1、. AC1 446-3R-ENGL Finite EI international“ 1i997 m Obb2949 0537950 172 D AC1 446.3R-97 ernent 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 First printing, January 1998 Finite Element A
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12、of AC1 documents. , + - STD-AC1 446.3R-ENGL 1997 IPI Ob62949 0537952 T45 U AC1 446.3R-97 Finite Element Analysis of Fracture in Concrete Structures: State-of-t he-Art Farhadhsari denek P. Baant oral Buyukozhrk Ignaciocarol Rolf Eligehausen Shu-Jin Fangv3 Ravindra Mu Toshiaki Hasegawa Neil M. Hawkins
13、 Anthony R ngraffeal* Jeremy isenbea Reported by AC1 Committee 446 Waiter GerstleZ Secretary and Sucommittee Co-Chai- YeOU-Sheng Jens Mohammad T. Kazemi Neven Krstulovic vim C. Li Jacky NIazars Steven L. dabed Christian Meyer Hirou Mihashi Richard A. Miller Sidney Mindess C. Dean Nomian Fracture is
14、an importan! mode of deformntion and damage in both plain and reulfomd concrete structures. To accurutely predictfrecn U2 (which represents a line segment in STD.ACI qY6.3R-ENGL 1997 Ob62949 0539966 53T FINm ELEMENT ANALYSIS OF FRACTURE IN CONCRETE STRUCTURES 446.3R-15 E S Fig. 3.2-Heterogeneiiy of
15、concrete at the size scale of the aggregate I l I I I S lD, a circle in 2D, and a sphere in 3D). Alternatively, the nod distribution function has been used in place of Eq. 3.2 and found to work well enough, although its values are nowhere exactly zero (Bazant 1986). For points whose distance from al
16、l the boundaries is larger than p otherwise the averaging volume protdes outside the body, and V,.(x) must be Calculated for each point to account for the locally unique averaging do- main (Fig. 3.2b). In finite element computations, the spatial averaging inte- grals are evaluated by finite sums ove
17、r all integration points of all finite elements of the structure. For this purpose, the matrix of the values of a for all integration points is comput- ed and stored in advance of the finte element analysis. This approach makes it possible to refme the mesh as re- quired by structurai considerations
18、. Since the representative volume over which structural averaging takes place is treated as a material propew, convergence to an exact continuum solution becomes meaningful and the stress and strain distri- butions throughout the FPZ can be resolved. The nonlocal continuum model for strain-softening
19、 of Ba- zant et al. (1984) involves the nonlocal (averaged) strain i as the basic kinematic variable. This corresponds to a system of imbricated (i.e., overlapping in a regular manner, like roof tiles) finite elements, overiaid by a reglar finite element sys- tem. Although this imbricate model limit
20、s localization of straui softening and guarantees mesh insensitivity, the pro- gramming is complicated, due to the nonstandard form of the differential equations of quiiibrium and boundary conditions, ie., energy considerations involve the noniocai strain i . These problems led to the idea of a part
21、ially noniocal con- tinuum in which stress is based on noniocai strain, but local srrains are retained Such a nonlocal model, called “the non- local continuum with local strain” (Bazant, Pan, and Pijaud- ier-Cabot 1987, Bazant and Lin 1988, Bazant and Pijaudier- Cabot 1988, 1989) is easier to apply
22、in finte element pro- gramming. In this formulation, the usual constitutive relation for strain softening is simply modified so that all of the state variables that characterize strain softeningare calculated from nonlocal rather than local strains. Then, all that is nec- - STD-AC1 446-3R-ENGL 1997
23、E Obb2949 05379b7 Y76 E 446.3R-16 ACI COMMITTEE REPORT essary to change in a local finte element program is to pro- vide a subroutine that delivers (at each integration point of each element, and in each iteration of each loading step) the value of for use in the constitutive model. Practically spea
24、king, the most important feature of a non- local fite element model is that it can correctly represent the effect of structure size on the ultimate capacity, as well as on the post-peak slope of the load-deflection diagram. Nonid models can also offer an advantage in the overali speed of solution (B
25、azant and Lin 1988). Aithough the nu- merical effort is higher for each iteration when using a non- local model, the formulation lends a stabilizing effect to the solution, allowing convergence in fewer iterations. b) Micromechanical Approach Another noniocal model for solids with interacting mim- r
26、acks, in which noddty is introduced on the basis of mi- crocrack interactions, was developed by Bazant and Jirasek (Bazant 1994, Jirasek and Bazant 1994, Bazant and Jmk 1994) and applied to the analysis of size effect and localization of cracking damage (Jirasek and Bazant 1994). The model represent
27、s a system of intemcting cracks using an integral equation that, unlike the phenomenological noniocal model, involves a spatial integral that represents microcrack intmc- tion based on fracture mechanics concepts. At long range, the integral weighting function decays with the square or cube of the d
28、istance in two or three dimensions, respectively. This model, combined with a mimplane model, has provided con- sistent results in the finite element analysis of fracture and structurai test specimens (Ozbolt and Bazant 1995). 323. Gradient models Another general way to introduce a localization limi
29、ter is to use a constitutive relation in which the stress is a function of not only the strain but also the first or second spatial de- rivatives (or gradients) of strain. This idea appeared original- ly in the theory of elasticity. A special form of this idea, called Cosserat continuum and characte
30、rized by the presence of couple stresses, was introduced by Cosserat and Cosserat (1909) as a continuum approximation of the behavior of crystal lattices on a small scale. A generalization of Cosserat continuum, involving rotations of material points, is the mi- mplar continuum of Eringen (1965,1966
