVY and TRI DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF ILLINOIS Contract N6ori-71, Task Order VI Project NR-035-183 Urbana, Illinois February 15, 1951 A NU'MERICAL PROCi1DUBE FOR TEE ANALYSIS 01 PRES SURE VESSEL HEADS. LI ST OF li'IGURES LI ST OF TA13LES SYNOPSIS I. -. INTRODUCTION. CONTENTS. 10 2. Object and Scope of Investigation Acknowledgment. . ; II. BESUME OF FUNDAMENTAL BASED ON LOVEIS THEORY OF THIN SHELLS. 30 Nomenclature . . . . . 4. Fundamental . . i. Page iii iv v 1 2 3 6 50 Development of Love-Meissner Equa.tions 11 6. Reduction of Equations for Shells of Special Geometrical Shape 13 III. GENERAL CONSIDERATIONS IN PBESSUBE VESSEL ANALYSIS. Basic Assumptions of the Analysis Types of Heads under Investigation Treatment of Boundary Conditions Nature of Critical Stresses . . . Analysis of the Cylindrical Shell IT 0 MmTHOD OF FINITE DIFFEEENCES. 12. General Concepts . . 13. Spherical Shell . . . . 140 Conical Shell 15. Toroidal Shell 16. Flat Circular Head. . . . 17. Application of the Procedure 19 20 . . . 22 . . . . 26 27 31 34 37 . . . 41 45 49 v. COMPARISON OF METHODS OF SOLDTION. 18. Statement of the Test Problem 190 Illustrative Example of the Method of ii. Page 53 Finite Differences. 54 2 0 ~ Comparison of Results by Different Methods of Solut ion . 58 VIo NUMERICAL ANALYSES OF VARIOUS TYPES aIr PRESSITBE VESSEL HEADS. 210 Dimensions of the Pressure Vessel Heads . . 22. Analyses of Six Types of Heads 0 67 6s 230 Oomparison of Results for Various Types of Heads 0 0 70 VII. OONCLUSIONo Summary o 0 . . Concluding Remarks APPENDIX I. ANNOT ATED 13 113 L1 OGRAPHY 10 Brief Review of Literature 20 Bibliography APPmmIX 110 CLASSICAL .AND APPROXIMATE SOLUTIONS FOR SPHERICAL-SEGMENT HEAD. Classical Analysis Approximate Method o . 85 It 87 . 89 93 . . . . . .107 0113 LIST CIB' T.A:BLES .. Table No. 1. Comparison of Results Obtained by Different Methods of Solution. A. Normal Forces, Momentsand.Shears at Various iii. Page Points on the Spherical Head 59 ]. Total Stresses, Direct and ]endingp at Various Points on the Spherical Head 60 2. Stresses in Spherical Head. of Pressure Vessel 3 ~ Stresse.s in Conical Head. of Pressure Vessel 4. Stresses in Torispheric'al Head of Pressure Vessel . '. 5. Stresses in Toriconical Head of Pressure Vessel 6. ,Stresses in Hemispherical Head of Pressure Vessel 72 73 74 75 76 7. Stresses in Flat Circular 'Head. of Pressure Vessel 77 LIST OF FIGURES. Fig. No. 1. Equilibrium of a Shell Element 20 Deformation of a Shell Element 3. Types of Heads under Investigation 4. Classification of Boundary Oonditions 5. Basic Elements in NUmerical Solution 6. Oomparison of Methods of Solution. iVa Page g 10 .21 23 32 A. Normal Forces at Various Point s on Spherical Head. 61 B. Moments at Various Points on Spherical Head C. Shear at Various Points on Spherical Head . D. Total Stresses along Outer Surface of Spherical Head E. T.otal Stresses along Inner Sur.f'ace of Spherical Head 7. Stress Distribution in Pressure Vessel. A. S];)herical Head B. Conical Read o. Torispherical Head Do Torieonical Head E. Hemispherical Head Fo Flat Circular Head 62 63 64 65 78 79 80 81 82 83 v. SYNOPSIS. The need for more nearly exact methods of analysis of pressure vessel heads is supplied in this report by a numerical technique of wide applicability. The analysis is based on the general theory of thin elastic shells due to A. E. H. Love. While the method is approx-imate in the sense that the governing mathematical equations are sat-isfied only in finite difference form, it has the merit of simplicity and of directly providing numerical values of normal forces, moments and shears needed in design. The general finite-difference method is first developed and then applied to the particular cases of pressure vessels having spherical heads, conical heads, torispherical heads, toriconical heads, hemispherical heads and flat circular heads. Close agreement is gotten between the results obtained by "classical" analy-sis and those given by the numerical method;. for the spherical-segment head chosen as a test problem. A NUMERICAL PROCEDURE FOR THE ANALYSIS OF PRES SURE VES SEL HEAD S . I. INTRODUCTION. 