1、ON I AC1 SP-134 92 E 0662949 0504779 “1T 9 CONCRETE DESIGN BASED I FRACTURE MECHANICS a a COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-134 i2 = Obb2949 0504780 b71 CONCRETE DESIGN BASED COPYRIGHT ACI International (American Concrete Institu
2、te)Licensed by Information Handling ServicesAC1 SP-134 92 W 0662949 0504783 508 5 DISCUSSION of individual papers in this symposium may be submitted in accordance with general requirements of the AC1 Publication Policy to AC1 headquarters at the address given below. Closing date for submission of di
3、scussion is March 1,1993. Ail discussion approved by the Technid Activities Committee along with closing remarks by the authors wiil be published in the September/October 1993 issue of either AC1 Structural Jou mal or AC1 Materials Journal depending on the subject emphasis of the individual paper. T
4、he Institute is not responsible for the statements or opinions expressed in its publications. Institute publications are not able to, nor intended to, supplant individual training, responsibility, or judgment of the user, or the supplier, of the information presented. The papers in this volume have
5、been reviewed under Institute publication procedures by individuals expert in the subject areas of the papers. Copyright o 1992 AMERICAN CONCRETE INSTITUTE P.O. Box 19150, Redford Sation Detroit, Michigan 48219 All rights reserved including rights of reproduction and use in any form or by any means,
6、 including the making of Copies by any photo proces, or by any electronic or mechanical device, printed or written or oral, or recording for sound or visual reprodudion or for use in any knowledge or retrieval system or device, unless permission in writing is obtained from the qyright proprietors. P
7、rinted in the United States of America Editorial production Victoria Lunick Library of Congress cataiog card number 92-73233 COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-I134 92 0bb2949 0504782 444 W PREFACE At the Fall meeting of the Ameri
8、can Concrete Institute in Phila- delphia in 1990, AC1 Committee 446 sponsored a technical paper session entitled “Design Based on Fracture Mechanics.“ In its meeting directly fol- lowing the session, the Committee agreed that the presented papers had been excellent, and voted to publish a proceeding
9、s. This book contains eight of the nine papers presented at that session, as well as one invited paper. Only one of the authors was a practitioner, the rest being academicians. About one-half of the papers are by authors from the United States, the other half are by authors from other countries. Man
10、y dollars and much human effort has been spent, as evidenced by hundreds of journal publications, on research into the fracture behavior of concrete structures. It is now time to make this massive research effort productive by changing design practice so that more economical, more func- tional concr
11、ete structures will be built. It is hoped that this volume will be a step in the right direction. The purpose of the session was to present recent advances in our un- derstanding of fracture in concrete in such a way that practitioners could understand and use it, and also to identi ways in which pr
12、actitioners can make use of fracture mechanics in design of concrete structures. Currently, designers in the United States use the AC1 318 Building Code, which cur- rently makes absolutely no use of fracture mechanics concepts. To enable designers to use fracture mechanics, a logical next step would
13、 be to incor- porate these concepts into a revised building code. All of the papers included in this proceedings address the cracking phenomenon in concrete structures. Two of the papers (by Bahnt and by Gerstle et al.) identie the importance of fracture mechanics in concrete structures and suggest
14、explicit (although different) recommendations for revision of the AC1 318 Building Code. The paper by Buyukozturk and Lee discusses mixed-mode fracture concepts in structural concrete design, with specific applications to joints between segments in prestressed segmental concrete bridge girders. The