31、). Bazant et al. (1984) and Schreyer (1990) have pointed out that spatial gradients of strains or strain-related variables can serve as a localization liters. It has been shown (Bazant et al. 1984) that expansion of the averaging integral (Eq. 3.1) into a Taylor series generally yields a constitutiv
32、e relation in which the stress depends on the second spatial derivatives of strain, if the averaging domain is symmetric and does not protrude outside the body, and on the first spatial derivatives (gradients) of strain, if this domain is unsymmetric or pro- trudes outside the domain, as happens for
33、 points near the boundary (Fig. 3.2b). Since the dimension of a gradient is length- times that of the differentiated variable, introduction of a gradient into the constitutive equation inevitably re- quires a characteristic length, L, as a material property. Thus, the use of spatial gradients can be
34、 regarded as an approxima- tion to nonlocal continuum, or as a special case. In material research of concrete, the idea of a spatial gradi- ent appeared in the work of LHermite et al. (1952), who found that, to describe differences between observations on smali and large specimens, the formation of
35、shrinkage cracks needs to be assumed to depend not only on the shrinkage stress but also on its spatial gradient see also LHermite et al. (1952) and a discussion by Bazant and Lin (1988)l. This was pmba- bly the first appearance of the noniocal concept in fracnire. The idea that the stress gradient
36、(or equivalently the strain gra- dient) influences the material response has also been demon- strated by Sturman, Shah, and Winter (1965) and by Karsan and Jim (1%9) for combined flexure and axial loading. The nonlocal averaging integral is meaningful oniy if the fi- nite elements are not larger tha
37、n about one third of the repm sentative volume, V, raz and Bazant 1989). The gradient approach, on the other hand, offers the possibility of using fi- nite elements with volumes as large as the representative vol- ume, V, Thus, be gradient approach offers the possibility of using a smaller number of
38、 nite elements in the analysis. It ap pears, however, that the programming may be more compii- cated and less versatile than for the spatial averaging integrals. The problem is that interelement continuity must be enforced not only for the displacements but also for the strains. This re- quires the
39、use of higher-order elements, or aiternatively, if first-nier elements are preferred the use of an independent strain field with separate first-order finite elements. Noniocal constitutive models for concrete are reviewed in AC1 446.1R. CHAPTER MITERATURE REVIEW OF FEM FRACTURE MECHANICS ANALYSES 4.
40、1-Generai To help provide an overview of the state-of-the-art in fi- nite element modeling of plain and reinforced concrete struc- tures, a number of representative analyses are summarized in this chapter. Emphasis is placed on easily available, pub- lished analyses that attempt to address the fract
41、ure behavior of concrete structures. Wherever possible, problems solved using the discrete cracking approach are compared to solu- tions using the smeared crack approach. Symposium pro- ceedings (Mehlhorn et al. 1978, Computer-Aided 1984, 1990, Firrao 1990, Fracture Mechanics 1989, van Mer et al. 19
42、91, Concrete Design 1992, Size Effect 1994, Computa- tional Modeling 1994, Fracture and Damage 1994, Fracture Mechanics 1995) provide the readers with the changing fla- vor of the state-of-the-art over the years. Many more pub- lished FEM fracture mechanics analyses exist than are presented in this
43、chapter; however, it is fair to say that those referenced here are representative of the state-of-theart. The term “size effect” used throughout this chapter, is a term that has taken on a special meaning for quasibnttle ma- terials, such as concrete and rock. It describes the decrease in average st
44、ress at failure with increasing member ske that is directly attributable to the well-established fact that frac- ture is governed by a fracture parameter(s) that depends on the dimensions of the crack (which is tied to structure size) as well as some measure of stress. For concrete, it is not clear
45、if the fracture parameter that governs failure is a material property or if it also depends on the structure size. However, this last point does not alter the general meaning of the term “size effect.” In contrast, in the field of fracture of metais, the - STDmACI 446-3R-ENGL 1997 0662949 05379b 302
46、 FINITE ELEMENT ANAiYW OF FRACTURE IN CONCRETE STRUCTURES 446.3-17 term “size effect” is understood to apply only to the depen- dence of fracture parameter (not the average stress) on the structurai dimensions. 4.2-Plain concrete Unreinforced concrek suctures are the most fracture sen- sitive. We us
47、ually reinforced with steel in design practice to provide adequate tensile strength, many structures (or parts of smctures), for one reason or another, are unreinforad. It makes sense to study the analysis of unreinforced concrete suuctures, because these provide the most severe tests of frac- ture
48、behavior, and because the results of these analyses can add insight into the more complex behavior evidenced in rein- forced concrete structures. In what follows, short descriptions (iisted in chronological order) of finite element analyses of un- reinfod concrete structures are presented. 4.2.1 Ten
49、sile failure The plain concrete uniaxial tension specimen is probably more sensitive to fracture than any other type. For this rea- son, it provides a good test of the fracture-sensitivity of a fi- nite element model for concrete. Aithough a number of such analyses have been reported, the following references are representative of first a discrete, then a local smeared, then a nonlocal smeared, and finally a micromechanics (random partiCie) approach. Gustaffson (1985) used the fictitious crack model (discrete crack) to
