Finite EI Ernent Analysis of Fracture in Concrete Structures

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AC1 446.3R-97

Finite EIernent Analysis of Fracture in Concrete Structures: State-of-the-Art

Reported by AC1 Committee 446

american concrete institute P.O. BOX 9094 FARMINGTON HILLS, MI 48333 international"

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First printing, January 1998

Finite Element Analysis of Fracture in Concrete Structures: State-of -the-Art

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AC1 446.3R-97

Finite Element Analysis of Fracture in Concrete Structures: State-of-t he-Art

Reported by AC1 Committee 446

Waiter Gerstle'Z David Secretary and Suòcommittee Co-Chai- Subcommittee Co-Chainnan

Farhadhsari YeOU-Sheng Jens Philip C. Perdis ïdenek P. Ba~ant'~ Mohammad T. Kazemi Gilles Pijaudier-Cabot oral Buyukozh~rk~'~ Neven Krstulovic victor E. saouma'" Ignaciocarol' vim C. Li Widsuaris Rolf Eligehausen Jacky NIazars Stuart E. swart^',^ Shu-Jin Fang'v3 Steven L. ~d~abe'd Tianxi Tang Ravindra Mu Christian Meyer Tatsuya Tsubaki Toshiaki Hasegawa Hirou> Mihashi' Cumaraswamy Vipuianandan Neil M.Hawkins Richard A. Miller Methi Wecharataua Anthony R ~ngraffeal*~ Sidney Mindess Yunping Xi Jeremy isenbea C. Dean Nomian Former member: Sheng-Taur

Fracture is an importan! mode of deformntion and damage in both plain Keywords: Concrete; crack; cracking; damage; discrete cracking; finite and reulfomd concrete structures. To accurutely predictfrecn

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4.2-Plain concrete The two main approaches used in FEM analysis to represent 4.3-Reinforced concrete cracking in concrete structures have been to 1) model cracks 4.4-Closure discretely (discrete crack approach); and 2) model cracks in a smeared fashion by applying an equivalent theory of con- Chapter !j-Conclusions, p. 446.3R-26 tinuum mechanics (smeared crack approach). A third ap- 5.1-General summary proach involves modeling the heterogeneous constituents of 5.2-Future work concrete at the size deof the aggregate (discrete particle approach) (Bazant et ai. 1990). Chapter &References, p. 446.3R-27 Kaplan (1961) seems to have been the fmt to have performed physical experiments regarding the fracme mechan- AppendixGlossary, p. 446.3R-32 ics of concrete structures. He applied the Chiffith (1920) fracture theory (modified in the middle of this century to be- CHAPTER 14NTRODUCTION come the theory of linear elastic fracture mechanics, or in this report, the state-of-the-art in hnite element modeiing LEW to evaluate experiments on concrete beams with of coneis viewed from a fracture mechanics perspective. crack-simulating notches. Kaplan concluded, with some res- Although finite element methais for modeling fmcture are un- ervations, that the Griffith concept (of a critical potential en- dergoing considemble change, the reader is presented with a ergy release rate or critical stress intensity factor being a snapshot of current mgand selected iiterature on the topic. condition for crack propagation) is applicable to concrete. His reservations seem to have been justified, since more re- 1.l-Background cently it has been demonstrated that LEFM is not applicable early turn of the 19th century, engiuetxs As as the realized to typical concrete structures. In 1976, Hillerborg, Modeer that of concrete behavior ddnot be ciescriw certain aspects and Petersson studied the process zone (FPZ) in based upon classical strength of materials tech- fracture or predicted front of a crack in a concrete structure, and found that it is niques. As the discipline of fracture mechanics has developed long and narrow. led to the development of the fictitious over the course of this century indeed, is stili developing), This (and crack model (FCM) (Hillerborg et al. 1976), which is one of it has become clear that a corn analysis of many concrete the simplest nonlinear discrete fracture mechanics models structures must include the ideas of fracaue mechanics. applicableto concrete structures. The need to apply fracture mechanics results from the fact that classical mechanics of materials techniques are inade- Finite element analysis was first applied to the cracking of quate to handle cases in which severe discontinuities, such as concrete structures by Clough (1962) and Scordelis and his cracks, exist in a material. For example, in a tension field, the coworkers Nilson and Ngo (Nilson 1967, Ngo and Scordelis stress at the tip of a crack tends to infinity if the material is 1967, Nilson 1968). Ngo and Scordeiis (1967) modeled dis- assumed to be elastic. Since no material can sustain infinite crete cracks, as shown in Fig. 1.1, but did not address the stress, a region of inelastic behavior must therefore surround problem of crack propagation. Nilson (1967) modeled pro- the crack tip. Classical techniques cannot, however, handle gressive discrete cracking, not by using fracture mechanics such complex phenomena. The discipline of fracture me- techniques, but rather by using a strength-based criterion. chanics was developed to provide techniques for predicting The stress singularity that occurs at the crack tip was not crack propagation behavior. modeled. Thus, since the maximum calculated s-ss near the Westergaard (1934) appears to have been the fmt to apply kip of a crack depends upon element size, the results were the concepts of fracture mechanics to concrete beams. With mesh-dependent (nonobjective). Since then, much of the re- the advent of computers in the 1940s, and the subsequent search and development in discrete numerical modeling of rapid development of the fuite element method (FEM) in the fracture of concrete structures has been carried out by in- 1950s, it did not take long before engineers attempted to an- graffea and his coworkers (hgra€fea 1977, hgraffea and alyze concrete structures using the FEM (Clough 1962, Ngo Manu 1980, Saouma 1981, Gerstle 1982, Ingraffea 1983, and Scordelis 1967, Nilson 1968, Rashid 1968, Cervenka Gerstle 1986, Wawrzynek and Ingraffea 1987, Swenson and and Gerstie 1971, Cervenka and Gerstle 1972). However, ingraffea 1988, Wawrzynek and Ingraffea 1989, Ingraffea even with the power of the EM, engineers faced certain 1990, Martha et ai. 1991) and by Hiilerborg and coworkers problems in trying to model concrete smctures. It became (Hillerborg et ai. 1976, Petersson 1981, Gustafsson 1985). apparent that concrete structures usuaiiy do not behave in a Another important approach to modeling of fracture in way consistent with the assumptions of classical continuum concrete structures is called the smeared crack model (Rash- mechanics (Bazant 1976). id 1968). in the smeared crack model, cracks are modeled by , Fortunately, the EMis sufficiently general that it can changing the constitutive (stress-strain) relations of the solid model continuum mechanical phenomena as weil as discrete continuum in the vicinity of the crack. This approach has phenomena (such as cracks and interfaces). Engineers per- been used by many investigators (Cervenka and Gerstle . forming finite element analysis of reinforced concrete struc- 1972, Darwin and Pecknold 1976, Bazant 1976, Meyer and tures over the past thirty years have gradually begun to Bathe 1982, Chen 1982, Balakrishnan and Murray 1988). recognize the importance of discrete mechanical behavior of Bazant (1976) seems to have been the fmt to realize that, be- concrete. Fracture mechanics may be defined as that set of cause of its strain-softening nature, concrete cannot be mod- ideas or concepts that describe the transition from continu- eled as a pure continuum. Zones of damage tend to localize ous to discrete behavior as separation of a material occurs. to a size scale that is of the order of the size of the aggregate.

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FiNilE ELEMENT ANALYSIS OF FRAWRE IN CONCRETE STRUCTURES 4463R-3

STCCL CLEMENT Fig. 1.1-îñejrstjnite element model of a cracked reinforced concrete beam (Ngo and Scordelis 1967)

Therefore, for concrete to be modeled as a continuum, ac- 1.2-Scope of report count must be taken of the size of the heterogeneous struc- Several previous state-of-the-art reports and symposium ture of the material. This implies that the maximum size of proceedings discuss finite element modeling of concrete finite elements used to model strain softening behavior structures (ASCE Task Committee 1982, Elfgren 1989, should be linked to the aggregate size. If the scale of the Computer-Aided 1984,1990, Fracture Mechanics 1989, Fir- structure is small, this presents no particular problem. How- rao 1990, van Mer et ai. 1991, Concrete Design 1992, Frac- ever, if the scale of the structure is large compared to the size ture 1992, Finite Element 1993, Computational Modeling of its internal structure (aggregate size), stress intensity fac- 1994, Fracture and Damage 1994, Fracture 1995). This ie tors (fundamental parameters in LEFM) may provide a more port provides an overview of the topic, with emphasis on the efficient method for modeling crack propagation than the application of fracaire mechanics techniques. The two most smeared crack approach (Griffith 1920, Bazant 1976). Most commonly applied approaches to the FEM analysis of frac- structures of interest are of a size between these two ex- ture in concrete structures are emphasized. The first ap- tremes, and controversy currently exists as to which of these proach, described in Chapter 2, is the discrete crack model. approaches (discrete fracnire mechanics or smeared cracking The second approach, described in Chapter 3, is the smeared continuum mechanics) is more effective. This report de- crack model. Chapter 4 presents a review of the literature of scribes both the discrete and smeared cracking methods. applications of the finite element technique to problems in- These two approaches, however, are not mutually exclusive, volving cracking of concrete. Finally, some general conclu- as shown, for example, by Elices and Planas (1989). sions and recommendations for future research are given in When fmt used to model concrete structures, it was expect- Chapter 5. ed that the F'EM could be used to solve many problems for No attempt is made to summarize all of the literature in the which classid solutions were not available. However, even area of FFM modeling of fracture in concrete. There are sev- this powerful numericai tool has proven to be difficult to apply eral thousand references deaiing simultaneously with the when the sîrength of a structure or structural element is con- FEM,fracture, and concrete. An effort is made to crystaiiize trolled by cracking. When some of the early fmite element the codusing amy of approaches. The most important ap- analyses are studied critically in light of recent developments, proaches are described in detail sufficient to enable the reader they are found to be nonobjective or inwrre-ct in terms of the to develop an overview of the field. References to the litera- current understanding of fracture mechanics, aithough many ture are provided so that the reader can obtain further details, produced a close match with experimental results. It is now as desired. The reader is referred to AC1 44ó.lR for an intro- clear that any lack of success in these models was not due to a duction to the basic concepts of fhcture mechanics, with spe- weakness in the FEM, but rather due to incorrect approaches cial emphasis on the application of the field to concrete. used to model cracks. in many cases, success can be achieved only if the principles of fracniremechanics are brought to bear CHAPTER 2-DISCRETE CRACK MODELS on the problem of cracking in plain and reinforced concrete. A discrete crack model treats a crack as a geometrid entity. These techniques have not only proven to be powerful, but In the FEM,unless the crack path is known in advance, dis- have begun to provide explanations for materiai behavior and crete cracks are usually modeled by altering the mesh to ac- predictions of structurai response that have previously been commodate propagating cracks. In the past, this remeshing poorly or inconectly understood. process has been a tedious and Micult job, relegated to the While some preliminary research has been performed in the analyst. However, newer software techniques now enable the finite element modeihg of cracking in three-dimensional stnic- remeshing process, at least in two-dimensional problems, to tures (Gerstle et ai. 1987, Wawrzynek and Jngraffea 1987, In- be accomplished automatically by the computer. A zone of in- @ea 1990, Martha et al. 1991), the state-of-theart in the elastic material behavior, cailed the fracture process zone fmcture analysis of concrete structures seems cmentiy to be (FPZ), exists at the tip of a discrete crack, in which the two geriemily to two-dimensional models of smctures. sides of the crack may apply tractions to each other. These limited --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

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tractions are generally thought of as nonlinear functions of the (about 2000 degrees of freedom are required to obtain 5 per- relative displacements between the sides of the crack cent.accuracy in the stress intensity factor solution). At this the, singular finite elements had not yet been developed 2.1 -Historical background (singular elements exactly model the stress state at the tip of Finite element modeling of discrete cracks in concrete a crack). in their paper, Chan et ai. pointed out that there beams was fmt attempted by Ngo and Scordeiis (1967) by were then three ways to obtain stress intensity factors from a introducingcracks into the frnite element mesh by separating finite element soiution: (1) displacement correlation; (2) * elements along the crack trajectory, as shown in Fig. 1.1. stress correlation; and (3) energy release rate methods (he They did not, however, attempt to model crack propagation. integrai or potentiai energy derivative approaches). Had they done so, they would have found many problems, Wilson (1969) appears to have been the first to have devel- starting with the fact that the stresses at the tips of the cmcks oped a singular crack tip element. Shortly thereafter, Tracey increase without bound as the element size is reduced, and (197 1) developed a triangular singular crack tip finite ele- no convergence (of crack tip stresses) to a solution would ment that required far fewer degrees of freedom than analy- have been obtained. Also, in light of the findings of Hilier- sis with regular elements to obtain accurate stress intensity borg et al. (1976) that a crack in concrete has a gradually factor. At about the same time, Tong, Pian, and Lasry (1973) softening region of significant length at its tip, it was inaccu- developed and experimented with hybrid singular crack tip rate to model cracks with traction-free surfaces. It is notable elements (including stress-intensity factors, as well as dis- that Ngo and Scordelis also grappled with the theoretically placement components, as degrees of freedom). difficult issue of connecting the reinforcing elements with Jordan (1970) noticed that shifting the midside nodes the concrete elements via “bond-link“ elements. along adjacent sides of an eight-noded quadrilateral toward Nilson (1967,1968) was the fmt to consider a FEM model the comer node by one-quarter of the element’s side length to represent propgation of discrete cracks in concrete stNc- caused the Jacobian of ttansformation to become zero at the tures. Quoting from his thesis Wilson 1967): comer node of the element. This led to the discovery by Hen- shell and Shaw (1975) and Barsom (1976) that the shift ai- The present anaiysis includes consideraiion of progressive lowed the singular stress field to be modeled exactly for an cracking. The uncracked member is loaded incrementally until elastic material. Thus, standard quadratic element with mid- previously defined cracking criteria are exceeded at one or side nodes shifted to the quarter-points can be used as a r-ln more. locations in the member. Execution teminates, and the singularity element for modeling stresses at the tip of a crack computer output is subjected to visuai inspection. If the average in a linear elastic medium. value of the principai tensile stress in two adjacent elements The virtuai crack extension method for calculating Mode I the ultimate tensile strength of the concrete, then a exceeds stress intensity factors was developed independently by between those two elements along their crack is defined com- Hellen (1975) and Park (1974). In this method, G, the rate mon edge. This is done by establishing two disco~e~tednodal of change of potential energy per unit crack extension, is cal- points at their common comer or comers where there formerly culated by a finite difference approach. This approach does was only one. When the principai tension acts at an angle to the not require the use of singular elements to obtain Mode I boundaries of the element, then the crack is &fined along the (opening mode) stress-intensity factors. Recently, it has been side most nearly nodto the principai tension direction. found that by decomposing the displacement field into sym- The newly defined member, with cracks (and perhaps partid metric and antisymmetric components with respect to the bond failure), is then re-loaded from zero in a second loading crack tip, the method may also be extended to calculate stage, also incrementally applied to account for the noniinear- Mode II (sliding) and Mode Ul (tearing) energy release rates ities involved. OnCe again the execution terminated is if and stress-intensity factors (Sha and Yang 1990, Shumin and cracking criteria are exded. The incremental extension of Xing 1990, Rahulkumar 1992). the crack is recorded, and the memkr loaded incrementally in the third stage, and so on. in this way, crack propagation Having developed the capability to compute stress intensity may be studied and the extent of cracking at any stage of factors using the FEU, the next big step was to model heax loading is obtained. elastic crack propgution using fmcture mechanics principles. This was started for concrete by inflea (1973, and continued The problems associated with this approach to discrete by hgraffea and Manu (1980), Saouma (1981), Gerstle (1982, crack propagation analysis are three-fold (1) cracks in con- 1986), Wawnynek and Ingraffea (1986), and Swenson and In- crete structures of typical size scale develop gradually (Hill- graffea (1988). These attempts were primany aimed at facilitat- erborg et al. 1976), rather than abruptly; (2) the procedure ing the process of discrete crack propagation thtough automatic forces the cracks to coincide with the predefined element crack trajectory computations and semi-automatic remedung to boundaries; and (3) the energy dissipated upon crack propa- allow discrete crack propagation b be modeled. Currently, the gation is unlikely to match that in the actual structure, result- main techaical difficulties involved in modeling of discrete , ing in a spurious solution. LEFM crack propagation are in the 3D regime. In 2D applica- in the 1970s, great strides were made in modeling of tions, automatic propagation and remeshing algorithms have LEFM using the FEM. Chan, Tuba, and Wilson (1970) been reasonably successíul and are improving. In three-dimen- pointed out that a large number of triangular constant stress sional modeling, automatic remeshing algorithms are on the frnite elements are requiredto obtain accurate stress intensity verge of king sufficiently developed to model general crack factor solutions using a displacement correlation technique propagation, and computers are just becomuig powerfui enough --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