15、papers by Elfgren and Swartz, and by Pukl et al., address the problem of pullout of headed anchor bolts. McCabe et al., investigated the bond behavior of epoxy-coated reinforcing bars from a fracture mechanics viewpoint. Practical methods for designing reinforced concrete beams and slabs, including
16、the effects of cracking are reported by Jan, and by Ashmawi et al., respectively. Finally, the response of plain and I 111 COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-134 92 W Obb2949 050Y783 380 W reinforced concrete structures under cycl
17、ic loadings, including progressive microcracking and crack closure effects, is investigated by La Borderie et al. The hard work and patience of the contributors and of the reviewers is gratefully acknowledged. Also, the support of the Session diagonal tension; failure; fracture mechanics; pullout te
18、sts; punching shear; shear properties; size effect; standards; structural design; torsion COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-134 92 Obb2949 0504787 T2b = 2 BaZant Zdeni5k P. Baiant is a Walter P. Murphy Professor of Civil Engineer
19、ing at the Center for Advanced Cement-Based Materials, Northwestern University, Evanston, Illinois, U.S.A. 1 Introduction It has long been known that ultimate loads of concrete structures exhibit size effect. The classical explanation has been Weibulls weakest-link theory which takes into account th
20、e random nature of concrete strength 1,2,3,4,5. However, for reasons given elsewhere 6 and briefly explained in the Appendix, it now appears that the statistical theory does not suffice to describe the essence of the size effect observed in brittle failures of reinforced concrete structures and play
21、s only a secondary role. The main mechanism of the size effect in this type of failure is deterministic rather than statistical, and is due to the release of the stored energy of the structure into the front of the cracking zone or fracture. This phenomenon is properly described by fracture mechanic
22、s in its recently developed nonlinear formulation which takes into account the distributed nature of cracking at the fracture front. The purpose of this review paper is to summarize the existing evidence and also present some recent experimental results obtained at Northwestern University. 2 Mathema
23、tical Description of Size Effect The size effect is defined by comparing the ultimate loads (maximum loads), Pu, of geometrically similar structures of different sizes. This is done in terms of the nominal stress UN at failure. For two-dimensional similarity (e.g., panels), UN = c,P,/bd, and for thr
24、ee-dimensional similarity (e.g., cylinders), UN = c,Pu/d2. Here b = thickness of a two-dimensional structure; d = characteristic dimension (size), which may be defined as any dimension of the structure, e.g., the depth of a beam or its span, since only the relative values of UN matter; and c, = chos
25、en dimensionless coefficient introduced for convenience. One may either set c, = 1 or use c, to make UN coincide with some convenient stress formula. E.g., for a simply supported beam of span L and a rectangular cross section of depth H, with load P at midspan, one may set d = H and c, = 3L/2H, in w
26、hich case UN = 3PL/2bH2 = maximum elastic bending stress (c, is constant because L/H is constant for geometrically similar structures); or one may set d = L and c, = 3L2 /2H2, with the same result for UN. When the UN- values for geometrically similar structures of different sizes are the same, one s
27、ays that there is no size effect. The size effect represents a dependence of UN on the structure size (characteristic dimension), d. According to plastic limit analysis, as well as elastic analysis with allowable stress or any theory that uses a failure criterion in terms of stresses or strains, UN
28、is independent of the structure size. This can be illustrated, e.g., by the elastic and plastic formulas for the strength of beams in bending, shear or torsion 7. Another theory of failure, conceived by Griffith SI and introduced to con- crete by Kaplan 9, is fracture mechanics. It was Reinhardt 10,
29、11 who pro- COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-134 92 O662949 0504788 b2 Fracture Mechanics 3 posed that fracture mechanics should be used to describe the size effect in concrete structures, particularly in diagonal shear failure.
