Available online at www.sciencedirect.com

International Journalof Fatigue

International Journal of Fatigue 29 (2007) 2138–2148

www.elsevier.com/locate/ijfatigue

Residual fatigue strength assessment of concrete considering tension softening behavior Trisha Sain, J.M. Chandra Kishen

*

Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India Received 19 September 2006; received in revised form 13 December 2006; accepted 14 January 2007 Available online 27 January 2007

Abstract In this study, the residual strength of plain concrete beams under fatigue loading is assessed. The quasi-brittle nature of the material is considered by including the effect of tension-softening taking place in the fracture process zone. A two step approach is followed. In the first step, the effective critical crack length for unstable fracture to occur is determined by using two different methods, namely a modified LEFM based fatigue crack propagation law and the crack resistance method. In the second step, the moment carrying capacity as a function of increasing effective crack length is obtained in order to assess the residual strength of the member. A parametric study is performed by considering three different softening laws: linear, bilinear and power laws. It is seen that the bilinear softening law matches close to the experimental predictions of other investigators.  2007 Published by Elsevier Ltd. Keywords: Fatigue; Fracture; Strain softening; Residual strength

1. Introduction The prediction of remaining life, residual strength and reliability of structures under realistic load histories are of primary interest to structural engineers. The factor of safety against failure cannot be assessed without the knowledge of the remaining life, which in turn depends on the level of damage suffered during the service period. Under fatigue loading, the repetitive nature of loads may increase the degree of damage induced in the member by accelerating the crack opening or the stiffness degradation process. During the past few decades, several attempts, both numerical and experimental, have been carried out to improve the knowledge about the fatigue behavior of materials. Interest in the fatigue behavior of concrete arises because structures such as concrete bridges, offshore elements, and concrete pavements are loaded by cyclic loads. It is important to consider the fatigue performance of these structures since

*

Corresponding author. Tel.: +91 80 2293 3117; fax: +91 80 2360 0404. E-mail address: [email protected] (J.M. Chandra Kishen).

0142-1123/$ - see front matter  2007 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2007.01.011

the failure under cyclic loads occurs well below the yield limit. The current approaches used to evaluate fatigue performance of concrete members are mainly empirical. Fatigue equations have been developed and used based on the well-known S-N concept [1–4]. The residual fatigue strength and other stiffness parameters are assessed based on these equations [5]. But implementation of the conventional S-N approach requires the collection of timeconsuming experimental data. The resulting information is not valid for other cases with different loading configurations or boundary conditions. A severe limitation of this approach is that the method does not use fundamental material parameters [6]. Mechanistic approaches that utilize the concepts of fracture mechanics to study crack propagation for fatigue loading have also been proposed [6]. Most of these methods have been developed by modifying the well known Paris law that was developed for ductile materials. For quasibrittle materials such as concrete, several researchers [7–12] have attempted to apply the Paris law to describe fatigue behavior. As the crack propagation depends on

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

2139

Nomenclature B C D E F F1 Gf Kmax DK KIc K ini Ic KR L Msoft MUt N

width of beam specimen fatigue law parameter depth of beam specimen elastic modulus overload function geometry factor fracture energy maximum stress intensity factor stress intensity factor range fracture toughness fracture initiation toughness crack extension resistance ligament length moment of resistance due to process zone moment of resistance due to uncracked concrete fatigue load cycle

the inherent properties of concrete, such as the tensile strength, fracture energy and size of the specimen, suitable modifications are included in the Paris law to obtain a better coherence with the experimental fatigue data. Perdikaris et al. [13] have shown that compliance measurements provide a convenient method for estimating the traction-free crack length of concrete specimens, when subjected to fatigue loads. Studies by Zhang et al. [14] have revealed that the crack propagation in concrete members is greatly influenced by the aggregate bridging mechanism that occurs inside the fracture process zone. This bridging phenomenon can be described through the post-peak softening response of the material. LEFM based fatigue law does not incorporate the presence of process zone in front of the crack tip, thereby underestimating the fatigue strength of the member. Hence, the following points may be noted with respect to the fatigue behavior considering tension softening: (1) After a dominant fatigue crack is created, the bridging behavior within the fracture process zone governs the rate of crack advancement. (2) The stress at the crack tip remains constant and is equal to the material tensile strength; material properties outside the fracture zone are unchanged during fatigue loading. (3) LEFM based fatigue analysis neglects the softening response after the post-peak, therefore underestimates the fatigue resistance of the member. Therefore, to analyze the fatigue life of a concrete member, its tension softening behavior should be considered. Recently, as a primary work, a method has been proposed by Xu and Reinhardt [15] to determine the crack extension resistance according to the cohesive force on the fictitious crack zone that is described by the softening

