FINITE ELEMENT ANALYSIS OF CONCRETE FRACTURE SPECIMENS .
FINITE ELEMENT ANALYSIS OF CONCRETEFRACTURE SPECIMENSbyLinda D. LeibengoodDavid DarwinRobert H. DoddsA Report on Research SponsoredbyTHE NATIONAL SCIENCE FOUNDATIONResearch Grant PFR 79-24696UNIVERSITY OF KANSASLAWRENCE, KANSASMay 1984
50272 JotREPORT DOCUMENTATIONPAGE12.L 1:. REPORT NO.i'3. Recic::uenfs Accession No.4. Title and SubtitleS. Report OateFinite Element Analysis of Concrete Fracture Specimens---7. AutMor(s)I May 1984--·------Linda D. Leibengood, David Darwin, and Robert H. Dodds9.8. SMrtoR ;p 1"N t o"liept. No.10. Project/Task/Work Unit No.P!!!rforming Orgi!llnlzation Name and AddressUniversity of Kansas Center fo·r Research, Inc.2291 Irving Hill Drive, 'lest CampueLawrence, KS 6604511. Contract(C) or Grant(Gl No.(Cl(G NSF Pf'R 79-2469613. Type of Report & Period Covered12. Sponsoring OriJaniutlon Name and AddressNa tiona l Science FoundationWashington, D.C. 20550. .14.-- -15. Supplementary Notes-115. Abstract (Limit: 200 wards)---- ---·---·-----·-.The effects of the descending branch of the tensile stress-strain curve, fracture energy,grid refinement, and load-step size on the response of finite element models of notchedconcrete beams are studied. The width of the process zone and constraint of crack anglesare investigated.Nonlinearity is limited to cracking of the concrete. A limiting tensile stress criteria ngoverns crack initiation. Concrete is represented as linear elastic prior to cracking.Cracks are modeled using a smeared representa ion. The post-crackir.g behavior is controlled by the shape of the descending branch, fracture energy, crack angle, and elementsize. Unloading occurs at a slope equal to the initial modulus of the material.Load-deflection curves and cracking patterns are used to evaluate the beam's response.Com pari sons of the process zone size are made. All analyses are performed on a 200 X200 X GOO mm concrete beam, with an initial notch length of 80 mm.The fracture energy, tensile strength, and shape of the descending branch interact todetermine the stiffness and general behavior of the specimen. The width of the processzone has a negligible influence on the beam's response. The importance of proper crackorientation is demonstrated. The model is demonstrated to be objective with respect togrid refinement and load-step size.17. Document Analysisa. Descriptorsconcrete, crack localization, cracking (fracturing), finite elements, fracturemechanics, fracture process zone, load-deflection, s tructura 1 engineering, tensionsoftening, unloadingb. Identifiers/Open-Ended Termsc. COSATI Field/Group18. Availability Statement19. Security Class (This Report)20. Security Class (This Page)Unclassified(See ANSJ-Z39.18)21. No. of PagesUnclassifiedRelease unlimitedSee /nstructrons on RI!!Yerse22. Price--OPTIONAL FORM 272 (4 77)(Formerly NTIS-35)Department of Commerce
iiACKN6WLEDGEMENTSThis report is based on a thesisleibengoodtosubmittedbylindaD.the Civil Engineering Department, University of Kan-sas, in partial fulfillment of the requirements forthedegreeofMaster of Science in Civil Engineering.The research was supported by the National Science PFR79-24696.KansasAdditional support wasGeneralResearchAllocation3131-X0-0038.Numerical computations were performed oncomputerattheComputerAidedEngineering, University of Kansas.EngineeringtheHarris500Facility, School of
iiiTABLE OF CONTENTSCHAPTER 1INTRODUCTION. . . . . . . . . .11.1 General.11.2Work.51.2.1 Stress Controlled, Smeared Cracking Models71.2.2 Fracture Mechanics Models . . . .10Objective and Scope.16CHAPTER 2 NUMERICAL PROCEDURES.181.3CHAPTER 3Prev1ous2.1General . . . . . . . . . . 182.2Concrete Mater1al Model . . . . 182.3Finite Elements.252.4Solution Procedures.262.4.1 General.262.4.2 Special Techniques . . . . .29NUMERICAL RESULTS AND DISCUSSION . 353.1General . . . . . . . . . .353.2Notched Beam Properties and Modeling Details 363.3Numerical Examples . . . . . . . 383.3.1 Effect of Tension Softening Representation 383.3.2 Discrete vs. Smeared Crack Representation.463.3.3 Fracture Energy Effects 513.3.4 Effects of Load Increment Size 543.3.5 Effects of Grid Refinement . 60