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FINITE ELEMENT ANALYSIS OF FRACTURE IN CONCRETE STRUCTURES 446.3R-5

to acculately solve problems with complex geometries caused a free surface. Fig. 2.1@) shows the case where the least di- by the pmpagation of a number of discrete cracks. mension is the crack length itself. Fig. 2. i@) shows the me Another development in discrete crack modeling of con- where the least dimension is controlled by the crack tip pass- crete structures has been the realization that LEFM does not ing close by a reiuforcing bar. [Of come, if the reinforcement apply to structural members of normai size, because the FPZ is considered as a smeared (rather than discrete) constituent of in concrete is relatively large compared to size of the mem- the reinforced concrete composite, then it need not be mcd- ber. "his has led to the development of finite element modeled discretely, and the constitutive reiations and the FPZ must eling of nonlinear discrete friicture-usuaiiy as the comspondingly include the effect of the smeared reinforcing implementation of the fictitious crack model (FCM) (Hilier- bars.] Fig. 2.l(d) shows the case where the least dimension is borg et al. 1976), in which the crack is considered to be a controlled by the size of the iigament (the remaining un- strain softening zone modeled by cohesive nodai forces or by cracked dimension of the member). In Fig. 2.1(e), the least di- interface elements [first developed by Goodman, Taylor, and mension is governed by a kink in the crack. Fdy,Fig. 2.10 Brekke ( 1968)l. shows an example of the least dimension being controiied by Finally, there appear to be situations in which even the the radius of curvature of the crack. FCM seems inadequate to model realistic concrete behavior As explained in Chapter 2 of AC1 446.1& one of the funda- in the FPZ. In this case, a smeared crack model of some kind, mental assumptions of LEFM is that the size of the FPZ is neg- as described in Chapter 3, becomes necessary. ligible (say, no more than one percent of the least dimension associated with the crack tip). It is this assumption that aliows 2.2-Linear Elastic Fracture Mechanics (LEFM) for a theoretical stress distnbution near the crack ti in linear Linear elastic fracture mechanics (LEFM) is an important elastic materials, in which the stress varies with i$ in which approach to the fracture modeling of concrete structures, ris the distance from the crack tip. Stress-intensity factors KI, even though it is only applicable to very large (say several KI[, and Kill are defineû as the magnitudes of the singular meters in length) cracks. For cracks that are smailer than this, stress fields for Mode i, Mode II, and Mode IíI cracks, respec- LEFM over-predicts the load at which the crack will propa- tively. if the FPZ is not smaü compared to the least dimension, then singular stress fields may not be assumed to exist, and

gate. To determine whether LEW may be used or whether --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- nonlinear fracture mechanics is necessary for a particular consequently, KI,KI[, and Killare not defined for such a crack problem, one must determine the size of the steady state frac- tip. in such a case, the FPZ must be modeled expiicitly and a ture process zone (FPZ) compared to the least dimension as- noniinear frachue model is required. sociated with the crack tip (AC1 446.1R). The FPZ size and As mentioned earlier, fracture process zones in concrete the crack tip least dimension are discussed next. can be on the order of 1 ft (0.3 m) or more in length. For the The FPZ may be defined as the area surrounding a crack tip great majority of concrete structures, least dimensions are within which inelastic material behavior occurs. The FPZ size less than severai feet. Therefore, fracture in these types of grows as load is applied to a crack, until it has developed to the structures must be modeled using nonlinear fracture me- point that the (traction-free) crack begins to propagate. If the chanics. Only in very large concrete structures, for example, size of the FPZ is small compared to other dimensions in the dams, is it possible to apply LEFM appropriately. For dams structure, then the assumptions of LEFM lead to the conclu- with large aggregate, possibly on the size scale of meters, sion that the Fpz will exhibit nonchanging characteristics as LEFM may not be applicable because of the correspondingly the crack propagates. This is calied the steady state FPZ. The larger size of the FPZ. size of the steady state FPZ depends only upon the material Even though it is recognized that LEM is not applicable properties. In concrete, as opposed to metais, the FPZ can of- to typical concrete structures, it is appropriate to review the ten be thought of as an interface separation phenomenon, with details of the finite element analysis of LEFM.Then, in Sec- little accompanying volumetric damage. The characteristics of tion 2.3, the finte element analysis of nonlinear discrete the steady state FPZ depend upon the aggregate size, shape fracture mechanics will be presented. and strength, and upon microstructurai details of the particuiar 2.2.1 Fracture criteria: k; G, mixed-mode models concrete under consideration. The FPZ was first studied in de- Stress-intensity factors KI, KI~,and KIIIor energy release tail by Hiilerborg, Moder, and Petersson (1976). The size of rates GI,GI[, and GIIImay be used in LEFM to predict crack the FPZ depends on the model used in the study. For example, equilibrium conditions and propagation trajectories. There in the anaiysis carried out by ïngrafíea and Gerstle (1985) for are several theories that can be used to predict the direction normal strength concrete, the steady state FPZ ranged from 6 of crack propagation. These include, for quasistatic prob- in. (150 mm) to 3 ft (1 m) in length. lems, the maximum circumferential tensile stress theory (Er- The least dimension (L.D.) associated with a crack tip is dogan and Sih 19631, the maximum energy release rate best defined with the aid of Fig. 2.1 (Gerstle and Abdalla theory (Hussain et al. 1974), and the minimum strain energy 1990). The least dimension is used to calculate an approxi- density theory (Sih 1974). These theories all give practically mate radius surrounding the crack tip within which the singu- the same crack trajectories and loads at which crack exten- lar stress field can be guaranteed to dominate the solution. The sion takes place, and therefore the theory of choice depends least dimension can be defined as the distance from the crack primarily upon convenience of implementation. Each of tip to the nearest discontinuity that might cause a local distur- these theories may also be applied to dynamic fracture prop- bance in the stress field. Fig. 24a) shows the case where the agation problems (Swenson 1986). As in metais, cyclic fa- crack tip L.D. is controlled by the proximity to the crack tip of tigue crack propagation in concrete may be modeled with the

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446.3R-6 AC1 COMMITTEE REPORT

L.D. = Least Dimension

Reinforcing-

(e)

L.D. = Radius of Curvature

Fig. 2. I-Examples illustrating the concept of “leastdimension (LD.)” associated with a crack tip (Gerstle and Abdalla I990)

Paris Model (Barsom and Rolfe 1987) in conjunction with cial-purpose hybrid elements that are not usually included in the mixed-mode crack propagation theories just mentioned. standard displacement-based finite element codes, and will However, it is rare that an unreinforced concrete structure is not be discussed in further detail here. The most successful both (1) large enough to merit LEFM treatment and (2) sub- displacement-based elements are the Tracey element ject to fatigue loading. (Tracey 1971) and the quarterpoint quadratic triangular iso- in most of the literature on discrete crack propagation in parametric element (Henshell and Shaw 1975, Barsoum concrete structures, it has been considered necessary to mod- 1976, Saouma 1981, Saouma and Schwemmer 1984). Most el the stress singularity at a crack tip using singular elements. general purpose finte element codes unfortunately do not in- However, accurate results can also be obtained without mod- clude the Tracey element, but they do include six noded tri- eling the stress singularity, but rather by calculating the en- angular elements, which can then be used as singular ergy release rates directly (Sha and Yang 1990, Rahulkumar quarterpoint crack tip elements. 1992). However, for a comprehensive treatment, we discuss After a finite element analysis has been completed, stress- modeling of stress singularities next. intensity factors can be extractedby several approaches. The 2.2.2 FEM modeling of singuhüies and stress intensity most accurate methods are the energy approaches: the J-in- . factors tegral, virtual crack extension, or stiffhess derivative meth- Special-purpose singular finite elements have been creat- ods. However, these approaches are not as easy to apply for ed with stress-intensity factors included explicitly as de- the case of mixeû-mode crack propagation, and have been grees-of-freedom (Byskov 1970, Tong and pian 1973, Atiun applied only rarely to three-dimensional problem (Shivaku- et ai. 1975, Mau and Yang 1977). However, these are spe- mar et al. 1988). Simpler to apply (for mixed-mode fracture

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Fig. 2.2-Nomenclature for 20quarter point singular isoparametric elements --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- mechanics) are the displacement correlation techniques. Be- the near-field displacements (u,v), in terms of polar coordi- cause these techniques sample local displacements at various nates r and 9, shown in Fig. 2.2, are given by: points, and correlate these with the theoretical displacement field associated with a crack tip, they are generally not as ac- 2K1(1 +v) curate as the energy approaches, which use integrated infor- (2.1) mation. The displacement correlation techniques are usually only when singular elements are employed, while the used 2KIl(l+V) &,e 2-2v+cos2 e energy approaches are used for determining energy release E 2[ 21 rates for cracks that may or may not be discretized with, the

help of singular elements. The displacement and stress correlation techniques assume y = zK,(t-~,in~~-2v-c~2E 2x 2 B21 + (2.2) that the fite element solution near the crack tip is of the same form as the singuiar neat-field solution predicted by LEFM

2Kll(' + v)&cos:[- 1 + 2v + sin 2 (Broek 1986). By matching the (known)finite element solu- E tion with the (known, except for KI,Ki[, and KIIl)theoretical near-field LEFM solution, it is possible to caldate the stress- in which u and are and perpendicular to the crack intensity factors. only equations are needed to ob- Since three face, respectively. tain the three stress-intensity factors, while many points that Now a crack tip node smmdd by guaaerpoint can be matched, there are many possible schemes for correla- tn.pk elementsshown in Fig. 2.2. radial These 'Odal revnsesOnly On the coo&&, r, dong &e si& AC, by ushg quadrafic shape func- Crack surfaces and least-wUares Of all Of the 'Odal re- tions nodm A, B, C, and solving for he mt- sponses associated with the singular elements. UIZil trianguiar area coordinate Ei in tem of r, we obtain: The displacement correlation approach is more accurate than the stress correlation approach because displacements converge more rapidly than stresses using the FEM. There- (2.3) fore only the displacement correlation approach is discussed in detail here (Shih et al. 1976). where L is the length of the side AC. Now interpolating the Consider a linear elastic isotropic material with Young's displacements along the side AC by using the computed dis- modulus E and Poisson's ratio v. For the case of plane strain, placement components at nodes A, B, and C, and using Eq.

446.3R-0 AC1 COMMITTEE REPORT

(2.3), we obtain the displacements along the crack surface strain. Here, GI,GI[, and GIIIare the potential energy release AC in terms of r. These are given by: rates created by collinear crack extension due to Mode I, Mode II, and Mode III deformations, respectively. in the simplest approach, the total energy release rate, G = GI+ Glf + GllI can be caiculated by performing an analysis, calculating the total potential energy, xA,colíinearly extending the crack by a small amount da, repeifoming the analysis to obtain xg, and then using a finite difference to approximate G as G = (xA- ICB)/ aa. If GI,GI[, and Ci11are quired separate- Simiiarly the displacements alongside AE can be written as: ly, they can be calculated by decomposing the crack tip displacement and the stress fields into Mode I, Mode II, and Mode Ill components (Rahuikumar 1992). The stiffness derivative method for determination of the stress-intensity factor for Made I (2D and 3D) crack pmb- lems was introduced by Parks (1974). The method is equiv- v = vA+(-3vA+4v,-vE) -+(2vA-4v,+2vE)L (2.7) L L alent to the J-integral approach (described later). With referenœ to Fig. 23, any set of fínite elements that Subtracting Eq. 2.6 from 2.4 and subtracting 2.7 from 2.5, forms a closed path around the crack tip may be chosen. The the crack opening displacement (COD)and crack sliding dis- simplest set to choose is the set of elements around the crack tip. placement (CSD) are computed as: The stiffness derivative method involves determination of the stress-intensity factor from a calculation of the potential energy decrease per unit crack advance, G. For plane strain COD = (4v,- vc-4v, + v,)&+ (- (4v,+ 2vc) + 4v,- 2vE)i (2.8) and unit thickness, the relation between KI and G is

Analyticai solutions for COD and CSD can be obtained by in which P is the potential energy, a is the crack length, E is evaìuating the displacement components u and v given by Young's modulus, and v is Poisson's ratio. Eqs. 2.1 and 2.2 for 8 = +ICand 8 =-II:and subtracting the val- Parks (1974) shows that the potential energy, K, in the ues at 8 = -x from the values at 8 = +x. Equating the like problem is given by: terms in the finite element and the analytical COD and CSD profiles, the stress intensity factors are given by:

E = g2(i+v)(3-4v)~4~vB~"D~~vE~vc1 (2*10) inwhich [fi istheglobaistiffnessmatrix,and isthevec- tor of prescribed nodai loads. Eq. 2.13 is differentiated with respect to crack length, a, to obtain the energy release rate as E [4(u,-u,)+uE-u,l (2.11)

(2.14) Thus--```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- by meshing the crack tip region with quarter-point quadratic triangular elements and solving for the displacements, the stress intensity factors can be computed by using The matrix '2 represents the change in the structure stüï- Eqs. 2.10 and 2.1 1. This technique does not require any spe- ness matrix per unit of crack length advance. The term aman is cial subroutines to develop the stiffness matrix for the singu- nil if the crack tip area is unloaded. The key to understanding lar elements. A single subroutine can be written to calculate the stiffhess derivative method is to imagine representing an the length L of the sides AC and AE, retrieve the displace- increment of crack advance with the mesh shown in Fig. 2.3 ment components at the nodes A, B, C, D,and E and thereby by rigidiy translating ail nodes on and within a contour r, (see compute the stress-intensity factors using Eqs. 2.10 and 2.1 1. Fig. 2.3) about the crack tip by an infinitesimal amount Au in ingrafíea and Manu (1980) have developed similar equa- the x-direction. Ail nodes on and outside of contour rl remain tions for the computation of stress-intensity factors in three in their initial position. Thus the global stiffness matrix [fi, dimensions with quarterpoint quadratic elements. In three which depends on only individual element geomemes, dis- dimensions, the crack tip is replaced by the crack front, the placement functions, and elastic material pmperties, remains crack edge by the crack face. unchanged in the regions interior to r, and exterior to ri,and Energy approaches for extracting stress-intensity factors the only Contributions to the fust term of Eq. 2.14 come from make use of the fact that KI = [EGI]la,KII = [EGII]':, KjiI the band of elements between the contours r, and ri. The = [EGII+(1 + v)]IRfor plane stress and KI = [EGI/l - v )] , structure stiffness matrix [KI is the sum over ali elements of K~~= [EG~~/(~- v 2)I 1/2 , K~~~= [EG~~+(I+ V)I'/~ for plane the element stiffness mamces [Ki]. Therefore,

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r '1

4 c - a+Aaa * 'L --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

Fig. 2.63-Integral mmnchture (Rice 1968)

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elements should be related to the size of the region within (2.15) which near-field solution is valid. For 5 percent accuracy in stress-intensity factors, the singular elements should be about 1/5 of the size of the least dimension associated with in which [K;]is the element stiffriess matrix of an element be- the crack tip. For one percent accuracy, the singular elements the contours and is the number of such el- tween r, ri,and N, should be about of the size of the least dimension asso- ements. The derivatives of the element stiffness matrices can ciated with the crack tip (&=tie and Abdalla 1990). be caiculated numericaily by taking a finite difference: 4. Reguiar quadratic elements should be limited in size, s, by their clear disrance, b, from a crack tip. The ratio of sí& (2.16) should not exceed unity to achieve 30 percent error, and should not exceed 0.2 to achieve 1 percent error in the near field solution (Gerstle and Abdalla 1990). The method may be extended to mixed-mode cracks. The meshing criteria given above show that a large num- The J-Integrai method (Rice lW8) for determining the en- ber of elements are required at a crack tip to obtain accurate ergy release rate of a Mode I crack is useful for determinhg near-field stresses. Experience shows that 300 degrees of energy release rates, not only for LEFM crack propagation, Momare required per crack tip to reliably obtain 5 per- but also for nonlinear fracture problems. For a two-dimen- cent accuracy in the near field stresses (Gerstle and Abdalia sional problem, a path r is traversed in a.counter-clockwise 1990). This becomes prohibitive from a computational sense between the two crack surfaces, as shown in Fig. 2.4. standpoint for problems with more than one crack tip. The J-integrai is defined as: Fortunately, it is not necessary to accurately model near- field stresses to calculate accurate stress intensity factors. in (2.17) fact, using no singular elements, energy methods can be used, as described above, to obtain accurate stress intensity factors with far fewer than 300 degrees of freedom per crack tip.

where summation over the range of repeated indices is un- --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- derstood. 2.3-Fictitious Crack Model (FCM) Here, w=~o,&~,,i,j = 1,2, 3 is the strain energy density, Since 1%1, there has been a growing realization that s is the arc length, and pi is the traction exerted on the body LEFM is not applicable to concrete stnictures of normal size bounded by r and the crack surface. The J-integral is equal and material properties (Kaplan 1961, Kesler et al. 1972, Ba- to the energy release rate G of the crack (Rice 1968). zant 1976). The Fpz ranges from a few hundred millimeters The J-integrai method can be relatively easiiy applied to a to meters in length, depending upon how the FPZ is defmed crack problem whose stress and displacement solution is and upon the propemes of the particular concrete being con- known, and is not limited to linear materials. However, elas- sidered (Hillerborg et al. 1976; ingraffea and Gerstle 1985; ticity or pseudoelasticity along the contour, ï, is a require- Jenq and Shah 1985). The width of the Fpz is small com- ment (Rice 1968). pared to its length (Petersson 1981). LEM, although not ap- Alternate energy approaches for extraction of stress-inten- plicable to small structures, may stili be applicable to large sity factors from three-dimensional problems have been de- structures such as dams (Elfgren 1989). However, even for veloped (Shivakumar et al. 1988). Bittencoun et al. (1992) very large structures, when mixed-mode cracking is present provide a single reference that compares the displacement the FPZ may extend over many meters; this is due to shear correlation, the J-integral, and the modified crack closure in- and compressive normal forces (tractions) caused by fnc- tegrai techniques for obtaining stress-intensity factors. tion, interference, and dilatation (expansion) between the When using triangular quarter-point elements to model the sides of the crack, far behind the tip of the FPZ. To clarify singularity at a crack tip, meshing guidelines have been sug- this notion, Gerstle and Xie (1992) have used an “interface gested by a number of researchers (Ingraffea 1983, Saouma process zone (IPZ)” to model the F’PZ. and Schwemmer 1984, Gerstle and Abdaila 1990). When us- The fictitious crack model (FCMJ has become popular for ing the displacement correlation technique to extract stress- modeling fracture in concrete (Hillerborg et al. 1976, Peters- intensity factors, the guidelines are summarized as follows: son 1981, Ingmffea and Saouma 1984, Ingraffea and Gerstle 1. Use a 2 x 2 (reduced) integration scheme (Saouma and 1985, Gustafsson 1985, Gerstle and Xie 1992, Feenstra et ai. Schwemmer 1984). 1991a, 1991b, Bocca et al., 1991, Yamaguchi and Chen 2. To achieve 5 percent maximum expected error in any 1991, Klisinski et al. 1991, Planas and Elices 1992, 1993% stress component due to any mixed-mode problem, use at 19938). Fig. 2.5 shows the terminology and concepts associ- least eight approximately equiangular singular elements ad- ated with the FCM. This model assumes that the Fpz is long jacent to the crack tip node. For 1 percent error, 16 singular and infinitesimally narrow. The FEZ is characterized by a elements should be used (Gerstle and Abdalla 1990). “nodstress versus crack opening displacement cwe,” 3. There is an optimal size for the crack tip elements. If which is considered a material property, as shown in Fig. 2.5. they are too small they do not encompass the near-field re- The FCM assumes that the FPZ is collapsed into a line in gion of the solution, and surrounding regular elements will 2D or a surface in 3D. A natural way to incoxporate the mod- be ‘’wasted“ modeiing the near field. If they are too big, they el into the finite element analysis is by employing interface do not model the far-field solution accurately. The singular elements. The first interface element was formulated by