P Ts TUt a0 a f ft n r1, w1 k lp w wc a tp tu

upper load limit in fatigue cycle tensile force in softening zone tensile force provided by uncracked concrete initial notch length crack length frequency of external loading tensile strength of concrete power-law exponent parameters in bilinear softening neutral axis depth factor fracture process zone length crack opening displacement critical crack tip opening displacement relative crack depth (a/D) tensile strain corresponding to elastic limit failure tensile strain of concrete

traction-separation law. They have determined the critical instability point and the initiation point of crack propagation for a large number of concrete beam specimens. In the present study, one of the methods for determining the critical crack length at unstable fracture is based on the crack resistance concepts. In this work, a method is proposed to determine the residual strength of concrete beams under fatigue loading by considering the quasi-brittle behavior of the material. A two step approach is followed. In the first step, the effective critical crack length for unstable fracture to occur is determined by using two different methods, namely a modified LEFM based fatigue crack propagation law and the crack resistance method. In the second step, the moment carrying capacity as a function of increasing effective crack length is obtained in order to assess the residual strength of the member. A parametric study is performed by considering three different softening laws: linear, bilinear and power laws. The proposed methods are validated using the experimental results of other investigators that are available in the literature. 2. Determination of critical crack length In this study, damage in concrete beam is assumed to be in the form of a discrete crack. Thus, in order to assess the residual strength, the critical crack length defined as the crack length at which unstable crack propagation takes place or at which complete failure of the beam takes place, needs to be determined. This is done in this investigation using two methods. In the first method, an improved fatigue crack propagation law is proposed by modifying the one proposed earlier by Slowik et al. [16] wherein they have proposed an LEFM based fatigue crack propagation law based on the effective crack concept that includes parameters such as fracture toughness, loading history and

2140

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

specimen size. This law does not account for the frequency of the applied loading. This fatigue law has been improved by including the effect of frequency of applied loading and by developing a closed form expression for computing the sudden crack increment due to overloads. In the second method, the critical crack length is determined using the crack extension resistance approach originally proposed by Xu and Reinhardt [15]. In this method, the effect of strain softening is incorporated as a crack resistance force. Since the main objective of our study is to consider the effect of tension softening on the fatigue strength of concrete beams, the critical crack length is determined using the two methods wherein the quasi-brittle nature of concrete is considered. The two methods for determining the critical crack length are detailed below after a short note on tension softening of concrete.

the effect of linear, bilinear and power law softening behavior on the fatigue strength of concrete beams. Amongst these, the simplest approximation is the linear softening relation as proposed by Hillerborg et al. [17], and stress at any point in the process zone is considered as a function of the crack opening only. Mathematically, the linear softening expression can be written as follows [17]:   w ð1Þ r ¼ ft 1  wc where ft is the tensile strength, w the crack opening displacement and wc the critical crack opening displacement, as is shown in Fig. 1a. Similarly, the bilinear softening behavior (Fig. 1b) can be mathematically expressed as follows: r ¼ ft  ðft  r1 Þw=w1

w 6 w1

¼ r1  r1 ðw  w1 Þ=ðwc  w1 Þ

2.1. Tension softening behavior of concrete In the last three decades, one of the main developments in the application of fracture mechanics to concrete is the discovery of the existence of cohesive force in the fracture process zone (FPZ) in front of the stress-free crack. As a consequence, the fictitious crack model was proposed by Hillerborg et al. [17] and the crack band model by Bazant and Oh [18]. 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) [17] or the tensile strain in the post-peak region [18]. Since tensile strain measurement in the post-peak region is extremely difficult, thereby crack opening displacement (w) is measured in the experiments, and converted into equivalent strain measured over a gage length h. At the tip of the FPZ, tensile stress is equal to tensile strength ft of the material, corresponding crack opening is zero and equivalent strain is ultimate tensile strain. The stress gradually reduces to zero at the tip of the true crack, which corresponds to critical crack opening wc with failure strain tu. In the available literature, the post-peak softening behavior has been mathematically modeled by different investigators using linear, bilinear, power-law or other relationships, as is shown in Fig. 1 depending on the trend followed by experimental results. In this work, we consider

where n is an index which is assumed to be 0.248 based on experimental calibrations. 2.2. Improved fatigue crack propagation law In metals, fatigue is a well understood phenomenon, causing irreversible material damage [20]. Unlike metals, the fatigue mechanism in concrete is different due to its quasi-brittle nature. In plain and reinforced concrete structures, fatigue may lead to excessive deformations, excessive crack widths, de-bonding of reinforcement and rupture of the reinforcement or matrix leading to structural collapse [8,21]. Based on linear elastic fracture mechanics concepts, the fatigue crack propagation law originally proposed by Slowik et al. [16] includes parameters such as fracture toughness, loading history, specimen size etc., except the frequency of externally applied load and is described by

