ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 104-S59

Prediction of Fatigue Strength in Plain and Reinforced Concrete Beams by Trisha Sain and J. M. Chandra Kishen A fatigue crack propagation model for concrete is proposed based on the concepts of fracture mechanics. This model takes into account the loading history, frequency of applied load, and size effect parameters. Using this model, a method is described based on linear elastic fracture mechanics to assess the residual strength of cracked plain and reinforced concrete (RC) beams. This could be used to predict the residual strength (load carrying capacity) of cracked or damaged plain and reinforced concrete beams at a given level of damage. It has been seen that the fatigue crack propagation rate increases as the size of plain concrete beam increases indicating an increase in brittleness. In reinforced concrete (RC) beams, the fracture process becomes stable only when the beam is sufficiently reinforced. Keywords: fatigue; fracture toughness; residual strength; stress.

INTRODUCTION In recent years, condition monitoring, repair, and retrofitting of existing structures such as buildings and bridges have been among the most important challenges in civil engineering. The primary reasons for condition assessment and consequent maintenance/strengthening of structures include enhancement of resistance to withstand underestimated loads, increase in the load-carrying capacity for higher permit loads, restoration of lost carrying capacity due to corrosion of structural steel or reinforcing bars, and cracking of concrete or other types of degradation caused by aging. Whereas a lot of research has been done in the area of repair and retrofitting of aged structures, not much has been reported on the assessment of current structural condition so as to estimate the residual capacity at present-day enhanced load levels. The assessment requires that any damage in the structure be detected before it has developed to a dangerous size. The concepts of fracture mechanics may be used as a mathematical tool for assessment of residual strength for providing equations that can be used to determine how cracks grow and how cracks affect the fracture strength of a structure. It is well known to designers that fatigue accounts for a majority of material failures. In metallic structural components, fatigue is a well understood phenomenon, causing irreversible material damage (Paris and Erdogan 1963). In the case of concrete, the fatigue mechanism is different from that in metals due to dissimilar fracture behavior. In plain and reinforced concrete structures, fatigue may lead to excessive deformations, excessive crack widths, debonding of reinforcement, and rupture of the reinforcement or cement mortar matrix leading to structural collapse (Perdikaris and Calomino 1987). Fracture of concrete is characterized by the presence of a fracture process zone (FPZ) at the crack tip as shown in Fig. 1. In this figure, the effective crack length aeff is longer than the true crack but shorter than the true crack plus the FPZ. The FPZ is a zone where the cement mortar matrix is intensively ACI Structural Journal/September-October 2007

cracked. Along the FPZ, there is a discontinuity in displacements but not in the stresses. The stresses are themselves a function of the crack opening displacement (COD). At the tip of the FPZ, tensile stress is equal to tensile strength ft′ of the material and it gradually reduces to zero at the tip of the true crack. It is assumed that under low-cycle fatigue loading, the resulting damage, that is, the decrease in load-carrying capacity and stiffness, occurs primarily in the FPZ and not in the undamaged material (Foreman et al. 1967). In most of the nonlinear material models for fatigue of concrete, it is assumed that only the FPZ is responsible for the variation of material properties during cyclic loading. If the fracture process zone exhibits greater sensitivity to fatigue loading than the surrounding material, then the fatigue behavior can be considered to be dependent on loading history (Slowik et al. 1996). Furthermore, the size, shape, and fatigue behavior of the FPZ are dependent on specimen size and geometry (Zhang et al. 2001). Thus, loading history is of paramount importance in fatigue behavior of concrete and only a nonlinear fracture mechanics model can rigorously explain it. Therefore, the cumulative damage theory based on Palmgren-Miner’s hypothesis is not applicable for the fatigue behavior of concrete specimens (Oh 1991). A method for residual life prediction of plain concrete has been proposed by Zhang and Wu (1997), but it is based on the S – N curve approach, where S is the cyclic stress level and N is the number of cycles to failure.

