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Numerical studies on load-carrying capacities of notched concrete beams subjected to various concentrated loads
Construction and Building Materials 18 (2004) 173–180
Numerical studies on load-carrying capacities of notched concrete beams subjected to various concentrated loads Zihai Shi*, Masaki Suzuki Research and Development Center, Nippon Koei Co., Ltd., 2304 Inarihara, Tsukuba, Ibaraki 300-1259, Japan Received 15 September 2003; received in revised form 12 October 2003; accepted 12 October 2003
Abstract In solving various fatigue problems, the time-varying characteristics of cyclic loads on structures due to winds, waves, vehicles and so forth are often oversimplified by assuming the loading positions as fixed. In order to clarify spatial effects of loading conditions on cracking behaviors and the maximum loads, numerical studies on the load-carrying capacities of notched concrete beams subjected to various concentrated loads are conducted. The fracture processes of a simply supported beam with multiple cracks are analyzed by using the extended fictitious-crack model. In the cases where loading positions are restricted to at two locations of four-point bending and at the midspan, it is shown that the flexural strengths of notched beams vary significantly, depending on particular loading conditions. The effects of initial notch configuration and lengths on the load-carrying capacities of the beam are clarified. Based on these results, the strength reduction due to various fatigue failures is discussed. 䊚 2003 Elsevier Ltd. All rights reserved. Keywords: Plain concrete beam; Fatigue; Fracture; Multiple cracks; Load-carrying capacity
1. Introduction In the case of testing and analyzing fatigue problems, the time-varying characteristics of cyclic loads on structures induced by winds, waves, vehicles and so forth are mostly oversimplified by assuming the loading positions as fixed during cyclic loading. Experimental studies have shown that during fatigue tests of reinforced concrete beams and plates, a moving cyclic load which simulates traffic loads could result in a reduction in the maximum load-carrying capacity, compared with fixedpoint monotonic or cyclic loading w1,2x. Consequently, the effects of loading positions during cyclic loading on the load-carrying capacity of a structural member are investigated. Due to inherent variations of materials and the random distribution of initial defects in tested specimens, even under the same test conditions the strengths of members could vary. Therefore, it is very difficult to single out the effect of the loading positions on the flexural strengths of concrete beams experimentally. This can be, *Corresponding author. Tel.: q81-29-871-2032; fax: q81-29-8712022. E-mail address: [email protected] (Z. Shi).
in contrast, readily done by a numerical analysis in which the same configuration and size of initial defects are modeled with different loading conditions. From this point of view, numerical studies are carried out on the load-carrying capacities of notched concrete beams subjected to various concentrated loads. Cracking behaviors and the maximum loads are clarified under various loading positions. As for a model, a round-robin test of un-notched concrete beams under four-point bending is selected w3x. The fracture process of beams involving multiple cracks is analyzed using the extended fictitious-crack model w4–8x. Introducing three initial notches of different sizes to the beam and restricting the loading points to at those of the four-point bending tests and at the midspan, structural responses of the notched beams under various loading conditions are studied. In the following, the crack equations for analyzing the beam models containing three discrete cracks are formulated, and a numerical scheme proposed for modeling a curvilinear crack is introduced. Numerical analyses of the notched beams are performed, and the engineering implication derived from results is discussed.
0950-0618/04/$ - see front matter 䊚 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2003.10.006
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Fig. 1. Crack-tip-controlled modeling of three discrete cracks. (a) Forces and displacements at the cracks due to unit external loads; (b) forces and displacements at the cracks due to a pair of unit cohesive forces at crack A; (c) forces and displacements at the cracks due to a pair of unit cohesive forces at crack B; (d) forces and displacements at the cracks due to a pair of unit cohesive forces at crack C; (e) load condition for the growth of crack A; (f) load condition for the growth of crack B; (g) load condition for the growth of crack C.
2. Numerical formulation Fig. 1 shows three cracks of the mode-I type, crack A, crack B and crack C, propagating in the direction normal to the tensile force at the tip of the crack. Note that a crack and its fictitious extension are modeled by separating a pair of two nodal points having the same coordinates and rigidly connected to each other up to separation. For clarity, hereinafter, the cohesive forces
and crack-opening displacements (CODs) of the restrained cracks are marked by asterisks. Initially, crack A is assumed as active with the tensile force at its tip reaching the nodal force limit Qla (the tensile strength of concrete times the surface area apportioned to a nodal point), as shown in Fig. 1e, N1
N2
N3
is1
js1
ls1
j l QlasCRaPaq8CIai Fai q8CIab FbjUq8CIac Fcl*
(1)
Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
where N1, N2 and N3 are the number of nodes inside the three fictitious cracks, respectively. Here, CRa, CIia, CIjab and CIlac represent the tensile forces at the tip of crack A due to a unit external load, a pair of unit cohesive forces at the i-th node of crack A, the j-th node of crack B and the l-th node of crack C, respectively. These coefficients are obtained by finite-element calculation by applying the models from Fig. 1a through d. Note that Pa is the external load required to propagate crack A, while crack B and crack C remain inactive. The CODs along the three fictitious cracks are given by N1 i a
i a
N2 ik a
k a
ij ab
jU b
js1
ing the number of unknowns (2N1q2N2q2N3q1), and thus the problem is solved uniquely. Alternatively, in the case that crack B is assumed as active (Fig. 1f), the crack equations become N1