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Goodman et ai. (1968) and was used in the modeling of rock continuities (ortiz et ai. 1987, Fish and Belytschko 1988, joints. Since then, many types of interface and thin layer el- Belytschko et al. 1988, Dahlblom and ûttosen 1990, Usin- ements have been developed and are widely used in geotech- ski et ai. 1991) or displacement discontinuities (Dvorkin et nicai engineering (Heme and Barbour 1982, Desai et ai. al. 1990, Lotn 1992, Lo~iand Shing 1994, 1995) within 1984). Zero-thickness elements are the most widely used continuum elements. type of interface, with normal and shear stresses and relative The FCM has been incorporated into finite element codes displacements across the intedace as constitutive variables. with the use of interface elements. Inpffea and coworkers Unrealistic jumps in the results of adjacent integration points (IngrafTea et al. 1984, Ingraffea and Saouma 1984, Ingraffea of contiguous interfaces have been reported by some authors and Gerstie 1985, Bittencourt, Ingraffea and Llora 1992) depending on initiai stiffness and load conditions, although extended the FCM to simulate mixed-mode crack propaga- most of these problems seem to disappear with the appropri- tion analysis employing six-noded interface elements. Swen- ate selection of integration points and integration rule (Gens son and Ingraffea (1988) used six-noded interface elements et al. 1988, Hohberg 1990, Rots and Schellenkens 1990, to model mixed-mode dynamic crack propagation. Bocca, Schellenkens and De Borst 1993). Other investigators have Carpinteri, and Valente (199 1) have published similar work. implemented a semidiscrete FCM by including strain dis- Gerstie and Xie (1992) used a simple four-nod4 liiear dis- k

Shear and normal tractions are functions of crack opening and shearing displacements and other state variables

Crack Opening Displacement --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

Fig. 2.5-Terminology and concepts associated with the fictitious crack del(FCM) (Hilierborg et al. 1976)

placement interface element that was modified to allow an tween crack surfaces. None of the models, however, arbitrary distribution of tractions along its length. provides secant unloading in pure tension, as is usual in the Other references that implement the FCM include Rots classic FCM, because this would complicate the model con- (1988), Stankowski (199û), Stankowski et al. (1992), Hoh- siderably, as discussed by Carol and Willam (1994). berg (1992a), Lotfi (1992), Vonk (1992), Lotñ and Shing An approach to nonlinear mixed-mode discrete crack (1994). Garcia-Alvarez et al. (1994), Lopez and Carol propagation analysis was proposed by Ingraffea and Gerstíe (1995), and Bazant and Li (1995). (1985). h this approach the FPZ is modeled by interface el- in the KM,the stiffness of the interface element is a non- ements, with the singuiar elements used in LEFM placed linear function of the crack opening displacement, so that a around the fictitious crack tip to predict the direction of the nonlinear solution procedure is required. As with any other crack propagation. However, singuiar elements are not nec- kind of nonlinear constitutive relation, FEM calculations essary for this purpose if energy release rates are caicuiated with interface elements behaving in accordance the FCM re- directly (Rim 1968, Parks 1974, Sha and Yang 1990, Rahul- quire a nonlinear solution strategy. The various existing kumar 1992). techniques such as classic Newton iteration, dynamic relax- Another potential approach to determining the direction of ation, and arc-length procedures have been used, with satis- a crack is based on the very reasonable assumption that the factory results repod in the literature (Swenson and crack will propagate when the maximum tensile principal Ingraffea 1988, Gerstle andxie 1992, Papadrakis 1981, Un- stress at the crack tip reaches the strength of the material (Pe- derwood 1983, Bathe 1982.) tersson 1981, Gustafsson 1985, Hiilerborg and Rots 1989, When using interface elements to model the FPZ, the ele Bocca et al. 1991, Gerstie and Xie 1992). The direction of ments must be very stiff prior to crack initiation to represent crack propagation is assumed to be perpendicular to the max- an uncracked material (i.e., to keep the two sides of the po- imum tensile principai stress. The problem with this approach tential crack together). However, care must be taken not to is that when the FPZ becomes small compared to the crack tip use a stiffness so high as to cause nonconvergent numerical element size, objectivity of the results is lost. lñerefore it behavior in the ftnite element solution. Brown et ai. (1993) makes more sense to use an energy-based approach to deter- successfully used interface elements with an axial stiffness mine crack propagation. A basis for such an approach has equal to 50 times the stiffness of the adjacent concrete ele- been developed by Li and Liang (1992). Promising results ments without numerical difficulties. Gedeand Xie (1992) have been obtained by using energy release rate approaches to suggested using a precrack stiffness equal to the secant stiff- determine both the direction and the load level at which a fic- ness to a point on the descending nodtraction-COD curve titious crack will propagate (Xie et al. 1995). (Fig. 2.5) equal to a COD of '/ZO to '/30 of the COD at which the normal traction drops to zero. 2.4-Automatic remeshing algorithms Some investigators have dispensed with interface ele- in 1981, work was completed on a two-dimensional frac- ments and have instead simulated the Fpz using an influence ture propagation code that used simple interactive computer function approach (Li and Liang 1986, Planas and Elices graphics to interactively model crack propagation (Saouma 199 1) in which cohesive forces are applied to the crack faces 1981). Many of the tasks that Ingraffea (1977) had per- (Gopalaratnam and Ye 1991). Weighted multipliers are used formed by editing files manually were now performed inter- in the superposition of FEM solutions to satis8 overall equi- actively. These tasks included semi-automatic remeshing to librium, compatibility and stress-crack width relations with- allow the crack to advance, limited post-processing to view in the FPZ. This approach results in the solution of a set of stresses, deformed mesh, and stress-intensity factors, and nonlinear algebraic equations to determine the multipliers. predictions based upon the mixed-mode crack propagation In some cases, it may be appropriate to linearize the relation- theories of the crack trajectory. ship between the COD and the tractions on the FPZ. Then Subsequently, Wawnynek and hgraffea (1987) devel- linear equations can be solved to obtain the solution effi- oped a second generation two-dimensional interactive ciently (Gopalaratnam and Ye 1991, Li and Bazant 1994). graphical fuiie element fracture simulation code based upon Extension of the FCM with interface elements to mixed a winged-edge topological data structure. This program mode cracking requires a constitutive relation for the inter- demonstrated the value of using a topological data structure face, in which nodand shear stresses and relative dis- in fracture simulation codes. More recently, Gerstle and Xie placements are fully coupled. Crack opening and closing (1992), in collaboration with others, developed an interac- conditions are expressed with a biaxial failure surface in the tive graphical finite element code that is capable of repre- normai-shear stress space. Crack surface displacements, senting and automatically propagating cracks in two thus, have two components: opening and sliding. Several dimensional problems. models of this kind have been proposed recently, all based Procedures have also been developed to handle numerical on the framework of non-associated work hardening plastic- discretization and arbitrary fracture simulation in three di- ity, to obtain a formulation that is fully consistent and con- mensions (Martha 1989, Sousa et al. 1989, Ingraffea 1990, tains fracture energy parameters (Stankowski 1990, Martha et al. 1991). Stankowski at al. 1993, Lotfi 1992, Lotfi and Shing 1994, A number of algorithms have been introduced for auto- Hohberg 199% 1992b, Vonk 1992, Garcia-Alvarez et al. matic meshing of solid models (Shepard 1984, Cavendish et 1994). After the crack is completely open, the models pre- aí. 1985, Schroeder and Shepard 1988, Perucchio et ai. vent interpenetration provide Coulomb-type friction be- 1989). These algorithms can be categorized into three broad and --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

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families: element extraction, domain triangulation, and re- There are, however, serious problems with the classical cursive spatial decomposition. Aithough substantial differ- smeared crack models. They are in principle nonobjective, ences exist between these families, the algorithm involve since they can exhibit spurious mesh sensitivity (Bazant the development of a geometric representation of the struc- 1976), i.e., the results may depend significantiy on the ture, which provides the basis for the construction of the fi- choice of the mesh size (element size) by the analyst. For ex- nite element mesh (Sapadis and Perucchio 1989). Automatic ample, in a tensioned rectangular plain concrete panel with a modeling of discrete crack propagation in three dimensions rectangular finite element mesh, the cracking localizes into a remains a challenge. one element wide band. The crack band becomes narrower It is worth noting that remeshing may not be required 1) if and increases in length as the mesh is refined, as shown in the crack path is known in advance due to symmetry or due Fig. 3.1 (Bazant and Cedoiin 1979, 1980, Bazant and Oh to previous experimental or analyticai experience with the 1983, Rots et al. 1984, Darwin 1985). Consequently, if not same geometry; or 2) if interface elements are placed along accounted for in the crack model, the load needed to extend all possible crack paths. the crack band into the next element is less for a finer mesh (for a crack model controlled by tensile strength alone, it de- CHAPTER 3-SMEARED CRACK MODELS creases roughly by a factor of fi if the element size is Early in the application of finite element analysis to con- halved, and tends to zero as the element size tends to zero). crete structures (Rashid 1%8), it became clear that it is often Thus, the maximum load decreases as the mesh is refined much more convenient to repment cracks by changing the (Bazant and Cedolin 1980). Furthermore, the apparent ener- constitutive properties of the finite elements than to change gy that is consumed (and dissipated) during structurai failure the topography of the finite element grid. The earliest proce- depends on the mesh size, and tends to zero as the mesh size dure involved dropping the material stiffness to zero in the tends to zero. Such behavior, which is encountered not only direction of the principal tensile stress, once the stress was for cracking with a sudden stress drop but also for graduai calculated as exceeding the tensile capacity of the concrete. crack formation with a finite slope of the post-peak tensile --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- Simultaneously, the stresses in the concrete were released sîrain-softening stress-strain diagram that is independent of and reapplied to the structure as residual loads. Models of element size (Bazant and Oh 1983), is nonobjective. These this type exhibit a system of distributed or “smeared” cracks. problems make the classical smeared crack approach unac- Ideally, smeared crack models should be capable of repre- ceptable, although in some structures such lack of objectivity senting the propagation of a single crack as well as a system might be mild or even negligible [this latter behavior occurs of distributed cracks, with reasonable accuracy. especially when the failure is controlled by yielding of rein- Over the years, a number of numericai and practical prob- forcement rather than cracking of concrete (Dodds, Darwin, lems have surfaced with the application of smeared crack and Leibengood 1984)l. models. Pnncipal among these involve the phenomenon of To avoid the unobjectivity or spurious mesh sensitivity, a “strain localization.” When microcracks form, they often mathematical device called a “localization limiter” must be tend to grow nonuniformly into a narrow band (called a introduced. “crack”). Under these conditions, deformation is concentrated in a narrow band, while the rest of the structure experienc- 3.2-Types of localization limiters es much smaller strains. Because the band of localized strain Several types of localization limiters have been proposed. may be so narrow that conventional continuum mechanics 3.2.1. Crack band model no longer applies, various “localization limiters” have been The simplest localization limiter is a relationship between developed. These localization limiters are designed to deal the element size and the constitutive model so that the total with problems associated with crack localization and spuri- energy dissipated will match that of the material being mod- ous mesh sensitivity that are inherent to softening models in eled. This can be done by adjusting the downward slope of general and smeared cracking in particular. Critical reviews the stress-strain curve, or, equivalently, the value of b, of the practical aspects of smeared crack models are present- shown in Fig. 3.1, as the element size is altered. %m is in- ed by ASCE Task Commitîee (1982) and Danvin (1993). creased as the element size is decreased. This procedure, known as the crack band model, has a limitation for coarse 3.1-Reasons for using smeared crack models meshes (large elements-, cannot be convenientiy re- The smeared crack approach, introduced by Rashid duced below the value of strain corresponding to the peak (1968), has become the most widely used approach in prac- stress, o-. tice. Three reasons may be given for adopting this approach More advanced constitutive models based on the theories 1. The procedure is computationally convenient. of plasticity and damage may not exhibit the simple stress- 2. Distributed damage in general and densely distributed strain relationship shown in Fig. 3.1. Crack band modes can parailel cracks in particular are often observed in structures still be applied as long as a local fracture energy is included (measurements of the locations of sound emission sources among the model’s parameters (Pramono and Willam 1987, provide evidence of a zone of distributed damage in front of Carol et al. 1993). a fracture). Practically, the most important feature of the crack band 3. At many size scales, a crack in concrete is not straight model is that it can represent the effect of the structure size but highly tortuous, and such a crack may be adequately repon: 1) the maximum capacity of the structure (Bazant 1984); resented by a smeared crack band. and 2) the slope of the post-peak load-deflection diagram.

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Constitutive Law:

Crack Band --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`---

Fig. 3.1-lfthe constitutive model is independent of element size, the crack band becomes longer and narrower as the mesh is refined

However, hmthe physical viewpoint, the width of the tative volume of the material, as shown in Fig. 3.2, under- cracking zone at the front of a continuous fracture (i.e., the stood to be the smallest volume for which the heterogeneous i FPZ) is represented by a single, element-wide band and can- materiai can be treated as a continuum (the size of V, is de- not be subdivided further, consequently, possible variations termined by a characteristic length, L, which is a material in the process zone size, which cause variation of effective property; a is a weighting function, which decays with dis- fracture energy (i.e., R-curve behavior) cannot be captured tance from point x and is zero or nearly zero at points suffi- and the stress and strain states throughout the FPZ cannot be ciently remote from x; and the superimposed bar denotes the resolved. The procedure, however, has been widely and suc- averaging operator. The dummy variable s represents the cessfully applied. spatial coordinate vector in the integrai. Nonlocal continuum 3.2.2. As the simplest form of the weighting function, one may a) Phenomenological Approach consider a = 1 within a certain representative volume Vo A general localization limiter is provided by the nonld centered at point x and 01 = O outside this volume. Conver- continuum concept and spatial averaging (Bazant 1986). gence of numerical solutions, however, is better if a is a A noniocal continuum is a continuum in which some field smooth bell-shaped function. An effective choice is variables are subjected to spatial averaging over a f~te neighborhood of a point. For example, average (nonlocal) sttain is defined as (3.2)

1 E(x) = - a(x-s)a(s)dv(s) = J a’(x,s)e(s)dv(s) (3.1) V,dV Y in which r = Lx-SI = distance from point x, L = Characteristic length (materiai property), and po = coefficient chosen in in which V,(x) = a(x - s)dV(s) and a‘(x, s) = ab- s)l(V,xx) ; such a manner that the volume under function a given by Eq. e(x) is the strain 2 the point in space defined by coordinate 3.2 is equal to the volume under the function a = 1 for r < U Copyright American Concrete Institute = r (which represents a line segment in Provided by IHSvector under license x; withV ACIis the volume of the structure; V, is the represen- Licensee=University2 and a of TexasO for Revised > Sub U2 Account/5620001114, User=yuyuio, rtyru No reproduction or networking permitted without license from IHS Not for Resale, 01/26/2015 03:34:12 MST Daneshlink.com daneshlink.com STD.ACI qY6.3R-ENGL 1997 Ob62949 053’9966 53T FINm ELEMENT ANALYSIS OF FRACTURE IN CONCRETE STRUCTURES 446.3R-15

I l I I I S S

Fig. 3.2-Heterogeneiiy of concrete at the size scale of the aggregate

lD, a circle in 2D, and a sphere in 3D). Alternatively, the The nonlocal continuum model for strain-softening of Ba- noddistribution function has been used in place of Eq. zant et al. (1984) involves the nonlocal (averaged) strain i as 3.2 and found to work well enough, although its values are the basic kinematic variable. This corresponds to a system of nowhere exactly zero (Bazant 1986). imbricated (i.e., overlapping in a regular manner, like roof For points whose distance from all the boundaries is larger tiles) finite elements, overiaid by a reg~~larfinite element sys- --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- than p&, V,.(x) is constant; otherwise the averaging volume tem. Although this imbricate model limits localization of prot~~desoutside the body, and V,.(x)must be Calculated for straui softening and guarantees mesh insensitivity, the pro- each point to account for the locally unique averaging do- gramming is complicated, due to the nonstandard form of the main (Fig. 3.2b). differential equations of quiiibrium and boundary conditions, In finite element computations, the spatial averaging inte- ie., energy considerations involve the noniocai strain i . grals are evaluated by finite sums over all integration points These problems led to the idea of a partially noniocal con- of all finite elements of the structure. For this purpose, the tinuum in which stress is based on noniocai strain, but local matrix of the values of a‘ for all integration points is comput- srrains are retained Such a nonlocal model, called “the non- ed and stored in advance of the finte element analysis. local continuum with local strain” (Bazant, Pan, and Pijaud- This approach makes it possible to refme the mesh as re- ier-Cabot 1987, Bazant and Lin 1988, Bazant and Pijaudier- quired by structurai considerations. Since the representative Cabot 1988, 1989) is easier to apply in finte element pro- volume over which structural averaging takes place is treated gramming. In this formulation, the usual constitutive relation as a material propew, convergence to an exact continuum for strain softening is simply modified so that all of the state solution becomes meaningful and the stress and strain distri- variables that characterize strain softening\are calculated butions throughout the FPZ can be resolved. from nonlocal rather than local strains. Then,’all that is nec-

