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A FINITE ELEMENT PROCEDURE for NONLINEAR PREBUCKLING and INITIAL POSTBUCKLING ANALYSIS by SAC Man and R. He Gazzugher

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A FINITE ELEMENT PROCEDURE for NONLINEAR PREBUCKLING and INITIAL POSTBUCKLING ANALYSIS by SAC Man and R. He Gazzugher https://ntrs.nasa.gov/search.jsp?R=19720008222 2020-03-11T18:48:08+00:00Z View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NASA Technical Reports Server NASA CONTRACTOR REPORT *o m m w I PL U LOAN COPY R€TURN TO 4 &A AFh’L bout) 4 KIRTLAND AFB, N. M. z A FINITE ELEMENT PROCEDURE FOR NONLINEAR PREBUCKLING AND INITIALPOSTBUCKLING ANALYSIS by SAC Man and R. He GaZZugher Prepared by CORNELL UNIVERSITY Ithaca, N.Y. 1.4850 for Langley ResearchCenter NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, 0. C. JANUARY 1972 1. Rrpat No. 2. Government Acnsrion No. 3. Recipient's Catalog No. "~ NASA~" CR-1936i.~ .. ~ ~~ .. ... ."~_____ .. ~~ 7. Author(r) 8. Pkforming Orgsnization Report No. S.-T. Mau -and R. H. Gallagher "~ 10. Work Unit No. 9. Performing Orpaniution Name and Addren 126-14-16-01 Cornell University 11. Contract or Gnnt No. Ithaca, New York NGR 33-010-070 ~ ~~ ~~ . .. 13. Type of Report andPeriod Covered 2. SponroringAgency Name and Address Contractor Report National Aeronautics and Space Administration 14. SponsoringAgnncy Code Washington, D.C. 20546 5.' Supplementary Notes .". .". .~ 6. Absbact A procedure cast in a formappropriate to thefinite elementmethod is pre- sented for geometrically nonlinearprebuckling and postbuckling structural analysis, includingthe identification of snap-through type of buckling. The principal features of this procedure are the use of directiteration for solution of thenonlinear algebraicequations in the prebuckling range, an interpolation scheme for deter- mination of theinitial bifurcation point, a perturbation method in definition of the load-displacement behavior through the postbuckling regime, and extrapolation in determination of thelimit point for snap-through buckling. Three numerical examples are presented in illustration of the procedure and in comparison with alternativeapprmches. ~ .. "- .. ~~~~ " ~" - .* .. - ~ 7. Kay Words (Sug&ted brAuthor(r)) 18. DistributionStatement Finite element methods, huckling analysis, .nonlinearanalysis, imperfection Unclassified sensitivity " " ." "_ -~ . ~ . ~~~~ IS. .*urity &f. (of ths rapktt 20. Security Umif. (of this pqp) 21. NO. of P- zz. Rid Unclassified .Unclassified 56 . '9.00. " For sale by tho Nmtiorul Tmchnial Infomution Wi,Springfirld, Virginia 22161 - I TABLE OF CONTENTS Page LIST OF SYMBOLS V I. INTRODUCTION 1 11. ELEMENT AND SYSTEM FORMULATIONS 5 111. PREBUCKLING ANALYSIS 7 IV. DETERMINATION OF BIFURCATION a V. POSTBUCKLING ANALYSIS 9 VI. EFFECT OF IMPERFECTION 15 VII. EXTRAPOLATION METHOD FOR CALCULATION OF LIMIT POINT 18 VIII. ILLUSTRATIVE EXAMPLES 19 1. Clamped Thin Shallow Circular Arch.Under Uniform Load 19 2. Beam on Nonlinear Foundation 20 3. Flat Plate Post-buckling 22 IX. CONCLUDING REMARKS 25 APPENDIX: SOLUTION DETAILS FOR ILLUSTRATIVE EXAMPLES27 1. Clamped Thin Shallow Circular Arch Under Uniform Load 27 2. Beam on Nonlinear Foundation 30 3. Flat Plate Post-buckling 33 REFERENCES 35 FIGURES 38 iii LIST OF SYMBOLS A Cross-sectionalarea of arch. bo,bl.. .bm Coefficients in theextrapolation formula of X and Det. Det, Deti Determinant of thetotal stiffness matrix and Determinant atthe ith load level. 9 D7 Coefficientsin the parametric formulas of D31'D51 load and displacement. Derivatives of e w.r.t. the perturbation el~ell~elll parameter. E1 Flexural rigidity of the beam and arch cross-section. Field functionof lateral displacement of beam and its derivatives w.r.t. element coordinate 5. {F} ,Fi Element nodal force vector and component. h Depth of the cross-sectionof the circular arch and plate thickness. €13,1i Pattern of imperfection and component. Element linear stiffness matrix and coef- ficients. System linear stiffness matrix and coef- ficients. Spring constants of the non-linear foundation. Matrices associated with the pre-buckling solution. R Length of an element of the beam. L Length of the beam. M Applied moment. [nl(A)3,Cn,fA2) J Element geometric stiffness matrices. and coefficients. Nijk'Nijkk Coefficients associated with the prebuckling 'ijk' 'ljk displacements and their derivatives respec- tively. CNI Aggregategeometric stiffness matrix PO Arch uniform load intensity. V Structure nodal force vector and component. Circular arch radius. Displacement components. Beam initial displacement. Imperfection parameter. Displacements. Prebuckling displacements. Additional post-buckling displacements. Prebuckling displacement at bifurcation load and their derivatives. Components of the post-buckling displacements. Components of Ai . 