Fig. 1—Fracture process zone. ACI Structural Journal, V. 104, No. 5, September-October 2007. MS No. S-2006-274 received July 6, 2006, and reviewed under Institute publication policies. Copyright © 2007, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the July-August 2008 ACI Structural Journal if the discussion is received by March 1, 2008.

621

Trisha Sain is a Graduate Student in the Department of Civil Engineering at the Indian Institute of Science, Bangalore, India. Her research interests include fatigue and fracture behavior in concrete, damage detection, and residual life assessment of concrete structures. J. M. Chandra Kishen is an Associate Professor in the Department of Civil Engineering at the Indian Institute of Science. He received his PhD from the University of Colorado at Boulder, Boulder, Colo. His research interests include fracture of interfaces, damage detection, and residual life assessment of aged concrete structures and structural rehabilitation.

In this study, a fatigue crack propagation model is proposed by modifying the one proposed by Slowik et al. (1996). The proposed modification includes the development of a closed form expression to compute the sudden increase in crack growth due to overloads. Furthermore, the crack growth rate can be computed for any frequency of applied loading that was not possible in the original model. This improved fatigue law is used in the assessment of residual strength for plain and reinforced concrete beams. RESEARCH SIGNIFICANCE The purpose of this work is to develop a method for assessing the residual fatigue strength of damaged plain and reinforced concrete beams. This is done by proposing an improved fatigue crack propagation law that takes into account the loading history, frequency of the applied load, and the size effect parameters. The residual strength of plain and reinforced concrete beams are assessed in terms of the number of load cycles that are needed for a structure to fail by unstable crack propagation. IMPROVED FATIGUE CRACK PROPAGATION MODEL FOR CONCRETE Based on linear elastic fracture mechanics concepts, the fatigue crack propagation law proposed by Slowik et al. (1996) includes parameters such as fracture toughness, loading history, and specimen size, except the frequency of externally applied load, and is described by m

respectively. It should be noted herein that the stress intensity factor be expressed in MNm–3/2. These values were determined for a particular loading frequency of 3 Hz. Because parameter C gives an estimation of crack propagation rate in fatigue analysis, it should also depend upon the frequency of loading. Further, the fatigue crack propagation takes place primarily within the fracture process zone; hence, C should be related to the relative size of the fracture process zone, which itself is related to characteristic length. Therefore, C should depend on the characteristic length lch and ligament length L, where lch = EGf /ft′2, and E is the elastic modulus of concrete, ft′ is the tensile strength of the concrete, and Gf is the specific fracture energy. Slowik et al. (1996) proposed a linear relationship between parameter C and the ratio of ligament length L to characteristic length lch given by –3 L C = ⎛ – 2 + 25 ------⎞ × 10 mm/cycle ⎝ l ch⎠

This equation does not account for the frequency of fatigue loading. Hence, in this study, a modified equation to include the effect of loading frequency has been proposed. This is established through a regression analysis using the experimental results of Slowik et al. (1996) and Bažant and Kangming (1991). While Slowik et al. (1996) have used compact tension specimens of two different sizes with loading frequency 3 Hz and interrupted by spikes, Bažant and Kangming (1991) have tested a series of geometrically similar three-point beams under fatigue with a loading frequency of 0.033 Hz. In a compact tension specimen, tensile force is applied in a direction perpendicular to the notch, thereby causing the propagation of the notch through the opening mode. The geometrical properties of these compact tension and beam specimens are shown in Table 1. The C values for the compact tension specimens are reported by Slowik et al. (1996), whereas for the beam specimens used by Bažant and Kangming (1991), the C values are computed by fitting the experimentally obtained

n

K Imax ΔK I da- = C --------------------------------- + F ( a, Δσ ) -----p dN ( K IC – K Isup )

(1)

where KIsup is the maximum stress intensity factor ever reached by the structure in its past loading history; KIC is the fracture toughness; KImax is the maximum stress intensity factor in a cycle; N is the number of load cycles; a is the crack length; ΔKI is the stress intensity factor range; and m, n, and p are constants. These constant coefficients are determined by Slowik et al. (1996) through an optimization process using experimental data and they obtained 2.0, 1.1, and 0.7, respectively. Parameter C represents the crack growth rate per fatigue load cycle, and function F(a, Δσ) describes the effect of sudden overload onto the crack propagation. In the subsequent sections, the procedures for estimating C and F(a, Δσ) are explained.