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446.3R-16 ACI COMMITTEE REPORT

essary to change in a local finte element program is to pro- found that, to describe differences between observations on vide a subroutine that delivers (at each integration point of smali and large specimens, the formation of shrinkage cracks each element, and in each iteration of each loading step) the needs to be assumed to depend not only on the shrinkage stress value of Ë for use in the constitutive model. but also on its spatial gradient [see also L‘Hermite et al. (1952) Practically speaking, the most important feature of a non- and a discussion by Bazant and Lin (1988)l. This was pmba- local fite element model is that it can correctly represent bly the first appearance of the noniocal concept in fracnire. the effect of structure size on the ultimate capacity, as well The idea that the stress gradient (or equivalently the strain gra- as on the post-peak slope of the load-deflection diagram. dient) influences the material response has also been demon- Nonid models can also offer an advantage in the overali strated by Sturman, Shah, and Winter (1965) and by Karsan speed of solution (Bazant and Lin 1988). Aithough the nu- and Jim (1%9) for combined flexure and axial loading. merical effort is higher for each iteration when using a non- The nonlocal averaging integral is meaningful oniy if the fi- local model, the formulation lends a stabilizing effect to the nite elements are not larger than about one third of the repm solution, allowing convergence in fewer iterations. sentative volume, V, @raz and Bazant 1989). The gradient b) Micromechanical Approach approach, on the other hand, offers the possibility of using fi- Another noniocal model for solids with interacting mim- nite elements with volumes as large as the representative vol- racks, in which noddty is introduced on the basis of mi- ume, V, Thus, begradient approach offers the possibility of crocrack interactions, was developed by Bazant and Jirasek using a smaller number of ñnite elements in the analysis. It ap (Bazant 1994, Jirasek and Bazant 1994, Bazant and Jmk pears, however, that the programming may be more compii- 1994) and applied to the analysis of size effect and localization cated and less versatile than for the spatial averaging integrals. of cracking damage (Jirasek and Bazant 1994). The model The problem is that interelement continuity must be enforced represents a system of intemcting cracks using an integral not only for the displacements but also for the strains. This re- equation that, unlike the phenomenological noniocal model, quires the use of higher-order elements, or aiternatively, if --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- involves a spatial integral that represents microcrack intmc- first-nier elements are preferred the use of an independent tion based on fracture mechanics concepts. At long range, the strain field with separate first-order finite elements. integral weighting function decays with the square or cube of Noniocal constitutive models for concrete are reviewed in the distance in two or three dimensions, respectively. This AC1 446.1R. model, combined with a mimplane model, has provided consistent results in the finite element analysis of fracture and CHAPTER MITERATURE REVIEW OF FEM structurai test specimens (Ozbolt and Bazant 1995). FRACTURE MECHANICS ANALYSES 323. Gradient models 4.1-Generai Another general way to introduce a localization limiter is To help provide an overview of the state-of-the-art in fi- to use a constitutive relation in which the stress is a function nite element modeling of plain and reinforced concrete struc- of not only the strain but also the first or second spatial de- tures, a number of representativeanalyses are summarized in rivatives (or gradients) of strain. This idea appeared original- this chapter. Emphasis is placed on easily available, pub- ly in the theory of elasticity. A special form of this idea, lished analyses that attempt to address the fracture behavior called Cosserat continuum and characterized by the presence of concrete structures. Wherever possible, problems solved of couple stresses, was introduced by Cosserat and Cosserat using the discrete cracking approach are compared to solu- (1909) as a continuum approximation of the behavior of tions using the smeared crack approach. Symposium pro- crystal lattices on a small scale. A generalization of Cosserat ceedings (Mehlhorn et al. 1978, Computer-Aided 1984, continuum, involving rotations of material points, is the mi- 1990, Firrao 1990, Fracture Mechanics 1989, van Mer et al. mplar continuum of Eringen (1965,1966). 1991, Concrete Design 1992, Size Effect 1994, Computa- Bazant et al. (1984) and Schreyer (1990) have pointed out tional Modeling 1994, Fracture and Damage 1994, Fracture that spatial gradients of strains or strain-related variables can Mechanics 1995) provide the readers with the changing fla- serve as a localization liters. It has been shown (Bazant et vor of the state-of-the-art over the years. Many more pub- al. 1984) that expansion of the averaging integral (Eq. 3.1) lished FEM fracture mechanics analyses exist than are into a Taylor series generally yields a constitutive relation in presented in this chapter; however, it is fair to say that those which the stress depends on the second spatial derivatives of referenced here are representative of the state-of-theart. strain, if the averaging domain is symmetric and does not The term “size effect” used throughout this chapter, is a protrude outside the body, and on the first spatial derivatives term that has taken on a special meaning for quasibnttle ma- (gradients) of strain, if this domain is unsymmetric or pro- terials, such as concrete and rock. It describes the decrease trudes outside the domain, as happens for points near the in average stress at failure with increasing member ske that boundary (Fig. 3.2b). Since the dimension of a gradient is is directly attributable to the well-established fact that frac- length-’ times that of the differentiated variable, introduction ture is governed by a fracture parameter(s) that depends on of a gradient into the constitutive equation inevitably re- the dimensions of the crack (which is tied to structure size) quires a characteristic length, L,as a material property. Thus, as well as some measure of stress. For concrete, it is not clear the use of spatial gradients can be regarded as an approxima- if the fracture parameter that governs failure is a material tion to nonlocal continuum, or as a special case. property or if it also depends on the structure size. However, In material research of concrete, the idea of a spatial gradi- this last point does not alter the general meaning of the term ent appeared in the work of L‘Hermite et al. (1952), who “size effect.” In contrast, in the field of fracture of metais, the

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FINITE ELEMENT ANAiYW OF FRACTURE IN CONCRETE STRUCTURES 446.3ñ-17

term “size effect” is understood to apply only to the depen- displacement and anaiyzed using two different meshes (Ba- dence of fracture parameter (not the average stress) on the zant and Ozbolt 1990, Droz and Bazant 1989). The model structurai dimensions. can represent tensile cracking in multiple directions, as well as nonlinear triaxial behavior in compression and shear, in- 4.2-Plain concrete cluding post-peak response. The calculated load-displace- Unreinforced concrek súuctures are the most fracture sen- ment diagrams were obtained by local and nonlocal analysis sitive. Weusually reinforcedwith steel in design practice to for two meshes. While the local results differed substantially provide adequate tensile strength, many structures (or parts of (spurious mesh sensitivity), the nonlocal results were nearly smctures), for one reason or another, are unreinforad. It mesh independent. while producing excellent results, the makes sense to study the analysis of unreinforced concrete nonlocal microplane model is very costly in terms of com- suuctures, because these provide the most severe tests of frac- puter disk space and computer time. ture behavior, and because the results of these analyses can Schlangen and van Mer (1992) employed a mimme- add insight into the more complex behavior evidenced in rein- chanics approach to model tensile fracture using a triangular forced concrete structures. In what follows, short descriptions lattice of bride beam elements with statistically varying (iisted in chronological order) of finite element analyses of un- properties to represent the heterogeneity of the material. reinfod concrete structures are presented. Fracturing of the materiai takes place by removing, in each 4.2.1 Tensile failure load step, the beam element with the highest stress (relative The plain concrete uniaxial tension specimen is probably to its strength). The input parameters are, however, not di- more sensitive to fracture than any other type. For this rea- rectly observable material properties. Thousands of degrees son, it provides a good test of the fracture-sensitivity of a fi- of freedom are required for the analysis, and larger speci- nite element model for concrete. Aithough a number of such mens would have required an unreasonable number of de- analyses have been reported, the following references are grees of freedom. representative of first a discrete, then a local smeared, then a Another lattice model, developed by Bazant et al. (1990) nonlocal smeared, and finally a micromechanics (random and refined by Jmsek and Bazant (1995a), involves lattices partiCie) approach. with only centrai force interactions between adjacent parti- Gustaffson (1985) used the fictitiouscrack model (discrete cles (avoiding the use of beams that are subject to bending crack) to study the uniaxial tension specimen tested by Pe- and do not realistically reflect deformation modes on the mi- tersson (1981). Except for the fictitious crack, he assumed crostructural level). The representation was further extended the continuum was hear elastic. He included the influence by Jmek and Bazant (1995b) with a very efficient numeri- of initial stresses in his model. Good agreement with test re- cai algorithm of the central difference type. With this algo- sults was obtained. Gustaffson (1985) also studied a number rithm, it was possible to solve the nonlinear response of a of other specimen geometries using the ñnite element meth- lattice model with over 120,000 degrees of freedom by ex- od. only Mode I cracks, with trajectories known in advance, plicit dynamic integration on a desk-top workstation. The were modeled. Interestingly, for larger specimens this anal- studies demonstrate that the transitional size effect can be re- ysis approach would still be expected to yield fracture-cor- produced by a lattice model. rect results (as long as a sufficient number of degrees of Other examples of EManalysis of tensile specimens are freedom are used to accurately represent the distribution of presented by Roelfstra et al. (1983, Rots (1988), Stankows- tractions dong the FPZ). This type of model is extremely ki (WO), Stankowski et ai. (1992), Vonk (1992), and Lopez powerful for the class of problems in which the crack trajec- and Carol (1 995). tory is known in advance, and the failure mechanism in- 4.2.2 Compressivefdure volves a single discrete crack. Although plain concrete compression specimens do not Bazant and Prat (1988% 1988b) developed and used a exhibit as significant a fracture mechanics size effect as (smeared crack) microplane model for brittle-plastic materi- notched tension specimens, a good model for Concrete als. The results were compared with Petersson’s experimental should be capable of predicting the behavior of such a spec- uniaxial tension results (petersson 1981 ), along with eight oth- imen. A significant fracturemechanics size effect is found in er series. The term microplane model is used to describe a the descending branch of the compression stress-straincurve class of constitutive models in which the material behavior is (van Mer and Vonk 1991). Some fracture mechanics-based specified independently for planes of many orientations models have been used to attempt to predict the compressive (called micropianes); the macroscopic stresses and strains are strength of compression specimens. More test results exist obtained through the use of a suitable superposition tecbnique, for this type of specimen than for any other. All but one of such as the principle of virtual work. In the study by Bazant the following examples used a smeared crack approach to and Prat, the agreement between the numerical and experi- analyze this specimen type. mental results was good. However, the analysis was conduct- Bazant and Prat (1988b) compared their local smeared ed only at the constitutive level that., by its nature, is size- crack microplane model for bnttle-plasticmaterials with ex- independent. The panmeters obtained for these tests were ap perimental uniaxial compression test results. Excellent plicable only to the pEuticular specimen size analyzed. agreement was found (although, once again, the five mi- A nonìd microplane model (see AC1 446.1R) was used croplane model parameters were selecteù to provide the best to analyze a rectangular concrete tension specimen in plane fit to the experimental data). As with the tensile experiments,

strain.--```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- The specimen was loaded by a prescribed uniform calculations with this model were purely constitutive, and

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4463R-18 AC1 COMMITïEE REPORT

therefore size independent and fracture insensitive. To ob- sion, including hysteresis loops. This modei, probably the tain some size effect in the softeningbranches with the mod- most physically complete mechanical model developed for el as described (without going into a non-local plain concrete to this time, requires a large computational ef- implementation), a new set of parameters would be needed fort for the analysis of even simple engineering structures. for each specimen size. Additional microstructural analyses of uniaxial compres- ûajer and Dux (1990) used a “simplified nonorthogonal sion specimens are reported by Stankowski (WO), crack model” to analyze a plain concrete prism in compres- Stankowski et al. (1!292), and Vonk (1992,1993). sion. The model was based on the crack-band concept, fea- 4.23 Fracture spechens turing simplified representations of both nonorthogonal The unreinforced *-point bend specimen is an impor- cracking and the influence of cracking on local material stiff- tant bench mark problem for several reasons. First, LEFM nesses. At an integration point, only one primary crack of solutions for some span-to-depth ratios are to better faed direction and one rotating allowed to devel- known crack are than 0.5 percent accuracy (Tada et al. 1973). it is a op. The model makes use of the band concept to Second, crack assure fracture toughness testing geometry, which means constant energy release for elements of different sizes during standard that expenmental results are available (Kaplan 1961, Catai- the frachue process. The model would be expected to give ano and Ingraffea 1982, Jenq and 1985). Third, it is a correct mechanics results as long the element size Shah fracture as symmetrical Mode I problem, and thus, the crack trajectory is larger than the width of the FPZ and smaller than the is in advance. Symmetry can be used to reduce length of the known the FPZ. cost of analysis. Many finte element analyses of this speci- Tasdemir, Maji, and Shah (1990) modeled mixed-mode men are reprted in the literature. It should be noted that crack initiation and propagation in the cement paste matrix, most of the reported analyses were of relatively small speci- starting from the interface of a single rigid rectangular inclu- mens, and therefore LEFW would not be expected to apply. sion in a Concrete specimen under uniaxial compression, using both a discrete LEFM approach and a FCM approach, The first analysis using the fictitious crack model of a with singularhies modeled at the crack tips in both approach- three-point bend specimen was reported by Hillerborg, es. The FEM results were compared with experimental re- Modeer, and Petersson (1976). Finite element results regard- sults and satisfactory local cracking results were obtained. ing the effect of beam depth on flexural smngth were pre- The approach is applicablefor the study of micromechanical sented. These theoretical results indicated that the flexural behavior in the neighborhood of a single aggregate particle, strength decreases with increased depth of the beam and that but inappropriate as a method for determining the gross re- nonuniform shrinkage strains have a greater effect on the sponse of the compression specimen. strength of deep beams than on the strength of shallow The original mimplane model of Bazant and ht(1988a, beams. A comparison between the theoretical predictions and a large number of experimental results showed good 1988b) was reformulated as a continuum damage model (Car- agreement. Similar analyses were presented by Modeer ol et al. 1991) and modified to make it computationaily more (1979), Petersson and Gustafsson (1980) for a discrete crack efficient (Carol et al. 1992). In both cases, the original results of uniaxial Compression specimens were reproduced with model and by Leibengood, Darwin, and Dodds (1986) for a similar accuracy. The reformulation of the model in terms of smeared crack model. a damage tensor, completely independent of the material rhe- Gerstle (1982) performed a series of discrete cracking ology, allowed the authors to then replace elasticity with linear analyses, employing a nonlinear FCM analysis, together aging viscoelasticity and obtain a realistic approach to Rush with an LEFM fictitious crack tip assumption.Kfc(tb,. Non- CW~S (-1 and Bmt1991, Car01 et al. 1992). AS dis- liiear interface elements were used to model the fictitious cussed earlier, these results were purely constitutive and there- crack. A secant stiffness iterative scheme was used to solve fore &-independent and fracture-insensitive. the nonlinear equations. Satisfactory agreement with experi- , Bazant and Ozbolt (1992) used the nodocal microplane mental results (Cataiano and Ingraffea 1982) was found us- model to investigate the effects of boundary conditions and ing a pure FCM. When it was assumed that KI,.+,, was non- size on the strength of a plane strain compression specimen. zero, however, the strength of the beam was significantly Meshes with 180 four-noded finite elements, enough to cap- over-predicted. --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- ture strain localizations, were employed. Many interesting Gustafsson (1985) analyzed a three-point bend specimen and plausible damage localization patterns were displayed. using the FCM. He used a supposition of nonlinearly vary- Although their model is capable of predicting a fracture me- ing cohesive nodal forces to represent the fictitious crack, chanics size effect, they found no size effect for this speci- and a substructuring technique to reduce the number of un- men type. The model would require great computational knowns in the iterative solution. He found the results to be effort for realistic engineering structures. consistent with theoretical expectations regarding the state Ozbolt and Bazant (1992) employed a nodocal mi- of initial stress and specimen size. croplane model to simulate a uniaxial compression specimen De Borst (1986) analyzed a pre-notched, three-point bend subject to cyclic load. The rate effect was introduced by specimen. The example was used mainly to demonstrate the combining the damage model on each microplane with the efficiency of quasi and secant-Newton methods in the solution Maxwell rheologic model. Nine four-noded isoparamemic of nonlinear finite element equations. A local. smeared crack plane stress elements were employed. The model was shown analysis was performed. He discussed severai interesting to produce qualitatively correct results for cyclic compres- points regarding the stability of the solution, and developed