N2
N3
is1
js1
ls1
QlbsCRbPbq8CIibaFiaUq8CIjbFjbq8CIlbcFclU N1
N2
N3
ks1
js1
ls1
(8)
kU ij j il lU WiaUsBKiaPbq 8 AKik a Fa q8AKabFbq8AKacFc
(9)
N3 il ac
W sBK Paq 8 AK F q8AK F q8AK F ks1
175
lU c
N1
N2
N3
is1
ks1
ls1
jl WjbsBKjbPbq8AKjibaFiaUq 8 AKbjkFbkq8AKbc FclU
ls1
(2) N1
N2
N3
is1
ks1
ls1
(10)
kU jl lU WjbUsBKjbPaq8AKjibaFiaq 8 AKjk b Fb q8AKbcFc
N1
N2
N3
is1
js1
ks1
lj WlcUsBKlcPbq8AKlicaFaiUq8AKcb Fbj q 8 AKclkFckU
(3) N1
lU c
l c
N2 li ca
i a
lj cb
jU b
(11)
N3 lk c
W sBK Paq8AK F q8AK F q 8 AK F is1
js1
kU c
ks1
(4) where is1,«, N1; js1,«, N2; ls1,«, N3. Here, BKia at crack A, BKjb at crack B and BKlc at crack C are the compliances at nodes i, j and l, respectively, due to the external load Pa. The influence coefficients AKika , AKijab and AKilac are the displacements at the i-th node of crack A due to a pair of unit cohesive forces at the k-th node of crack A, the j-th node of crack B, and the l-th node of crack C, respectively. Similarly, the influjl ence coefficients AKjiba, AKbjk and AKbc represent the displacements at the j-th node of crack B and AKlica, AKljcb and AKlkc stand for the displacements at the l-th node of crack C, respectively, due to a pair of unit cohesive forces at the corresponding locations. FE models to compute these coefficients are given in Fig. 1a– d. Along each fictitious-crack, a relationship between the cohesive forces and CODs follows the strain-softening law of concrete i a
i a
F sfŽW . jU
jU
Fb sfŽWb . FlcUsfŽWlcU.
FiaUsfŽWiaU.
(12)
FjbsfŽWjb.
(13)
FlcUsfŽWlcU.
(14)
where Pb is the load required for the propagation of crack B, while crack A and crack C remain inactive. Finally, when crack C is assumed as active (Fig. 1g), the crack equations are obtained as N1
N2
N3
i j QlcsCRcPcq8CIca Fai*q8CIcb Fbj*q8CIcl Fcl is1
js1
N1
N2
N3
ks1
js1
ls1
kU ij jU il l WiaUsBKiaPcq 8 AKik a Fa q8AKabFb q8AKacFc
(16) N1
N2
N3
is1
ks1
ls1
WjbUsBKjbPcq8AKjibaFiaUq 8 AKbjkFbkUq8AKjlbcFcl
(5) (6) (7)
Eqs. (1)–(7) form the so-called crack equations, with the number of equations (2N1q2N2q2N3q1) match-
(15)
ls1
(17) N1
N2
N3
is1
js1
ks1
li iU lj WlcsBKlcPcq8AKca Fa q8AKcb FbjUq 8 AKclkFkc
(18)
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Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
FiaUsfŽWaiU.
(19)
FjbUsfŽWjbU.
(20)
FlcsfŽWlc.
(21)
where Pc is the required load for the extension of crack C, while cracks A and B are inactive. The three sets of crack equations from Eqs. (1)–(21) are employed to solve interactive extensions of three cracks, assuming each crack individually as an active crack while restraining the growth of others. Based on the minimum load criterion that allows the propagation of only one active crack at the minimum load, crack paths are determined and the stress and displacement fields are obtained. When an assumed cracking mode is irrelevant to the problem, invalid solutions are encountered. These invalid solutions are manifested either by the tip tensile stress exceeding the tensile strength at the tip of an assumed inactive crack, or by the overlapping of the crack surfaces with negative CODs. Resetting the tips of the corresponding cracks by releasing the stress or closing the crack surface, a variety of cracking modes could result in including simultaneous crack propagation of several cracks and crack growth accompanied by crack closure. When a crack is to be closed, the disconnected dual nodes next to the tip of the crack are simply reconnected without residual deformation, while the previous cohesive forces acting at these nodes are referred to as the transient tensile strength of the cracked material. Further details on the numerical procedure of the crack analysis were published in the previous papers w7,8x. 3. Numerical scheme for modeling curvilinear crack propagation In general, precise locations of crack paths in concrete are unknown in advance and remeshing is inevitable in the numerical analysis. To minimize the computational burden, a simple scheme has been proposed to automatically remesh crack trajectories, without increasing the total number of finite meshes as below w8x. To simplify the rules for remeshing, triangular elements are connected in a regular manner. As shown in Fig. 2a–c, when a new stress field is obtained, the nextstep of crack growth is set in the direction normal to the tip tensile force Q, and then the presumed crack path marked by a row of dual nodes ahead of the crack is shifted parallel by moving the nearest nodes of corresponding elements to the new positions. Except a few interchanges between normal nodes and dual nodes in the cases of (b) and (c), no new nodes and meshes need to be generated in this process. Though Fig. 2a–c illustrate only the cases when the crack curves to the left, the same rule is applicable when the crack curves
Fig. 2. Remeshing scheme. (a) Curving to the left: r(1y2u; (b) curving to the left: 1y2u-r(u; (c) curving to the left: r)u; (d) curving to the right: forming new meshes.
to the right. As demonstrated in Fig. 2d, reconstructions of the two meshes on the right side of the crack are required in the equivalent cases of (b) and (c), prior to constructing the new crack path. After remeshing at each crack propagation step, the new coefficients of the crack equations are generated and the crack equations are solved. Note that this simple scheme has limitations because it may result in ill-shaped elements, and therefore should be used with caution. By applying Eqs. (5)–(7), (12)–(14), (19)–(21) to curvilinear cracks, the tangential cohesive stress components along the crack surface are ignored. Regardless of whether the crack is straight or curved, it is assumed that the normal cohesive stress component is a function of the component of the displacement discontinuity normal to the crack surface. A similar approach was reported elsewhere w9x. 4. Numerical studies 4.1. Numerical models and material properties Fig. 3 illustrates the four-point bending test of the plain concrete beam and numerical cases with various loading conditions. As seen, three initial notches are introduced in each numerical model, i.e. two of notch
Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
177
Fig. 3. Four-point bending test and numerical cases of concentrated loads.