σ

ft

ft

ft

σ1

wc

w

w1

ð2Þ

where w1 is the opening displacement when the softening curve changes slope due to bi-linearity and the corresponding stress is r1. The power function suggested by Reinhardt [19] is given by   n  w r ¼ ft 1  ð3Þ wc

σ

σ

w > w1

wc w

Fig. 1. Tension softening laws for concrete.

wc

w

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

ð4Þ

where C is a parameter which gives us a measure of crack growth per load cycle, KIsup is the maximum stress intensity factor ever reached by the structure in its past loading history, KIc 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, DK is the stress intensity factor range, and m, n, p, are constants. These constant co-efficients are determined by Slowik et al through an optimization process using the experimental data and are 2.0, 1.1, 0.7, respectively. Although the fatigue crack propagation law given in Eq. (4) is based on LEFM, the effect of the quasi-brittle nature of concrete and the presence of FPZ is accounted for by the parameter C. In concrete members this parameter indicates the crack growth rate for a particular grade of concrete and is also size dependent. Slowik et al. [16] have determined the value of C to be equal to 9.5 · 103 and 3.2 · 102 mm/cycle for small and large size specimens, respectively. It should be noted here that the stress intensity factor is expressed in M N/m3/2. These values were determined for a particular loading frequency of 3 Hz. The parameter C should also depend upon the frequency of loading. Further, the fatigue crack propagation takes place primarily within the fracture process zone and 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/ft2, 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. [16] proposed a linear relationship between parameter C and the ratio of ligament length (L) to characteristic length (lch), given by   L C ¼ 2 þ 25 ð5Þ  103 mm=cycle lch 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. [16] and Bazant and Xu [10]. While Slowik et al have used compact tension specimens of two different sizes with loading frequency 3 Hz and interrupted by spikes, Bazant and Xu have tested a series of geometrically similar three point beams under fatigue with loading frequency of 0.033 Hz. 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. [16], whereas for the beam specimens used by Bazant and Xu [10], the C values are computed by fitting the experimental a versus N data into Eq. (4). The values so obtained are tabulated in Table 2 along with the frequency of loading used in the tests. Fig. 2 shows the plot of C times f (Cf) against the ratio of ligament length L to characteristic length lch. The

0.1

Exact values Curve fitted 0.09 0.08

C*f (mm/sec)

da K mImax DK nI ¼C p þ F ða; DrÞ dN ðK Ic  K Isup Þ

2141

0.07 0.06 0.05 0.04 0.03 0.02 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Ligament length / Characteristic length (L/lch) Fig. 2. Relation between Cf and the ratio of ligament length to characteristic length.

Table 1 Geometry and loading details Specimen

Depth (mm)

Width (mm)

Span (mm)

Initial notch (mm)

KIc (M N/m3/2)

Peak load (N) (experimental [10])

Large Medium Small

152.4 76.2 38.1

38.1 38.1 38.1

381 190.5 95.3

25.4 12.7 6.35

1.41 1.51 1.66

5184 2986 1815.6

Table 2 C values and material parameters lch

L lch

Gf (N/m)

C (mm/cycle)

f (Hz)

Cf (mm/s)

Size/reference

238.74 172 242.66 278.54 337.52

1.38 0.872 0.52 0.23 0.09

206 158 73.19 84.01 101.8

32 · 103 25.33 · 103 1773 · 103 1170 · 103 840 · 103

3 3 0.033 0.033 0.033

0.096 0.0285 0.0585 0.0386 0.0277

Small [16] Large [16] Large [10] Medium [10] Small [10]

2142

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

resulting best fit curve represents a quadratic polynomial given by,  2   L L þ 0:0809 Cf ¼ 0:0193 þ 0:0209 mm=s ð6Þ lch lch From this equation one can obtain the value of parameter C for any loading frequency, grade of concrete and size of specimen. Unlike in metals wherein an overload causes an increase in the plastic zone size thus retarding the rate of subsequent crack growth, the size of process zone in concrete is increased due to overload, and the rate of crack propagation also increases suddenly. In Eq. (4), the function F(a, Dr) describes the sudden increase in equivalent crack length due to an overload [16]. It should be noted here that the function F(a, Dr) 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. [16] have concluded that overloads cause a sudden propagation of the fictitious crack tip. They have obtained the values of function F(a, Dr), 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, Dr) 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. Since 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   DK I F ¼ Da ð7Þ K Ic where DKI, indicates the instantaneous change in stress intensity factor, from the normal load cycle, to a certain overload cycle, i.e. DK I ¼ K Ioverload  K Inormal-load