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FINITE ELEMENT ANALYSIS OF FRACTURE IN CONCRETE STRUCTURES 446.3R-19

arc length control algorithms to increase solution stabiïity for (13 mm). The maximum values of the nominal stress o, problems in which the post-peak response is important. (load divided by beam depth and thickness) obtained by non- A three-point bending test according to a RILEM recom- local finite element analysis with step-by-step loading are mendation was numerically simulated using a discrete FCM plotted as a function of the beam depth relative to the size of with a bilinear stress versus crack opening displacement the aggregate, dda,for both linear and exponential softening curve by Carpinteri et al. (1987) and Carpinteri (1989). The laws. Their work demonstrates that a noniocal f~teelement fictitious crack was represented by closing forces computed model can represent the size effect quite well. through the solution of a set of noniinear equations. Many Bazant and Lin (1988) also compared results obtained with loaddefidon curves for beams of various size were calcu- slanted square meshes to resuits obtained with aiigned mesh- lated and reported. es. Men tbe element size was less than about of the char- Yamaguchi and Chen (1990) used a smeared crack, crack- acteristic length L,they observeù no mesh bias with regard to band-like model to Simulate a notched three-point bend spec- crack direction. In contrast, with the crack band modei, it was imen. Eight-node isoparametric quadrilateral elements with produce --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- impossible to anaiyiicaîly fractures in specimens with 2 x 3 Gauss quadrature (3 in direction of crack propagation) slanted meshes that run vertically, as they should. were used in this analysis, and the notch was modeled as an Additional,analyses of three-point bend specimens are re- element-wide gap. An extremely coarse mesh was used, with ported by Rots (1988), Carol et al. (1993), Garcia-Alvarez et only two elements to model the 100 mm (3.9 in.) deep liga- al. (1994), andlotñ and Shing (1994). ment. agreement was found with previous analysis Good re- 4.2.4 Shear f&e in pluin concrete beams suits by others. The f~teelement analysis of the single-notched beam Gopalaratnam and e (199 1) a fictitious crack Y performed subject to four-point loading is a benchmark problem that analysis of a three-point bend specimen to determine the has been investigated by many researchers. It provides a characteristics of the nonlinear FPZ in concrete. They used a good test of the ability of a finite element code to model linearized solution process to solve the nonlinear equations mixed-mode fracture. Tests have been perfomed by Arrea obtained in the FCM. The fictitious crack was modeled and Ingraffea (1982). No accurate closed-form solu- through appropriate application of cohesive point loads LEFM tion to this problem has been reported. The following are de- along the fictitious crack. Using a very refined mesh, they scriptions of several of these analysis attempts. were able to obtain results that demonstrate the fracture mechanics size effect. They report extensively upon the process Ingraffea and Gentle (1985) perfomed severai mixed- zone chamcteristics calculated. mode FCM analyses of the beam tested by Arrea and Ingrafîea (1982) using interface elements. They used a discrete aggre- Liang and Li (1991) used a boundary element program gate interlock model to simulate shear stress transfer across based on a discrete FCM to study the size effect in a plain con- the fictitious crack. They also used quadratic elements, requir- crete beam with a bending crack. The size effect was observed ing a totai of approximately 600 degrees of freedom. The pre- and veriñed. Justification for R-curve analysis (see AC1 dicted crack trajectory and load-displacement curve were in 446.1R) for specimens was given. A double cantilever such close agreement with the experimental results. beam specimen is also analyzed using the same techniques. Bolander and Hikosaka (1992) used a nonlocal smeared IngraiTa and Panthaki (1986) performed a FCM mixed- crack finite element model to simulate hcture of a three- mode fracture analysis of the four point shear beam tested by point bend specimen. As shown by their finite element qesh, Bazant and Pfeiffer (1985). They concluded that, although a large number of finite elements is required near the crack shear fracture (€facture forming by shear slip, rather than by tensile opening) can occur under certain conditions, tensile tip to adequately represent the development of the FPZ. They were able to obtain good agreement with experimental re- and not shear fracture occurred in the specimens. sults using this approach, but it should be noted that, in their A smeared crack analysis of the four point shear beam test- analysis, the FPZ was of significant size compared to the size ed by Arrea and Ingraffea (1982) was performed by de Borst of the beam specimen. Had the specimen been larger, the ap (1986). A rotating crack band model was used, with adjust- proach might not have been feasible because of the large ment of the constitutive model with element size to maintain number of elements required. a constant fracture energy. Loading was controlled by the A noniocal smeared crack analysis of a beam was reported arc-length method to prevent solution instabilities. The anal- by Bazant and Lin (1988). Geometrically similar notched ysis required approximately 600 degrees of freedom, using beams of three different sizes of ratio 1:2:4 were analyzed eight-noded elements. The fuiite element results agree close- using the meshes of four-node quadrilaterals. The concrete ly with the experimental results. was described by the nonlocal smeared crack model, which A smeared crack band model that covers tensile softening is the same as the classical smeared crack model, except that in Mode I and shear softening in Mode II fracture was used the cracking strain was calculated from the maximum prin- by Rots and de Borst (1987) to analyze the specimen tested cipai value of the spatiaiiy averaged noniocal strain, E,rather by Arrea and ingraffea (1982). Interesting snapback behav- than local strain, E. The stress was assumed to decrease, ei- ior was modeled using an arc-length control procedure. ther linearly or exponentially, with the maximum nonlocal Carpinten (1989) used a FCM to simulate a double-edge principal straia The characteristiclength was taken in the notched four-point loaded shear Specimen. Snapback instabd- first case as 2.3 in. (58 mm) and in the second case as 3.2 in. ity was modeled for some cases, using a crack length control (81 mm), while the maximum aggregate size is da = 0.5 in. approach. A number of other problems were also analyzed.

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446.3R-20 AC1 COYMilTEE REPORT

A discrete cracking FCM was used by Gerstle and Xie concepts is presented in Saouma, Ayari and Boggs (1987). (1992) to model the single-edge notched four-point loaded The work includes the results of discrete LEW fracture me- specimen of Arrea and Ingraffea (1982). A dynamic relax- chanics analyses. The effect of various forms of loading, ation solution scheme was employed, with stress-based cri- concrete age and anisotropy on the stress intensity factors, teria for crack propagation and an automated remeshing direction of crack profiles, crack lengths and stress redistri- algorithm. The anaiysis is significant in that it uses a very bution is considered. coarse mesh (200 degrees of freedom), yet seems to capture A plane strain LEFM discrete cracking finite element the significantbehavior. analysis of the Koinbrein dam, a doubly curved arch dam in Malvar (1992) modeled the Arrea and IngmfYea (1982) Ausaia, is presented by Wawrzynek and Ingraffea (1987) beam using a local smdcrack approach with and without and Linsbauer et al. (1989% 1989b). Cracks are loaded by considering the transfer of shear stresses across the crack. A water pressure, as well as by remotely applied stress result- better prediction of the experimental results was obtained ants from a three-dimensional arch analysis. The cracks are when shear stress msfer was modeled. However, up to the approximately 14 m long, and therefore the assumption of peak load, the results were not sensitive to presence or absence LEFM seems justifted. Actual fractures in the dam seem to of sheax stress uansfer across the crack. Finally, it was shown have been accurately modeled using LEFM. that inadmissible results are obtained if both tensile and shear Mera et al. (1990) describe a smeared crack approach to stresses are assumed to completely vanish upon cracking. the analysis of progressive cracking in iarge áams. They ac- Schlangen and van Mer (1992) accurately modeled shear munt for crack pressurizafion by the water by including an ef- fracture with the triangular lattice model of brittle beam ele- fective stress in the model. Thermal and water pressure effects ments described in Section 4.2.1, which represent the heter- are also accounted for. 2D and 3D models are presented. ogeneity of the material. Ayari and Saouma (1990) developed discrete LEFM in conclusion, the four-point loaded shear beam has been cracking models for the efficient simulation of transient dy- analyzed by numerous investigators. The discrete cracking, namic discrete crack closure and crack propagation in dams. the nonlocal smeared cracking, and the triangular lattice After presenting simple validation problems, +ese models model all performed well in simulating the fracture behavior are integrated into an interactive graphical program that was of the beam. All of the methods have particular advantages used to analyze the Koyna Dam. and limitations, and it is therefore difficult to decide which Ingraffea (1990) describes the analyses of the Fontana of the approaches is preferable although the discrete crack- Dam, a generic gravity dam, and the Kolnbrein Dam. The ing approach appears to be more efficient for beams that ap- studies employ mixed-mode LEFM implemented within a proach--```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- the size at which LEFM becomes applicable. discrete crack model, automatic rezoning, finite element 4.25 Dams method. The study of the Fontana dam elucidated the mech- Dams belong to an imporrant class of concrete structures anisms for crack initiation, accurately reproduced the ob- that are fracture sensitive, and may in some cases even be an- served trajectory, and evaluated the effectiveness of interim alyzed using LEFM.A collection of nine papers presented at repair measures. The study of the generic gravity dam has as the International Conference held in Vienna in 1988 con- its objective the evaluation of usefulness of LEMfor design cerning fracture of dams is given in a special issue of Engrg. and quality control during construction. An envelope of safe Fmc. Mech., V. 35, 112/3,1990. ûther valuable sources in- lengths, heights, and orientations of cracks potentially grow- clude Dam Fracture (1991,1994) and six papers on fracture ing from cold lift joints on the upstream face is derived. It is of dams in Fracture (1987). also shown that, by neglecting the toughness of the founda- Several finite element studies of dams (considering crack- tion contact, classical design methods predict a conservative ing, but not using fracture mechanics concepts) were con- factor-of-safety against sliding, and that, when toughness is ducted in the 1960's, notably by Clough (1962) and set to zero, LEFM predictions are in good agreement with Zienkiewicz and Cheung (19644). the classical method. Apparentiy, the fitst true FEM fracturemechanics analysis Carpinteri et al. (1992) performed testing and analysis of a of a dam is presented by Ingrafíea and Chappe1 (1981). The gravity dam. Two scaled-down (1:40) models of a gravity Fontana Dam had developed a crack, and finite element anal- dam were subjected to equivalent hydraulic and self-weight yses using LEFM were used to model the discrete crack A loading. From an initial notch, a crack propagated during the rational explanation for the crack was thus obtained. loading process towards the foundation. The numerical sim- A discrete crack FEM approach was employed by Skxiker- ulation of the experiments used a FCM. The structural be- ud (1986) with a simple remeshing scheme to analyze a dam havior of the models and the crack trajectories are under dynamic excitation. However, despite his use of a dis- reproduced satisfactorily. crete crack approach, no fracture mechanics based criteria 4.2.6 Bond-~ly, were used to predict crack extension, and tensile stresses at Modeling of bond-slip is one of the most difficult and con- the crack tip were simply compared with the tensile strength. troversial aspects of fite element analysis of reinforced con- This approach would produce nonobjective results, because crete structures. Much of the controversy lies in the fact that the stress magnitudes produced by the crack tip elements many finte element models of reinforced concrete have pro- would be strongly dependent upon their size. vided excelient representations of experimental behavior A comprehensive investigation into the behavior of while allowing no bond-slip to occur &e., perfect bond) @ar- cracked concrete gravity dams using fracture mechanics win 1993). While it is known that bond-slip behavior arises at

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FINITE ELEMENT ANAìYSiS OF FRACTURE IN CONCRETE STRUCTURES 446.3R-21

least partiaüy from fracture of the concrete, other types of non- not based on fundamental or experimentally obtainable me- linearity such as crushing in front of ribs, chemical adhesion, chanical properíies (such as tensile strength and frachue

--```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- and fiction between the concrete and the steel play a role. The toughness). Additionally, mesh sensitivity would be expected detailed modehg of bond-slip associated with even one rein- because of the (Id)softening model used to describe the forcing bar is an extremely complex problem. concrete contjnuum. However, very close agreement between The problem of modeling the mechanical interaction be- analysis and experiment was obtained. tween a ribbed steel reinforcing bar and the surrounding con- Kaiser and Mehlhom (1987) studied and compared the crete has been addressed in a number of different ways. bond-link and the interface element approaches for modeling When the steel is modeled as smeared, the reinforced con- bond-slip between concrete and a single steel reinforcing bar. crete is considered as a homogeneous composite material They analyzed a reinforcd tensile specimen and compared with constitutive relations that include (impiicitiy or expiic- the numerical solution with tests and the numerid solutions itiy) the bond-slip behavior. These deisoften represent of others. They made the questionable assumption that bond- perfect bond. At a more detailed level, individual reinforcing siip can be considered to be a property of the interface, rather bars may be represented by tniss elements. in this case, ei- than a property that results from damage to the stnicture. ther the truss elements are connected directly to the concrete Kay et al. (1992) developed a smeared crack model to rep- elements, where bond-slip is zero, or nonlinear link elements resent accumulated damage in plain concrete and bond be are used to connect the steel tniss elements to the concrete to tween concrete and steel reinforcing bars. The model used a represent the effect of bond-slip. This is a cumbersome mod- continuum approach to describe mi-cks and crack coa- eling approach, theoretically unsatisfactory from a fracture lescence in plain concrete for the prediction of concrete ten- mechanics viewpoint, and nearly always ignores the splitting sile hcture. Concrete-reinforcing bar interaction was effects of bar movement. At a much more detailed scale, achieved through the use of onedimensionai beam elements both the steel and the concrete can be modeled as continua, that interact with three-dimensional continuum elements with interface elements placed between the concrete and the through a one-dimensional contact algorithm. SticWsiip in- steel. Rècognizing bond-slip as a fracture mechanics prob- teractions between the concrete and the reinforcing bars lem, the cracking of the concrete can be modeled using either were expressed in terms of the interface stress or the internal discrete or smeared fracture mechanics. In this approach, the damage variables of the concrete damage model. concrete can be considered as an unreinforced continuum Ozbolt and Eligehausen (1992) analyzed the puilout of a smundingthe reinforcing bar. Several analyses of this type deformed steel bar embedded in a concrete cylinder under are described next. monotonic and cyclic loading. The analysis was performed In pioneering work, a singly reinforced, axisymmetric, using axisymmetric finite elements and a 3D microplane “tension-pull” specimen was analyzed wing the FEM by (smeared crack) model for concrete. instead of the classical Bresler and Bertero (1968). To simulate cracking and other interface element approach, a more general approach with nonlinear effects, a soft “homogenized boundary layer” of spatial discretization modeling the ribs of a deformed steel prespecifed width was assumed surrounding the steel. As bar was employed. The pull-out failure mechanism was ana- expected, the existence of the boundary layer significantíy lyzed. Comparison between numerical results and test re- decreased the overall stiffness of the specimen. However, sults indicated good agreement. Their approach was able to this approach did not use fracture mechanics principles. The correctly predict the monotonic as weil as cyclic behavior, approach was of limited predictive utility. as both the width including friction and degradation of pull-out resistance due of the boundary layer and its constitutive properties were not to the previous damage. determined from any fundamentai theoretical or experimen- Brown et al. (1993) and Darwin et al. (1994) modeled the tal information. bond-slip/strength behavior of individual bars in beam-end The first fracture mechanics approach to the analysis of specimens. The 3D analyses represented the individual ribs bond slip was taken by Gerstie (1982) and Ingraffea et ai. on the bar, and used the FCM to represent concrete fracture (1984). An axisymmemc discrete FCM analysis of a single and a Mohr-Coulomb model to represent steel-concrete in- concentrically reinford axisymmetric tension-pull speci- teraction at the face of the ribs. The analyses, aimed at better men was presented. No slip was assumed between the con- understanding bond behavior, gave an excellent match with crete and the steel; instead, “bond-slip” occurred as the result empirical relationships between bond strength and the ef- of the development of radial cracks propagating outward fects of embedded length and concrete cover. from each of the ribs on the reinforcing bar. The effect of More work is required to develop credible methods for the longitudinal spiitting cracks was neglected due to the axi- numerical modeiing of bond-slip. symmetric modeling assumptions. 4.2.7 Other types of purin concrete símetures Vos (1983) reported on severai smeared crack FEM bond- Severai other types of plain concrete structures have been siip calculations. To account for the complex cracking/crush- extensively analyzed using fracture mechanics and FEM ap ing situation surrounding a ribbed steel bar, the bar was as- proaches. hilout of anchor bolts is an important fracture-con- surned to be surrounded by a softening layer of concrete, îrolìed problem. Analyses of this problem has been reported approximately one bar diameter in width. The elements within by Heliier et al. (1987), Bittenmurt, In@ea and Llorca this softening layer were assumed to be equal in length to the (1992), Eligehausen and ûzbolt (199û), Cervenka et al. rib spacing on the bar. This approach suffers from the fact that (1991), Eligehausen and Ozbolt (1992). Elfgren and Swam the input parameters used to describe the softening layer are (Elfgren 1992) have reported resuits of a round-robin analysis

