Determination of the Equation of Equilibrium and the Boundary Conditions of a Circular Cylindrical Shell by the Method of Variations PDF
Preview Determination of the Equation of Equilibrium and the Boundary Conditions of a Circular Cylindrical Shell by the Method of Variations
DmwmmicB m m ^umms ot m tm mmùmï Gominmê w x ommum mm* BT m» mmm as* fmmmn pBsia Wmltteà là partial fulflimeat of IBs r#%àlremoàt$ for $h# 4#gy#@ of iu s « or àmoNAUfxojo» maimaim at the mnwm or momts Peter M. Hose* #%me 10#O %g>rove&# __ f<k./ ’oï 'ïïÿSSSÎr ProQuest Number: 27591402 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 27591402 Published by ProQuest LLO (2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 48106- 1346 $0 % Beloved Barents AQmomsDQMjm fhe writer Is deeply indebted and gratefnl to Dr. 7ito D. Salerno^ his adviser, and ifer. Bernard Devine, as well as their oolleaqnos in the Department of deronaw* tieal &igineering and implied Mechanise far their most helpfnl eug^etions and assistance which made possible the writing of this paper. smmï t% la the purpo^ of thia paper to derive the differential equati^s and boimdary ooMition of a elreuXar cylindrical sh^X by means of the oalculim of variations. $he solutions found ly the above method are applicehle to a cXo^d circular shell as well as s# open shell, furthermore, the equations are equally valid for any shell element with a constant radius of curvature in one direction. She above solutions are based on a ^oeed circular shell supported with rings at its boundaries. nfx writer was bom in Germany in 1923 and came to the Ihiited Sates in 193S. Where, graduatim from Brooklyn technical High School, he served in tM drmed Borces for three years. After his release from the services the writer entered Brooklyn Bolytechnis Institute and received a bachelors degree in Aeronautical jfegtn?- eering in 1949. !&e title of his thesis, written under the guid ance of Dr. Vito D. Salerno, was "Gorrelation of all Methods of Solving Thin flats BudtiLiag Problems by the Use of the Sbthod of Variations". It was during the writing of the above paper that the potentialities of the calculus of variations were realised and therefore the work was extended to tylindrical shells, hence the subject of this paper. TôBDS Of oommfs Dist of î%^ol«........................ ............ 1 lâi1aroducti<m ..... ....... 4 Thoo3% S The Total Potential 8 Varlatl<Mi of the Total Botential 12 The GalmAdü» Integral and Associated Dine Integrals 10 The Boundary Conditions 19 the Strain Energy of the Stiffener Bings 25 Variation of the Bing Ibergy* . . . ..... . ... ... .. .. 28 Brtension of IWhlem .... 88 Conclusions,.......... 33 Appendix I ..... 35 Extensions and Curvatures Appendix II...... .. .... ... ... 38 Potential of the Sadload Appendix III...,*..,.......... .......... . . . 40 Potential of the lateral Load %pmdix IV.......... 42 Rogges Boundary Conditions Appendix 45 Strain Energy of the Bings Beferez^sa. ..... 48 1. sùsméS A* (bross-s^tlonsl area of rings A" . Càross-seotlonai area of edge beam a Bad lus of shell Trigonometric coefficients Central angle for shell element angle of twists D a......... 2(1 -9 3) E Youngs modulns of Elasticity e eecentrlcity € Axial strain f Shear Modulus h thickness of ^ell wall 1 Moment of inertia h2 I2a^ l^gth of shell between rings m,n number of WLf waves for shall configuration B axial pressura p radial pressuré shear force consonants q Horiaal shear S Total Hormal Shear f Shear flow Ü Strain Energy 7 External pot<mtlal u,v,w displacement» X, y coordinates of shell X change in curvsturs resultant moment about x^constant resultant moment aboutconstant resultant twisting moment n^ resultant normal force'at x » constant a (y resultant normal force at y « constant resultant shear at x or y « constant 9 Boissons Batlo é Variational Dotation ^ noordlmensicaiallzed coordinate = ^ ' a shear strain
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