Table 1—Geometry and material properties of specimens Specimen

Depth, Width, Span, Initial E, mm mm mm notch, mm MPa

Discussion on parameter C Parameter C in empirical Eq. (1) basically gives a measure of crack growth per load cycle. In concrete members, this parameter indicates the crack growth rate for a particular grade of concrete and is also size dependent. Slowik et al. (1996) have determined the value of C to be equal to 9.5 × 10–3 and 3.2 × 10–2 mm/cycle for small and large size specimens,

KIc, MN/m3/2

Compact tension

900

400

—

230

17,000

Compact tension

300

100

—

50

16,000

0.95

Beam

152.4

38.1

381

25.4

27,120

1.41

1.48

Beam

76.2

38.1

191

12.7

27,120

1.51

Beam

38.1

38.1

95

6.35

27,120

1.66

Note: 25.4 mm = 1 in.; 1 MPa = 145 psi; 1 MN/m3/2 = 907 lbs/in.3/2.

Table 2—C values and material parameters GF, L/lch N/m

C, mm/cycle

238.74 1.38 206

–3

lch

32 × 10

172 0.872 158 25.33 × 10–3

622

(2)

f, Hz

Cf, mm/s

Size and reference

3

0.096 Small (Slowik et al. 1996)

3

0.0285 Large (Slowik et al. 1996)

242.66 0.52 73.19 1773 × 10–3 0.033 0.0585

Large (Bažant and Kangming 1991)

278.54 0.23 84.01 1170 × 10–3 0.033 0.0386

Medium (Bažant and Kangming 1991)

337.52 0.09 101.8 840 × 10–3 0.033 0.0277

Small (Bažant and Kangming 1991)

Note: 1 N/m = 5.7 × 10–3 lb/in.; 1 mm/s = 0.04 in./s.

ACI Structural Journal/September-October 2007

a – N data into Eq. (1). The values so obtained are tabulated in Table 2 along with the frequency of loading used in the tests. Figure 2 shows the plot of C times f (Cf ) versus the ratio of ligament to characteristic length (L/lch). The resulting best-fit curve represents a quadratic polynomial given by L L 2 Cf = – 0.0193 ⎛ ------⎞ + 0.0809 ⎛ ------⎞ + 0.0209 mm/second (3) ⎝ l ch⎠ ⎝ l ch⎠ From this equation, one can obtain the value of parameter C for any loading frequency, grade of concrete, and size of specimen. Effect of overload Unlike in metals where an overload causes an increase in the plastic zone size, thus retarding the rate of subsequent crack growth, in concrete, the size of process zone is increased due to overload and the rate of crack propagation also increases suddenly. In Eq. (1), the function F(a, Δσ) describes the sudden increase in equivalent crack length due to an overload (Slowik et al. 1996). It should be noted herein that the function F(a, Δσ) is not directly related to fatigue, but takes care of the structural response due to overloads only. Based on a nonlinear interpretation, Slowik et al. (1996) have concluded that overloads cause a sudden propagation of the fictitious crack tip. They have obtained the values of function F(a, Δσ) for compact tension geometries by unloading and reloading at several load levels in the prepeak region and calculating the equivalent crack lengths from the corresponding compliances. No closed-form equation for computing F(a, Δσ) has been developed by them. In this work, a closed form analytical expression to compute the sudden increase in crack length due to overloads is developed. Because the rate of crack propagation due to overload depends on the inherent property of concrete and stress amplitude, the function F accounting for these parameters is proposed as ΔK F = ⎛ ---------I⎞ Δa ⎝ K IC ⎠

ΔK I = K Ioverload – K Inormal load

(5)