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446.3R-22 AC1 COMMITTEE REPORT

of anchor bolts organized by RILEM TC 9û-FMA. Fifteen of nisms of shear wall specimens were examined. The excellent the sixteen analyses were based on fracnire mechanics con- agreement with the monotonic results probably resulted cepts. Variations in the analysis assumptions included: LEFM, from the fact that these panels had a reasonable amount of re- FCM, and smeared crack models. The responses included inforcing steel, and therefore were insensitive to fractureme- some 82 solutions. Punching shear in slabs and anchorage of chanics and did not exhibit the fracture mechanics size tendons in prestressed members can also often be considered effect. The cracking of concrete in these panels was of a dis- as the fracture of a piain concrete structure. tributed, rather than a localized, nature. The lack of agree- ûther types of plain concrete structuresthat have been an- ment under cyclic load was due to other (noncrack) aspects alyzed include thick walled rings (pukl et al. 1992) and many of their modeling scheme. analyses of the double cantilever beam @CB) fracture Another (locai) smeared crack analysis of several Cerven- toughness testing specimen (Gustafsson 1985, Sluys and de ka and Gerstle (1972) shear panels was conducted by Darwin Borst 1992, Liang and Li 1991, Liaw et al. 1990%Liaw et al. and Pecknold (1976), using both smeared crack and smeared 199Ob, Dahiblom and ûttosen 1990, Du et al. 1990, Bert- modeling of steel. The observed behavior of the panels was holet and Robert 1990). replicateú very well by the analysis for both monotonic and cyclic loading. ûther (locai) smeared crack anaiyses of the 4.3-Reinforced concrete same shear panel are presented by Schnobrich (1977) and The fracture mechanics analysis of reinforced concrete Bergan and Holand (1979), giving excellent agreement be- structures is often much more difficult than the analysis of tween experimental and analytical results. plain concrete structures. The reinforcing steel may be mod- Bazant and Ceúolin (1980) performed a smeared crack eled discretely (usuaily using truss elements) or in a smeared analysis of a center cracked reinforced concrete membrane fashion, with the steel included in the constitutive model in uniaxial tension. They did not, however, compare the an- used to calculate the element stiffness matrices. In both cas- alytical results to tests. They used the crack band model to es, it is difficult to correctly model the effect of bond-slip be- investigate the effect of mesh size on the results. They ob- tween the concrete and the steel, although modeling bond- served that, if bond-slip is not correctiy modeled, lack of ob- slip does not appear to be critical to the solution of many jectivity results. In a subsequent paper, Bazant and Cedolin problems @arwin 1993). (1983) investigated the same problem, this time with respect ûften, but not always, the addition of steel reinforcement to to meshes at an angle to the crack propagation direction. a structure makes it fracture insensitive. Because of this fact, They concluded that the blunt crack band approach is valid early finite element anaiyses of reinforced concrete stnicaires only if fine meshes are used, although correct convergence were often successful, even without realistically modehg appears to occur as the mesh size tends to zero. fracaire of the concrete. Sometimes, however, structures wiii Gentle (1982) and Ingraffea et al. (1984) employed a dis- fail by fracniring in unanticipated ways. This is the reason for crete FCM model to simulate a mkedmembrane in uniaxial developing finite element methods that are capable of predict- tension. Bond-slip was modeled using special ‘’tension-soft- ing fracturing modes of behavior. A summary of recent tech- ening” elements which bridge the primary crack. The effect of niques used to model reinfod concrete siruCaires using varying assumptions regarding bond-slip were studied. finite element analysis is presented by Darwin (1993). Bedard and Kotsovos (1985) used a (locai) smeared crack in the remainder of this section, the literature is surveyed to approach to model four reinforced concrete panels subjected illustrate various methods of analysis, including hcture me- to loading in pure shear, shear and uniaxial compression, and chanics effects, of reinforced concrete structures. In the course shear and biaxial compression. The panels were reinforced of the chapter, the term ”tension stiffening” will be used to de- with different steel ratios in each direction, varying from scribe some models. The term refers to a numerical device 0.00713 to 0.01785. Good correlation was obtained with ex- used to limit the rate at which the stress across a smeared crack perimental results. In this case, fracture was not a controlling drops to zero once a crack forms. Tension stiffening represents mechanism, and therefore correct modeling of fracture was the physical behavior of cracks crossed by reinforcing steel unnecessary to obtain reasonable results. and serves to stabilize the numerical solution. Models that in- Chang, Taniguchi and Chen (1987) applied a (local) clude tension stiffening usually do not take into account the smeared crack model to the analysis of reinforced concrete fact that the methoú inherently represents fracture energy. panels tested and analyzed by Vecchio (1986). The “fracture Theefore, appiication of tension stiffening does not provide criteria” in the sophisticated constitutive model were based an accurate fracture mechanics representation and serves prin- upon limiting stresses and strains, and therefore are not true cipaiiy to slow the rate at which residual saesses are reim- fracture criteria. However, very good agreement between ex- posed on the smchue being modeled. periment and numericai results was obtained. The panels 43.1 Reinforced concrete membranes were heavily reinforced, and therefore fracture criteria, Shear walls in buildings are usually considered as mem- again, probably did not control the behavior of the panels. brane elements because the major loads are in the plane of Channalceshava and Iyengar (1988) presented a (local) the wall. Cervenka and Gerstle (1972) reported on an early smeared crack constitutive model used to model shear panels smeared crack analysis of reinforced concrete panels that tested by Vecchio (1986). An elasto-plastic crackhg model compared very favorably with their experimental results un- for reinforced concrete was presented. The model included der monotonic loading but not under cyclic loading. Load- concrete cracking in tension, plasticity in compression, aggre- displacement diagrams, crack patterns and failure mecha- gate interlock, tension softening, elasto-plastic behavior of the

steel, bond-slip, and tension stiffening. A procedure for incor- to modeling discrete cracks by remeshing to represent the porating bond-slip in smeared steel elements was described. development of cracks. His criteria for cracking were stress- Good agreement with exprimeniai resuits was obtained based, rather than fracnire mechanics-based, and yet, for the Barzegar (1989) examined the effect of skew, anisotropic example that he studied (eccentrically reinforced tension- reinforcement on the post-cracking response and ultimate pull specimen), reasonable results were obtained. me be- capacity of reinforced concrete membrane elements under havior of the tension-pull specimen that he studied was not monotonically increasing proportional loading. Appropriate primarily controlled by fracture of the concrete, but rather by constitutive models, including post-cracking behavior, were the stiffness of the reinforcement) discussed. Several shear panels were analyzed and compared Valliapan and Doolan (1972) discussed a smeared crack with experimental results. approach to the finite element analysis of reinforced con- Crisfield and Wills (1989) used smeared crack models to crete. Their model assumed elastoplastic response of the analyze the concrete panels tested by Vecchio and Collins concrete with a tension cut-off, discrete modeling of the steel (1982). Different models involved both fixed and rotating using truss elements, and no bond-slip. They analyzed sever- cracks, with and without allowance for the tensile strength of al reinforced concrete beams with reasonable success, even concrete. A simple plasticity model was also applied. Be- without using correct fracture mechanics principles. cause the numerical solutions were obtained using the arc- Colville and Abbasi (1 974) employed an approach very length analysis procedure, the complete (including post- similar to that of Valiiapan and Doolan (1972), with the ex- maximum) load-deflection responses were traced, and con- ception that the steel was considered to be smeared. For the sequently it was possible to clearly identify not only the examples presented in the paper, reasonable results were ob- computed collapse load but also the collapse modes. tained. Again, their examples appear to have been fracture Gupta and Maestrini (1989a, 1989b) used a (local) insensitive. smeared approach to model the post-cracking behavior of re- Nam and Salmon (1974) reported upon the smeared crack inforced concrete panels tested by Vecchio (1986). Good analysis of reinforced concrete beams. Their approach was agreement was obtained. similar to that of Colville and Abassi (1974). One of their Massiwte, Elwi and MacGregor (1990) presented a practical most significant conclusions was that an incremental tangent twodimensional hypcelastic model for reinfored conrete, stiffness rather than an iterative initial stiffness method of with emphasis on a new approach for the descripiion of tension solution was necessary to solve such problems. While cracks stifíening, using a smeared crack band model. The model pre were predicted, fracture mechanics principles were not in- dictions were compared with test results of reinfod concrete cluded in their model. members in uniaxial tension, reinfod concrete panels in Sal& El-Din and El-Adawy Nassef (1975) were among shear, and reinfo& comme plates, Simpiy supported along the first to combine fracture mechanics and the finite ele- four sides and loaded axially and mversely. ment method to analyze reinforced concrete beams. They as- Wu, Yoshikawa and Tanabe (1991) presented a complex sumed (incorrectly) that the conditions of LEFM apply to smeared crack and damage mechanics approach used to relatively small concrek beams. Discrete cracks were repre- model the tension stiffening effect for cracked reinforced sented by setting the stiffnesses of the concrete elements concrete. Several membranes in various states of shear and equal to zero. They modeled the reinforcement discretely, tension were analyzed, and the results were compared with and connected the steel elements directly to the concrete el- the experimental results. Nonlocal effects appear to have ements, modeling zero bond-slip. They used the compliance been ignored. derivative method to calculate the energy release rate of a 4.3.2 Beams andfiames single vertical crack. Their derivation predicted the fracture A number of finite element analyses of beams and frames mechanics size effect. By comparing their numerical results include correct fracture mechanics effects. In general, early to the results of an experimental investigation, they found models did not account for the fracture effects correctly, but that the fracture mechanics-based approach was more accu- since about 1980, most of the work has been quite sophisti- rate than the stress-based approach for the prediction of cated in this regard. Neither the smeared nor the discrete ap cracking in reinforced concrete beams. proach to cracking is clearly dominant. Some of the studies Although they recognized that their smeared crack ap described below have provided useful insights into the struc- proach to the modeiing of shear-criticai reinforced concrete tural behavior of beams. However, it is clear that the fracture beams was not theoretically correct from a fracture mechan- analysis of beams and frames is not yet at a state where it can ics viewpoint, Cedoïm and Dei Poli (1977) concluded that be used routinely in the design offíce. their local noniinear constitutive model €or concrete was ad- Ngo and Scordelis (1967) reported the first discrete crack- equate to predict most of the significant responses. Studying ing FEM analysis of a reinforced concrete beam. Cracks a reinforced concrete beam 22 in. (560 mm) deep with ap were modeled but not propagated, their positions were as- proximately 1.8 percent flexural steel, they compared the nu- sumed a priori. The analysis was linear elastic, and stresses merical and experimental results. Their finite element at the crack tips were not accurately modeled. This pioneer- model, which incorporated a nonlinear representation for ing paper did not make use of fracture mechanics principles. concrete under biaxial stress, was able to predict the load-de- Nilson (1967,1968) discussed the use of the finite element flection curve, crack pattern, and failure load of the rein- method to represent reinforced concrete beams, including forced concrete beams. The model, however, was not able to bond-slip and discrete cracking. He suggested an approach represent dowel action and crack propagation at failure.

Bergan and Holand (1979), in a comprehensive discussion large crack strains was observed prior to failure. Their model of finite element analysis of reinforced concrete structures, took into account the width of the elemenî/Gauss point over presented an analysis using local smeared crack techniques which a crack is assumed to be smeared, and thus attempted similar to those just described by Cedolin and Dei Poli to use a form of the crack band model to objectively rep= (1 977). The computed load-deflection curve agreed with the sent the fracture energy associated with discrete cracking. experimental results, except that the ultimate load was over- They concluded that an accurate response was obtained us- predicted, and the actual beam failed by a rapid diagonal ten- ing the technique. sion faiiure mechanism that could not be captured by their Gustafsson and Hülerborg (1988) used the finite element smeared crack model. method to model size effects due to discrete cohesive diago- A smeared crack approach was also employed by Bedard nal tension cracks in singly reinforced beams without shear and Kotsovos (1985), who analyzed a reinforced concrete reinforcement. They found a significant fracture mechanics deep beam with web openings and web reinforcement, using size effect. They carried out studies of the sensitivity of shear plane stress analysis. Comparison with experimental results strength to reinforcement ratio and depth-to-span ratio. Their was good. They stated that the smaliest fínite element size work, aside from being correct from a fracture mechanics should be greater than two to three times the maximum ag- viewpoint, serves as an example of how the finite element gregate size. However, there is no evidence that such a rule method can be used to great advantage in developing design would produce universally successful analyses. code equations. More recently, Ma, Niwa, and McCabe Bazant, Pan, and Pijaudier-Cabot (1987) and Bazant, Pi- (1992) performed similar fictitious crack analyses. Their re- jaudier-Cabot, and Pan (1987) analyzed the softening post- sults show conclusively that a size effect exists for shear crit- peak loaci-defiection relation for reinforced concrete beams id beams. They therefore recommend that the ACI-318 and frames using layered finite elements. Concrete was mod- code be modified to reflect this size effect. eled as a strain softening material in both tension and in com- Cervenka, Eligehausen, and Puki (1991) used a smeared pression; the steel reinforcement was modeled as elastic- crack approach to model shear failure in simply supported, plastic. Standard bending theory assumptions were used and reinforced and unreinforced concrete beams without shear bond-slip of reinforcement was neglected. The model could reinforcement, using a crack band model. They performed approximate existing test results for beams and frames. At two plane stress analyses of a fracture-sensitiveunreinforced the same time, constitutive laws with strain softening, in- beam. One analysis used a fine mesh, and the other used a cluding those of continuum damage mechanics, were shown, coarse mesh. They found that the same peak load was pre- in general,--```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- to lead to spurious sensitivity of results with redicted by both meshes, but that the post-peak behavior of the spect to the chosen finite element size, similar to that docu- beam was sensitive to mesh density. mented for other strain-softening problems. in analogy to the Feenstra, de Borst, and Rots (1991b) used the fictitious finite element crack-band model (Section 3.2.1), they rec- crack approach, including a representation for aggregate in- ommend the use of a minimum admissible element size, terlock, to model a moderately deep beam with several pre- specified as a cross sectional property. defined discrete crack locations. Eight-noded continuum One of the major problems in the analysis of reinforced elements, six-noded interface elements and three-noded concrete structures is the representation of reinforcing bars. truss elements were used to model steel, with interface ele- Typically, either the steel is represented in a smeared man- ments used to model bond-slip. A Newton-Raphson method ner, in which case the concrete and steel are combined and with indirect displacement control was used. The computed treated as a composite material, or the steel is represented response of the beam matched the experimental results rea- discretely, in which case nodes must be collocated with the sonably well. This, of course, was not a completely predic- steel bars, which severely constrains the finite element mesh. tive analysis because they inserted fictitious cracks at Aliwood and Bajarwan (1989) presented an interesting ap- predetermined locations. proach for modeling reinforcing steel that allows the steel to Gajer and Dux (1990,1991) analyzed two reinforced con- be represented independently of the concrete (using separate crete beams using a crack band model, featuring simplified nodes and elements). An iterative solution scheme is used to representations of both nonorthogonal cracking and the in- bring the steel elements into equilibrium with the concrete fluence of cracking on local materiai stiffnesses. At an inte- elements. gration point, one primary crack of fmed direction and one Balakrishnan and Murray (1988) described a practical rotating crack are allowed to develop. The beams analyzed stress-strain relationship for concrete for use in a smeared by de Borst and Nauta (1985) and Bedard and Kotsovos crack model, which was effective in predicting the behavior of (1985) were analyzed, with satisfactory results. reinforced concrete beams and panels. The model includes ho- Ozbolt and Eiigehausen (1991) studied the shear resistance mogenized material properties that incoprate the estimated of four similar reinforced concrete beams of different sizes effects of strain localization. Methods of estimahg these ho- without shear reinforcement using a nonld microplane mogenized properties are presenteù in their paper. model and plane stress finite elements to study the size effect. Channakeshava and Iyengar (1988) used a smeared crack The calculated results demonstrated a decrease in average model for a singly reinforced concrete beam without shear strength with an inmase in member size. It was demonstrated reinforcement. The shear critical beam failed by diagonal that the results were insensitive to mesh size and load path, in cracking. in the analysis, although the exact mode of failure contrast to the results of local continuum anaíyses. They stated could not be assessed, extensive diagonal cracking with that element size must not be larger than one-half of the char-

acteristic length of the nonlocal continuum. ûtherwise, the the effects of reinforcement and cracking, as the foliowing ref- analysis is equivalent to the classical local continuum anaiy- erences demonstrate- In general, a close match has been ob- sis. Thus, large numbem of elements were required in these tained between analyticai and experimental results, without analyses, especially for the larger beams. spialregard for fracture concepts. The lack of importance of Pagnoni, Slam, Ameur-Moussa, and Buyukozturk (1992) fracture mechanics is likely due to the dominant role played by formulated a finite element procedure for the nonlinear analy- reinforcement in the structures analyzed. sis of general three-dimensional reinforced concrete struc- Hand, Pecknold, and Schnobrich (1973) used layered ele- tures. They employed a threeimensionai version of the crack ments, in which material properties changed from layer to band model to objectively represent fracme. The model was layer, to model several reinforced concrete plates under uni- demonstrated using àetaüed analyses of a singly reinfod form stress, a comer-propped reinforced concrete plate un- deep beam and a presfressed concrete reactor vessel. der a concentrated centrai load, and a reinforced concrete 43.3 Containment vessels shell under uniform load. The numericai results closely Concrete containment vessels present challenging analy- match the experimeatal results for all problems. Cracking sis problems. However, because of the severe consequences was governed by the principal tensile stress. of a failure, detailed analysis is required to assure a safe de- Lin and Scordelis (1975) took a layered shell element ap sign. These vessels are usually heavily reinforced, and there- proach to model both the reinforcement and the concrete. fore, fracture mechanics issues are of secondary importance. The concrete was assumed to behave plastically in compres- Rashid (1968), in the first application of the smeared crack sion, and as a gradually softening material in tension. Three approach, analyzed a reinforced concrete pressure vessel. examples, including a circular slab, a square slab, and a hy- The steel elements were assumed to be elastic-perfectly- perbolic paraboloid shell, were analyzed, with the predicted plastic. Concrete and steel were modeled using separate fi- results closely matching the experimental data. nite elements and an axisymmetric condition was assumed. Schnobrich (1977) used a (local) smeared crack and Limited experimental cornpansons were made. Because the smeared steel model to solve several plate and shell prob- concrete cracking was expected to be distributed, rather than lems. The agrement between analysis and experiment was localized, due to the presence of heavy reinforcement, no excellent. Specimens modeled include a plate subject to fracture mechanics issues arose from this approach. In es- twist and uniaxial moment, a comer-supported two-way sence, the composite reinforced concrete material was frac- slab, a three-hole conduit, a shear panel, and a hyperbolic pa- ture insensitive. raboloid shell roof subjected to vertical loading. Meyer and Bathe (1982) used smeared mck and smeared Bashur and Darwin (1978) used a (iocal) smeared crack steel models to address some of the questions facing the engi- model to represent cracking as a continuous process through neer charged with assessing the safety of concrete strums the depth of plate elements. Reinforcing steel was represented for which conventional linear analysis is deemed inadequate. as a smeared material at the appropriate position(s) through Guidelines were given to aid the engineer in selecting the ma- the depth. Element stiffness was determined using numerical terial model most appropriate for this purpose. Questions re- btegration, removing the need to use layered elements. A lated to finite element analysis were discussed, including close match was obtained with experimental results for a se- numexical solution techniques, their accuacy, efficiency and ries of oneway and two-way reinforced concrete slabs. applicability. An example of a prestressed concrete reactor Bergan and Holand (1979) analyzed reinforced concrete vessel illustrated the application of nonlinear finite element plates, using a smeared crack and smeared reinforcing mod- analysis to concrete structures in engineering practice. el. Initially, zero tensile strength was assumed, which pro- Clauss (1987) presented a summary of the analyses of a duced deflections that were too large. A gradually softening one-sixth scale model of a reinforced concrete containment tensile behavior, however, produced a good comparison with --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- vessel. The analyses were conducted by ten organizations in experimental data. Neither of the plates analyzed was frac- the United States and Europe. Each organization was sup- ture sensitive. plied with a standard information package, which included Barzegar ( 1988) introduced the use of a layering technique construction drawings and actual material properties for in general nonlinear FEM analysisof reinforced concrete flat most of the materials used in the model. Each organization shell elements. A stack of simple four-node plane stress and worked independently using its own analytical methods. The Mindlin plate elements were used to simulate membrane and report includes descriptions of the various analytical ap- bending behavior. The method, together with nonlinear con- proaches and pretest predictions. Significant milestones that stitutive models for concrete and steel, was described and occur with increasing pressure, such as cracking and crush- implemented. Bond degradation between concrete and steel ing of the concrete, yielding of the steel components, and the was simulated. Several examples were presented. failure pressure and failure mechanism were described. Hu and Schnobrich (199 i) proposed plane stress constitu- 4.3.4 Pcates unà shells tive models for the nonlinear FEh4 analysis of reinforced Plates and shells are common components of reinforced concrete structures under monotonic loading. An elastic concrete súuctws. Cracking can significantly influence both strain hardening plastic stress-strain relationship with a non- the stiffness and strength of these members. To model such a associated flow rule was used to model concrete in the com- structure as a three-dimensionai solid is computationally ex- pression dominated region and an elastic brittle fracture pensive. It is therefore necessary to use spiai plate and shell behavior was assumed for concrete in the tension dominated fuiite elements. Various appxuaches have been taken to model area. After cracking took place, the smeared crack approach,