A and notch C below the two loading points, and notch B at the midspan. By varying the sizes of these initial notches and the loading positions, six types of the notched beam models are chosen, in order to demonstrate the sensitivity of the load-carrying capacity to a given loading condition. In cases 1-1 and 1-2, the two side-notches A and C are assumed to be small with 10mm-depth, while the central notch B with the depth of 30 mm is three times larger. As shown, two loading
conditions are studied, i.e. a four-point bending and a three-point bending. In cases 2-1 and 2-2, notch A and notch B are both 10 mm in depth, and notch C with the depth of 50 mm is five times larger. Two loading conditions are considered, as a single load over notch A in case 2-1 and a single load over notch C in case 22. In addition, to investigate a relation between the maximum load and notch size, the depths of notch C in cases 2-1 and 2-2 are reassigned from 10 to 150 mm as cases 2-3 and 2-4. In solving the crack equations, a bi-linear strainsoftening relation as shown in Fig. 4 is assumed w10x. The material properties of the test specimen are summarized in Table 1, which shows the elastic modulus E, the tensile strength f t, the compressive strength f c, Table 1 Material properties of plain concrete beam
Fig. 4. Bi-linear strain-softening relation of concrete w10x.
E (GPa)
n
fc (MPa)
ft (MPa)
Gf (Nymm)
Wc(s5Gf yf t) (mm)
27.50
0.20
33.00
2.80
0.10
0.18
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178
4.2.1. Cases 1-1 and 1-2 In view of the symmetrical feature of the models, the two problems are solved by actually modeling a half portion of the beam. As a result, the nodal displacements at notch B are modified to be CODs before solving the crack equations. As shown in the propagation charts of Fig. 5, a single crack propagates from the central notch under both the four-point and the three-point bending, and eventually leads to final failure. The maximum load is reached at the sixth step in both cases, and no cracks propagate from the two smaller side-notches. From the load–displacement relations, a large difference between the two maximum loads is observed. As the loading is changed from the four-point bending in case 1-1 to the three-point bending in case 1-2, the maximum load decreases from Pmaxs33.8 to 25.3 kN, which amounts to a reduction of 25% in terms of the load-carrying capacity of the simple beam. It is worth emphasizing that these two cases clearly illustrate the extent of the strength variation.
Fig. 5. Numerical results of cases 1-1 and 1-2.
Poisson’s ratio n, the fracture energy Gf, and the limit COD Wc w3x. Numerical studies are carried out under the plain–strain condition. 4.2. Results and discussion Figs. 5 and 6 represent numerical results of cases 11 and 1-2, and cases 2-1 and 2-2, including the load vs. midspan displacement relations and crack propagation charts. In these charts, circled numbers along crack paths denote the tip position of the particular crack at the designated step of the crack-tip-controlled computation. The maximum loads obtained under the given loading conditions are generally lower than the experimental results of the four-point bending. This may result from the fact that the round-robin tests were performed on un-notched specimens. In the present analysis, the crack increment is set to as 10 mm, which is one twentieth of the beam depth. According to the preliminary study on mesh-size sensitivity, as the incremental size was changed from 10 to 5 mm, the decrease of the maximum loads was within 3%.
4.2.2. Cases 2-1 to 2-4 Because the notch arrangement and loading conditions are not symmetric anymore, all the cases are analyzed as a whole model, as seen in Fig. 6. When the size of notch C reaches one-fourth of the beam depth, the others are only one-twentieth. Then, the fracture of the beam readily takes place at notch C in case 2-2, keeping notch A and notch B intact. In case 2-1, a single load is applied directly above notch A. Whereas the stress concentration at notch A also becomes critical, two cracks emerge from notch A and notch C and compete for simultaneous propagation. Before reaching the peak load at the sixth step, the growth of crack C loses its momentum as crack A becomes the dominant crack and propagates throughout the post-peak regions. As seen in the propagation charts, a small crack appears at notch B from the fourth to the seventh steps, which is followed by the eventual closure of crack C after the eighth step. As seen in Fig. 6, the maximum loads differ greatly in these two cases. Compared with Pmaxs37.7 kN in case 2-1, the maximum load in case 2-2 is reduced by 40% to Pmaxs22.7 kN. (For brevity of presentation the numerical cases of the four-point and the three-point bending are omitted, of which the maximum loads are 28.8 and 25.2 kN, respectively.) Next, cases 2-3 and 24 are analyzed, assuming various sizes for notch C. Relations between the maximum load and the size of notch C are presented in Fig. 7. As the ratio of notch size to beam depth increases, the peak load PC in case 2-4 decreases monotonically. On the other hand, the peak load PA in case 2-3 seems to be unaffected by notch C until it reaches the depth of 50 mm. As its size increases beyond 50 mm, PA drops quickly at faster rate than that of PC. This indicates that a transition in the fracture process occurs at this stage in case 2-3, as the
Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
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Fig. 6. Numerical results of cases 2-1 and 2-2.
beam is no longer broken at notch A but at notch C. Comparing these results, the reduction rates of PC to PA are calculated and presented in Fig. 8. With an extremely large notch of 150 mm at C (three-fourths of the beam depth), the ratio reaches approximately 50%. In other words, the flexural strength of the beam predicted by PA could be twice as large as that of PC.