Herein, KIoverload represents the maximum stress intensity factor due to overload and KInormal load is the maximum stress intensity factor due to normal load just before the overload. The value Δa is the increase in crack length with respect to its initial value before the application of overload and KIC is the fracture toughness of concrete. Validation of proposed fatigue model The proposed fatigue law given by Eq. (1) together with Eq. (3) and (4) are validated using the experimental results of Slowik et al. (1996) for small and large compact tension specimens and Bažant and Kangming’s (1991) results for small, medium, and large beam specimens. The specimen details are mentioned in the previous section and are tabulated in Table 1. In Fig. 3, the proposed model is compared with the experimental results of Bažant and Kangming (1991) under the constant amplitude loading. The maximum load applied in the fatigue cycles is 80% of the monotonic failure load, whereas minimum load level is kept as zero. It is seen that, initially, the crack growth rate is moderate, and as the stress intensity factor approaches the fracture toughness, crack growth becomes faster, finally leading to failure, which is represented by the asymptotic nature of the crack

(4)

where ΔKI, indicates the instantaneous change in stress intensity factor from the normal load cycle to a certain overload cycle, that is

Fig. 3—Fatigue crack propagation for three-point bend specimens.

Fig. 2—Relationship between Cf and ratio of ligament length to characteristic length.

Fig. 4—Fatigue crack propagation for compact tension specimens (G05).

ACI Structural Journal/September-October 2007

623

propagation curve. A good agreement is seen between the proposed model and the experimental results. Figures 4 and 5 show the fatigue crack propagation curves for the experimental results reported by Slowik et al. (1996) for their compact tension specimens together with the proposed fatigue law, considering the variable amplitude loading. Two sets of experimental results having different fatigue load cycles are validated herein under the same fatigue loading as reported by Slowik et al. (1996) in their experiments. One set is represented by G05 and the other set is represented by G06, G07, G13, G15, and G17 by Slowik et al. (1996). For Specimen G05, it is seen from Fig. 4 that at 1800 cycles and 3700 cycles, there is a sudden increase in crack length. This is due to the presence of spikes/overload in the load spectrum at 1800 cycles and 3700 cycles. For the second set of specimens, two spikes were introduced in the experiments at 1800 cycles and 2700 cycles and a sudden increase in crack length is predicted at these load cycles by the proposed analytical model as seen in Fig. 5. Hence, under variable amplitude loading, a sudden increase in load magnitude causes a rapid propagation of the effective crack. A good agreement is seen between the experimental results and the proposed model validating the same. RESIDUAL STRENGTH ASSESSMENT OF PLAIN CONCRETE BEAMS In this study, an analytical procedure is developed to assess the residual strength in terms of load/moment carrying capacity of plain and reinforced concrete beams by considering damage in the form of a dominant discrete crack. Failure of the member takes place when a dominant discrete crack propagates and becomes unstable at some critical crack size. To determine the critical crack length, the energy criteria is used according to which unstable crack propagation takes place when the energy release rate GI reaches the critical energy release rate GIC. Strength of cracked plain concrete beams According to linear elastic fracture mechanics, the basic equation that relates the stress intensity factor with the applied load, specimen geometry, and crack size are given by (Bažant and Kangming 1991) Pf ( α ) f(α) K I = -------------- = σ n d ---------Cn b d

(6)

Fig. 5—Fatigue crack propagation for compact tension specimens (G06, G07, G13, G15, and G17). 624

where α is the relative crack depth equal to a/d, a is the crack length, d is the characteristic dimension of the structure such as depth of beam, b is the specimen thickness, P is the applied load, and f(α) is a function depending on specimen geometry and for a three-point bend beam is given by –3 ---

2

3

4

f ( α ) = ( 1 – α ) 2 ( 1 – 2.5α + 4.49α – 3.98α + 1.33α )(7) The value Cn is a coefficient chosen for convenience to generalize the stress expression. For a three-point bend beam specimen having an initial notch length a0, Cn = 3L/{2(d – a0)}. In terms of energy release rate, Eq. (6) can be written as 2