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together with the rotating crack concept, was employed. To- It does not appear that the discrete approach is "better" gether with layered elements, these material models were than the smeareü approach or vice versa. On the contrary, it used to model the flexural behavior of reinforced concrete appears that it would be wise to move toward a marriage of plates and shells. Good agreement with test results was ob- these two approaches to better model the entire regime of tained. Fracture mechanics issues were not considered. concrete behavior from multiaxial compression of small Lewinski and Wojewodzki (1991) introduced a noniinear specimens to LEFM in large structures. This is surely one of (local, smeared crack) procedure to solve the elasticity prob- the major challenges of the coming years. lem for a doubly reinforced concrete slab was found. Ten different cracking patterns were assumed in this formulation. CHAPTER HONCLUSIONS Use of each pattern implied the division of the slab thickness --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- into several layers (maximum three layers of concrete). Nu- 5.1-General summary merical integration was used through the total thickness. Re- The purpose of this report has been to describe the state- sults obtained with this approach compared favorably with of-the-art of finite element analysis of fracture in reinfo& experimental results for deep beams and plates. concrete structures. Chapter 1 provides general background 435 @her reinforced concrete structures and historical perspective, and highlights the two most pop Gentle (1987) used a plane strain (iocal) smeared crack fi- ular methods used in fullte element analysis to represent nite element analysis technique for the analysis of reinforced fracture: the discrete crack approach and the smeared crack concrete culverts. Since the steel was smeared, the rein- approach. Chapters 2 and 3 present the two approaches in forced concrete was treated as a composite material. No greater detail. in each case, the goal has been to provide a post-cracking strength was included in the model. The FEM broad view of the field and to provide references to the liter- analyses matched the experimental results reasonably well. ature so that the interested reader may go directly to the pri- However, the analyses tended to underestimate the deflec- mary sources for additional information. tions and overestimate the strength of the ring sections. Chapter 4 provides examples (with special attention to Swknson and hgraffea (1991) used discrete cracking fracture mechanics) of finite element analyses. of concrete models to analyze the Schoharie Creek Bridge. The bridge structures that are available in the archival literature. Atten- collapsed in 1987 due to a flood that scoured the soil under a tion is focused on twelve types of structures, consisting of pier, which led to unstable cracking of the minimally rein- both plain and reinforced concrete. The review includes al- forced pier section and precipitated the actual collapse. The most one hundred papers that describe finite element anaiy- analyses included linear and nonlinear fracture mechanics ses of these structural types. While most of the early analyses evaluations of the initiation, stability, and trajectory of the of concrete were inadequate from a theoretical fracture me- crack that precipitated failure. A principal motivation of the chanics viewpoint, the results proved more than adequate, effort was to apply the results of recent concrete fracture me- probably because of the nature of the problems selected. chanics research to a problem of practical significance. The These early models were principaiiy research tools, not ap- authors concluded that cracking of the bridge foundation plicable to design. The more recent efforts, although largely could be rationally explained through the use of fracture me- theoretically correct from a fracture mechanics viewpoint, chanics principles. generally describe analysis methods that remain too cumbersome to apply in routine design environments. For the most 4.4-Closure part, only the most readily available archival literature has Over the past 30 years, research into finite element model- been referenced. ing of plain and reinforced concrete structures has been in- From Chapter 4, it is clear that concrete structures can be tense. The early models that did not take fracture mechanics categorized into two classes: those with behavior that is not into account proved quite satisfactory to solve the problems governed by fracture mechanics and those with behavior that to which they were applied. While the possibility exists that is. It is clear that special fracture mechanics considerations the problems that have been presented in the literature were are not necessary for a large class of reinforced concrete only the most successful (and fracture sensitive structuresfor structures. However, it is also clear that some reinforced which correspondence between analysis and test results was concrete structures are fracture sensitive. To predict fracture poor were simply not reported), it is clear that the published behavior, a finite element code must include the capability to problems that were solved without the proper application of model fracture. As far as the engineer is concerned, if the íkacture mechanics were largely fracture insensitive. Fur- structure being analyzed is known to be fracture insensitive, ther, nearly all of the anaiyses fail to include a convergence then the analysis may proceed without resorting to fracture study to determine the effect of mesh refmement upon the mechanics. On the other hand, for structures that are known solution. to be fracture sensitive, the engineer is well-advised to use fi- Discrete and smeared crack FEM analyses have success- nite element codes that correctly model fracture. There are, fully solved problems that are known to be fracture sensitive. of course, times when an engineer is uncertain whether or However, most of the problems solved to date have been al- not a structure is ftacture sensitive. In these cases, the use of most trivial in their geometric simplicity. The challenge now a finite element code capable of predicting fracture is advis- is to apply correct fracture mechanics principles to the e=- able. cient analysis of nontrivial structures, such as reinforced Fracture sensitivity is similar to buckling in that both be- concrete solids, frames, plates, and shells. haviors may or may not exist, but each can only be detected

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through the use of analytical tools capable of predicting such 446.1R Fracture Mechanics of Concrete: Concepts, effects. It seems likely that, in the next ten years, reliable. Models and Determination of Material codes with the capability to predict fracture of concrete Properties smctures will be developed, just as reliable codes with geometrically nonlinear capabilities have developed over the 6.2-Cited references past ten years. Allwood. R J.. and Bajarwan, A. A., 1989. “A New Method for Modeling Reinforcement and Bond in Finite Element Analyses in Reiiorced Con- crete,” Int. J. NU^ Meth Yl Engg., V. 28, p~.833-844. 5.2-Future work Arna. M., and in- A. R,1982, ”Mi~ed-ModeCmk Ropagata in Very few successfui commercial finite element codes can Mortar and Coa-” Report NO.81-13, Depaamnt Of S- Engineer- handle the nonlinear analysis of concrete structures. Those ing, Come11 University, Ithaca, N.Y. commercial codes that can handle the analysis and design of ASCE Tssk Cornminee on %te Elment Analysis of Reinfoioed Con- Structures, 1982, Eiemeru Analysis OfReinfirceàGmcme, k H. reinforced concrete structures typicaily perform hear anal- crete F* Nikm,chmn.,Ameraan Society of Civii Engineers, New York 545 pp. yses, or at most nonlinear geometric analyses, and then use Muri, S. N.; Kobayashi, A. S.; and Nakagaki, M., Apr., 1975, “An the results of these analyses, together with design provisions Assumed Displacement Hybrid Element Model for Linear Fracmm Mechan- (ACI 318), to determine the adequacy of the members. In ics,”Int. 1. Fmrrure, V. 11, No. 2, pp. 257-271. other words, current design practice does not account for AyaM L,and Saouma.V. E., 1990, “A Fracme M~~ Bad seis- mic Adysis of Concrete Gravity Dams Using hteCraclrs.” Enge. noniinear material behavior, much less fracture behavior, Fmc. Mech, V. 35. No. 1#3, pp. 587-598. during the analysis phase. Balaluishnan, S., and Murray, D. W.,1988, “Concxete Constitutive Model It therefore seems that the research community is far ahead for NLFE Analysis of Structures.” L Smcf. ha.,ASCE, V. 114, No. 7, of (and out of step with) the practicing community in the July. pp. 1449-1466. Balakrishm S.; Elwi A. E.; and M-y, D.W., 1988, “‘Effect Of Model- analysis of concrete structures. Since practicing engineers do ing on NLFE Analysis of Concme Seuctures,” L Star. Engrg., ASCE, V. not have a routine need to predict fracture behavior, they are 114, NO. 7, July, p~.1467-1487. not willing to pay for such an analysis capability. When they Barsom, J. U,and Rolfe, S. T., 1987, Fmcture and Fatigue Contml in do have a need to predict or evaluate the fracturebehavior of Szrucrures,2nd Edition, Rentice-Hall, New York Barsoum, R S., 1976, “ûn the Use of Isoparamenic Finite Elements in a structure, it usually involves a special application requiring Linear Fracnue Mechanics:’ Int. L Nwn Meth in Enge.. V. 10, pp. 25-37. so much of their time and energy that they often go to re- Banegar, E, 1988, “Layering of RC Membrane and Plate Elements in searchers to provide the solution. Nonlinear Analysis,” J. Smcrr Engtg., ASCE, V. 114, No. 1 1, Nov., pp. The research community should continue to develop gen- 24742492. Banegar F., 1989, “Analysis of RC Memlnane Elements with Anisotropic erally accepted methods for numerically modeling fracture Reinforcement,” J. Smc~Enge., ASCE, V. 115, No. 3, Mar., pp. 647-665. behavior. But it is clear that, at least for now, these analysis Bashur, F. K. and Danvin, D., 1978, “Nonlinear Model for Reinforced methods will not be used directly by practicing design engi- Concrete Slabs,” 1 Stnrcr. Div., ASCE, V. 104, No. STI, Jan., pp. 157-170. neers, but rather by other researchers who may use the codes Bathe, K.-J., 1982, Fiiire Element Pmedwes in Engineering Analysis. F’rentice-Hail, Englewood Cliffs. NJ. to study aspects of concrete structural behavior. These codes Bazant, Z. I?, 1976, “instability, Ductility and Si EneCt in Sttain-Soh- may be used, for instance, to develop new and better design hg Concrete,mJ. Engrg. Mech Div., ASCE. V. 102, No. EM2, Apr.. pp. 331- expressions for the AC1 Building Code (AC1 3 18). 344. A miitfui area of research involves combining the Barant, 2. P., 1984, “Size Effect in Blunt Fracture: CMiCrete, Rock, Metal.”J. fige. Mech. ASCE, V. 110. No. 4, Apr., pp. 518535. smeared crack and the discrete crack approaches. Both ap Barant, 2 P., 1986. “Mechanics of Distributed Cracking:’ MME AppL proaches have limitations, and by combining the two ap MEhRev.. V. 39, NO. 5, May, pp. 675-705. proaches, a high-fidelity numerical representation of Barant, Z. P., 1994, “Nonlocal Damage Theory Based on Mimmechanics concrete behavior may be achieved. of Crack Inredons,” J. Engrg. Mech, ASCE,V. 120, No. 3, Mar., pp. 593- 617. Addendum and Emta, V. 120. No. 6, June, pp. 1401-1402. The fracture analysis of three-dimensional structures is Bazant, Z. P.. and cedolin, L, 1979, “Blunt Crack Band Propapation in stili in its infancy. Both the smeared and discrete crack ap Finite Elemt Analysis,” 1. Enge. Mech, ASCE, V. 105, No. 2, Feb., pp. proaches require further development to be useful for the 297-315. analysis of three-dimensional solids, not to mention shells, Bazant, Z P., and cedolin, L., 1980, ‘‘Fracture Mechanics of Reiid Coimete,” J. fige. Mech, ASCE, V. 106, No. 6, Dec., pp. 1287-1306. plates, and beams. The numerical approaches to the repre- BazmG 2 I?, and Cedolin, L., 1983, ‘Finite Element Modeling of Crack sentation of the interaction between reinforcement and con- Band prOpagation,”J. Suuct. Enge., ASCE, V. 109, No. 2, Feb., pp. 69-92. crete (bond-slip) are currently unsatisfactory and primitive. Bmt, 2. i?, and Oh, B. H, 1983, ‘’Crack Band ?heoiy for Fracture of Also, the effects of concrete creep, shrinkage, and cyclic be- Concrete,” Morei: Sm.RILEM, Paris, Fxance, V. 16, Feti., pp. 155-177. havior usually ignored in numencal models of concrete Bazar& 2. P.; Belytschko T. B.; and ctiang, T.-P., 1984, “Continuum The- are ~ryfor Strain-Softening,” J. Enge. Mech, ASCE, V. 1IO, NO. 12, pp. 1666- behavior, even though they may have large effects upon 1692. structural response. Although much progress has been made, Bmt,Z I?, and Pfeiffer, P. A., 1985, ‘Tmof Shear Fractun and Suain the goal of developing accurate and general numerical mod- Softening in Concrete,” Froc., 2nd Symposium on the lnteraction of Non- els for plain and reinforced concrete structures remains. Nuclear Munitions on Stnicnires, PanamaCity Beach Florida, Apr., 15-19. Bazant, 2. P.; Pan, J.; and Pijaudier- G., 1987, “Softening in Rein- forced Concrete Beams and Frames,” J. Sfrucf. Ekgrg.., ASCE, V. 113, No. CHAPTER ”REFERENCES 12. Dec., pp- 2333-2347. BmL2 P.; Pijaudier-cabot,G-; and Pan, J., 1987, “Ductility, Snapback 6.1 -Recommended references Size Effw and Redistribution in Softening Beams or Fmnes.’’ L Sm American Concrete Institute fige..,ASCE V. 113, NO. 12, Dec., pp. 2348-2364. Bazant, 2. i?, and Lin, F. B., 1988, ‘Nonlocal Smeared Crack Model for 3 18 Building Code Requixments for Reinforced Comte Fracture:’ J. Enge._- Mech, ASCE, V. 114, No. Il, Nov.. -_pp. 2493- Concrete 2510.