It is noted that PA and PC should stand for the same peak load if the beam is devoid of any initial defects.
Fig. 7. Relations between the maximum load and the ratio of notch size to beam depth (cases 2-3 and 2-4).
Fig. 8. Relations between the reduction rate and the ratio of notch size to beam depth (cases 2-3 and 2-4).
5. Engineering implication The purpose of this study is to investigate the effect of loading positions on load-carrying capacity, as a cause of the strength reduction during fatigue failure. If
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this strength discrepancy is found to be fairly large, commonly observed strength reduction due to fatigue could be legitimately attributed to the proposed mechanisms on fatigue failure. Numerical results presented clarify that the flexural strengths of notched beams vary significantly depending on specific loading conditions, such as types of loading (four-point bending or three-point bending as in cases 1-1 and 1-2) and loading positions (a single load at A or at C as in cases 2-1 and 2-2). It is realized that the extent of the strength variation is comparable to the strength reduction of a typical fatigue failure. The dependence of the beam’s strength on initial notch configuration and length is also demonstrated. It is known that initial defects exist in all structural members as only the degree of imperfection varies. Under actual cyclic loading, not only the amplitude but also the loading position may change. Eventually, the material weakening process of a structural member caused by repeated loading involves crack propagations in multiple cracking modes, and the strength is to be determined by its most critical mode of fracture, such as those observed in cases 1-2 and 2-2. Under fixedpoint cyclic loading, however, the cracking modes involved in the fracture process are unlikely to be as diversified as those under actual cyclic loading. This leads to a rational conclusion that the latter is more likely to contain the most critical mode of fracture than the former. This may explain the aforementioned strength reduction phenomena as observed by Okada et al. w1x and Kawaguchi and coworkers w2x in their fatigue tests of reinforced concrete beams and plates, in which they employed moving cyclic loads to replace fixedpoint monotonic and cyclic loads. Because the obtained numerical results are specific to the particular beam configurations considered in the study, this paper is not intended to imply that these results are readily applicable to all other situations. By presenting some of the numerical evidence, it is hoped that the approach stimulates further study for fatigue problems. 6. Conclusions This paper presents numerical studies on the loadcarrying capacities of notched concrete beams subjected to various concentrated loads, focusing on the change of cracking behavior and the maximum load as the loading position varies. The fracture process of a simple beam with multiple cracks was analyzed numerically
using the extended fictitious-crack model. It is found that the flexural strength of a notched beam can vary significantly depending on loading conditions, and the extent of the strength variation seems to be comparable to the strength reduction level of fatigue. The dependence of the beam’s strength on initial notch configuration and length is also clarified. In general, initial defects exist in all structural members as only the degree of imperfection varies. Under actual cyclic loading, not only the amplitude but also the loading position could change. Eventually, the material weakening process caused by repeated loading involves crack propagations in multiple cracking modes. Rationally, the strength of a cracked member is to be derived from its most critical mode of fracture. References w1x Okada K, Okamura H, Sonoda K, Shimada I. Cracking and fatigue behavior of bridge deck RC slabs. Proc JSCE 1982;321(5):49 –61. w2x Taniguchi M, Kawaguchi T, Harada K, Takahashi M. Fatigue tests of reinforced concrete slab models of a highway bridge and an attempt to diagnose their residual lives. Proc JSCE 1987;380(I-7):283 –292. w3x Uchida Y, Rokugo K, Koyanagi W. Flexural tests of concrete beams. JCI Com. Rep. Applications of Fracture Mechanics to Concrete Structures. Tokyo: Japan Concrete Institute; 1993;p. 346–349. w4x Hillerborg A, Modeer M, Peterson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 1976;6(6):773 –782. w5x Hillerborg A. Numerical method to simulate softening and fracture of concrete. In: Sih GC, DiTommaso A, editors. Fracture mechanics of concrete. The Netherlands: Martinus Nijhoff Publishers, 1985. p. 141 –170. w6x Ohtsu M. Tension softening properties in numerical analysis. JCI Com. Rep. (JCI-C19). JCI Colloquium on Fracture Mechanics of Concrete Structures. Tokyo: Japan Concrete Institute; 1990;p. 55–65. w7x Shi Z, Ohtsu M, Suzuki M, Hibino Y. Numerical analysis of multiple cracks in concrete using the discrete approach. J Struct Eng ASCE 2001;127(9):1085 –1091. w8x Shi Z, Suzuki M, Nakano M. Numerical analysis of multiple discrete cracks in concrete dams using extended fictitious crack model. J Struct Eng ASCE 2003;129(3):324 –336. w9x Carpinteri A, Valente S, Ferrara G, Imperato L. Experimental and numerical fracture modeling of a gravity dam. In: Bazant ZP, editor. Fracture mechanics of concrete structures. New York: Elsevier Applied Science, 1992. p. 351 –360. w10x Rokugo K, Iwasa M, Suzuki T, Koyanagi W. Testing methods to determine tensile strain softening curve and fracture energy of concrete. In: Mihashi H, Takahashi H, Wittmann FH, editors. Fracture toughness and fracture energy—test method for concrete and rock. Balkema Publishers, 1989. p. 153 –163.