ð8Þ

Here, 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. Da 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. Using Eqs. (4) and (6), fatigue crack propagation curve (a versus N) can be computed for the applied fatigue load cycles. The authors [22] have earlier verified the modified law by comparing with the experimental results available in the literature. 2.3. Crack extension resistance associated with the softening traction-separation law In order to determine the crack extension resistance, a method has been proposed by Xu and Reinhardt [15] incorporating the cohesive force on the fictitious crack zone. The

basic principle of the approach is that the total crack extension resistance is composed of two parts. One part is the inherent crack initiation toughness K ini Ic , whereas the other is the crack extension resistance offered by the cohesive forces distributed on the fracture process zone and is a function of the cohesive force distribution f(r), the tensile strength ft and the length of the propagating crack, a. Hence, the total crack extension resistance can be expressed through the following equation: c K R ðDaÞ ¼ K ini Ic þ K ðft ; f ðrÞ; aÞ

ð9Þ K ini Ic

The inherent initiation toughness in a standard threepoint bending beam can be computed using the following expression: 3PL pffiffiffiffiffiffiffi a0  K ini pa0 g1 ð10Þ ¼ KðP ; a Þ ¼ 0 Ic D 2BD2 where P is the upper fatigue load limit; a0 is the initial notch length; L, B, D is the span, width and depth of the beam, respectively, and g1(a0/D) is the geometric factor. The general expression of the crack extension resistance due to cohesive force is given by [15,23], Z a x a  pffiffiffiffiffiffi c pa dx ð11Þ K ðft ; f ðrÞ; aÞ ¼ 2rðxÞF 1 ; a D a0 where F1 is the geometry factor defined by x a  3:52 1  x 4:35  5:28 x a a F1 ; ¼

3=2 

1=2 a D 1  Da 1  Da 2 3

x3=2 x7 61:3  0:3 a þ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 þ 0:83  1:76 a5 1 a h  x a i  1 1 a D

ð12Þ

In Eq. (11), r(x) is the assumed softening distribution as given by Eqs. (1)–(3). In the present study, the three idealizations for the traction-separation law as defined earlier are considered in order to determine the crack extension resistance and the corresponding critical crack length for which unstable fracture takes place. To calculate the traction distributions over the fracture process zone for a given crack length, one has to know the crack opening displacement. Through the application of the linear asymptotic superposition assumption [23], the nonlinear properties on the load (P) – CMOD can be approximately taken into account using compliance Cs given by Cs ¼

6La g2 ðaÞ BD2 E

ð13Þ

where g2(a) is given by g2 ðaÞ ¼

1:73  8:56a þ 31:2a2  46:3a3 þ 25:1a4 ð1  aÞ

3=2

ð14Þ

Therefore, crack mouth opening displacement is computed using the following expression:

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

CMOD ¼ C s P

ð15Þ

Assuming linearity of the crack profile, the crack tip opening displacement for a beam under three-point bending can be computed from CMOD using, 2

CTOD ¼ CMOD½ð1  b0 Þ þ ð1:081  1:149b0 Þðb0  b20 Þ

1=2

ð16Þ where b0 = a0/a. The condition for unstable crack propagation is considered when KR(Da) equals KI(P, a). KI(P, a) is the mode I stress intensity factor under the load P which, for a beam under three-point bending, is given by Eq. (10) replacing a0 by crack length a. The critical crack length, thus, corresponds to the crack length when either KR(Da) equals KI(P, a) or when the critical CTOD is reached, whichever is minimum. 2.4. Residual strength assessment: analytical formulation The objective of the foregoing study is to examine the effect of tension-softening on the moment carrying capacity of a concrete beam. Different methods [17,24] have been reported in the available literature on the determination of the response of concrete beams that include the strain softening behavior. In this work, an inverse method is used to determine the moment or load carrying capacity of concrete beams for a given crack size (or tensile strain or crack opening at the tip of crack). The criterion used for computing the ultimate moment capacity is the crack tip opening displacement, w at the tip of each incremental crack length reaching the critical crack tip opening displacement, CTODc, which is a material parameter. The following assumptions are made in the analysis regarding the stress–strain distribution along the cracked section:

within an effective softening zone width hs. In this way, an effective stress–strain constitutive relationship can be adopted in the spirit of the nonlocal continuum concept [18,25]. As an example, Fig. 3 represents the equivalent stress–strain relation, corresponding to linear softening. In this figure, ET represents the tension softening modulus. The crack opening displacement in the discrete crack model and the post-peak strain in the continuum model are related by w = hst. In the present study, hs is taken as 0.5D, where D is the beam depth. To assess the residual moment carrying capacity of a pre-notched beam as a function of increasing crack length under fatigue loading, the stress–strain distribution corresponding to a notch length a0 is shown in Fig. 4. It follows the linear tension-softening law. The corresponding compressive stress–strain behavior is assumed to obey the following bilinear relation, which is graphically shown in Fig. 5: r ¼ Ec

c 6 cp ¼ fck

σ

f

t

ET

ε

Fig. 3. Tensile stress–strain relation in equivalent continuum concept.

σ

cm

kD

ε

cm

C

c1

εt1

D l

αD

εtu

εtp

ε

N.A

ð17Þ

cp 6 c 6 cu

As is mentioned earlier, the criterion for determining the ultimate capacity is the crack tip opening displacement reaching the critical value (w = wc). By fixing the limiting tip opening displacement, corresponding equivalent strain is calculated following assumption (3). The ultimate tensile strain corresponding to w = wc is denoted as tu, and the strain corresponding to elastic limit (in other words  for w = 0 or for r = ft) is represented as tp. The fracture process zone of length lp is assumed to form in front of the

(1) Strain varies linearly across the depth of beam, i.e. plane section remains plane before and after bending. (2) The crack opening profile is linear. The softening behavior is known in terms of cohesive force versus crack opening law. Alternatively, an average strain t on the continuum scale may be defined as representative of the opening displacement of the microcracks

2143

ε

tp

p

f

t

ε

tu

L Fig. 4. Bending stress–strain distribution for proposed model (linear softening).

T T

s

2144

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

and the tensile forces ft1, ft2, . . . , ftm are calculated incrementally for each segment as the product of the stress value for the considered segment with its area (Bdx). The depth of the neutral axis is calculated such that the total compressive force (C = fc1 + fc2 +  + fcm) equals the total tensile force (T = Ts + ft1 + ft2 +  + ftm). Once the neutral axis depth factor is known, the moment carrying capacity can be computed for that equilibrium configuration. The moment of resistance can be computed as

σ

f

ck

M R ¼ M soft þ M UT εcu

εcp

where Msoft is the moment of resistance provided by the softening zone and equals

ε

M soft ¼ T s  lever  arm

Fig. 5. Constitutive law for concrete in compression.

crack tip. It comprises of the zone starting from the crack tip, where (w = wc) or equivalently t = tu, and extending until w = 0 or t = tp. To compute the moment carrying capacity for the assumed strain distribution, an incremental procedure [26] is adopted. The method is based on the fundamental equilibrium equation for the progressive failure of concrete beams. The uncracked ligament portion (D  aD  lp), as is shown in Fig. 4, is divided into a number of segments (say 10,000), each having a segment of depth dx = [(1  a)D  lp]/10,000. To calculate the neutral axis depth factor k (see Fig. 4), a trial and error procedure is adopted. Knowing k, by the linearity assumption, lp can be computed as follows:   tp lp ¼ 1  ð1  k  aÞD ð18Þ tu Hence, the resistance provided by the softening zone (assuming linear softening behavior) can be expressed as follows: 1 T s ¼ Blp ft ð19Þ 2 where B is the width of the beam. To calculate the strain and stress distribution in the uncracked portion of the beam, the compressive and tensile zones are divided into m and n numbers of segments, respectively, each having a width of dx as mentioned earlier in this section. Hence, it results into (m + n = 10,000) in the present case. Based on the linearity assumption of the strain distribution, the strains in the compressive and tensile zone are computed as follows: c;x

tp ðx  dxÞ ¼ ; Dð1  k  aÞ  lp

¼ T s ½ð1  a  k=3ÞD  2=3lp 

M UT ¼ T Ut ½ð1  a  k=3ÞD  xx=3  lp 

In Eq. (25), X1 represents the distance from the neutral axis to the location where crack opening w is equal to w1, as is shown in Fig. 6, and xx represents the uncracked concrete ε

