A Six-Node Curved Triangular Element And A Four-Node .
NASA/CR-2004-210725A Six-Node Curved Triangular Element and aFour-Node Quadrilateral Element for Analysisof Laminated Composite Aerospace StructuresC. Wayne Martin and David M. BreinerMartin Engineering Inc.Lincoln, NebraskaUnder NASA Contracts NAS4-97007,NAS4-50079, NCA2-318, and NCA2-497July 2004
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NASA/CR-2004-210725A Six-Node Curved Triangular Element and aFour-Node Quadrilateral Element for Analysisof Laminated Composite Aerospace StructuresC. Wayne Martin and David M. BreinerMartin Engineering Inc.Lincoln, NebraskaPrepared forNASA Dryden Flight Research CenterEdwards, CaliforniaUnder NASA Contracts NAS4-97007,NAS4-50079, NCA2-318, and NCA2-497National Aeronautics andSpace AdministrationDryden Flight Research CenterEdwards, California 93523-0273July 2004
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CONTENTSPageSUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ELEMENT PERFORMANCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2MATHEMATICAL DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4CONCLUDING REMARKS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8ACKNOWLEDGEMENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9APPENDIX A: CURVED ELEMENT DEVELOPMENT. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19APPENDIX B: MODIFICATIONS TO Bo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27APPENDIX C: SHEAR PERPENDICULAR TO SURFACE IN FLAT ELEMENTS . . . . . . 28APPENDIX D: STRAIN CALCULATION AT NODES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36APPENDIX E: OUT-OF-PLANE ROTATION STIFFNESS . . . . . . . . . . . . . . . . . . . . . . . . . 39APPENDIX F: SIMULATED ANTI-SYMMETRIC BENDING MODE . . . . . . . . . . . . . . . . 42APPENDIX G: 6-NODE FLAT SHEAR PANEL (USING 10 NODE SHAPE FUNCTIONS). . 45TABLES1.NAFEMS LE2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.NAFEMS LE3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.Scordelis-Lo Roof Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.Morley Skew Plate Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.Plunkette’s Vibrating Wedge Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.NAFEMS T1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.Results for NAFEMS Laminated Strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168(a). NAFEMS LE2 Curved Shell Patch Test 1989 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 178(b). NAFEMS LE3 Pinched Hemisphere Shell 1989 Results . . . . . . . . . . . . . . . . . . . . . . . . 179.Error Summary for 6 and 8 Node Elements on Six Critical Test Cases. . . . . . . . . . . . . . . . . 18iii
FIGURESPage1.NAFEMS LE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.NAFEMS LE3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113(a). Scordelis-Lo Roof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123(b). Scordelis-Lo Roof Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124(a). Morley Skew Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134(b). Morley Skew Plate Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.Plunkett’s Vibrating Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.NAFEMS T1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.NAFEMS Laminating Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16C.1. Sign Conventions and Shear Deformation Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29(a). Deformation in x-z plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29(b). Deformation in y-z plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29(c). Shear deformation without bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29C.2.Nodal Configurations and Interpolation Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30G.1. 10-Node Triangle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45iv
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Form ApprovedOMB No. 0704-0188REPORT DOCUMENTATION PAGEPublic reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington,VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.1. AGENCY USE ONLY (Leave blank)2. REPORT DATE3. REPORT TYPE AND DATES COVEREDJuly 2004Contractor Report4. TITLE AND SUBTITLE5. FUNDING NUMBERSA Six-Node Curved Triangular Element and a Four-Node QuadrilateralElement for Analysis of Laminated Composite Aerospace Structures710-55-24-E8-RR-00-0006. AUTHOR(S)C. Wayne Martin and David M. Breiner7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)8. PERFORMING ORGANIZATIONREPORT NUMBERNASA Dryden Flight Research CenterP.O. Box 273Edwards, California 93523-0273H-24699. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)10. SPONSORING/MONITORINGAGENCY REPORT NUMBERNational Aeronautics and Space AdministrationWashington, DC 20546-0001NASA/CR-2004-21072511. SUPPLEMENTARY NOTESThis work was done as partial fulfillment for contract numbers NAS4-97007, NAS4-50079, NCA2-318, and NCA2-497ranging from 1991–1998. NASA Technical Monitor – Kajal K. Gupta, NASA Dryden Flight Research Center.12a. DISTRIBUTION/AVAILABILITY STATEMENT12b. DISTRIBUTION CODEUnclassified—UnlimitedSubject Category 05This report is available at http://www.dfrc.nasa.gov/DTRS/13. ABSTRACT (Maximum 200 words)Mathematical development and some computed results are presented for Mindlin plate and shell elements,suitable for analysis of laminated composite and sandwich structures. These elements use the conventional3 (plate) or 5 (shell) nodal degrees of freedom, have no communicable mechanisms, have no spurious shearenergy (no shear locking), have no spurious membrane energy (no membrane locking) and do not requirearbitrary reduction of out-of-plane shear moduli or under-integration. Artificial out-of-plane rotationalstiffnesses are added at the element level to avoid convergence problems or singularity due to flat spots inshells. This report discusses a 6-node curved triangular element and a 4-node quadrilateral element. Findingsshow that in regular rectangular meshes, the Martin-Breiner 6-node triangular curved shell (MB6) isapproximately equivalent to the conventional 8-node quadrilateral with 2 2 integration. The 4-nodequadrilateral (MB4) has very good accuracy for a 4-node element, and may be preferred in vibration analysisbecause of narrower bandwidth. The mathematical developments used in these elements, those discussed in theseven appendices, have been applied to elements with 3, 4, 6, and 10 nodes and can be applied to other nodalconfigurations.14. SUBJECT TERMS15. NUMBER OF PAGESAerospace engineering, Composites, Curved elements, Finite element, Structuraldesign17. SECURITY CLASSIFICATIONOF REPORTUnclassifiedNSN 7540-01-280-550018. SECURITY CLASSIFICATIONOF THIS PAGEUnclassified19. SECURITY CLASSIFICATIONOF ABSTRACTUnclassified5816. PRICE CODE20. LIMITATION OF ABSTRACTUnlimitedStandard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102
show that in regular rectangular meshes, the Martin-Breiner 6-node triangular curved shell (MB6) is approximately equivalent to the conventional 8-node quadrilateral with integration. The 4-node quadrilateral (MB4) has very good accuracy for a 4-node element, and may be prefe
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