Numerical studies on load-carrying capacities of notched concrete beams subjected to various concentrated loads Zihai Shi*, Masaki Suzuki Research and Development Center, Nippon Koei Co., Ltd., 2304 Inarihara, Tsukuba, Ibaraki 300-1259, Japan Received 15 September 2003; received in revised form 12 October 2003; accepted 12 October 2003
Abstract In solving various fatigue problems, the time-varying characteristics of cyclic loads on structures due to winds, waves, vehicles and so forth are often oversimplified by assuming the loading positions as fixed. In order to clarify spatial effects of loading conditions on cracking behaviors and the maximum loads, numerical studies on the load-carrying capacities of notched concrete beams subjected to various concentrated loads are conducted. The fracture processes of a simply supported beam with multiple cracks are analyzed by using the extended fictitious-crack model. In the cases where loading positions are restricted to at two locations of four-point bending and at the midspan, it is shown that the flexural strengths of notched beams vary significantly, depending on particular loading conditions. The effects of initial notch configuration and lengths on the load-carrying capacities of the beam are clarified. Based on these results, the strength reduction due to various fatigue failures is discussed. 䊚 2003 Elsevier Ltd. All rights reserved. Keywords: Plain concrete beam; Fatigue; Fracture; Multiple cracks; Load-carrying capacity
1. Introduction In the case of testing and analyzing fatigue problems, the time-varying characteristics of cyclic loads on structures induced by winds, waves, vehicles and so forth are mostly oversimplified by assuming the loading positions as fixed during cyclic loading. Experimental studies have shown that during fatigue tests of reinforced concrete beams and plates, a moving cyclic load which simulates traffic loads could result in a reduction in the maximum load-carrying capacity, compared with fixedpoint monotonic or cyclic loading w1,2x. Consequently, the effects of loading positions during cyclic loading on the load-carrying capacity of a structural member are investigated. Due to inherent variations of materials and the random distribution of initial defects in tested specimens, even under the same test conditions the strengths of members could vary. Therefore, it is very difficult to single out the effect of the loading positions on the flexural strengths of concrete beams experimentally. This can be, *Corresponding author. Tel.: q81-29-871-2032; fax: q81-29-8712022. E-mail address: [email protected] (Z. Shi).
in contrast, readily done by a numerical analysis in which the same configuration and size of initial defects are modeled with different loading conditions. From this point of view, numerical studies are carried out on the load-carrying capacities of notched concrete beams subjected to various concentrated loads. Cracking behaviors and the maximum loads are clarified under various loading positions. As for a model, a round-robin test of un-notched concrete beams under four-point bending is selected w3x. The fracture process of beams involving multiple cracks is analyzed using the extended fictitious-crack model w4–8x. Introducing three initial notches of different sizes to the beam and restricting the loading points to at those of the four-point bending tests and at the midspan, structural responses of the notched beams under various loading conditions are studied. In the following, the crack equations for analyzing the beam models containing three discrete cracks are formulated, and a numerical scheme proposed for modeling a curvilinear crack is introduced. Numerical analyses of the notched beams are performed, and the engineering implication derived from results is discussed.
0950-0618/04/$ - see front matter 䊚 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2003.10.006
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Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
Fig. 1. Crack-tip-controlled modeling of three discrete cracks. (a) Forces and displacements at the cracks due to unit external loads; (b) forces and displacements at the cracks due to a pair of unit cohesive forces at crack A; (c) forces and displacements at the cracks due to a pair of unit cohesive forces at crack B; (d) forces and displacements at the cracks due to a pair of unit cohesive forces at crack C; (e) load condition for the growth of crack A; (f) load condition for the growth of crack B; (g) load condition for the growth of crack C.
2. Numerical formulation Fig. 1 shows three cracks of the mode-I type, crack A, crack B and crack C, propagating in the direction normal to the tensile force at the tip of the crack. Note that a crack and its fictitious extension are modeled by separating a pair of two nodal points having the same coordinates and rigidly connected to each other up to separation. For clarity, hereinafter, the cohesive forces
and crack-opening displacements (CODs) of the restrained cracks are marked by asterisks. Initially, crack A is assumed as active with the tensile force at its tip reaching the nodal force limit Qla (the tensile strength of concrete times the surface area apportioned to a nodal point), as shown in Fig. 1e, N1
N2
N3
is1
js1
ls1
j l QlasCRaPaq8CIai Fai q8CIab FbjUq8CIac Fcl*
(1)
Z. Shi, M. Suzuki / Construction and Building Materials 18 (2004) 173–180
where N1, N2 and N3 are the number of nodes inside the three fictitious cracks, respectively. Here, CRa, CIia, CIjab and CIlac represent the tensile forces at the tip of crack A due to a unit external load, a pair of unit cohesive forces at the i-th node of crack A, the j-th node of crack B and the l-th node of crack C, respectively. These coefficients are obtained by finite-element calculation by applying the models from Fig. 1a through d. Note that Pa is the external load required to propagate crack A, while crack B and crack C remain inactive. The CODs along the three fictitious cracks are given by N1 i a
i a
N2 ik a
k a
ij ab
jU b
js1
ing the number of unknowns (2N1q2N2q2N3q1), and thus the problem is solved uniquely. Alternatively, in the case that crack B is assumed as active (Fig. 1f), the crack equations become N1