K G I = -----IE

(8)

where E is the elastic modulus. Equations (6) and (8) are used to determine the load carrying capacity of a cracked plain concrete member characterized by the critical energy release rate GIC for a given size. Procedure for computing residual life of plain concrete At the beginning of this paper, a model for computing the rate of crack propagation with respect to the number of cycles of fatigue load is discussed. This model, explained by Eq. (1) and (4) is used to determine the number of load cycles required for a dominant crack to reach a critical size. In addition, the strength or the load carrying capacity at a particular crack length can be determined using Eq. (6). These three equations are used in this study to assess the residual strength of a plain concrete beam using the following procedure: 1. The crack length a versus the number of load cycles N relationship is plotted for a given concrete member using the model discussed in Eq. (1) together with Eq. (3); 2. The strength or the load carrying capacity of the member as a function of continuously increasing crack length a is determined using Eq. (6); and 3. Using the plot obtained in Step 1, the number of load cycles Nc required for the existing crack to become critical (at the point when the curve becomes asymptotic) is determined. At Nc, using the plot in Step 2, the current strength of the beam is determined, which gives the residual strength. RESIDUAL STRENGTH ASSESSMENT OF REINFORCED CONCRETE BEAMS In a conventional analysis of reinforced concrete members, the tensile strength of concrete is completely neglected. A conventional design of reinforced concrete structures includes the ultimate limit and service performance analysis. Validity of the ultimate limit analysis may require a structure showing some type of yield plateau before failure on its loaddisplacement relationship (ductile failure). Failures of many concrete structures are often quasi-brittle, however, and are characterized by a softening load-displacement curve after reaching a maximum load. Hence, the concepts of fracture mechanics may be conveniently used to assess the load carrying capacity of the member. In this study, the residual strength of a reinforced concrete beam is obtained in terms of the ultimate moment carrying capacity using the fracture mechanics model originally proposed by Carpinteri (1984) and later modified by Baluch et al. (1992). Figure 6 shows a ACI Structural Journal/September-October 2007

cracked reinforced beam subjected to moment M. The effect of steel bar is simulated by a crack closing force Fs acting at the centroid of the reinforcement. Based on linear elastic fracture mechanics, the stress intensity factors for the beam subjected to M and Fs are given by MY M ( a ⁄ d ) K IM = -------------------------3⁄2 d t

(9)

YM ( a ⁄ d ) Fs YF ( a ⁄ d ) d - – F s ⎛ --- – d s⎞ --------------------K IF = – -------------------------3⁄2 1⁄2 ⎝2 ⎠ d t d t

(10)

In the aforementioned equations, KIM and KIF are the stress intensity factors due to M and Fs, respectively, t is the width of the beam, d is the total depth of the beam, ds is the effective cover to tensile reinforcement, a is the crack length, and YM and YF are geometric factors given by YM ⎛ a ---⎞ = ⎝ d⎠ 1---

3 ---

5 ---

7 ---

9 ---

(11)

a 2 a 2 a 2 a 2 a 2 6 1.99 ⎛ ---⎞ – 2.47 ⎛ ---⎞ + 12.97 ⎛ ---⎞ – 23.17 ⎛ ---⎞ + 24.8 ⎛ ---⎞ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠

YF ⎛ a ---⎞ = ⎝ d⎠ 1---

3 ---

5 ---

7 ---

9 ---

(12)

a 2 a 2 a 2 a 2 a 2 1.99 ⎛ ---⎞ – 0.41 ⎛ ---⎞ + 18.7 ⎛ ---⎞ – 38.48 ⎛ ---⎞ + 53.85 ⎛ ---⎞ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠ ⎝ d⎠