Bazam, Z F?, and Pijaudier-cabor, G., 1988, “Nonlccal Continuum Dam- Elsevier, London, pp. 299-304. age, Localization Instabüity and Convergem,” 1. AppL MA.ASME, V. Carol, I;prat. F?; and Bazant, 2,1992, “New Explicit Microplane Model 55, NO. 2 p~.287-293. for Concrete: Theoetid Aspeas and Numerical ïmplemntation:’ I& 1. BW2 P., and Rat, P. C.. 198% “Microplaw Model for Brittle-Plastic Solids Srnrctures, V. 29, No. 9, pp. 1173-1 191. Material: I. Themy,” 1. Enge. Mech, ASCE, V. 114, No. 10, Oct. pp- 1672- Carol, I.; Rat, P.; and Geoy R., 1993. ‘Numaical Analysis Of Mured- 1688. Mode Fnicnire of Quasibiittie Materiais Using a Mulucrack Constitutive Bmt, 2 P.. and Rat. P. C., 1988b. “Microplane Model fur Brittle-Plastic Model,”Mired-Mode Fmgue and Fracture, edited by H. Rosmanith and K Material: IL Verification,” 1. Enge. Mech, V. 114, No. 10, Om, pp. 1689- Miller, BIS PubL 14, Mechanical Enginmring Publications Ltd.. London, 1702. pp. 319-332. Bazant Z. 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N., and P-hio, R, 1989, “Advanced Techniques for Auto- 1365-1386. matic Finite Element Meshing Erom Solid Models.“ Compyer-AidedDesign, ozbolt, J., and Bazant, Z. P., 1996, “Numerical Smeared Fracaue Analy- V. 21. pp. 248-253. --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- sis: Nonlocal Micawrack Interaction Approach” Int. J. Num Meth Engrg., Schellekens, J., and de Borst, R,1993, “On the Numerical Integration of V. 39, pp. 635-661. Interface Eiemnts,” Inr. J. Num Meth Engrg., V. 36, pp. 43-66. ammi, T; SW,I.; AmeUr-Mous~a.R; and B~yuk~~t~rk,O., 1992, “A Schlangen, E., and van Mim, J. G. M.. 1992, “Shear Frachue in Cementi- Nonlinear ïke-Dimensionai Analysis of Reinforced Concrete Based on a tious Composites Part E Numerical Simulations,” Fmchcre Mechanics of Bounding Surface Model,” Computen undSmccrures,V. 43, No. 1, pp. 1-12. Concrere Smares, edited by Z. P. Bazant. Elsevier, London, pp. 671-676. Papaddcis, M., 1981, “A Method for the Automatic Evaluation of the Schnobrich, W. C., 1977, “Behavior of Reinfod Concrete Stnicaires Dynamic Relaxation Parameters,” Comp. Meth AppL Mech Engrg., V. 25, F’redicted by the Fnite Element Method,” Computers and Strucrure~,V. 7, No. 1, pp. 3148. pp. 265-376. Parks,D. M.,1974, “A Stiffness Derivative Finite Element Technique for Schreyer, H. L., 1990. “Analytical Solutions for Nonlinear Strain-Gradient Determination of Cmck lip SWSS Intensity Factors,” Inr. J. of Fracr., V. 10, Softening and Localization,” J. AppL Mech,V. 57. pp. 522-528. NO. 4. pp. 487-502. Schroeder, W. J., and Shepard. M. S., 1988, ‘Yjeomeay-based Fully Aute Peruchio, R; Sanena, M.; and Kela, A., 1989. “Automatic Mesh Genera- matic Mesh Genedon and the Delanney Triangulation,” IN. J. Num Merk tion from Solid Models Based on Recursive Spatial Decompcsitions:’ IaJ. in Engrg, V. 26, pp. 2503-2515. Nwn. Methods m Engrg., V. 28, pp. 2469-2501. Sha, G. T., and Yang, C-T., 1990, “Use of Analytically houpled Near- Petersson, €!-E., and Gustafsson, P. J., 1980, “A Model for Calculation of rip Displacement Solutions for calculating the Decoupled Weight Functions Crack omwch in Conmte-Like Materials,” Niunerical Medtodr in Fmcnve of Mixed Fraaure Modes,” Engrg. Fracr. Mech., V. 37, No. 3, pp. 591-610. Mechanics, edited by Owen and Luxmoore,Pineridge F’ress, Swansea, U.K., Shepard, M. S., Oct., 1984, “Automated Analysis Model Generation,” pp. 707-719. Computer Aided Design in Civil Engineering, American Society of Civil petasson, P. E, 1981, ‘Crack Growth and Developmeat of Fracture Zones Engineers. New York pp. 92-99. in Plain CornCrete and Similar Materials,” Report TVBM-1006, Division of Shih, C. F.; deloremi, H G.; and German. M.D., 1976, “Crack Extension Building Mamials, Lund Institute of Technology, Sweden. Modelling with Siguiar Quadratic rsOparamemc E1ements:’ïm J. Fmt,V. Planas, J., and Elices. M., 1991, “Nonlinear Fran~reof Cohesive Materi- 12. pp. 647-651. als,” In?. J. Fmcr~re,V. 51, No. 2,pp. 139-158. Skvakmar, K. N.; Tan, €? W.; and Newman, J. C.. 1988, “A Vimial Planas, J.. and Elices, M., 1992, “Asymptotic Analysis of a Cohesive crack-closure Technique for Calculating Stress intensity Factors for Crack 1. Theoretical Background,” Im J. Fmcncre, V. 55, No. 2, pp. 15% Cracked Three-ûimensionai Bodies:’ Int. J. Frachue,V. 36, pp. R43-R50. 178. Shumin, C.,and Xing, II, 1990. ”Gmraiized Stiffness Mvative Method Planas, J.. and E)ices, M., 1993a. ‘The Equivalent Elastic Crack 1. Load- for Mixed Mode Crack Problems,’’ Mechunics RES.Com., V. 17, No. 6. pp. Y Equivalences,” Inr. J. Fruc~e,V. 61, No. 2, pp. 159-172. 43744. Planas, J., and Eli=, M., 1993b. ‘I Asymptotic Analysis of a Cohesive SiG. C., 1974, “Strain Energy Density Factor Applied to Mixed-Mode Crack 2. Muence of the SoMgCurve,” Int J. Fmcncre, V. 64,No. 3. pp. Crack Roblem,” Int. J. of Fmmre, V. 10, pp. 305-321. 221-237. Sue Effectin Concrete S-, 1994. H. Mihashi, H. 0- and 2 Pramono, E., and Wdam, K, 1987, “Fracture Energy-Based Plasticity P. Bazant, eds., E & F N Sp.London. Formulation of plain Concrete,” J. Engrg. Mech, ASCE,V. 115, No. 6, June, Sbikenid, P., 1986, “Discrete Crack Modeiüng for Dynamically Loaded, pp. 1183-1204. Uminfomd Concrete Stnictuffs:’ Emthquake Engrg. and Srm. Dyn, V. pukl, R,Schlottke, B.; ozbolt.J.; and Eligehausen, R, 1992, “Computer 14, pp. 297-315. Simulation: Splitting Tests of Chnuete Dick-Wailed Rings” Fraawe Sluys, L J., and de Bor% R. 1992, “‘Anaiysisof Impact Fracnire in a Dou- Mechanics OfConcrete Structures, edited by Z P. Bazanf E.lsev¡er, London. blsNotched Specimen Including Rate Effects,” Frricncre Mechunics of Con- Rahulkumar, P.. iS92, “FEM Modeling of Crack Propagation in Concrete crete Srrucrures, edited by 2 P. Bazant, Elsevier. London, pp. 610-615. Using an Energy Approach,” MS Thesis, Dept. of Civil Engineering, Univ. of Sousa, J. L.: Martha. L. E; Wawzynek, F! k; and ingdm A R.. 1989, New Mexico, Albuqueque. Simuhion of Non-Phr Crack Pmpgation in Three-Dimenswd Smc-

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twes m Concrere and Rock Fmctum of Comme and Rock Recent Devebp- Model far Cracked Reinforced Concrete,” J. Smct. ihgrg., ASCE, V. 117, ments, S.F! Shah, S. E. Swartz, and B. L G. Barr, eds., Elsevies Science No. 3, Mar., pp. 715-732. Publishing. New York, pp. 254-264. Xie. M.; ûerstle, W.; and Rahulkmar, P., 1995, “Energy-Based Aut; Stankowski. T.. 1990, “Numexical Simulation of Regressive Faiiure in Par- matic Mixed-Mode Crack Propagarion Modeling,” 1. Engrg. Meckanics, tick Composites,“ PhD ï%esis, Dept of CE& Univ. of colorado, Boulder. ASCE,V. 121,No. 8, Aug..pp. 914-923. Srankowski, T; Etse, G.; Runesson, K; Sture, S.; and Wdlam, IL, 1992, Yamaguchi. E., and chen, W.-F., 1990, “Ccracking Model for Finite Ele- “Composite Analysis with Discreie and Smeared Crack Concepts,” Fmcnue ment Analysis of concrefe Materials,‘‘ 1. Engrg. Mechanics, ASCE, V. 116, Mechrmicr of Concrere Srruenues, edited by Z P. Bazant, Elsevier, London, NO. 6, he, 1242-1261. pp. 269-274. Yamaguchi, E, and ChaW.-F., 1991, “Mi-k Propagation Study Of StankoWski T.; Rumssoa, IC, and Sture, S.. 1993, “Fiacnire and Slip of Concrete under Compression,” 1. Engrg. MediMiCs, ASCE, V. 117, No. 3, Iaterfaces in cnmititious I: characten‘stics, II: Impiementa- Mar., pp. 653-673. tim” J. bgrg. Me&. ASCE, V. 119, No. 2, Ftb., pp. 292-327. Zienkiewicz O. C., and Cheung, Y. IC, 1964, “Buttress Dams on Complex Struman, G. M.; Sbab, S. P.; and Wmter, G., 1965, ‘‘EíYect of Flexural Rock Foundaeons,”Warer Power, V. 16. p. 193. Strain ûradients on Micnicracking and Stress-Strain Behavior of concrete,” AU JOURNAL.Pn>ceedyigs V. 62No. 7, July, pp. 805822 Swenson, D. V., 1986. “Mcdeüng Mixed-Mode Dynamic Crack Pmpaga- APPENDN-GLOSSARY tion Using Finite Elements,” PhD DisserWion, School of Civil and Environ- Bond-slipThe force-displacement relationship describ- mentai Engineuing, cornell Univcrsity, Ithaca. ing the mechanical interactionbetween a reinforcing bar and Swwson, D. V.. and ingraf€ca,A. R,1991, ‘?he Coliapse of the Schoha- the sunnxuiding concrete. The bond-slip relationship is a rie creelr Bridge: A Case Study in Concrete Fracaue Mechanics,” Inr. J. of Fracnue, V. 51, pp. 73-92. structurai rather than materials phenomenon. Swenson, D. V., and ïngraffea A. R, 1988, “Modeüng Mixed-Mode Continuum mechanics-A view of mechanical behavior Dynamic Crack propagation Using Fkite Hemen& Theory and Applica- that assumes that all responses can be represented using con- tions,* Comp. Mech. V. 3, pp. 381-397. tinuous functions. Tada, H.; Pan$ F! C.; and huh,G. R. 1973. The Smss Anuiysk of Cracks Homibook, Del Research Corp., Heilertom. Continuum danurge mechanics (CDM)-A form of con- Tasdemir, M. k; Maji, A. E,and Shah, S. P.. 1990. “Crack hpagation in tinuum mechanics that assumes that all damage is in the form Concrete under Compression,” J. ihgrg. Me&, ASCE, V. 1 16, No. 5, May, of continuously distributed damage (microcracks). pp. 1058-1077. Tong, P.. and Pian, T. H. H.. 1973, ‘On the Convergence of the Finite Ele- Cosserat continuum-A particular view of continuum me- ment Methcd for Problems with Singularity,” lnt. 1. Solids SrruCrUres. V. 9, chanics that includes the assumption of couple stresses as a pp. 313-321. continuum approximation of the behavior of crystai lattices Tong, F!; Pian, T. H. It; and Lauy. S. J., 1973, “A Hybrid Element on a small scale (Cosserat and Cosserat 1909). Approach to Crack Roblems in Plane Elasticity,” Inr. 1. Num Merk in Engg., v. 7, pp. 293-308. Crack-A conceptual model of damage within a solid Tracq, D. M., 197 I, ‘‘Finite Elements for the Ddermination of Crack Tip body. A crack is usually understood as a surface within the Elastic Stress Intensity Factors,” Engrg. Frack Mech, V. 3, pp. 255-265. body, across which tensile tractions cannot be transmitted. Undenvood, P.. 1983, “Dynamic Relaxation,” Comprctarional MerhorLI in Mecf~m~icrfor ïmnsieni Analysis. edited by T. Belytschko and T. J. R Crack band model-A method for objectiveiy modeling Hughes, North-Holland Publishing. Amsterdam, pp. 245-265. cracks using the finite element method, in which the consti- Valliappan. S.. and Doolan, T., 1972. “NodiStress Analysis of Rein- tutive models for continuum finite elements are adjusted to forced Concrete,” 1. S:ruct. Div., ASCE, V. 98. No. ST4, Apr., pp. 885-898. correctly account for the formation of a discrete crack within Van Mier, J. G. M.;Rots, J. G.; and Bakker, A.. 1991, “Fmcture Processes the element (Bazant 1984). in Concrete, Rock and Ceramics,” Pm., ïntl. RILEM/EsS Conf., June, Noordwijk The Netherlands. Critical stress NltensiQfacror (KIc)-In LEFM, for a giv- Van mer, I. G. U,and Vonk, R, 1991, “Fracture of Concrete and Multi- en material, the mode I stress-intensity factor at which a axial Skess-Recent Dcveìopmaits,”Muret Snucr.. RILEM,V. 24, pp. 61-65. crack will propagate (Broek 1986). The critical stress-inten- Vccchio. E J., 1986, “Nonlinear Ffite Element Analysis of Reinforced ConcreteMembmes,”ACISmcct. 1.V. 86. No. 1, Jan.-Feb.. pp. 2635. sity factor is a material property. Vecchio, E I.. and Cobs, U P., 1982, ‘The Respome of Reimforced Damage-Breaking of bonds within a material. Damage Conmete to Inpiane Shear and NdStms,” Pubkath 82-03. Dtpt of can be manifested as continuum damage (distributed micro- Civil Engrg.. Univ. of Toronto. cracks), or as a single macrocrack. Other forms of damage Vonk, R. 1% “Softening of Concrete Loaded in Compression,” PN) Thes¡¡, Technische Univeriteh Eindhoven, Eindhoven, the Netherlands. are also conceivable. Voa R, 1993. “A Micromechanid Investigation of Softwing of Con- Deformation-Relative movement of particles within a crete Loaded in Compression,”HERON, V. 38. body, due to a variety of causes. Vos, E, 1983, “Influence of MingRate and Radial Pressure Bond in on Discrete cracking-Damage of a deformable body which Reinforced Concrete,“ PhD Thesis, Delft University pres$ the Netherlands. Wamzynek. P. A, and InnPranea,A. R, 1986, “Local Remeshing around a results in one or more distinct crackc. Discrete cracking Propagating Crack Tip Using ïnteiacfive Computer Graphics,” FteEie- models usually assume no dismbuted damage in the form of mcnr Metho4 Mciieimg, and New Appücaziom, E. M. Paaen et ai., eds., microcracks. ASMECEDI, PW-101,~~.33-38. Wawnpek, P. A,and Ingraffea, k R, 1987, ‘Intaactive FUiite Elemeat Dynamic relaxaton-A numerical solution methoâ, par- Analysis of Fnicture Rocesses: An Integrattd Approach,” Theo. and AppL ticularly suited to nonlinear statical problems, in which a Fmct. Meck, V. 8. pp. 137-150. static problem is viewed as a damped dynamic problem Wawnynek. F! A., and In@q A. R, 1989, “An interactive Approach to which converges to a steady-state static solution. The damp- Local Remeshing Around a Propagating Crack,” FieElementr in Andysis and Design. V. 5. pp. 87-96. ing coefficients are adaptively adjusted to achieve rapid con- Westergaard. H. M., 1934, “Stresses at a Crack, Size of the Crack, and the vergence (Papadrakis 1981). Bending of Reiifomd Concrete,’’ ACI JOURNAL, Proceedings V. 30, pp. 93- Energy release rate (G)-The rate at which potential en- 102. ergy is released with respect to change in area of a discrete Wilson, W. K. 1%9, “On Combined Mode Fmam Mechanics,” PhD Dissertation, Univwity of Pittsburgh. crack (Broek 1986). The rate of energy release may also be Wu, 2.; Yoshikawa, R; and Tanah, T.-A., 1991, ‘Tension Stiffness considered with respect to other variables.

Fictitious crack model (FCM)--A view of fracture me out of the crack plane, giving rise to tearing, or mode ìïI, chanics that assumes that a discrete crack forms through. crack deformation in the plane of the crack, parailel to the gradual breaking of bonds across its advancing path (Hiller- edge of the crack (Broek 1986). borg et ai. 1976). Mixed-&Type of loading, often resulting in a curved Finite element methal 0-A numerid technique for crack, due to the simultaneous application oftwo or more solving boundary value problems (such as sttmuraî analyses) loading modes. In most cases, mixed-mode cracks rapidly by representing a continuum (a strucíure or structural member) follow a direcrion that places them under mode I loading by a group of discrete regions (ñnite elements) for which the (Broek 1986). behavior is described by an assumed set of functions (which, in Nonlocal continuum-A form of continuum damage me- structurai analysis, relate displacements and forces). chanics in which localization iimiters are employed to pre- Fracture-A fracture is a synonym with “crack.” Each serve solution objectivity with respect to mesh reñnement word may be used as a noun or a verb. (Bazant 1986). Fracture mechanics-The study of the formation of Propagation-A tem used to describe the evolution or cracks (Broek 1986). Recently, fracture mechanics also in- growth of a discrete crack. cludes concepts of continuum damage mechanics. Size effect-A phenomenon associated with cracks in Fracíure-sensitive-A structure is fracture-sensitive if its which the apparent nominal strength of a structure decreases global mechanical response is sensitive to the presence of as a function of structure size (Bazant and Oh 1983). cracks. Smeared cracking-A synonym for “continuum damage Fracture toughness (GIC)-In LEFM, for a given materiai, mechanics.” Some smeared cracking models have been the Mode I energy release rate, above which a crack will found to be nonobjective with respect to mesh reñnement. propagate (Broek 1986). Fracture toughness is a material Singularity-A mathematical concept in which a field property. Sometimes confused with critical stress-intensity tends asymptotically to infmity at a point. factor. Singularcrack tip element-A fuite element with a strain Fractureprocess zone (FPZ)-The zone at the tip of a dis- singuiarity included in its shape function, for use in finite el- crete crack within which damage (or some contexts, any ement modeling of LEFM problems (Barsoum 1976). --```````,,,,,```,`,`````,````-`-`,,`,,`,`,,`--- nonlinear material behavior) occu~s. StijWss-derivative method-A finite-difference method Hybridfinite element-A fuiite element that includes both for obtaining the energy release rate of a discrete crack using stresses and displacements or strains primary responses as the finite element method (Parks 1974). (Tong et al. 1973). Stress intensity factor (KI, KIII)--Factors that charac- Zntet$ace-An idealized surface representing the mechan- KI], terize the stress field a the tip of a crack using the ical interaction between two adjacent bodies. assumptions of LEFM (Broek 1986). Integace element-A finite element designed to model an Strain gradient-Spatial rate of change of strain, in a interface (Goodman et al. 1968). used method of limiting localization in continuum damage me- Linear elastic fracture mechanics classical chanics (Schreyer 1990). branch of fracture mechanics, in which material behavior is assumed to be linear elastic, damage is considered to be only Strain localization-A phenomenon associated with con- in the form of discrete cracks, and the fracture process zone tinuum damage mechanics, in which damage tends to con- is considered to be negligibly small (Broek 1986). centrate along surfaces, resulting in the formation of cracks J-Integral-An integral method for determining energy (Bazant 1976)). release rates of discrete cracks (Rice 1968). Strain-sofenutg-A feature of some constitutive models, Least dimension-A parameter characterizing the geomet- in which stress decreases with increasing strain. Constitutive ricai size associated with a crack tip (Gentle and Abdalla modeis that include strain-softening must include the con- 1990). cept o€ nonid continuum to obtain objective solutions (Bazant 1976). Localization limiter-A mathematical device used to Traction-Force per unit area acting upon a specified sur- avoid unobjectivity or spurious mesh sensitivity when modeling continuum damage mechanics (Bazant and Pijaudier- face. Traction is a vector quantity. Cabot 1988). Virtual crack extension (VCE)-A finite-difference meth- Mode-Type of loading in which stresses at a crack tip are od for obtaining the energy release rate of a discrete crack us- (i) normal to the crack surface, giving rise to opening, or ing the finite element method (Hellen 1975). mode I, crack deformation perpendicular to the plane of the i crack; (2) shear stresses in the crack plane, giving rise to slid- This report was submitted to letter ballot by the commit- ing, or mode II, crack deformation in the plane of the crack, tee and was approved in accordance with AC1 balloting perpendicular to the edge of the crack or (3) shear stresses procedures.

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