σcm

cm

x ¼ 1; . . . n

C

ð20Þ εtp

tp ðx  dxÞ ; Dð1  k  aÞ  lp

ð24Þ

where TUt is the tensile resistance provided by the uncracked concrete and xx is the length of the corresponding uncracked portion. The procedure is repeated for different crack lengths a1, a2...an as long as the equilibrium is satisfied. For bilinear and power law softening models, the moment capacity is computed by incorporating the appropriate modifications in evaluating Ts. For bilinear and power law models, Eq. (19) takes the form, respectively, as follows: 1 1 T s ¼ Bðft þ r1 ÞðX 1  xxÞ þ Br1 ðlp  X 1  xxÞ ð25Þ 2 2 and   n T s ¼ Bft lp ð26Þ nþ1

kD

x ¼ 1; . . . m

ð23Þ

where lever-arm is the centroidal distance between the Ts and total compressive force C. Similarly, MUt is the moment of resistance provided by the uncracked tension concrete, which is given by,

and t;x ¼

ð22Þ

ð21Þ

The stresses are calculated using Eq. (17) for the compressive strains c1, c2 . . . cm that are obtained from Eq. (20). Similarly, for the tensile strains t1, t2 . . . tn, the stresses are computed using the relation rt = E for the uncracked tensile region. Then the compressive forces fc1, fc2, . . . , fcm

ft

X1 xx Ts lp

αD

εtu

Fig. 6. Bending stress–strain distribution for proposed model (bilinear softening).

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

2145

Table 3 Parameters of the softening laws Specimen

Fracture energy, Gf (N/m)

Tensile strength, ft (MPa)

wc (mm) Linear

Bilinear

Power

Large Medium Small

73.19 84.01 101.8

3.35 3.86 4.68

0.0436 0.0435 0.0435

0.0785 0.078 0.078

0.11 0.11 0.11

ligament that carries tension. In Eq. (26), the power law exponent n is considered to be 0.248 as mentioned earlier. 2.5. Numerical example and validation The proposed method is validated for three different sizes of geometrically similar beams under three-point bending with constant amplitude fatigue loading. Originally, these specimens were used by Bazant and Xu [10] in their experimental analysis, to study the size effect in bending fatigue of concrete. The specimen dimensions and fracture properties are listed in Table 1. The above method for predicting the residual strength of the beam under continuous application of fatigue loading is applied in a step by step manner for these three specimens. Initially, the softening law is defined and its parameters are computed. The area under the softening curve, denoted as fracture energy Gf as reported in Table 3, is considered to be a

0.5 small (proposed law) small (experiment) Medium (proposed law) Medium(experiment) Large(proposed law) Large(experiment)

Relative Crack Depth (a/D)

0.45 0.4 0.35 0.3 0.25 0.2 0

1

10

2

10

3

10

10

4

10

Number of Fatigue Load Cycle (N) Fig. 7. Fatigue crack propagation curves.

material property. In the present study, keeping Gf constant for all the three softening laws, other parameters such as wc, w1, r1 are computed. In this way, a comparison between the three softening models could be made. The ultimate tensile strength of the specimen is computed from the maximum load P as reported by Bazant and Xu [10]. The critical opening displacements wc computed for three different softening laws are reported in Table 3. For bilinear softening law the points where the slope changes are taken as, w1 = 2/9wc and r1 = ft/3, as proposed by Petersson [27]. The next step of the analysis is to determine the critical crack length using either of the two methods described earlier, for all the three softening laws. The fatigue crack propagation behavior under the applied loading is assessed using the fatigue law as described above. A constant amplitude fatigue loading of frequency 0.033 Hz is taken for the present case study in line with the experimental work of Bazant and Xu. The amplitude of the cyclic loading is 80% of the monotonic failure load P as given in Table 1. Fig. 7 shows the fatigue crack propagation curves for small, medium and large beam specimen together with the experimentally observed data of Bazant and Xu [10]. From these curves, the critical crack size is obtained as the crack length at which the curves become asymptotic. The number of fatigue loading cycles corresponding to this critical crack length gives the failure load cycles (for the specimens) which are 684, 1153 and 1456 cycles for large, medium and small specimens, respectively. The critical crack size is tabulated in Table 4 for the small, medium and large beams. Using the crack extension resistance method, the effect of cohesive force in the process zone is computed using Eq. (9). Fig. 8 shows the crack extension resistance curves as functions of relative crack size (a/D) together with the stress intensity factor plots considering linear, bilinear and power law softening models for large sized specimen. Since the applied loading and specimen geometry does not alter, KI(P, a) remains same. The resistance curve KR

Table 4 Critical crack depth aC Specimen size

Crack resistance method Linear softening

Large Medium Small

KR = KP

w = wc

a

0.58 0.58 0.6

0.43 0.52 0.6

0.43 0.52 0.6

D c

Bilinear softening KR = KP

w = wc

a

0.385 0.41 0.46

0.53 0.61 0.68

0.385 0.41 0.46

D c

Power softening KR = KP

w = wc

a

0.44 0.484 0.52

0.58 0.66 0.7

0.44 0.484 0.52

Fatigue law

Experimental value [10]