2 AS The displacement chosen to be the path parameter 8. Components of A: in the power series expansion. Value of Ai at load levelX j‘ Path parameter. Axial strain of the arch. Arch circumferential coordinate and total angle. Nondimensional load parameter. The lth load level. Bifurcation load, limit load. Coefficients in the perturbation equation. Radial distance measured from the arch middle surface. 91’@2 Nodal slopes of the beam element. E Element local coordinate. nn Potential energy and ith componentof P’ Pi potential energy (i=1,2,3). vi I I. INTRODUCTION The analysis of instability phenomena of complicated thin shells has drawn intensified interest due, in part,to the dev- elopment of finite element analysis proceduresfor such struc- tures. (1) It is well known that structuresof this class col- lapse at load levels which are less than those predicted by linear instability theory because of the role played by initial imperfections and geometric nonlinearities. The extensive ef- forts in the development of theoriesto cope with the latter considerations have been surveyed by Hutchinson and (2)Koiter Other noteworthy surveys have been written by Haftka,a1 (3jet and Bienek(4) . Although the various typesof instability phenomena which might occur in the complete range of load-displacement behavior prior to final collapse are not as yet fully understood, certain forms are known and areof considerable practical importance, especially those which occurin the earliest stages of loading. These are illustrated in Figure 1. Curve a applies to "perfect" structures and represents the casein which the structure first displaces along the path defined by OAB (the fundamental path) and bifurcates (or branches) at the PointA to another path, OC. In CoratrBat to a rising postbuckling path, asOC, a descending path OD (as pictured in Figure lb) may be encountered. When the structure possesses fabricational imperfections the load-displacement behavior follows the paths indicated by dotted lines. The structure with a rising postbuckling path will have strength exceeding the bifurcation load. The strength of an imperfect structure witha descending postbuckling path in the perfect state will not achieve strengthsas high as the bifurcation load. Such structures, under the appropriate load condition, are termed "imperfection sensitive" and the maximum load attained (PointE) is termed the "limit point". A 'non-bifurcating load-displacement behavior may also occur for a structure assumed to be devoidof imperfections and may take the form shownin Figure IC, which is similar in shapeto the curve OE (Figure lb) of the imperfection-sensitive structure. Thus, a limit point is again encountered, atG, and the buckling 2 phe-nomenon isof the 'snap-through' type. A landmark developmentof procedures for establishing the shape of the postbuckling path andfor determining the limit point for imperfection-sensitive structuresIs due to Koiter.(5) Using the conceptof perturbations from the bifurcation point, this approach enables an efficient definitionof load-displace- ment behavior in the immediate postbuckling range. Further con- tributions or alternative forms of these concepts, in the classl- cal vein, have been presentedby Budiansky and Hutchinson (6), Sewell('l), and Thompson (8,9) Extensions of Koiter's procedure to the formatof finite element analysis, aswell as other finite element approaches to the same physical problem, have appeared I. Morin (10) ap- plies a predictor-corrector scheme in calculation of non-linear prebuckling behavior, in which a perturbation approachis em- ployed as the predictor and Newton-Raphson iterationis employed as the corrector. The perturbation approach,in both the pre- and post-buckling computational phases, draws heavily upon earlier work by Thompson and Walker(I1). Thompson has also ad- vocated a new perturbation approach (Reference9) for the subject type of problem. Haftka, et a1 (3) propose the definitionof an "equivalent structure", onein which the nonlinear terms are treated as initial imperfections, in order to exploit the con- cepts derived by Koiter for imperfect structures. Dupuis, et a1 (12), attack the solutionof the nonlinear equations inan incremental-iterative manner. The work by Lang (I3) is a direct adaptation of Koiter's concepts, including retentionof the condition of a linear prebuckling state. Recent analyses for both idealized structures(14) and for thin shells(I5) have shown that the assumptionof a linear pre- buckling state may lead to inaccurate results. Oneof the principal aspects of the work describedin this report is the method of determination of the load, and displacement stateon the fundamental path,at the bifurcation point following upon a nonlinear prebuckling state. The information so-calculated 3 furnishes the necessary ingredientsfor an analysis of the post- buckling or limit point behavior. Additionally, a new method for calculating the llmit pointof a perfect structure is sug- gested by the method of
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