iv3.4Conclud1ng Remarks . . . . . . . . 62CHAPTER 4 SUMMARY AND CONCLUSIONS . . . . 65REFERENCES4.1Summary 654.2Cone 1 us ions 664.3Recommendations for Further Study 69.71APPENDIX A NOTATION 118
vLIST OF FIGURESFigure .1.1Stress Distribution in a Cracked ReinforcedConcrete Element (26)761.2Corner Supported, Center-Point Loaded Two-Way Slab,McNeice (23)771.3Load-Deflection Curves for Two-Way Slab Supportedat Corners, Hand, Pecknold, and Schnobrich (19), andBashur and Darwin (1,23)781.4Assumed Concrete Tensile Response. Scanlon (44):(a) Post-cracking Modulus Reduced to 20, 10, 5, or0 % of Initial Value: (bl Stepped Representation791.5Load-Deflection Curves for Two-Way Slab Supportedat Corners. Scanlon (23,44)801.6Load-Deflection Curves for Two-Way Slab Supportedat Corners, Lin and Scordelis (23,26)811.7Models Used by Gilbert and Warner (18) to Account forTension Softening in Concrete After Cracking:(a) Scanlon's Stepped Model: (b) Lin's GraduallyUnloading Model: (cl Discontinuous Model: Cdl ModifiedStress-Strain Diagram for Reinforcing Steel821.8Load-Deflection Curves for Two-Way Slab Supportedat Corners, Gilbert and Warner (18,23)832.1Interpretation of Smeared Crack Model: (a) SmearedRepresentation of Microcracked Element: (b) LumpedApproximation of Microcracked Element842.2Stress-Strain Relationships for Fracture ProcessZone: Cal Stress-Strain Relationship for MicrocrackedMaterial: (b) Equivalent Uniaxial Stress-StrainCurve for Tension Softening Material842.3Equivalent Uniaxial Stress-Strain Curve forTension Softening Material with Unloading85
vi 2.4Linear, Four Node, Isoparametric Element:(a) Parent Element: (b) Element in Structure862.5Quadratic, Eight Node, Isoparametric Element:(a) Parent Element: (b) Element in Structure862.6Determination of Envelope Strain from whichUnloading Occurs873.1Assumed Concrete Tensile Responses: (a) LinearSoftening: (b) Discontinuous Softening: (c) BilinearSoftening: (d) Dugdale Softening883.2Finite Element Model of Notched Beam893.3Nonlinear Portion of Finite Element Grid903.4Effect of Assumed Concrete Tensile Response onLoad-Deflection Curves913.5Effect of Assumed Concrete Tensile Response onFracture Process Zone Length923.6Crack Patterns for Beam with Linear Softening933.7Crack Patterns for Beam with Bilinear Softening943.8Crack Patterns for Beam with Discontinuous Softening953.9Crack Patterns for Beam with Dugdale Softening963.10Coordinate System Describing Region Ahead of aSharp Crack Tip973.11Stress Components Ahead of a Sharp Crack Tip983.12Comparison of Load-Deflection Curves of Discreteand Smeared Crack Models with Linear Softening993.13Comparison of Load-Deflection Curves of Discreteand Smeared Crack Models with Bilinear Softening1003 .14Comparison of Load-Deflection Curves of Discreteand Smeared Crack Models with Dugdale Softening1013.15Effect of Crack Angle Constraint and Width ofNonlinear Zone on Load-Deflection Curves ofBeam with Linear Softening102
vi i 3.16Effect of Crack Angle Constraint and Width ofNonlinear Zone on Fracture Process Zone Lengthin Beam with Linear Softening1033.17Crack Patterns for Beam with Linear Softening,1 Element Wide Nonlinear Zone, Unconstrained Cracks1043.18Crack Patterns for Beam with Linear Softening,l Element Wide Nonlinear Zone, Constrained CrackslOS3.19Effect of Fracture Energy on Load-Deflection Curvesof Beam with Discontinuous Softening1063.20Effect of Fracture Energy on Fracture Process ZoneLength in Beam with Discontinuous Softening1073 .21Crack Patterns for Beam with DiscontinuousSoftening, Fracture Energy SO N/m1083.22Crack Patterns for Beam with DiscontinuousSoftening, Fracture Energy 200 N/m1093.23Effect of Load Increment Size on Load-DeflectionCurves of Beam with Discontinuous Softening1103.24Crack Patterns for Beam with Discontinuous Softening,Load Applied in Small Increments (43 Steps)1113.2SNonrecoverable Energy after Crack Formation,Discontinuous Softening1123.26Effect of Load Increment Size on Load-DeflectionCurves of Beam with Linear Softening, 3 ElementWide Nonlinear Zone, Unconstrained Cracks1133.27Crack Patterns for Beam with Linear Softening,Load Applied in Small Increments (98 Steps)1143.28Cracking Sequence in Beam with Linear Softening:(a) Pattern in Side Column: (b) Pattern AlongBoundary between Side and Center Column11S3.29Effect of Grid Refinement on Load-Deflection Curvesof Beam with Linear Softening, l Element WideNonlinear Zone, Constrained Cracks116