0.376 0.438 0.410

0.394 0.414 0.45

D c

2146

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148 140 KR(linear) KI (a,P)

KR, KI(a,P) in MPamm 1/2

120

KR(bilinear) KR(Powerlaw)

100

80

60

40

20

0.2

0.3

0.4

0.5

0.6

0.7

Relative Crack Depth (a/D)

final step of the analysis involves the determination of moment carrying capacity of the beam for any given crack length. The moment capacity computed using the method described above is normalized with the maximum moment capacity (Mun) of an uncracked cross-section, given by Mun = ftBD2/6, where ft is the tensile strength of the member and plotted as a function of the relative crack length. Figs. 10–12 show the normalized moment plots for small, medium and large sized specimens, respectively, for all the three softening laws. It is clearly seen from these plots that the residual capacity varies substantially depending on the assumed softening law although the fracture energy was considered to be constant. For all the cases, the linear softening law gives the highest moment capacity followed by power-law and bilinear. The moment carrying capacity reduces along with increase in crack lengths, as expected.

Normalized Moment capacity (M/Mun)

Fig. 8. KR, KI(a, P) curves of large specimen for linear, bilinear and power-law softening.

0.12

CTOD in mm

0.1

Linear Bilinear Powerlaw

0.08

0.06

0.04

0.02

0.7

Linear Bilinear Powerlaw

0.6 0.5 0.4 0.3 0.2 0.1 0 0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Relative crack depth (a/D) (stress free) 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Relative Crack Depth (a/D)

Fig. 10. Normalized moment as a function of relative crack depth for small specimen.

Fig. 9. CTOD curves of large specimen for linear, bilinear and power-law softening.

Normalized Moment capacity (M/Mun)

depends on the softening approximations. The corresponding CTOD variation is also plotted as a function of relative crack sizes in Fig. 9. From these plot, the crack size corresponding to KR = KI(P, a) and the same for critical CTOD are noted. The critical crack length is selected as the minimum of the two crack sizes corresponding either to the crack size when KR = KI(P, a) or when the critical CTOD is reached. Table 4 shows the critical crack sizes obtained using these two criteria together with the experimental values as reported by Bazant and Xu [10]. It is seen that the critical crack size obtained using the crack resistance method varies considerably between the three softening laws, and the results obtained using bilinear softening model matches closely with the experimental results. Further, the results of critical crack size obtained using the fatigue law also agree well with the experimental ones. The

Linear Bilinear Powerlaw

0.6

0.5

0.4

0.3

0.2

0.1 0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Relative crack depth (a/D) (stress free) Fig. 11. Normalized moment as a function of relative crack depth for medium specimen.

Normalized Moment capacity (M/Mun)

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

assuming linear strain distribution at the critical section. The normalized moment carrying capacity shows that residual strength prediction also varies for a given crack length, depending on the choice of approximation of the softening law. It is seen that the bilinear softening law gives the best prediction for determining the critical crack length and the residual strength (force/moment). Thus, using the procedures described in this work, the critical crack size at failure, the fatigue load cycles corresponding to failure and the moment capacity of the beam at any levels of damage could be determined.

0.8 Linear Bilinear Powerlaw

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.15

2147

References 0.2

0.25

0.3

0.35

0.4

0.45

0.5

Relative crack depth (a/D) (stress free) Fig. 12. Normalized moment as a function of relative crack depth for large specimen.

It is observed, that the rate of reduction in the capacity is higher in case of linear softening. Throughout the crack propagation stage, power-law prediction gives us the lowest value of the normalized capacity. 3. Conclusions In the present work, a two-step method is proposed to predict the residual strength of the concrete beams under fatigue loading by incorporating the tension-softening effect of the fracture process zone. The effect of the cohesive forces due to aggregate bridging is approximated mathematically through standard traction-separation laws. Three different approximations such as linear, bilinear and power-law models are used. In the first step of the analysis, the critical crack size at which failure occurs is determined using two methods. The first is through a fatigue crack propagation law and the second is through a crack extension resistance technique. Knowing the critical crack length, the number of fatigue load cycles corresponding to failure is also computed using the fracture mechanics based fatigue law. The second step of the analysis describes a simple method for determining the residual moment/load carrying capacity of a cracked concrete beam. The effect of tension-softening behavior is studied for beams under three-point bending with constant amplitude loading. The critical crack lengths computed through crack extension resistance method show high dependence on softening approximations, although the fracture energy remains constant. It turns out that variation between experimental predictions and theoretical results of crack lengths is minimum in case of bilinear softening law, for all the three specimens of different sizes. It is also seen that, as the size of the specimens increases, unstable fracture occurs at a lower value of crack length, irrespective of the softening approximation. Secondly, the moment carrying capacity is computed as a function of crack length,

