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2 Stresses in thin Vessels Under Internal pressure (10 JANUARY 1964 Prepared by NOMAN N. AUSolid Mechanics DepartmengPrepared for COMMANDER SPACE SYSTEMS DIVISIONUNITED STATES .AIR FORCEI nglewood, CalifornirQENGINEERING DIVISION "CONTRACT NO. AF 04(695)-269tL~Report TDR- Z69(4304)-5(formerlyATN-63(3304)-1)STRES SES IN thin VESSELS UNDER INTERNAL PRESSUREP repared byNorman N. AuSolid Mechanics DepartmentAEROSPACE CORPORATIONEl Segundo, CaliforniaContract No. AF 04(695)-26915 January 1964 Prepared forCOMMANDER SPACE SYSTEMS DIVISIONUNITED STATES AIR FORCEI nglewood, CaliforniaReport TDR- 269(4304)- 5 STRESSES IN thin VESSELS UNDER INTERNAL PRESSUREP repared by:Norman N. Au, ManagerStructures SectionApproved by:ISteinman, HeadSdlid Mechanics DepartmentThis Technical documentary report has been reviewed and is approved forpublication and dissemination.)

3 The conclusions and findings containedherein do not necessarily represent an official Air Force Space Systems DivisionAir Force Systems Command:Major, USAFSSYCAEROSPACE CORPORATIONEl Segundo, CaliforniaFOREWORDThe number of publications in the field of pressure vessel stressanalysis has increased precipitously in recent years and the informationcontained therein, though pertinent, is, many times, lost in this sea ofliterature. In addition, a large proportion of the solutions, as presented,are not carried to the point where they can be used directly for determiningstress distributioms. The purpose of this report, then, is to make avail-able a compact and adequate summary of the formulas and to present themin a readily usable fashion for the stress analysis of pressure vessels thatare commonly encountered in missile design.

4 That such a report as thisderives largely from the work of others is self-evident, and it is theauthor'ts hope that due acknowledgement has been made of the immediatesources of all material here presented. However, while most publicationsdeal with head closures joined to long circular cylindrical shell sections,this report places equal emphasis on designs in which the circularcylindrical shell section is short and is integrally joined to head closuresof different shapes and thicknesses. The equations are organized suchthat interchangeability of head closures is a matter of algebraic manipu-lation. It is believed that the material contained in this report will behelpful to those involved in pressure vessel stress analysis and will beparticularly useful for members of the Structures Section, Solid MechanicsDepartment, who are often asked to appraise pressure vessel designspeedily and yet with sufficient accuracy as to insure a degree of confidencein their appraisal.

5 The material presented herein can also be used to spotcheck stress levels obtained from computer author would like to take this opportunity to express his thanks toMr. Walter Smotrys of the Structures Section for his assistance in checkingthe algebraic manipulations associated with the preparation of this also wished to thank Miss M. J. McNeil for her computations of thefrequently encountered parameters and the preparation of the , it should be said that, although every care has been taken toavoid errors, it would be over-sanguine to hope that none had escapeddetection; for any suggestions that readers may make concerning neededcorrection, the author will be grateful,-Vi-Report TDR-269(4304)- 5 STRESSES IN thin VESSELS UNDER INTERNAL pressure *ABSTRACTE lastic stresses in thin shells of revolution under the action of internalpressure are presented.

6 The formulas given are developed on the basisof Love's classical shell theory. The pressure vessel configurationsunder consideration consist of various commonly encountered head clo-sure designs integrally joined to circular cylindrical shell sections whichmay be classified as long where the characteristic lengthofP I > orshort where p 1c < In additiontothe membrane stresses, the bendingstresses resulting from forces and moments at the junctures of the headsand cylinders are also presented. The conceptof edge influence numbersis used where convenient to express the discontinuity forces and mo-ments at the junction. Many important parameters are expressed ingraphical forms to facilitate analysis.*This work completed March, OF CONTENTSITEM PAGEFOREWORD.

7 VABSTRACT .. vii1. INTRODUCTION .. 12. MEMBRANE THEORY SOLUTIONS .. 52. 1 thin Circular Cylindrical Shell underInternal pressure ..52. 2 thin Spherical Shell under InternalPressure .. thin Ellipsoidal Shell under InternalPressure .. thin Conical Shell under Internal pressure .92. 5 thin Elastic Simply Supported CircularPlate under Uniform Load over EntireSurface .. 113. BENDING THEORY SOLUTIONS .. 133. 1 Circular Cylindrical Shells .. 133. la Uniform Radial Shear Q Lb per LinearInch of Circumference ai Znd of LongCircular Cylindrical Shell .. 163. lb Uniform Radial Moment M Inch of Circumference at End ofLong Circular Cylindrical Shell .. 173. Ic Short Circular Cylindrical Shell Bent byForces Distributed along the Edges.

8 181) Equal Edge Forces .. 182) Unequal Edge Forces .. 19-11-TABLE OF CONTENTS (Cont'd)ITEM PAGE3. Id Short Circular Cylindrical Shell Bent byMoments Distributed along the Edges .. 20I) Equal Edge Moments .. 202) Unequal Edge Moments .. 213. 2 Shells of Surfaces of Revolution .. Uniform Normal Shear Q Lb per LinearInch of Circumference at End ..253. Zb Uniform Radial Moment Mo In. -Lb perLinear Inch of Circumference at End .. 263. Zc Uniform Radial Shear 0 Lb per LinearInch of Circumference at End of RightCircular Conical Shell .. 30-J3. Zd Uniform Radial Moment M0 In. -Lb perLinear Inch of Circumference at End ofRight Circular Conical Shell .. 313, ?e Bending Stresses in Cone Loaded byUniform Internal pressure .

9 Circular Plate with Uniform Edge MomentM In. -Lb per Linear Inch of Circumference ..334. SOLUTIONS FOR DISCONTINUITY SHEAR FORCESAND BENDING MOMENTS .. Two Long Cylinders of Unequal .354. Z Two Long Cylinders of Unequal Thicknessesand Mismatch of the Middle " Long Circular Cylinder with CircumferentialRing Stiffeners .. Long Circular Cylindrical Shell with ManyEquidistant Circumferential Ring Stiffeners ..39 JIITABLE OF CONTENTS (Cont'd)ITEM Long Circular Cylindrical Shell with aFlat Head Closure .. Short Circular Cylindrical Shell withEqual Thickness Flat Head Closures .. Short Circular Cylindrical Shell withUnequal Thickness Flat Head Closures .. Long Circular Cylindrical Shell with anEllipsoidal Head Closure.

10 Short Circular Cylindrical Shell with EqualThickness Ellipsoidal Head Closures .. 474. 10 Short Circular Cylindrical Shell withUnequal Thickness Ellipsoidal HeadClosui"es .. 494. 11 Long Circular Cylindrical Shell with aHemispherical Head Closure .. Short Circular Cylindrical Shell with EqualThickness Hemispherical Head Closures .. Short Circular Cylindrical Shell withUnequal Thickness Hemispherical HeadClosures .. Long Circular Cylindrical Shell with aConical Head Closure .. Short Circular Cylindrical Shell withUnequal Thickness Head Closures ofEllipsoidal and Conical Shape .. Short Circular Cylindrical Shell withUnequal Thickness Head Closures ofHemispherical and Conical Shape .. 585. GRAPHICAL PRESENTATION OF PERTINENTPARAMETERS.