viii3.30Effect of Grid Refinement on Fracture Process ZoneLength in Beam with Linear Softening, 1 ElementWide Nonlinear Zone, Constrained Cracks117
Chapter 1INTRODUCTION1.1 GeneralThe enonlinearconcrete make the development of a generalconstitutive model a difficult task.tension,toCracking oftheconcreteincompression softening, and the degradation of bond betweenthe reinforcement and concrete are a few of the nonlinearbehaviorsthat can be modeled.The selection of a constitutive model is influenced by thetypeofproblem beingconsidered.Ifcrackingdominatestheresponse of a structure while compressive stresses remain small, thenonlinearcompressive behavior of concrete may be neglected withoutadversely affecting predictions of structural response.mayPredictionsbe indicative of either the microscopic or macroscopic behaviorof the structure.crackresponse by estimating bond stresses, crack widths, dowel ef-fects, etc.actualMicroscopic analyses provide evaluations of localA discrete crack representation, whichgeometricdiscontinuityintroducedgenerally recommended for such analyses.byreplicatesthea crack, has beenSmearedcrackingmodelstreat cracks as stress discontinuities and Jess accurately model thecrack opening and the highThesedeficienciesconcerned with thedeflectionarestraingradientnearthecracktip.Jess important if the analyst is primarilygeneralcrackingresponse of a structure.patternsandoverallload-For such macroscopic analyses.
2the smeared cracking model is the method of choice.theNotonlydoessmeared model permit cracks to form and propagate in any direc-tion, but most importantly it requires no change in topology astheanalysis progresses.The interaction of finite element modeling parameters withtheconstitutivemodelmust also be understood.the predicted str.uctural response toload-stepsize,andelementchangestype?inHow sensitive isgridSystematic studies clarifyingthese interactions are required if an analyst is towhileobtainingrefinement,minimizecost,reasonable estimates of structural behavior.study is part of a continuing efforttoevaluatetheThiseffectsofvarious modeling parameters on the predicted macroscopic response ofa structure.descendingThe current work focuses on the effects of including abranchinthetensile stress-strain curve of concretewithin a smeared crack model.Early smeared cracking models reducedferredcrackingwasunrepresentativeThis suddeninclusionreinforcedreleasemethods.TheAs cracksformconcrete structure, the reinforcing steel carriesthe total load at the cracks.cracksenergyof a descending branch in the tensile stress-strain curveprovided one answer to these two distinct problems.atrans-of actual material behavior andposed numerical stability problems for some solutioninstressacross a newly formed crack to zero, as soon as the limitingtensile stress or strain was attained.attheHowever, the intact concretebetweenis still capable of transferring some tensile stress.although the concrete stress is zero at a crack,theaverageThus,con-
3crete stress over some length is non-zero (fig. 1.1).load is applied to the structure, and a continuouscracksform,thisAs additionaldistributionaverage stress will decrease to zero.ofMost in-vestigators (24,26,45) have modeled this behavior with adescendingbranch of the tensile stress-strain curve for concrete.In general,these more realistic models have predicted theresponseofstructures better than models without a descending branch.actualHowever,the data is far from conclusive.A number of researchers modeled the thin, simply supportedtwo-wayslab tested by McNeice (fig. 1.2).The shape and extent ofthe descending branch, element type, and degree ofmeshaffectedof the slab.Inthese early studies, the terminal point of the descending ementconcrete tensile stress-strain curve was empirically determinedand was held constant for changing branch shape orsize.Toaddvaryingelementto the confusion, some models without this "tensionsoftening" produced load-deflection curves that closely ashurcrackingtheand Darwin (1), using no tensionasacontinuousprocessnumerically integrating through the depth of the slab.over the entire load-deflection curve was achieved.byA good matchHand, Pecknold,and Schnobrich (19) used layered finite elements and represented ess.Althoughtheirpredictedfrom the experimental than Bashur s, theirresults were equally as good as those obtained with some modelsploying tension-softening (Fig. 1.3).em-