[1] ACI Committee 215. Damage of concrete in fatigue. Fatigue Concr Struct 1982;SP-75 118(11):2176–90. [2] RILEM Committee. Long term random dynamic loading of concrete structures. Mater Struct 1984;17(9):1–28. [3] Oh B. Fatigue life distributions of concrete for various stress levels. ACI Mater J 1991;88(2):122–8. [4] Pasakova T, Meyer C. Optimum number of specimens for low-cycle fatigue tests of concrete. J Struct Eng ASCE 1994;120(7):2242–7. [5] Zhang B, Wu K. Residual fatigue strength and stiffness of ordinary concrete under bending. Cem Concr Res 1997;27(1):115–26. [6] Subramaniam V, ONeil E, Popovics J, Shah S. Crack propagation in flexural fatigue of concrete. J Eng Mech ASCE 2000;126(9):891–8. [7] Baluch M, Quershy A, Azad A. Fatigue crack propagation in plain concrete. In: Shah S, Swartz S, editors. Proceedings of the SEM/ RILEM international conference on fracture of concrete and rock, SEM/RILEM; 1987. p. 80–7. [8] Perdikaris P, Calomino A. Kinetics of crack growth in plain concrete. In: Shah S, Swartz S, editors. RILEM international conference fracture of concrete and rock; 1987. p. 64–9. [9] Carpinteri A. Energy dissipation in R.C. beams under cyclic loadings. Eng Fract Mech 1991;39(2):177–84. [10] Bazant Z, Xu K. Size effect in fatigue fracture of concrete. ACI Mater J 1991;88(4):427–37. [11] Bazant Z, Schell W. Fatigue fracture of high strength concrete and size effect. ACI Mater J 1993;90(5):472–8. [12] Carpinteri A, Spagnoli A. A fractal analysis of size effects on fatigue crack growth. Int J Fatigue 2004;26(2):125–33. [13] Perdikaris P, Calomino A, Chudnovsky A. Effect of fatigue on the fracture toughness of concrete. J Eng Mech ASCE 1986;112(8):776–91. [14] Zhang J, Stang H, Victor C. Fatigue life prediction of fiber reinforced concrete under flexural load. Int J Fatigue 1999;21:1033–49. [15] Xu S, Reinhardt H. Crack extension resistance and fracture properties of quasi-brittle materials like concrete based on the complete process of fracture. Int J Fract 1998;92:71–99. [16] Slowik V, Plizzari G, Saouma V. Fracture of concrete under variable amplitude loading. ACI Mater J 1996;93(3):272–83. [17] Hillerborg X, Modeer M, Petersson P. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 1976;6:773–82. [18] Bazant Z, Oh B. Crack band theory for fracture of concrete. Mater Struct 1983;16:155–77. [19] Reinhardt H. Fracture mechanics of an elastic softening material like concrete. HERON 1984;29:1–44. [20] Paris P, Erdogan F. A critical analysis of crack propagation laws. J Basic Eng ASME 1963;85(3). [21] Carpinteri A, Spagnoli A, Vantadori S. An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading. Int J Solids Struct 2006;43:4917–36. [22] Sain T, Chandra Kishen J. Damage and residual life assessment using fracture mechanics and inverse method. In: Proceedings of the EM 2003, ASCE, Seattle, Washington; 2003.

2148

T. Sain, J.M. Chandra Kishen / International Journal of Fatigue 29 (2007) 2138–2148

[23] Reinhardt H, Shilang X. Crack extension resistance based on the cohesive force in concrete. Eng Fract Mech 1999;64:563–87. [24] Ananthan H, Raghuprasad B, Iyengar K. Influence of strain softening on the fracture of plain concrete beams. Int J Fract 1990;45:195–219. [25] Bazant Z, Chang T. Instability of nonlocal continuum and strain averaging. J Eng Mech ASCE 1984;10:1441–50.

[26] Raghuprasad B, Bharatkumar B, Ramachandra Murthy D, Narayanan R, Gopalakrishnan S. Fracture mechanics model for analysis of plain and reinforced high-performance concrete beams. J Eng Mech ASCE 2005;131(8):831–8. [27] Petersson P. Crack growth and development of fracture zones in plain concrete and similar materials. Tech Rep TVBM-1006, Lund Institute of Technology; 1981.