where NP is a nondimensional number, which is basically a function of material, geometric, and sectional properties. For a particular grade of concrete and steel, it is proportional to the size of the structure, indicating the increase in brittleness property along with the increase in size. Procedure for computing residual strength of reinforced concrete beams The procedure is the same as that explained for plain concrete beams except that Eq. (13) is used for obtaining the stress intensity factor for cyclic load as required in Eq. (1). The residual strength is determined in terms of the moment carrying capacity using Eq. (14). CASE STUDY: RESULTS AND DISCUSSIONS Plain concrete beams The residual life and strength of plain concrete beams are determined for three different sizes of concrete beams under three-point bending, as given in Table 1. A constant amplitude loading cycle with a frequency of 0.033 Hz and a minimum load of zero and a maximum of 80% of the monotonic failure load as given in Table 3 is considered. Figure 3, as described previously, shows the relative crack length plotted against the number of load cycles for these beams. The load-carrying capacity as a function of crack size is determined using Eq. (6) with KI = KIc, (KIc is the fracture toughness given in Table 1) and plotted as shown in Fig. 7. The linear elastic crack solution indicates an infinite stress at the crack tip when the crack length a approaches zero, and because no material can sustain infinite stress, the load carrying capacity is terminated at the ultimate strength of the material. This ultimate strength is obtained when the tensile stress at the

The previous two geometric factors are applicable for relative crack depth α < 0.7. The total stress intensity factor KI is obtained by superposing KIM and KIF. Thus Fs 1 d K I = ----------- Y ( a ⁄ d ) M – F s ⎛ --- – d s⎞ – ----------- Y ( a ⁄ d ) (13) 2⁄3 M 1⁄2 F ⎝2 ⎠ d t d t In this study, unstable crack propagation takes place when KI reaches the material fracture toughness KIc, corresponding to the moment M reaching the fracture moment Mf. Hence 3⁄2

d K Ic d t Fs d ---⎞ + Y M ⎛ a ---⎞ ⎛ 1 - Y ⎛a M f = ----------------------- – ----s⎞ (14) - + --------------------⎝ d⎠ ⎝ 2 d ⎠ Y M ( a ⁄ d ) Y M ( a ⁄ d ) F ⎝ d⎠ The aforementioned equation describes a relationship between Mf , Fs, and a. If the applied moment corresponds to yielding of steel, then Fs will be replaced by the plastic force FP in Eq. (14). In nondimensional form, the previous equation can be rewritten as Fig. 6—Cracked beam element model for reinforced concrete. Mf YF ( a ⁄ d ) 1 ds 1 - + --- – ---------------------- = ---------------------- + N P --------------------2⁄3 Y ( a ⁄ d ) 2 d Y M M(a ⁄ d) K Ic d t

(15)

Failure load, kN

in which 1⁄2

fy d As N P = ------------------K Ic A ACI Structural Journal/September-October 2007

Table 3—Failure loads for plain concrete beams

(16)

Beam size

Critical crack size, mm

Fatigue model

Experimental (monotonic)

Small

15.24

1.382

1.8

Medium

33.3

2.763

2.98

Large

59

4.32

5.18

Note: 25.4 mm = 1 in.; 1 kN = 224.5 lb.

625

bottommost fiber reaches the tensile strength of the material, which is determined using the ACI relationship 0.673√fck, where fck is the cylindrical compressive strength (32.8 MPa [4750 psi]). The vertical broken line in Fig. 7 shows the cut-off between the strength criterion and linear elastic fracture mechanics (LEFM). Thus, using Fig. 3 and 7 in conjunction, the residual strength at any crack size is obtained. Further, from Fig. 3, it is seen that unstable crack propagation takes place for the small, medium, and large beams at relative crack sizes (a/D) of 0.450, 0.414, and 0.394, respectively, from which the critical crack lengths are computed and tabulated in Table 3. A close match is obtained between the experimental critical crack lengths with the proposed model predictions. Corresponding to these relative crack sizes, the failure load is obtained from Fig. 7 and is tabulated in Table 3 together with the experimentally measured failure loads under monotonic loading as obtained by Bažant and Kangming (1991). It is seen that the failure loads obtained using the present fatigue model are smaller than the monotonic failure loads obtained experimentally as expected. This is due to the faster rate of stiffness degradation in fatigue loading as compared with monotonic loading. Further, it is seen that failure of small- and medium-sized beams are governed by the strength criterion, whereas failure of large beams is governed by LEFM. This is reflected in the size effect law proposed by Bažant (1984) wherein largesized structures fail due to propagation of a single dominant discrete crack.