4If tension-softening is to become a viable modeling optionfor the analyst, rational guidelines for its use are required.shapes are most effective for the descending branch and why?the terminal point of the descending branch determined?WhatHow isIn what in-stances is tension-softening applicable?The existence of a descending forplainconcretehas been repeatedly verifiedIt follows that tension-softening could also bemodelingplainconcrete structures.usefulShould tension-softeningprocedures vary with the amount of reinforcementpresent?Andifso, why?The work of Hillerborg, Modeer, and Petersson (21) was instrumental in providing answers to some of these questions.borg, et al. applied fracturemechanicstheir "fictitious crack" model.principlesdevelopingCracks were discretely modeled, andwere assumed to transfer some stress if onlyareainHiller-partiallyopen.Theunder the stress-displacement curve was shown to be equivalentto the energy required to form a unit area of crack surface, a quantity referred to as the fracture energy.Bazant and Oh (6) extended Hillerborg, Modeer, and Petersson'sworktoa smearedcracking model.Triaxial stress-strainrelations were derived which provide for a gradual reduction inPoissoneffectasa crackopens.theOnly Mode I behavior was con-sidered, and the crack front was constrained to be one element dOheliminated the need to consider unloading after crack formation.If
5theprocess zone is several elements wide, cracks may form and thenunload as the structure undergoes additional deformation.Theas-sumption of a one element wide process zone simplifies the constitutive model but is not entirely realistic.processina mortarIn studying thespecimen, Mindess and Diamond (28) noted theformation of branching cracks.As additional load was applied, onlyone of the cracks continued to open and propagate.If this observedbehavior is to be modeled, the width of the crack front canarbitrarilycrackingnotberestricted, rather the model must permit cracks to formand then unload during subsequent loading of the structure.The current work generalizes Bazant's approach ns of plain concrete are used to re.deflection curves are used to evaluate thegridrefinement,ofonthecomputedCrack patterns and loadeffectsof changesinload-step size, fracture energy, and the shape ofthe descending branch.effectsincludeimposingThe effects of nonlinear zone sizeconstraintsandtheon the crack angles are also ex-amined.1.2 Previous Any model attempting to represent concreteincludesomemethoddistinctiveandmustfor modeling crack formation and propagation.The brittle, linear elastic, tensile responsemostbehaviorofconcreteoften dominant nonlinear behavior.the precise load at which a crack forms isoftenisAlthoughunimportant,fact that a structure is cracked must be considered.itsthe
6Two approaches have been used cretecrackingmodels(35)andcon-introduce a geometricdiscontinuity in a mesh by separating elementsNilsonaand Scordelis (34) predefined discretecracks in a beam in an effort to study local bond, steel,creteinattheboundaries.permitted crack propagation by separating common nodesof adjacent elements when the average stressexceededthetensilestrength.Although conceptually simple, the use of discrete crackingmodelshasbeen limited by the problems associated with a changingstructural topology.As cracks propagate and nodes are addedmesh, the bandwidth of the stiffness matrix increases.tion process is to remain efficient, some methodminimizingquired.doublefortoaIf the soluautomaticallythe bandwidth or new equation solving algorithms are re-Both approaches (30,31)This solution is useful ifcrack paths are predictable or if the analyst is willing to restrictpotentialpaths to predefined element boundaries.Saouma (42) per-mitted cracks to propagate in any direction by addingaswell as nodes to the mesh.newelementsThe bandwidth was then automaticallyminimized.The early problems associated with therequiredtopologyby discrete cracking led to the development of the smearedcrack model.thotropicchangingRashid (39) treated concrete as a linear elastic,material.or-After cracking, the material stiffness normalto the crack was eliminated.Thiseffectivelysimulatedthein-