1. Object and Scope of Investigation The design and analysis of pressure vessels have for many years pro-ceeded along separate lines, On the one hand, a mathematical theory, due in the first instance to A. E. H. Love (15)1, has been applied to the analysiS of a few simple shapes. Industry, on the other hand, has been forced to rely on rule-of-thumb design formulas because the shapes of pres-sure vessels encountered in practice are not amenable to analysis by the exact theory. So long as large factors of safety could be used, this di-chotomy produced no great difficulties; but today, as the recrrlrementsof high pressures and economy of design become more stringent, the need for more nearly exact analyses of pressure ves'sel heads is pressing. This need is here supplied by a numerical technique of wide applicability. While it is approximate in the sense that the governing mathematical equations are satisfied only in finite difference form, it has the merit of Simplicity and of directly providing numerical values of normal forces9 moments and shears needed in design. The analysis is based on the general theory of thin elastic shells due to Love. Although Love I s theory has been re-examined by many others who have questioned its adequacy in some respects, it represents a atic first approximation for most practical :purposes. Reduction of Lovaas theory to two differential equations of second order for shells of revolu-tion having constant radius of curvature along the meridian has been made by E. Meissner(38)& Solutions of the Love-Meissner equations even for shells of simple geometrical shapes, with the notable exception of the cy-lindrical shell and flat circular pressure vessel head, involve the use of the hypergeometric series which usually exhibits slow convergence. Several simplified methods have been developed as to Love's theory. Moc:!t of these analyses follow the "approximate theoryl1 of J. W" Geckeler (49) which neglects some of the terms in the governing differential equa-tions. The idea of seeking solutions of the governing differential tiona by the finite difference is not entirely newj but little 1. Numbers in parentheses refer to the Annotated Bibliography at the end of the report. -2-progress has been made along this line since the pioneer work of P. Pasternak (52). The investigation consists in the development of a general procedure and its application to different types of pressure vessel heads including: 1. Spherical head 2.. Conical head 3. .Torispherical head 40 Toriconical head 5 Hemispherical head 6. Fl at circular head The accuracy of the numerical method is examined. by means of a compara-tive study of a test problem, the spherical-segment head, analyzed by three different methods: Love-Meissner's classical analysis, Geckeler's approximate solution, and the method of finite differences. 2. Acknowledgment This report is based on the doctoral dissertation in the Department of Civil Engineering, University of Illinois, prepared by T. Au, under the supervision of L. E. Goodman and the general direction of N. M. lrewmark.. Part of the material is the outgrowth of a research program "Numerical and Approximate Methods of Stress Analysi sn sponsored by the Office of Naval Research as Task VI of Contract N6ori-7l, Project DeSig-nation in the Engineering Experiment Station of the Univer-sity of Illinois. The writers also acknowledge the advantages of associa-tion the Design Division of the Pressure Vessel Research Committee of the Welding Research OO'Wlcl1--land the Copper and Erass Research Asso-ciation. -3-/ /. II. RESUMID OF FUNDAMENTAL E ~ M I O N S EASED ON LOVE' S THEORY OF THIN SHELLS. 3. Nomenclature The following nomenclature has been used in the derivation of equa-tions throughout the report. Insofar as practicable, this