30、 He also showed that the size effect of classical, linear elastic fracture mechanics agrees reason- ably well with some test results, although later it was found that nonlinear fracture mechanics is necessary in general. In the linear form of fracture mechanics, in which all the fracture process is
31、assumed to be happening at a point-the crack tip-the size effect is the strongest possible. In the plot of log UN vs. log d, it is described (regardless of the structure shape) by an inclined straight line of slope -1/2 (Fig. i), provided that the cracks at the moment of failure of geometrically sim
32、ilar structures of different sizes are also similar. The reason for stipulating this condition (which has been shown from tests 12,13,14,15,16,17 to be usually satisfied) will be briefly explained after Eq.1. Concrete structures in reality exhibit a transitional behavior between the size effect of s
33、trength or yield criteria (i ., no size effect), represented in Fig. 1 by a horizontal line, and the size effect of linear elastic fracture mechanics, represented by the straight line asymptotic of slope -1/2; see the curve in Fig. 1. This size effect is generally ignored by the current design codes
34、, but recent tests 12,13,14,15,16, as well as Eq.1, show it to be very strong, and thus important. The aforementioned transitional size effect can be most simply explained by considering uniformly stressed rectangular panels of different sizes d, loaded by uniform distributed load UN, as shown in Fi
35、g. 2. Each panel is assumed to have a weak spot in the middle of the left side, from which the fracture originates. For a brittle heterogeneous material such as concrete, it is impor- tant to take into account a relatively large zone of distributed cracking at the fracture front. The size of this zo
36、ne is not proportional to the structure size but is approximately related to the maximum aggregate size. In the simplest approximation, it may be assumed that the width, h, of the cracking band at the fracture front is approximately constant, independent of the structure size (when similar structure
37、s made from the same concrete are compared). Like- wise, it may normally be assumed that, at maximum load, the length of the fracture, u, is proportional to dimension d of the structure, Le., a/d = con- stant. (This is supported by many of the brittle failures of reinforced concrete structures, as w
38、ell as by finite element fracture studies.) Formation of a fracture with crack band of thickness h and length a may be imagined to release the strain energy of density for the panel, k = a/2, but the value of k. does not matter for the present argument, only the fact it COPYRIGHT ACI International (
39、American Concrete Institute)Licensed by Information Handling Servicesis a size-independent constant). Now it is crucial to realize that in a larger structure the energy that is released into a certain small extension Au of the fracture is larger if the UN-value is the same because it comes from a zo
40、ne of a larger volume. Since the energy dissipated by fracture per unit area of the fracture plane is approximately constant (being equal to the fractaure energy, Gf, which is a material property), the value of UN for a larger structure must be less so that the total energy release from a zone of a
41、larger volume would remain the same. Hence the size effect. The strain energy released from the aforementioned densely cross-hatched strip is AW = b(hAu + 2kaAa)u%/2E where b = panel thickness. Setting AW = GfbAa = dissipated energy, one obtains u%h + 2k(u/d)4 = 2EGj. Solving for UN, one can bring t
42、he resulting expression to the form of the size effect law 7: UN = B f:(l+ )-i, = d/do (1) in which the following notations have been made: B = (2EGj/hf,“)/2,do = hd/2ka, and fi, representing the direct tensile strength of concrete, is intro- duced to make B nondimensional. The ratio is called the b
43、rittleness number of the structure, for reasons explained later. Now it is important to note that parameters B and do are size-independent, i.e., constant, because d/a is con- stant if there is geometric similarity (see hypothesis 3 below), and h is also approximately size-independent, as already me
44、ntioned. It must be emphasized that EQ.1 is only approximate. But its accuracy is sufficient for a rather broad size range-from experience, up to about 1:20, which is adequate for most practical purposes. For a still broader size range, a more complicated formula would nevertheless be required. For
45、small enough structures (compared to t (3) the lailitte modes Le, fracture shapes uid lengtha) of gmmetrically similar structure of diffcrcnt sim are, at the moment of maximum lard, ateo gmtnetrically aimilar, and (4) the structura dom not fail at crack initiation. hm tCBthg 12,13,14,16?d,25,26, aa
46、well aa finite eknimt (and other) computatiatial models 2 errors by factors up to 2 may be tolerable). In view of the universality of the size effect law in Fig. 1 and the associated brittleness number, it appears that a simple adjustment can introduce the size effect into the existing code formulas
47、 based on limit analysis. It might suffice to take the existing code formula for the nominal stress due to concrete at COPYRIGHT ACI International (American Concrete Institute)Licensed by Information Handling ServicesAC1 SP-334 92 Ob62949 0504793 457 8 Ba2ant I ultimate load, vu, and replace it by t
48、he expression: vu( 1 + p2 (5) Note, however, that for some types of failure there may exist some limit v,mi“, since at very large sizes there can be a transition to some nonbrittle failure mechanism (this in fact is the case for the Brazilian split-cylinder test). An empirical expression for do need
49、ed for calculating in Eq.5 has been proposed in 24,26 but a rational method to calculate 34 for most types of brittle failure still awaits development. 4 Previous Tests of Brittle Structural Fail- ures I After the size effect law has been formulated, much effort has been devoted to comparing and validating it on the basis of the test data in the literature. The efforts were especially focused on the diagonal shear failure of reinforced concrete beams without and with stirrups 12,24,26. The latter study in- cluded essentially all the experimental dat