N2
N3
is1
js1
ls1
QlbsCRbPbq8CIibaFiaUq8CIjbFjbq8CIlbcFclU N1
N2
N3
ks1
js1
ls1
(8)
kU ij j il lU WiaUsBKiaPbq 8 AKik a Fa q8AKabFbq8AKacFc
(9)
N3 il ac
W sBK Paq 8 AK F q8AK F q8AK F ks1
175
lU c
N1
N2
N3
is1
ks1
ls1
jl WjbsBKjbPbq8AKjibaFiaUq 8 AKbjkFbkq8AKbc FclU
ls1
(2) N1
N2
N3
is1
ks1
ls1
(10)
kU jl lU WjbUsBKjbPaq8AKjibaFiaq 8 AKjk b Fb q8AKbcFc
N1
N2
N3
is1
js1
ks1
lj WlcUsBKlcPbq8AKlicaFaiUq8AKcb Fbj q 8 AKclkFckU
(3) N1
lU c
l c
N2 li ca
i a
lj cb
jU b
(11)
N3 lk c
W sBK Paq8AK F q8AK F q 8 AK F is1
js1
kU c
ks1
(4) where is1,«, N1; js1,«, N2; ls1,«, N3. Here, BKia at crack A, BKjb at crack B and BKlc at crack C are the compliances at nodes i, j and l, respectively, due to the external load Pa. The influence coefficients AKika , AKijab and AKilac are the displacements at the i-th node of crack A due to a pair of unit cohesive forces at the k-th node of crack A, the j-th node of crack B, and the l-th node of crack C, respectively. Similarly, the influjl ence coefficients AKjiba, AKbjk and AKbc represent the displacements at the j-th node of crack B and AKlica, AKljcb and AKlkc stand for the displacements at the l-th node of crack C, respectively, due to a pair of unit cohesive forces at the corresponding locations. FE models to compute these coefficients are given in Fig. 1a– d. Along each fictitious-crack, a relationship between the cohesive forces and CODs follows the strain-softening law of concrete i a
i a
F sfŽW . jU
jU
Fb sfŽWb . FlcUsfŽWlcU.
FiaUsfŽWiaU.
(12)
FjbsfŽWjb.
(13)
FlcUsfŽWlcU.
(14)
where Pb is the load required for the propagation of crack B, while crack A and crack C remain inactive. Finally, when crack C is assumed as active (Fig. 1g), the crack equations are obtained as N1
N2
N3
i j QlcsCRcPcq8CIca Fai*q8CIcb Fbj*q8CIcl Fcl is1
js1
N1
N2
N3
ks1
js1
ls1
kU ij jU il l WiaUsBKiaPcq 8 AKik a Fa q8AKabFb q8AKacFc
(16) N1
N2
N3
is1
ks1
ls1
WjbUsBKjbPcq8AKjibaFiaUq 8 AKbjkFbkUq8AKjlbcFcl
(5) (6) (7)
Eqs. (1)–(7) form the so-called crack equations, with the number of equations (2N1q2N2q2N3q1) match-
(15)
ls1
(17) N1
N2
N3
is1
js1
ks1
li iU lj WlcsBKlcPcq8AKca Fa q8AKcb FbjUq 8 AKclkFkc
(18)
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FiaUsfŽWaiU.
(19)
FjbUsfŽWjbU.
(20)
FlcsfŽWlc.
(21)
where Pc is the required load for the extension of crack C, while cracks A and B are inactive. The three sets of crack equations from Eqs. (1)–(21) are employed to solve interactive extensions of three cracks, assuming each crack individually as an active crack while restraining the growth of others. Based on the minimum load criterion that allows the propagation of only one active crack at the minimum load, crack paths are determined and the stress and displacement fields are obtained. When an assumed cracking mode is irrelevant to the problem, invalid solutions are encountered. These invalid solutions are manifested either by the tip tensile stress exceeding the tensile strength at the tip of an assumed inactive crack, or by the overlapping of the crack surfaces with negative CODs. Resetting the tips of the corresponding cracks by releasing the stress or closing the crack surface, a variety of cracking modes could result in including simultaneous crack propagation of several cracks and crack growth accompanied by crack closure. When a crack is to be closed, the disconnected dual nodes next to the tip of the crack are simply reconnected without residual deformation, while the previous cohesive forces acting at these nodes are referred to as the transient tensile strength of the cracked material. Further details on the numerical procedure of the crack analysis were published in the previous papers w7,8x. 3. Numerical scheme for modeling curvilinear crack propagation In general, precise locations of crack paths in concrete are unknown in advance and remeshing is inevitable in the numerical analysis. To minimize the computational burden, a simple scheme has been proposed to automatically remesh crack trajectories, without increasing the total number of finite meshes as below w8x. To simplify the rules for remeshing, triangular elements are connected in a regular manner. As shown in Fig. 2a–c, when a new stress field is obtained, the nextstep of crack growth is set in the direction normal to the tip tensile force Q, and then the presumed crack path marked by a row of dual nodes ahead of the crack is shifted parallel by moving the nearest nodes of corresponding elements to the new positions. Except a few interchanges between normal nodes and dual nodes in the cases of (b) and (c), no new nodes and meshes need to be generated in this process. Though Fig. 2a–c illustrate only the cases when the crack curves to the left, the same rule is applicable when the crack curves
Fig. 2. Remeshing scheme. (a) Curving to the left: r(1y2u; (b) curving to the left: 1y2u-r(u; (c) curving to the left: r)u; (d) curving to the right: forming new meshes.
to the right. As demonstrated in Fig. 2d, reconstructions of the two meshes on the right side of the crack are required in the equivalent cases of (b) and (c), prior to constructing the new crack path. After remeshing at each crack propagation step, the new coefficients of the crack equations are generated and the crack equations are solved. Note that this simple scheme has limitations because it may result in ill-shaped elements, and therefore should be used with caution. By applying Eqs. (5)–(7), (12)–(14), (19)–(21) to curvilinear cracks, the tangential cohesive stress components along the crack surface are ignored. Regardless of whether the crack is straight or curved, it is assumed that the normal cohesive stress component is a function of the component of the displacement discontinuity normal to the crack surface. A similar approach was reported elsewhere w9x. 4. Numerical studies 4.1. Numerical models and material properties Fig. 3 illustrates the four-point bending test of the plain concrete beam and numerical cases with various loading conditions. As seen, three initial notches are introduced in each numerical model, i.e. two of notch
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177
Fig. 3. Four-point bending test and numerical cases of concentrated loads.