Fig. 7—Strength of plain concrete beams for different sizes.

Fig. 8—Crack propagation curve for reinforced concrete beams with different reinforcement areas. 626

Reinforced concrete beams A parametric study is undertaken on a reinforced concrete beam, subjected to three-point bending, to study the residual strength by considering two parameters, namely reinforcement ratio and ultimate compressive design strength. The effective depth and width of the beam are taken as 355 and 102 mm (14 and 4 in.), with an initial notch length of 38 mm (1.5 in.). The percentage of reinforcement has been taken as 0.28%. The concrete is assumed to have a compressive strength of 23 MPa (3400 psi) and a fracture toughness of 31 MPa (4500 psi). A constant amplitude loading cycle with frequency 3 Hz producing a minimum zero and a maximum 11 × 10–3 MN-m (96.8 × 103 lb-in.) bending moment at the midspan is considered. Using Eq. (3), parameter C is computed to be 34.98 × 10–3 mm/cycle (1.78 × 10–3 in./ cycle). Using Eq. (1) and (13), the fatigue crack propagation curves as shown in Fig. 8 are obtained. Failure by unstable crack propagation takes place at a critical crack length when the curves become asymptotic. For the present beam with NP = 0.68, failure occurs at a crack length of approximately 110 mm (4.33 in.) when the load has undergone approximately 3,000,000 cycles. It is also seen from this figure that the slope of the curves decreases slightly with increasing reinforcement ratios indicating retardation of fatigue crack growth and hence an improvement in ductility. The next part of the residual strength analysis is concerned with the determination of strength as a function of crack length. The strength is determined using Eq. (14). The normalized moment carrying capacity is plotted against the relative crack depth for varying steel reinforcement (varying NP), as shown in Fig. 9. It is seen from this plot that the normalized moment carrying capacity increases with increasing reinforcement for any given crack size. Further, as the crack size increases, the normalized moment carrying capacity decreases. This decrease is quite substantial at the beginning when the crack begins to form and propagate. For low NP values, the normalized moment keeps decreasing with increasing crack lengths; but as NP increases, it is seen that the curves begin to rise up after the crack has propagated substantially. Physically, this rise indicates that crack propagation takes place in a stable manner. Using Eq. (15), the minimum value of NP for which a local minimum exists is computed to be 0.67. For NP values below 0.67, the normalized moment keeps on decreasing with an increase in crack length. For NP values greater than 0.67, the normalized

Fig. 9—Normalized moment-carrying capacity corresponding to different NP values. ACI Structural Journal/September-October 2007

moment increases after the crack has propagated to a certain depth indicating stable crack propagation. Thus, the fracture process in reinforced concrete members becomes stable only when the beam is sufficiently reinforced or the cross section of the beam is sufficiently large. This phenomenon is also observed by Carpinteri (1984). With increasing load, the stress in reinforcing bars gradually increases. At different values of the reinforcing bar stress fa, the moment carrying capacity is computed for the aforementioned beam with NP = 0.7 and is plotted in Fig. 10. Theoretically, the fracture moment obtained using Eq. (14) for a zero crack length is infinite, and the curves become asymptotic. In reality, because no material can resist infinite stress, the curves have to be terminated at the moment value corresponding to the ultimate strength of the material. This ultimate moment capacity depends upon the type of constitutive law used for concrete in compression. Different constitutive laws have been proposed and used by various investigators, for example, the bilinear law, parabolic-linear law, Hognested model, and Todeschini model (MacGregor 1988). Further, concrete is characterized by a softening type post-peak load deflection behavior (Hillerborg et al. (1976). Hence, it is assumed that concrete will resist some part of the tensile stress coming in the tension zone according to the softening law. Therefore, the stress-strain diagram in tension zone would be a combination of steel and concrete tension as shown in Fig. 11. In this figure, lp is the process zone length. The total tension T can be written as T = T1 +T2 +FS (Carpinteri 1984), where T1 is the total force coming from the prepeak zone, T2 is the tension

