Table Of Content

Unclassified SECURITY CLASSIFICATION OF THIS PAGE REPORT DOCUMENTATION PAGE FOoMrBmNAoppr07o0v4e-d01 1a REPORTSECURITY CLASSIFICATION lb RESTRICTIVE MARKINGS Unclassified 2a SECURITYCLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITYOF REPORT Approved for public release: 2b DECLASSIFICATION/DOWNGRADING SCHEDULE Distribution is unlimited 4 PERFORMING ORGANIZATION REPORT NUMBER(S) 5 MONITORING ORGANIZATION REPORT NUMBER(S) 6a NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION Naval Postgraduate School (Ifapplicable) Naval Postgraduate School ME 6c. ADDRESS (City, State, andZIPCode) 7b ADDRESS(City. State, andZIPCode) Monterey, CA 93943-5000 Monterey CA 93943-5000 . 8a NAME OF FUNDING/SPONSORING OFFICE SYMBOL 9 PROCUREMENT INSTRUMENT IDENTIFICATION NUMBEI ORGANIZATION (Ifapplicable) 8c ADDRESS(City. State,andZIPCode) 10 SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK WORK UNIT ELEMENT NO NO NO ACCESSION NO 11 TITLE (Include SecurityClassification) WEIGHT OPTIMUM ARCH STRUCTURES 12 PERSONAL AUTHOR(S) THOMAS JAMFS WTI I.TAMSON 13a TYPE OF REPORT 13b TIME COVERED 14 DATE OF REPORT (Year,Month.Day) 115 PAGE COUNT MasifiEls Thpsis FROM TO JUNE 1992 117 SUPPLEMENTARY NOTATION iThe views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. COSATI CODES 18 SUBJECT TERMS (Continue onreverse ifnecessaryandidentifybyblock number) arches, optimization, mechanical structures 19 ABSTRACT (Continue onreverseifnecessaryandidentifybyblock number) The goal of this investigation is to design minimum weight arch structures which spar the distance between two points in two-dimensional space. An arch of unknown shape and variable cross-sectional width is modeled as a series of straight bar-beam elements. Finite Element Methods are used to compute the stresses in each element. Automated Design Synthesis (ADS) software is then used to vary the slope of each element and the cross-sectional width to prevent the yield stress of the material from being exceeded as ADS minmizes the volume of the arch to arrive at the minimum weight structure. Results are presented for a number of different loadings and boundary conditions. 20 DISTRIBUTION/AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION BUNCLASSIFIED/UNLIMITED SAME AS RPT DTIC USERS assified 22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE(IncludeAreaCode) 22c OFFICE SYMBOL told ^i-j"^ (408) 646-3426 1 ME/Sa DDForm1473, JUN 86 Previouseditionsareobsolete SECURITY CLASSIFICATION OF THIS PAGE S/N 0102-LF-014-6603 Unclassified T259295 i Approvedforpublicrelease;distributionisunlimited. WeightOptimumArchStructures by ThomasJamesWilliamson Lieutenant UnitedStatesNavy , B.S.M.E. United StatesNavalAcademy,1985 , Submittedinpartialfulfillment oftherequirementsforthedegreeof MASTEROFSCIENCEINMECHANICALENGINEERING fromthe NAVALPOSTGRADUATESCHOOL JUNE1992 ABSTRACT The goal of this investigation is to design minimum weight arch structures which span the distance between two points in two-dimensional space. An arch of unknown shape and variable cross-sectional width is modeled as a series of straight bar-beam elements. Finite Element Methods are used to compute the stresses in each element. Automated Design Synthesis (ADS) software is then used to vary the slope of each element and the cross-sectional width to prevent the yield stress of the material from being exceeded as ADS minimizes the arch volume to arrive at the minimum weight structure. Results are presented for a number of different loadings and boundary conditions. 6.1 TABLE OF CONTENTS I. INTRODUCTION 1 A. MOTIVATION 1 B. PROBLEM STATEMENT 4 II. PROBLEM FORMULATION A. PROBLEM STATEMENT AND ASSUMPTIONS B. MATHEMATICAL MODEL C. OPTIMIZATION PROBLEM 10 III. OPTIMIZATION ANALYSIS 12 A. STRATEGY 12 B. OPTIMIZER 13 C. ONE-DIMENSIONAL SEARCH • 16 D. ADS PROGRAM PARAMETERS 17 IV. STRESS ANALYSIS 18 A. STRESS DEVELOPMENT 18 ... B. FINITE ELEMENT BEAM EQUATION DEVELOPMENT 21 .... C. FINITE ELEMENT BAR EQUATION DEVELOPMENT 24 D. THE ELEMENTAL STIFFNESS MATRIX 26 E. COORDINATE TRANSFORMATION OF THE ELEMENTAL SYSTEM OF EQUATIONS 28 F. SOLUTION 31 V. PROGRAM DESCRIPTION 32 VI. CASE STUDIES 36 A. CASE 1: HINGED-HINGED ARCH WITH DISTRIBUTED LOAD 37 B. CASE 1A: HINGED-HINGED BEAM WITH DISTRIBUTED LOAD 40 C. CASE 2: HINGED-HINGED ARCH WITH CONCENTRATED LOAD 43 D. CASE 3: HINGED-HINGED ARCH WITH CONCENTRATED LOAD 46 E. CASE 4: HINGED-HINGED ARCH WITH CONCENTRATED MOMENT 49 F. CASE 5: ROLLER-HINGED ARCH WITH DISTRIBUTED LOAD 52 G. CASE 6: ROLLER-FIXED ARCH WITH DISTRIBUTED LOAD 55 H. CASE 7: FIXED-FREE ARCH WITH DISTRIBUTED LOAD 58 I. CASE 8: FIXED-FIXED ARCH WITH DISTRIBUTED LOAD 61 J. CASE 9: HINGED-HINGED ARCH WITH DISTRIBUTED LOAD 64 K. CASE 10: HINGED-HINGED ARCH WITH DISTRIBUTED LOAD 67 VII. CONCLUSIONS 70 APPENDIX A: JUSTIFICATION FOR OMITTING SHEAR STRESSES 72 . APPENDIX B: PROGRAM ARCHOPT COMPUTER CODE 75 APPENDIX C: CASE STUDIES 85 LIST OF REFERENCES 107 INITIAL DISTRIBUTION LIST 108 VI