A and notch C below the two loading points, and notch B at the midspan. By varying the sizes of these initial notches and the loading positions, six types of the notched beam models are chosen, in order to demonstrate the sensitivity of the load-carrying capacity to a given loading condition. In cases 1-1 and 1-2, the two side-notches A and C are assumed to be small with 10mm-depth, while the central notch B with the depth of 30 mm is three times larger. As shown, two loading
conditions are studied, i.e. a four-point bending and a three-point bending. In cases 2-1 and 2-2, notch A and notch B are both 10 mm in depth, and notch C with the depth of 50 mm is five times larger. Two loading conditions are considered, as a single load over notch A in case 2-1 and a single load over notch C in case 22. In addition, to investigate a relation between the maximum load and notch size, the depths of notch C in cases 2-1 and 2-2 are reassigned from 10 to 150 mm as cases 2-3 and 2-4. In solving the crack equations, a bi-linear strainsoftening relation as shown in Fig. 4 is assumed w10x. The material properties of the test specimen are summarized in Table 1, which shows the elastic modulus E, the tensile strength f t, the compressive strength f c, Table 1 Material properties of plain concrete beam
Fig. 4. Bi-linear strain-softening relation of concrete w10x.
E (GPa)
n
fc (MPa)
ft (MPa)
Gf (Nymm)
Wc(s5Gf yf t) (mm)
27.50
0.20
33.00
2.80
0.10
0.18
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178
4.2.1. Cases 1-1 and 1-2 In view of the symmetrical feature of the models, the two problems are solved by actually modeling a half portion of the beam. As a result, the nodal displacements at notch B are modified to be CODs before solving the crack equations. As shown in the propagation charts of Fig. 5, a single crack propagates from the central notch under both the four-point and the three-point bending, and eventually leads to final failure. The maximum load is reached at the sixth step in both cases, and no cracks propagate from the two smaller side-notches. From the load–displacement relations, a large difference between the two maximum loads is observed. As the loading is changed from the four-point bending in case 1-1 to the three-point bending in case 1-2, the maximum load decreases from Pmaxs33.8 to 25.3 kN, which amounts to a reduction of 25% in terms of the load-carrying capacity of the simple beam. It is worth emphasizing that these two cases clearly illustrate the extent of the strength variation.
Fig. 5. Numerical results of cases 1-1 and 1-2.
Poisson’s ratio n, the fracture energy Gf, and the limit COD Wc w3x. Numerical studies are carried out under the plain–strain condition. 4.2. Results and discussion Figs. 5 and 6 represent numerical results of cases 11 and 1-2, and cases 2-1 and 2-2, including the load vs. midspan displacement relations and crack propagation charts. In these charts, circled numbers along crack paths denote the tip position of the particular crack at the designated step of the crack-tip-controlled computation. The maximum loads obtained under the given loading conditions are generally lower than the experimental results of the four-point bending. This may result from the fact that the round-robin tests were performed on un-notched specimens. In the present analysis, the crack increment is set to as 10 mm, which is one twentieth of the beam depth. According to the preliminary study on mesh-size sensitivity, as the incremental size was changed from 10 to 5 mm, the decrease of the maximum loads was within 3%.
4.2.2. Cases 2-1 to 2-4 Because the notch arrangement and loading conditions are not symmetric anymore, all the cases are analyzed as a whole model, as seen in Fig. 6. When the size of notch C reaches one-fourth of the beam depth, the others are only one-twentieth. Then, the fracture of the beam readily takes place at notch C in case 2-2, keeping notch A and notch B intact. In case 2-1, a single load is applied directly above notch A. Whereas the stress concentration at notch A also becomes critical, two cracks emerge from notch A and notch C and compete for simultaneous propagation. Before reaching the peak load at the sixth step, the growth of crack C loses its momentum as crack A becomes the dominant crack and propagates throughout the post-peak regions. As seen in the propagation charts, a small crack appears at notch B from the fourth to the seventh steps, which is followed by the eventual closure of crack C after the eighth step. As seen in Fig. 6, the maximum loads differ greatly in these two cases. Compared with Pmaxs37.7 kN in case 2-1, the maximum load in case 2-2 is reduced by 40% to Pmaxs22.7 kN. (For brevity of presentation the numerical cases of the four-point and the three-point bending are omitted, of which the maximum loads are 28.8 and 25.2 kN, respectively.) Next, cases 2-3 and 24 are analyzed, assuming various sizes for notch C. Relations between the maximum load and the size of notch C are presented in Fig. 7. As the ratio of notch size to beam depth increases, the peak load PC in case 2-4 decreases monotonically. On the other hand, the peak load PA in case 2-3 seems to be unaffected by notch C until it reaches the depth of 50 mm. As its size increases beyond 50 mm, PA drops quickly at faster rate than that of PC. This indicates that a transition in the fracture process occurs at this stage in case 2-3, as the
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Fig. 6. Numerical results of cases 2-1 and 2-2.
beam is no longer broken at notch A but at notch C. Comparing these results, the reduction rates of PC to PA are calculated and presented in Fig. 8. With an extremely large notch of 150 mm at C (three-fourths of the beam depth), the ratio reaches approximately 50%. In other words, the flexural strength of the beam predicted by PA could be twice as large as that of PC.