contributed by the softening branch of the curve, and FS is the steel force. Maintaining force equilibrium, C = T, where C is the resultant compressive force, the ultimate moment is evaluated. In Fig. 10, the ultimate moment carrying capacity is computed using the bilinear stress-strain model. The ultimate normalized moment values computed using bilinear, paraboliclinear, and Hognestad and Todeschini models are 0.545, 0.520, 0.547, and 0.540, respectively. Not much variation is seen in the ultimate moment values for these four different models. Further, from Fig. 8, it is seen that unstable crack propagation takes place for the beam considered (Np = 0.68) at crack size of 110 mm (4.3 in.). Corresponding to this crack size, the normalized moment is obtained from Fig. 10 as 0.3, which corresponds to a load of 28.6 kN (6430 lb). The experimentally measured failure load under monotonic loading as obtained by Carpinteri (1984) is 29 kN (6520 lb). Thus, as expected, the failure load obtained using the present fatigue model is smaller than the monotonic failure load obtained experimentally.

Fig. 10—Normalized moment carrying capacity with constitutive law for concrete in compression being bilinear.

CONCLUSIONS In this study, an improved fracture mechanics-based fatigue law for concrete is proposed and validated. This law takes into account the effects of size, loading history, grade of concrete, overloads, and the frequency of applied loading. According to the proposed law, sudden overload in a regular fatigue load cycle increases the crack propagation rate. The analytical prediction through present model matches closely with experimental results for plain concrete specimens. The proposed law can also be used to predict fatigue crack propagation in a reinforced concrete member by simulating the reinforcement effect as a closing force in the resultant stress intensity factor expression. Using this fatigue law, a method for computing the residual strength of reinforced concrete cracked beams is described and validated for already existing experimental results. From the case studies on plain and reinforced concrete beams, the following conclusions are made: 1. The frequency of external fatigue loading influences the rate of crack propagation; 2. The crack length and the rate of crack propagation increases due to an overload; 3. In a plain concrete beam, the rate of fatigue crack propagation increases as the size of the specimen increases, indicating an increase in the brittleness; and 4. The fatigue crack propagation rate decreases along with increase in percentage reinforcement. The fracture process becomes stable in a reinforced concrete beam only when the beam is sufficiently reinforced or the cross section of the beam is sufficiently large.

Fig. 11—Stress-strain variation in reinforced concrete section (compression and tension).

A As a aeff C d ds Fs f ft′ fy GF KIC KIF KIM

NOTATION

ACI Structural Journal/September-October 2007

= = = = = = = = = = = = = = =

cross sectional area reinforcement area crack length, mm effective crack length, mm fatigue law constant, mm/cycle characteristic dimension of structure, mm effective cover, mm force in steel reinforcement, N frequency of applied loading, Hz tensile strength of concrete, MPa yield stress in steel, MPa specific fracture energy, N/mm fracture toughness, MNm–3/2 stress intensity factor (SIF) due to axial force F, MNm–3/2 SIF due to moment M, MNm–3/2

627

KImax KInormal load KIoverload KIsup

= = = =

L lch MF N Nc NP T t α ΔK Δσ

= = = = = = = = = = =

maximum SIF in cycle, MNm–3/2 maximum SIF due to normal load maximum SIF due to overload maximum SIF ever reached by structure in its past loading history ligament length, mm characteristic length, mm unstable fracture moment, N-mm number of load cycles number of cycles at failure brittleness number total tension in reinforcing bars, N thickness or width of beam, mm relative crack depth, a/d stress intensity factor range, MNm–3/2 increase in stress amplitude due to overload, MPa

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ACI Structural Journal/September-October 2007

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