7troductionof many finely spaced cracks perpendicular to the direc-tion of maximum principal oducesnothe mesh, only a stress discontinuity.Because of this, cracks may form in any direction while theinitialgrid remains unchanged.Both the smeared and discrete models have been eformodelingconcreteaggregate-as well as·post-cracking behavior.comprehensive review of cracking models may be46.representfoundinAReferenceThe remainder of this review will focus on the methods used torepresent acture mechanics based models.1.2.1 Stress Controlled, Smeared Crackjng ModelsScanlon (44,45) introduced the use of a descendingofthetensilestress-strain curve to account for the stress car-rying capacity of the intact concreteFinitelyingbetweenconcreteslabs.cracks * toformofreinforcedThe steel was assumed to be linear elastic and theconcrete was treated as linear elastic until cracking.*twoelement models composed of rectangular, layered, orthotropicplate elements were used to estimate the deflectionsforcedbranchvertically,andCrackswerea secant solution method was em-With these early models, the use of a descending branchwasreferred to as "tension stiffening" rather than tensionsoftening. The "stiffening" term was used because models with adescending branch were relatively stiffer than models with a suddenreducti
Finite Element Analysis of Concrete Fracture Specimens I May 1984 . -----·-----7. AutMor(s) . Finite Element Model of Notched Beam Nonlinear Portion of Finite Element Grid Effect of Assumed Concrete Tensile Response on Load-Deflection Curves
Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
Figure 3.5. Baseline finite element mesh for C-141 analysis 3-8 Figure 3.6. Baseline finite element mesh for B-727 analysis 3-9 Figure 3.7. Baseline finite element mesh for F-15 analysis 3-9 Figure 3.8. Uniform bias finite element mesh for C-141 analysis 3-14 Figure 3.9. Uniform bias finite element mesh for B-727 analysis 3-15 Figure 3.10.
1 Overview of Finite Element Method 3 1.1 Basic Concept 3 1.2 Historical Background 3 1.3 General Applicability of the Method 7 1.4 Engineering Applications of the Finite Element Method 10 1.5 General Description of the Finite Element Method 10 1.6 Comparison of Finite Element Method with Other Methods of Analysis
2.7 The solution of the finite element equation 35 2.8 Time for solution 37 2.9 The finite element software systems 37 2.9.1 Selection of the finite element softwaresystem 38 2.9.2 Training 38 2.9.3 LUSAS finite element system 39 CHAPTER 3: THEORETICAL PREDICTION OF THE DESIGN ANALYSIS OF THE HYDRAULIC PRESS MACHINE 3.1 Introduction 52
Finite Element Method Partial Differential Equations arise in the mathematical modelling of many engineering problems Analytical solution or exact solution is very complicated Alternative: Numerical Solution – Finite element method, finite difference method, finite volume method, boundary element method, discrete element method, etc. 9
3.2 Finite Element Equations 23 3.3 Stiffness Matrix of a Triangular Element 26 3.4 Assembly of the Global Equation System 27 3.5 Example of the Global Matrix Assembly 29 Problems 30 4 Finite Element Program 33 4.1 Object-oriented Approach to Finite Element Programming 33 4.2 Requirements for the Finite Element Application 34 4.2.1 Overall .
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000 The Finite Element Method Vol. 2 Solid Mechanics by O.C. Zienkiewicz and R.L. Taylor, Oxford : Butterworth Heinemann, 2000 Institute of Structural Engineering Method of Finite Elements II 2
Tulang-tulang atau cadaver yang digunakan untuk mempelajari ilmu anatomi ini adalah bagian tubuh manusia , YANG TIDAK BOLEH DIPERMAINKAN. Previllage menggunakan cadaver dan tulang guna mempelajari ilmu anatomi hanya dapat dipertanggung jawabkan, jika kita menggunakan kesempatan itu dengan maksud dan tujuan yang suci. 2. Dalam mempelajari cadaver dan tulang kita harus selalu ingat bahwa .