It is noted that PA and PC should stand for the same peak load if the beam is devoid of any initial defects.
Fig. 7. Relations between the maximum load and the ratio of notch size to beam depth (cases 2-3 and 2-4).
Fig. 8. Relations between the reduction rate and the ratio of notch size to beam depth (cases 2-3 and 2-4).
5. Engineering implication The purpose of this study is to investigate the effect of loading positions on load-carrying capacity, as a cause of the strength reduction during fatigue failure. If
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this strength discrepancy is found to be fairly large, commonly observed strength reduction due to fatigue could be legitimately attributed to the proposed mechanisms on fatigue failure. Numerical results presented clarify that the flexural strengths of notched beams vary significantly depending on specific loading conditions, such as types of loading (four-point bending or three-point bending as in cases 1-1 and 1-2) and loading positions (a single load at A or at C as in cases 2-1 and 2-2). It is realized that the extent of the strength variation is comparable to the strength reduction of a typical fatigue failure. The dependence of the beam’s strength on initial notch configuration and length is also demonstrated. It is known that initial defects exist in all structural members as only the degree of imperfection varies. Under actual cyclic loading, not only the amplitude but also the loading position may change. Eventually, the material weakening process of a structural member caused by repeated loading involves crack propagations in multiple cracking modes, and the strength is to be determined by its most critical mode of fracture, such as those observed in cases 1-2 and 2-2. Under fixedpoint cyclic loading, however, the cracking modes involved in the fracture process are unlikely to be as diversified as those under actual cyclic loading. This leads to a rational conclusion that the latter is more likely to contain the most critical mode of fracture than the former. This may explain the aforementioned strength reduction phenomena as observed by Okada et al. w1x and Kawaguchi and coworkers w2x in their fatigue tests of reinforced concrete beams and plates, in which they employed moving cyclic loads to replace fixedpoint monotonic and cyclic loads. Because the obtained numerical results are specific to the particular beam configurations considered in the study, this paper is not intended to imply that these results are readily applicable to all other situations. By presenting some of the numerical evidence, it is hoped that the approach stimulates further study for fatigue problems. 6. Conclusions This paper presents numerical studies on the loadcarrying capacities of notched concrete beams subjected to various concentrated loads, focusing on the change of cracking behavior and the maximum load as the loading position varies. The fracture process of a simple beam with multiple cracks was analyzed numerically
using the extended fictitious-crack model. It is found that the flexural strength of a notched beam can vary significantly depending on loading conditions, and the extent of the strength variation seems to be comparable to the strength reduction level of fatigue. The dependence of the beam’s strength on initial notch configuration and length is also clarified. In general, initial defects exist in all structural members as only the degree of imperfection varies. Under actual cyclic loading, not only the amplitude but also the loading position could change. Eventually, the material weakening process caused by repeated loading involves crack propagations in multiple cracking modes. Rationally, the strength of a cracked member is to be derived from its most critical mode of fracture. References w1x Okada K, Okamura H, Sonoda K, Shimada I. Cracking and fatigue behavior of bridge deck RC slabs. Proc JSCE 1982;321(5):49 –61. w2x Taniguchi M, Kawaguchi T, Harada K, Takahashi M. Fatigue tests of reinforced concrete slab models of a highway bridge and an attempt to diagnose their residual lives. Proc JSCE 1987;380(I-7):283 –292. w3x Uchida Y, Rokugo K, Koyanagi W. Flexural tests of concrete beams. JCI Com. Rep. Applications of Fracture Mechanics to Concrete Structures. Tokyo: Japan Concrete Institute; 1993;p. 346–349. w4x Hillerborg A, Modeer M, Peterson PE. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concrete Res 1976;6(6):773 –782. w5x Hillerborg A. Numerical method to simulate softening and fracture of concrete. In: Sih GC, DiTommaso A, editors. Fracture mechanics of concrete. The Netherlands: Martinus Nijhoff Publishers, 1985. p. 141 –170. w6x Ohtsu M. Tension softening properties in numerical analysis. JCI Com. Rep. (JCI-C19). JCI Colloquium on Fracture Mechanics of Concrete Structures. Tokyo: Japan Concrete Institute; 1990;p. 55–65. w7x Shi Z, Ohtsu M, Suzuki M, Hibino Y. Numerical analysis of multiple cracks in concrete using the discrete approach. J Struct Eng ASCE 2001;127(9):1085 –1091. w8x Shi Z, Suzuki M, Nakano M. Numerical analysis of multiple discrete cracks in concrete dams using extended fictitious crack model. J Struct Eng ASCE 2003;129(3):324 –336. w9x Carpinteri A, Valente S, Ferrara G, Imperato L. Experimental and numerical fracture modeling of a gravity dam. In: Bazant ZP, editor. Fracture mechanics of concrete structures. New York: Elsevier Applied Science, 1992. p. 351 –360. w10x Rokugo K, Iwasa M, Suzuki T, Koyanagi W. Testing methods to determine tensile strain softening curve and fracture energy of concrete. In: Mihashi H, Takahashi H, Wittmann FH, editors. Fracture toughness and fracture energy—test method for concrete and rock. Balkema Publishers, 1989. p. 153 –163.